ff r <::^^ 
 
. ^. I^ha^oras '(aboU't 569-500 bc.) settled in Croton, a Dorian 
 • • (fcv^fiy^jVi'36utlteiriT'^ Italy, where he opened a school in which 
 philosophy and mathenriatics were taught. He founded a brother- 
 hood, the members of which were afterwards called Pythagoreans. 
 He is said to have taught that the foundation of the theory of the 
 universe was to be found in the science of numbers. The word 
 mathematics has been ascribed to him. 
 
ELEMENTARY ALGEBRA 
 
 BY . 
 
 GEORGE H. HALLETT, A.M., Ph.D. 
 
 PROFESSOR OF MATHEMATICS 
 THE UNIVERSITY OF PENNSYLVANIA 
 
 AND 
 
 EGBERT F. ANDERSON, A.M., Sc.D. 
 
 PROFESSOR OF MATHEMATICS 
 STATE NORMAL SCHOOL, WEST CHESTER, PA. 
 
 SILVER, BURDETT AND COMPANY 
 
 BOSTON NEW YORK CHICAGO 
 
aAi54 
 
 Copyright, 1917, 
 By silver, BURDETT AND COMPANY 
 
 SDUCATION DEFT^ 
 
Ol^ 
 
 PREFACE 
 
 This book is designed primarily for the use of those who 
 are beginning the study of algebra ; it is, however, sufficiently 
 extensive to serve as a text for a review of algebra during the 
 third or fourth year of the high school course in those schools 
 in which the curricula call for such a review. 
 
 In preparing this text the authors have kept in view the 
 fact that no substantial progress in algebra is possible for the 
 student unless due emphasis is placed on the fundamental prin- 
 ciples and processes of the subject, and that these essentials 
 must be provided for irrespective of whatever trend the teach- 
 ing of algebra may have had within the last few years, or may 
 have in the future. 
 
 Long experience in teaching has convinced the authors that 
 the technical terms employed and the principles involved in 
 problems of physics and engineering are not sufficiently under- 
 stood to warrant their inclusion in an elementary course in 
 algebra ; that practically the whole of such a course should be 
 devoted to the treatment of the elements of the subject itself, 
 and that the problems referred to may be. solved with little 
 difficulty by the student who has mastered the first course in 
 algebra before he begins the study of physics. However, they 
 believe that formulae drawn from various sources, including 
 physics and engineering, should be used extensively in ele- 
 mentary algebra; for there is, perhaps, no more important 
 practical exercise in the subject than that which comes from 
 determining the actual or approximate numerical value of a 
 literal number which occurs in a formula, when given numerical 
 values are substituted for the remaining letters of the formula. 
 
 To make effective provision for attaining the end in view, 
 
 54 5 -jftU 
 
iv PREFACE 
 
 namely, the furnishing of a textbook from which the student 
 may acquire a thorough grounding in the elements of algebra, 
 the authors have made its prominent features the following : 
 
 1. The simplest possible presentation of the topics of elemen- 
 
 tary algebra. 
 
 2. The use of the inductive method in developing fundamental 
 
 concepts and principles. 
 
 3. The use of illustrative problems to make clear the applica- 
 
 tion of the principles. 
 
 4. The actual application of the principles by means of nu- 
 
 merous, well-graded examples, both sight and written. 
 
 5. The extensive use of numerical checks to promote habits 
 
 of accuracy and to give the student confidence in the 
 results of his work. 
 
 6. The copious supply of problems designed to give the stu- 
 
 dent facility in applying the mechanics of algebra. 
 
 7. The reviews designed to make the students' knowledge 
 
 cumulative and coherent and thorough. 
 
 8. Such treatment of graphs as is necessary to render the 
 
 student familiar with the underlying principles of 
 graphical representation so that he may be able to 
 apply them wherever there may be need for their 
 application. 
 
 9. The conciseness and exactness of statement of definitions 
 
 and principles. 
 
 The author wishes to gratefully acknowledge the courtesy 
 of the Open Court Publishing Company in permitting the re- 
 production of the portraits contained in this book. 
 
CONTENTS 
 
 OHAPTEB PAGE 
 
 I. Introduction 1 
 
 n. Fundamental Processes 34 
 
 m. Simple Equations 81 
 
 rv. Type Products and Factors . . . . . 100 
 
 V. Fractions 153 
 
 VI. Fractional and Literal Equations . . . 191 
 
 Vn. Systems of Linear Equations 205 
 
 Vni. Katio, Proportion, and Variation . ... . 239 
 
 IX. Graphs 251 
 
 X. Powers, Roots, Radicals, and Exponents . . 262 
 
 XI. Involution and Evolution 300 
 
 XII. Quadratic Equations 312 
 
 XIII. Systems of Quadratic Equations .... 350 
 
 XIV. Progressions 365 
 
 XV. General Review 381 
 
Digitized by the Internet Archive 
 
 in 2008 with funding from 
 
 IVIicrosoft Corporation 
 
 http://www.archive.org/details/elementaryalgebrOOhallrich 
 
ELEMEI^TARY ALGEBRA 
 
 CHAPTER I 
 
 INTRODUCTION 
 
 The Notation of Algebra 
 
 1. The Notation of Arithmetic. In arithmetic numbers 
 are expressed by the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. 
 The numbers represented by these symbols are called 
 integers. The operations of addition, subtraction, multi- 
 plication, and division performed on these integers lead 
 either to integers or to fractions. Therefore, primarily, 
 arithmetic treats of integers and fractions and the opera- 
 tions which are indicated by the signs +, — , X, and -?-. 
 
 2. Algebra. Algebra, like arithmetic, treats of number. 
 The symbols which are used in arithmetic are retained in 
 algebra. In algebra, however, new symbols of number 
 and of relations between numbers are introduced, and, in 
 the written language of algebra, systematic use is made 
 of letters to represent numbers. A number which is repre- 
 sented by a letter is called a literal number. 
 
 3. Use of literal numbers. In arithmetic it is shown 
 
 2 4 2x4 
 that -x- = - -• This example illustrates a principle 
 
 d O O X O 
 
 which may be expressed in words as follows : 
 
 The product of two fractions is equal to a fraction 
 whose numerator is the product of the two given numerators 
 
 1 
 
2 ELEMENTARY ALGEBRA 
 
 and whose denominator is the product of the two given 
 denominators. 
 
 This statement may be expressed concisely in the writ- 
 ten language of algebra thus : 
 
 a^c_ axe 
 b d b X d 
 
 Here a denotes the integer which is the numerator of the first frac- 
 tion and b the integer which is its denominator ; also, c denotes the 
 integer which is the numerator of the second fraction and d that 
 
 which is its denominator. In the algebraic expression -, the letters 
 
 a and b represent any integers whatever, whereas in the arithmetical 
 expression l^, the symbols 2 and 3 denote definite integers. 
 
 4. Symbols. The language of algebra employs symbols 
 to represent: (1) Numbers themselves; (2) operations 
 on numbers; (3) relations between numbers. 
 
 5. Symbols of number. Numbers in algebra are ex- 
 pressed by Arabic numerals, and also by letters ; thus; 
 
 c may represent the number of cents in the cost of an orange ; 
 a dollars may denote any number of dollars ; 
 m may stand for the number of miles between two towns ; 
 X may stand for any number, which for the sake of brevity is called 
 the number x. 
 
 6. Symbols of operation. Symbols of operation are as 
 follows : 
 
 -}- is the sign of addition ; it is read plus. 
 7 + 3 denotes the sum of the two numbers 7 and S; a -\- h stands 
 for the sum of the two numbers represented by the letters a and b. 
 
 — is the sign of subtraction ; it is read minus. 
 
 a — b stands for the difference of the numbers represented by the 
 letters a and b. 
 
 X is the sign of multiplication; it is read times or mul- 
 tiplied hy» 
 
INTRODUCTION 3 
 
 a y. h stands for the product of the numbers represented by the 
 letters a and b. Multiplication is also indicated by a dot. 2 x a 
 may be written 2 • a. When numbers are represented by letters, the 
 sign of multiplication may be, and usually is, omitted. 
 
 Thus, 5 ab stands for 5 x a x &. 
 
 -f- or : is the sign of division; it is read divided hy, 
 
 3 -f- 5 or 3 : 5 denotes the quotient obtained from the division of 
 3 by 5. 
 
 Remark. Neither the sign -h nor the sign : is so frequently used 
 
 in algebra as formerly. For instance, a -4- & is usually written -. 
 
 b 
 
 7. Symbols of relation. Symbols of relation are as 
 follows : 
 
 = is the sign of equality and is read equals^ or is equal to. 
 
 a = b expresses the equality of the numbers represented by the 
 letters a and b. 
 
 > and < are signs of inequality. > is read greater 
 than ; < is read le%% than. 
 
 a'>b means that the number represented by a is greater than the 
 number represented by 6. a < 6 means that the number represented 
 by a is less than the number represented by b. 
 
 EXERCISE 1 
 
 1. In the following, c stands for cost (meaning, say, 
 the number of cents in the cost), s for selling price, g for 
 gain, and I for loss ; read in words : 
 
 c-{-g = 8. 8-hl=o. 8—c = g. 
 
 c— 8 = 1. 8 — g = e, c— I = 8. 
 
 2. 3 + 2 expresses the sum of 3 and 2 ; what then is 
 expressed by 3 + 5? 5 + a? 5 + 6? a+b? t 
 
 3. 6 — 2 expresses the difference of 6 and 2 ; what then 
 is expressed by 8-5? 7-a? 6-3? a-b? 
 
4 ELEMENTARY ALGEBRA 
 
 4. 3x5 expresses the product of 3 and 5 ; what then 
 is expressed by 4x6? 3xa? 4: x b? a xh? 
 
 5. What is expressed by3.c? a » b? I x w? ab? 
 
 f* ♦ 
 
 6. - expresses the quotient of 6 divided by 2 ; what 
 
 then is expressed by ?? ^? h ^? £? 4^? ^? 
 ^ '^42a2cJ5r^ 
 
 7. Using the + sign, express the sum of 6 and 7 ; 5 
 and a ; c and d. 
 
 8. Using the — sign, express the difference of 7 and 
 
 6 ; a and 7 ; 3 and b ; a and b. 
 
 9. Using the x sign, express the product of 2 and 5 ; 
 
 7 and a ; 3 and b ; a and 6. 
 
 10. Indicate in three ways the product of 3 and a ; 4 
 and b ; c? and c?. 
 
 11. Using the fractional form, indicate the quotient of 
 9 divided by 4 ; a divided by 3 ; 6 divided by 2 ; a 
 divided by c. 
 
 12. Find the number that is equal to a + 1 when a 
 stands in turn for 3 ; 5 ; 7 ; 12 ; J ; 1| ; ; .3. 
 
 13. Find the number that is equal to b — 1 when b 
 stands in turn for 2 ; 3; 7; IJ ; 1; 2^; 1.5; 2.25. 
 
 14. Find the number that is equal to 2 a when a stands 
 in turn for 2 ; 3 ; 5 ; | ; 1 J ; .5 ; 1.1. 
 
 15. Find the number that is equal to ^ when a stands 
 in turn for 2 ; 1 ; 3 ; 6 ; J ; 5 ; IJ. 
 
 16. When a stands for 2 and b stands for 1, what num- 
 ber is equal to a + 6 ? a — b? U'b? -? 
 
 
 
INTRODUCTION 5 
 
 17. When x= 6 and ?/ = 3, what number is equal to 
 
 1/1/ 2.y 
 
 18. When a; = |^ and i/ = ^i what number is equal to 
 x-\-i/? x-yl x-yl -? — ^? ? 
 
 y y ^-y 
 
 19. If goods cost c dollars and were sold at a gain of g 
 dollars, express the selling price. If c = 7 and ^ = 3, what 
 was the selling price ? 
 
 20. If goods cost p dollars and were sold at a loss of 
 t dollars, express the selling price. If jo = 9 and < = 2, 
 what was the selling price ? 
 
 21. Express the number that is 6 more than a. 
 
 22. Express the number that is c more than d. 
 
 23. Express the number that is 5 less than c. 
 
 24. Express the number that is n less than m. 
 
 25. Express the sum of a, 6, and c. 
 
 26. Express the sum of a and h diminished by c. 
 
 27. Find the cost of 3 pounds of sugar at a cents a 
 pound. 
 
 28. Find the cost of a yards of cloth at h cents a yard. 
 
 29. If p pounds of sugar cost 30 cents, what did one 
 pound cost ? 
 
 30. If h bushels of grain weigh p pounds, what does 
 one bushel weigh ? 
 
 31. What did a boy pay for 2 oranges at c cents each 
 and 3 apples at h cents each ? 
 
 32. How much change should a boy receive from c 
 cents given in payment for a ball that cost h cents? 
 
 33. Express the result of adding p times r to I. 
 
6 ELEMENTARY ALGEBRA 
 
 34. Express the result of adding t times r to p. 
 
 35. Express the result of dividing Vhy a times h. 
 
 36. Write the product of 3, a, 6, and c in three different 
 ways. 
 
 37. How many quarts are there in a gallons ? 
 
 38. How man^ quarts are there in h pecks ? 
 
 39. How many pecks are there in c bushels ? 
 
 40. How many quarts are there in m pints ? 
 
 41. How many units are there in c dozen ? 
 
 42. How many dozen are there in u units ? 
 
 43. If I stands for the length of one line in inches and 
 w for the length of another line in feet, what stands for 
 the sum of their lengths in inches ? What stands for the 
 difference of their lengths in feet ? 
 
 44. If I stands for the number of inches in the length 
 of a rectangle and w for the width in inches, what stands 
 for the number of inches in the perimeter ? 
 
 45. If 8 stands for the length of one side of an equi- 
 lateral triangle, what stands for the perimeter ? 
 
 8. Algebraic expressions. Any symbol or combination 
 of symbols used in algebra to express a number is called 
 an algebraic expression, or simply an expression. 
 
 Thus, 2 a, a + &, — , 3 ar — y + « are algebraic expressions. 
 
 9. Evaluation of an algebraic expression. The process 
 of substituting numbers for letters in an expression and 
 calculating tlie numerical value of the result is called the 
 evaluation of the expression for the given values of the 
 letters. 
 
INTRODUCTION 7 
 
 Thus, to evaluate 2 a -\- b when a = 2 and 6 = 3, substitute the given 
 values of a and b in the expression, obtaining 2-2 + 3=4 + 3 = 7. 
 
 Remark. It is often convenient in testing the accuracy of alge- 
 braic work to evaluate the expression for simple numerical values of 
 the letters. This is termed checking the result. 
 
 10. Order of operations. Algebraic expressions often 
 contain different signs of operation. In evaluating such 
 expressions it is understood that : 
 
 When operations of addition and subtraction are indicated 
 in an expression^ they are to he performed in the order of 
 their occurrence from left to right. 
 
 Thus, 4 + 5-2 + 3-6 = 9-2 + 3-6 = 7+3-6 = 10 -6= 4. 
 
 When operations of multiplication and division are indi- 
 cated in an expression^ they are to he performed in their 
 order from left to right and before the operations of addition 
 and subtraction are performed. 
 
 Thus, 6 X 8-^4 + 12 -6 X2-24 -4 -2=48-4 + 2 x2- 6-^2 
 
 = 12 + 4-3 = 13. 
 
 EXERCISE 2 
 
 1. What is the value of 2 a — 6 when a = l and 5 = 1? 
 
 2. What number is when a = 2 and 5 = 4? 
 
 2 
 
 When a = 1 and 5 = 3 ? When a = J and 5 = J ? 
 
 3. If a stands for the number of units in the altitude, 
 h for the number in the base, and A for the area of a 
 rectangle, read in words the statement, A = ah. 
 
 4. What is the area of a rectangle when the altitude 
 equals 5 ft. and the base equals 7 ft.? 
 
 5. What is the area of a rectangle when a = 3 and 
 6 = 4? 
 
8 ELEMENTARY ALGEBRA 
 
 6. A man walked 3 hours at the rate of 4 miles an hour. 
 How far did he walk ? If he walked x hours at the same 
 rate, how far did he walk ? 
 
 7. A man walked a hours at the rate of x miles an 
 hour. How far did he walk ? 
 
 8. What is the cost of 5 yards of cloth at x cents a 
 yard? 
 
 9. How many a's in a + a? In2a4-a? In3a-fa? 
 In3a + 2a? Tn4a-a? In 5a- 2a? In7a-3a? 
 
 In examples 10-16, supply the missing numbers. 
 
 10. 4x $10 + 2 X 110 + 3 X $10 = ( )x$10. 
 
 11. 4xa + 2xa + 3xa=( )xa. 
 
 12. 6 X 10 ft. + 5 X 10 ft. - 3 X 10 ft. = ( ) X 10 ft. 
 
 13. 6x6 + 5x6-3x6= ( )x6. 
 
 14. 6a; + 5a; + 4a; — 2aj=( )a:. 
 
 15. 7w+5m — 3m— 2m=( )w. 
 
 16. 9r + 5r— 3r-r=( )r. 
 
 17. A boy is a years old, his father is 2 a years old. 
 How many years are there in the sum of their ages ? 
 
 18. A person is x years old. How old will he be two 
 years hence? How old was he 2 years ago? How old 
 was he a years ago ? 
 
 19. One number is three times another. If n represents 
 the smaller number, what expression represents their sum ? 
 Their difference ? 
 
 20. Some sugar costing a dollars was sold at a gain of 
 50%. State the selling price. 
 
 21. The circumference of a circle is equal to 2 tt times 
 the radius; this truth may be briefly expressed thus: 
 
 c = 2 Trr. 
 
INTRODUCTION 9 
 
 Find the length of the circumference if 7r = ^ and r = 7; 
 if 7r=^andr = |. 
 
 22. The radius of a circle is x ft. What is the circum- 
 ference ? 
 
 23. How much greater than a is a + 2 ? How much 
 greater than a -{- 1 is a -^ 2? 
 
 24. Read the expression 2a - a; also, a * 2 a. 
 
 Assume that a = 5, 5 = 2, c = 1, a; = 3, y = 2, 2 = 0, 
 and evaluate the following : 
 
 25. 
 
 y x-z 
 
 4: ab-\- ac—bc 
 
 27. 2 a5tf — X1/Z, 28. 
 
 29. ^^ I ^^^ I 25^ 
 
 2ic^ 
 
 30. 
 
 a + b 2a;+l 2^-f-a; 
 2a. 35 x + l 20y 
 a4-5 + <? 4«/ 3a;— 1 
 
 11. Laws of combination in algebra. The laws of com- 
 bination of algebraic expressions are essentially the same 
 as the laws of combination of numbers in arithmetic; for, 
 in general, by substituting numbers for the letters, the 
 algebraic expressions become arithmetical. Constant ref- 
 erence to this principle will be made in developing the 
 fundamental laws of algebra. 
 
 12. Factors. Each of the numbers whose product is a 
 given number is called a factor of the given number. 
 
 Thus, since 12 = 2 x 6, each of the numbers 2 and 6 is a factor of 
 12. Since 2xy = 2 > x > y, each of the numbers, 2, x, and y, is a factor of 
 2xy. 
 
 Remark 1. The expression 2(x + y) means 2 times the sum of x 
 and^. 
 
10 ELEMENTARY ALGEBRA 
 
 Thus, the factors of 2(-c + y') are 2 and (x + y). 
 Remark 2. A number may have different sets of factors. 
 Thus, sets of factors of 24 are 4 and 6, 2 and 12, 3 and 8 ; 2, 2, and 
 6 ; 2, 3, and 4 ; and 2, 2, 2, and 3. 
 
 13. Coefficient. When a number is the product of two 
 factors, either of these factors is called the coefficient of 
 the other in the product. 
 
 . Thus, in 5 ah, 5 is the coefficient of ah, and 5 a is the coefficient of h. 
 
 Note 1. A numerical coefficient is usually written first. 
 
 Thus, the product of 2 and a is written 2 a, not a 2. 
 
 Note 2. The numerical coefficient 1 is usually omitted. 
 
 Thus, 1 X a is usually written a. 
 
 Remark. When the term coefficient is used, numerical coefficient 
 is usually meant. 
 
 Thus, in 3 a&, 3 is understood to be the coefficient, unless otherwise 
 implied. 
 
 EXERCISE 3 
 
 1. Give factors of 18 ; 14 ; 96 ; 23. 
 
 2. Omitting the factors 1 and 3 a6, name six factors of 
 
 3. Name the factors of 3(a -|- 5). 
 
 4. Name three sets of factors of abc. 
 
 5. What is the coefficient of ah in 3 ai ? 
 
 6. What is the coefficient oi 2 xy m ^ ah - '^^xyl 
 
 7. Each factor in 2xyz has a coefficient. Name each 
 factor and give its coefficient. 
 
 8. Write the factors of a(h + c); 2 a(h-{- c). 
 
 9. Write the product a x 2 x 6 in its usual form. 
 
 10. Write 1 . a 4- 1 • 5 in a better form. 
 
 11. Is the following a true statement: l.a+2.a 
 = (H.2)a? 
 
INTRODUCTION 11 
 
 14. Powers. The product of two equal factors is called 
 the square, or second power, of one of the factors. Similarly, 
 the product of three equal factors is called the cube, or 
 third power, of one of the factors; and the product of four 
 equal factors is called the fourth power, etc. 
 
 Thus, 2 X 2 is the square, or second power, of 2 ; a x a is the 
 square of a; aaa is the third power of a; and aaaaaa is the sixthi 
 power of a. 
 
 15. Base and exponents. For convenience, a x a is 
 usually written a^, read a square ; a x a x a is written a^ 
 read a cube ; ax a x a x a is written a*, read a to the 
 fourth power or a with an exponent Jf. In an expression 
 similar to a*, the number a is called the base, and the 
 number 4 is called the exponent, or index. When the ex- 
 ponent is an integer it shows how many times the base is 
 used as a factor. 
 
 Thus, in 2^ = 64, the number 64 is the sixth power of 2, and the 
 exponent., 6, shows that the ftcwe, 2, is used six times as a factor in ob- 
 taining 64. 
 
 Note. The exponent 1 is usually omitted. Thus, a}- is usually 
 written a. 
 
 Remark 1. In a subsequent chapter other numbers than integers 
 will be introduced as exponents. Their use will then be explained. 
 
 Remark 2. Care should be taken not to confuse an exponent 
 with a coefficient. 
 
 Thus, 3 a means a + a + «, while a^ means a x a x a. 
 
 Remark 3. It should be emphasized that such expressions as 
 2 ah^ mean 2 dbh and not* 2ab x2 ah. The latter expression is written 
 (2 ahy. 
 
 EXERCISE 4 
 
 Write examples 1-5 in another form, using exponents 
 and coefficients : 
 
 1. 2.2.2.2.2. 
 
 2. a + a + a-f-a + a. 
 
12 ELEMENTARY ALGEBRA 
 
 3. aa 4- CL(X" 
 
 4. XXX -\- XX -{- XX + 1, 
 
 5. aaa + aaa — aa -{• a + a -\- a. 
 
 6. Evaluate each of the following when x = 2: 
 
 a^; ar^; a:*; a^. 
 
 7. Evaluate each of the following when a; = 3 : 
 
 2x^; bx^; ^x^; 2 3^; ^ x^. 
 
 8. Evaluate a^ + 2x^ — x-\-2 when a; = 3. 
 
 9. Evaluate a^— b^ + 2 ab when a = 3 and 6 = 2. 
 
 10. Evaluate a^ — h^ when a = 2 and 6 = 1. 
 
 11. Evaluate 3 a^ 4. 2 6 — 5 when a = 2 and 6=1. 
 
 12. Evaluate y^—2y-{-l when y = 2. 
 
 Rewrite these expressions, using exponents when possible : 
 
 13. 2 a6 X 2 a6 X 2 ah. 
 
 14. 3 a26 X 3 a^h x 3 a%. 
 
 15. 3 aa - 2 66 ; aaa - 2 x 3 66 ; 2 • 2 . 2 aa - 3 x 3 66. 
 
 16. 2 x2aa-3 X 3666; 2 x 2 x 2a;a; - 2 x 3yy ; 25aa 
 - 49 66. 
 
 16. Parentheses. Parentheses, ( ), are used to indi- 
 cate that the expression inclosed is to be treated as a single 
 number. When different parentheses are used in the same 
 expression, other forms, as brackets, [ ], braces, | j, and 
 the vinculum, , are employed. 
 
 Thus, a +{1) — c) means that the number remaining after subtract- 
 ing c from h is to be added to a. (5 + 4)(6 — 2) means the product 
 
 of 9 and 4. In — -^, the vinculum groups a + 6 into one number, and 
 
 c + a 
 c -\- d into one number. 
 
 Note. The various forms of parentheses are called signs of 
 aggregation ; their use is illustrated by the following expression : 
 [2 + {3 - 2} + 3^ +(2 + 5) + (4 - 2)] = [2 + 1 + 2 + 7 + 2] = 14. 
 
INTRODUCTION 13 
 
 EXERCISE 5 
 
 Simplify : 
 
 1. (2 + 3)(4 + 5). 
 
 Thus, (2 + 3)(4 + 5) = 5 X 9 = 45. 
 
 2. (5 + 3)(8-3). ^ 3. 5(3 + 4-5). 
 
 4. (18-12)^3. 5. 15-(3x2 + 4). 
 
 6. (12 + 6)H-(8-5). 7. (15-5)-f-(3 + 2). 
 
 8. (2 4-4)(6-4)^(5-hl). 9. (8 + 4) -5- (10 -6) (15 -5). 
 
 ^^ (12 + 3)x(12-7), ^^ 2 + [3+(l + 15)]. 
 
 2x5 
 12. 3 + (2-l) + (3 + 2). 13. 8 + [4-f.2 + 3]. 
 
 14. (3 + l)(2 + l)+2. 
 
 15. 5[2 + (3-2) + j3 + 2-|-in. 
 
 16. 15^^ X 2 H- 8T6"-J- 2. 
 
 17. Monomial. An algebraic expression which does 
 not contain an addition or a subtraction sign is called a 
 monomial expression, or simply a monomial. 
 
 Thus, 3 a, —, and ^ are monomials. 
 b d 
 
 Note. An expression within parentheses must be regarded as a 
 
 monomial, since it is to be taken as a whole. 
 
 Thus, 2(a + b) and (a -f- a) are monomials ; but a + 6 is not a 
 monomial. 
 
 18. Terms. The monomials of an algebraic expression 
 which are connected by -f and — signs are called the 
 terms of the expression. 
 
 Thus, the terms of aa; + 3 6 + 2 c are ax, 3 b, and 2 c. 
 
 19. Binomial. An algebraic expression of two terms is 
 called a binomial. 
 
 Thus, a + 2, 3 a - X, and 2(a -{- b) -\- 'd(x + y) are binomials. 
 
14 ELEMENTARY ALGEBRA 
 
 20. TrinomiaL An algebraic expression of three terms 
 is called a trinomiaL 
 
 Thus, X + y + z, 2a + 3b - c, and 2(b + c) + 3(a; + y) + c(a + b) 
 are trinomials. 
 
 21. PolynomiaL An algebraic expression of two or 
 more terms, is, in general, called a polynomial. 
 
 Thus, a binomial is a polynomial of two terms, and a trinomial is 
 a polynomial of three terms. 
 
 22. Like terms. Terms which do not differ except in 
 their coefficients are called like terms. 
 
 Thus, 3ab% 6ab% and ^ab^ are like terms; but 3ab^ and 5a% are 
 unlike terms. 
 
 Remark. Terms may be regarded as like with respect to a cer- 
 tain factor or certain factors, when the remaining factors are regarded 
 as coefficients. Thus, ay and by are like terms with respect to y ; also 
 abx and cbz are like with respect to bx. 
 
 EXERCISE 6 
 
 1. How many terms are there in 2x— 3^ + «? 
 
 2. Name the terms inx — 4i/ — Sz-\-l. 
 
 3. Name the numerical coefficient in each of the 
 monomials 2a%,, ah^ Sabx, mn\ Zxyz^ 4(a;-|-y). 
 
 4. Of the following monomials name those that are 
 like: 
 
 2xy, Sx^y, Zxy, 2xy\ \^y, bxy\ xy, xy\ x^y, bxy^. 
 
 5. State with respect to what letter 4 rw, mw, and cm are 
 like. What is the coefficient of each of these monomials ? 
 
 6. With respect to what factor are axy^ bxy, and cxy 
 like, and what is the coefficient of each monomial ? 
 
 7. With respect to what factor are 2(w + w), c{m -f- w), 
 and d(m + n) like, and what is the coefficient of each 
 monomial ? 
 
INTRODUCTION 15 
 
 8. If a stands for one number and h for another, a-\-h 
 represents their sum ; a — b, their difference ; a6, their 
 
 product ; and ^, the quotient of a divided by b. Which 
 
 of these expressions are binomials, and which monomials ? 
 Evaluate each when a = 2 and 5=1. 
 
 Evaluate the following trinomials when a: = 4, ^ = 2, 
 and 2 = 1. 
 
 9. X -{- 1/ -{- z. 10. X — y -\- z. IX. x — y — z. 
 
 12. X — 2 y -\- z, 13. X — y — 2z. 14. x + y —'^z, 
 
 23. The equation. Such statements as, 
 
 a = b^ 2 a; = 3, and x-\- y = 4: 
 
 are called equations. The equation a = b means that the 
 number represented by the letter a is equal to the number 
 represented by the letter b. In the equation a=b^ a is 
 called the first member of the equation and b is called 
 the second member. 
 
 24. Application of the equation. 
 
 Problem. $ 500 is to be divided between two persons, A and B, in 
 such a way that B shall receive $ 100 more than A. How many dol- 
 lars should each receive ? 
 
 Solution. Let x = the number of dollars in A's share. 
 
 Then, x -)- 100 = the number of dollars in B's share. 
 
 Hence, x -\- x -\- 100 = the number of dollars both are to receive. 
 
 meaning that two times the number 
 of dollars that A receives plus 100 is 
 equal to 500, the number of dollars to 
 be divided. 
 -100, 
 
 .'. X = 200, the number of dollars that A is to receive. 
 x + 100 = 300, the number of dollars that B is to receive. 
 Note. The symbol .-. is read therefore. 
 
 That is. 
 
 2a: + 100=.500< 
 
 Hence, 
 
 2x = 500 
 
 or, 
 
 2a; = 400; 
 
16 ELEMENTARY ALGEBRA 
 
 In the foregoing solution, the equation 2x = 500 — 100 
 was obtained from the preceding equation *2x-\- 100 = 500, 
 by means of the principle that if equal numbers he subtracted 
 from equal numbers the resulting numbers are equal. Also, 
 the equation x = 200 was obtained from the equation 
 2x = 400 by means of the principle that if equal numbers 
 be divided by equal numbers (the number zero excepted} the 
 resulting numbers are equal. 
 
 25. Assumptions. The simplification of all equations 
 depends on such assumptions, or principles, as those 
 stated in section 24. These assumptions are : 
 
 1. ^ equal numbers be added to equal numbers^ the sums 
 will be equal. 
 
 2. If equal numbers be subtracted from equal numbers^ 
 the differences will be equal. 
 
 3. If equal numbers be multiplied by equal numbers^ the 
 products will be equal. 
 
 4. If equal numbers be divided by equal numbers (zero 
 excepted) .> the quotients will be equal. 
 
 5. Numbers which are equal to the same number are equal 
 to each other. 
 
 26. The generalizing spirit of algebra may be illus- 
 trated by further consideration of problems similar to 
 that of section 24. 
 
 ILLUSTRATION 
 
 Instead of dividing $ 500 between two persons, let the number of 
 dollars divided be represented by the letter w, which may represent 
 any number of dollars, and let the second person receive a dollars 
 more than the first. 
 
 Solution. Let x = the number of dollars in A's share. 
 
 Then, x -\- a = the number of dollars in B's share. 
 
 Hence, x -^ x -\- a — the number of dollars they both receive. 
 
INTRODUCTION 17 
 
 That is, 2x -{■ a = n, 
 
 or, 2x = n-a, [§25,2] 
 
 .•.x = ^. [§25,4] 
 
 Hence, n — a _ ^j^^ number of dollars in A's share, 
 
 and n — a _j_ ^ _ ^j^g number of dollars in B's share. 
 
 27. The resulting value of x in the solution, section 26, 
 namely, x = "" , is an illustration of what in algebra is 
 
 termed a formula. This formula gives the solution of a 
 
 great number of particular problems which are all of the 
 
 same kind and which differ only in the numerical values 
 
 assigned to the letters. 
 
 Thus, when n = 500 and a = 100, [§ 24] 
 
 500 - 100 
 
 2 
 
 and x + 100 = 300 ; 
 
 which were the results obtained in § 24. 
 
 = 200, 
 
 28. As a general definition we have the following : 
 
 A formula is a rule of calculation expressed in algebraic 
 
 symbols. 
 
 Remark. Problem 3, Exercise 2, page 7, contains an important 
 
 geometrical formula, that for the area of a rectangle. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. When a: + 3 = 7, what is the value of x ; that is, for 
 what number does x stand ? 
 
 Solution. ar + 3 = 7. 
 
 Subtracting 3 from each member, a? = 7 — 3. [§ 25, 2] 
 
 .*. X = 4:. 
 
 2. When 3 a: = 12, what is the value of a;? 
 Solution. 8 ar = 12. 
 
 Dividing both members by 3, x = 4. [§ 25, 4] 
 
18 
 
 ELEMENTARY ALGEBRA 
 
 3. When Jo; = 2, what is the value oi x? 
 
 Solution. ' J a: = 2. 
 
 Multiplying both members by 4, x = 8. 
 
 [§ 25, 3] 
 
 4. Solve 3 a; + 5 = 11 and check the resulting value 
 of X. 
 
 Solution. 3 a; + 5 = 11. 
 
 3a; = 11 -5. 
 
 3a; = 6. 
 x = 2. 
 Check. 3 X 2 + 5 = 6 + 5, or 11. 
 
 5. Solve 3a; + 2a; — a;4-7 = 2a; + 15 and check. 
 
 Solution. 3a; + 2a;-a; + 7 = 2a; + 15. 
 
 4a; + 7 = 2x + 15. 
 
 [See Example 9, Exercise 2] 
 4a;-2a; + 7 = 2a;-2a;+15. [§25,2] 
 2a; + 7 = 15. 
 2a; = 8. 
 a; = 4. 
 Check. 3x4 + 2x4-4 + 7 = 2x4 + 15. 
 
 12 + 8 - 4 + 7 = 8 + 15. 
 23 = 23. 
 
 EXERCISE 7 
 
 Solve the following equations and check : 
 
 1. a; + 3-5. • 2. a; + 7 = 9. 
 
 3. a; H- 2 = 3. 4. a; +- 1 = 5. 
 
 5. a; +- 1 = 4. 6. a; +- 7 = 8. 
 
 7. 3 + a; = 4. 8. 2 + a; = 4. 
 
 9. 4+-a; = 6. 10. 2a; + 3 = 7. 
 
 11. 3a; + l==10. 12. 5a:H-2 = 12. 
 
 13. 3a;4-3 = 9. 14. 5a; +-1 = 11. 
 
 15. 2a;-h4 = 6. l^. 3a;+-4 = 5, 
 
INTRODUCTION 19 
 
 17. 42^ + 8 = 12. 18. 72:4-6 = 20. 
 
 19. 32:+ 22: = 10. 20. 32;-f-2; = 12. 
 
 21. 52:+ 22: -2^=12. 22. 62: =22: + 3. 
 
 23. 72:- 2: + 1 = 22: + 9. 24. 2x-\-Sx — 4:X=B, 
 
 25. ^2: = 3. 26. ^2: = 3. 
 
 27. ^2:=1. 28. 52: + 1=32; + 7. 
 
 29. 52: + 2 = 62:+l. 30. 4x-x -\- 2x = ^x -{-4:. 
 
 In problems 31-35, denote the unknown number by x. 
 Form the equation and find the value of 2:. 
 
 31. Twice a certain number is 20. What is the 
 number ? 
 
 32. The sum of 10 and twice a certain number is 50. 
 What is the number ? 
 
 33. If three times a certain number is added to 9, the 
 sum is 15. What is the number ? 
 
 34. The sum of two consecutive integers is 11. Find 
 the integers. 
 
 Suggestion. Consecutive integers are those which differ by 1. 
 
 35. If a pound of butter and one of lard together cost 
 60 cents, what was the price of each if the butter cost 
 three times as much as tlie lard ? y 
 
 36. The perimeter of the rectangle 
 in the accompanying diagram is given 
 by the formula P = 2 2: + 2 y. Find 
 X when P = 100 and «/ = 30. Find t/ 
 when P = 50 and 2: = 10. 
 
 When A stands for the area of a rectangle, h the num- 
 ber of units in its base, and a the number of units in its 
 altitude, A = axb. 
 
20 ELEMENTARY ALGEBRA 
 
 Using this formula, find the value of the missing letter, 
 given : 
 
 37. 6 = 6, a = 4. 38. 6 = 10, a =2^. 
 
 39. ^ = 20, a = 4. 40. A = 60, ,b = 10. 
 
 When A stands for the area of a triangle, b the number 
 of units in its -base, and a the number of units in its 
 altitude, ^ _ a x 6 
 
 From this formula find the value of the missing letter, 
 given: 
 
 41. a = 6,b = 8. 42. a=9,b = 4. 
 
 43. ^ = 12, a = 6. 44. A = 20,b = S. 
 
 When r stands for the number of units in the radius of 
 a circle, c the number in the circumference, and A the 
 area, then, (j) ^ ^ 2 irr; (2) A = irr^. 
 
 Using the proper formula, 
 
 45. Find (?, given r = 3. 
 Thus, c = 2 TT X 3, or 6 TT. 
 
 46. Find A, given r = 5, 
 Thus, ^ = -TT X 52, or 25 tt. 
 
 47. Find r, given c = 8 tt. 
 
 Thus, 2 7rr=8ir; 2r = 8; r = 4. 
 
 Remark. Observe that the equation 2 r = 8 was derived from the 
 equation 2 irr = 8 tt by dividing both members of the latter by x. 
 
 48. Find r, given A = 9 tt. 
 
 Thus, irr2 = 9 TT ; r2 = 9 ; r = 3. 
 
 49. Find c, given r = 6. 50. Find A, given r = 2. 
 51. Find A, given r = 10. 52. Find r, given c = 16 tt. 
 53. Find r, given ^ = 647r. 54. Find r, given -4 = | tt. 
 
Introduction 21 
 
 55. Using the letters c and n, write a formula for the 
 cost of any number of dozen oranges when one dozen 
 costs 25 cents ; when one dozen costs a cents. 
 
 56. Write as a formula the rule for finding the simple 
 interest for a given number of years i on a given sum of 
 money s at a given rate of interest per cent per annum r. 
 
 57. I am twenty-five years younger than my father, 
 whose age is a years. Write in a formula the rule for 
 finding my age when his age is known. 
 
 58. Write a formula for the weight of a bottle of milk, 
 given the weight of the bottle, the weight of a cubic inch 
 of milk, and the quantity in the bottle. (Use the letters 
 Tf, b, if, and v.) 
 
 59. Write a formula for the number of cents in a 
 dollars + h dimes -h e nickels. 
 
 60. Write a formula for the number of inches in a yards 
 b feet c inches. 
 
 61. Write an expression for the rate of a train which 
 runs m miles in 5 hours ; which runs m miles in h hours. 
 
 62. If n articles cost d dollars, find an expression for 
 the number of articles which can be bought for x dollars. 
 
 63. Construct a formula for the number iV which, when 
 divided by d, gives the quotient q and the remainder r. 
 
 64. The sum of three consecutive integers is 18. Find 
 the integers. 
 
 65. Write a formula for the volume Fof a beam, I feet 
 long, w feet wide, and d feet thick. Apply the formula 
 to find the thickness of a beam which contains 10 cubic 
 feet of timber and is 2 feet wide and 5 feet long. 
 
 66. What is the area of a triangle whose base is 5 feet 
 and whose altitude is 10 feet ? 
 
22 ELEMENTARY ALGEBRA 
 
 67. A room is a feet long and h feet wide. Construct 
 a formula for the cost of carpeting the room with lino- 
 leum which costs $2 per square yard. 
 
 68. The rule for making tea is : " One teaspoonful of 
 tea for each person and one for the pot." Express this 
 rule by a formula letting t denote the number of tea- 
 spoonfuls of tea and p the number of persons. 
 
 69. A formula for the time in hours required to cook a 
 joint of beef of given weight in pounds is t = \w + \. 
 From this formula state the rule. 
 
 Positive and Negative Numbers 
 
 29. Algebra makes use of all the numbers of elementary 
 arithmetic and in addition to these it introduces certain 
 other numbers called positive and negative numbers. 
 
 ILLUSTRATION 
 
 Let a point on a horizontal line be selected and marked ; let the 
 point one unit to the right of be marked + t, the point two units 
 to the right of be marked + 2, and so on. Let the point one unit 
 to the left be marked — 1, the point two units to the left of be 
 marked — 2, and so on ; thus : 
 
 I I I I I I I I I 
 
 _4 _3 _2 -1 +1 +2 +3 +4 
 The points marked +1, + 2, + 3, etc., represent the positive 
 numbers, and those marked — 1,-2, — 3, etc., represent the negative 
 numbers. 
 
 30. Two positive or two negative numbers are said to 
 have like signs, A positive number and a negative 
 number are said to have unlike signg. Positive and 
 negative numbers are called algebraic numbers, 
 
 31. Whenever quantities exist which are exact opposites, 
 as illustrated in section 29, these quantities may be repre- 
 sented by positive and negative numbers. 
 
INTRODUCTION 23 
 
 Thus, an ordinary thermometer scale may be divided in the 
 manner indicated by the diagram in section 29. When so divided 
 the point marked indicates zero degrees ; one degree above zero is 
 marked + 1, two degrees above zero, -f 2, onjp degree below zero, 
 — 1, two degrees below zero, — 2, and so on. 
 
 32. The scale of positive and negative numbers, section 
 29, has various practical applications, such as indicating 
 degrees of latitude north ' and south from the earth's 
 equator, marked 0°; degrees of longitude west and east 
 from some chosen meridian, as that of Greenwich, which 
 is marked 0° ; intervals of time after and before a certain 
 event ; gains and losses in business transactions. 
 
 Note. Negative numbers are introduced into algebra by a simple 
 convention. In a - 6 it is convenient to call the expression — 6 a 
 number and to say that a — b is obtained by adding — 6 to «. Thus, 
 a — b is written a -\-{— b). This is a new use of the word number. 
 A negative number is, therefore, simply a number which is to be sub- 
 tracted. In the same way a positive number is a number which is 
 to be added. The numbers of arithmetic are neither positive nor 
 negative. 
 
 EXERCISE 8 
 
 1. Using the signs + and — , write : 
 
 5 positive units ; 6 negative units ; a positive units ; 
 h negative units ; 2 a positive units ; Sx negative units. 
 
 2. State how many and what kind of units there are in 
 each of the following : 
 
 -1-3; -1; +c; -b; ■j-2x; -Sb. 
 
 3. If 10° north latitude is represented by + 10°, what 
 number will represent 25° north latitude ? 10° south 
 latitude ? 30° south latitude ? ^ 
 
 4. If north latitude is marked -f- and south latitude — , 
 write, using the signs + and — instead of N. and S. : 
 
 6° N.; 6° S.; 9° 30' N.; 7° 30' 12" S. 
 
24 ELEMENTARY ALGEBRA 
 
 5. If west longitude is marked -|- and east longitude 
 — , write, using the signs + and — instead of W. and E. : 
 20° E.; 5° W.; 4° 15' W. ; 7° 10' 20'' E. 
 
 6.' If the year 1916 a.d. is represented by +1916, what 
 will represent the year 2000 A.D.? The year 399 B.C.? 
 The year 646 B.C.? 
 
 7. If gains are marked + 'and losses — , write, using 
 the signs + and — : 
 
 15 gain; $6 loss; 18 loss: $4 gain. 
 
 8. If temperature above zero is marked -|- and tempera- 
 ture below zero — , state what temperature is indicated by 
 each of the following : 
 
 ^5°; ^lo. +60°; -7°; +80°. 
 
 Addition 
 
 33. A gain of f 10 together with a gain of $6 makes a 
 total gain of f 16 ; also a loss of -f 10 together with a loss 
 of $6 makes a total loss of $16. Hence, regarding gain 
 as positive and, therefore, loss as negative^ we may infer that : 
 
 1. (+10) + (+6)= + 16. 
 
 2. (_10)4.(-6) = -16. 
 
 Remark. If it is necessary to distinguish a sign of an algebraic 
 number from a sign of operation, the algebraic number is put into 
 parentheses ; otherwise, in writing such numbers the positive sign is 
 usually omitted. 
 
 34. A gain of $10 combined with a loss of $6 is equiva- 
 lent to a net gain of f 4 ; also, a loss of -f 10 combined with 
 a gain of $6 is equivalent to a net loss of |4. Hence, we 
 may infer that : 
 
 1. (+10) + (-6) = -f-4. 
 
 2. (_10)-h(+6) = -4. 
 
INTRODUCTION 25 
 
 35. The absolute value of an algebraic number is its 
 value without regard to sign. 
 
 Thus, the absolute value of + 2 is 2 and the absolute value of — 3 is 3. 
 Remark. The expressions arithmetical value and numerical value are 
 sometimes used instead of absolute value. 
 
 36. Positive and negative numbers are also called 
 opposite numbers. See section 31. 
 
 37. The result obtained by combining (adding) two or 
 more algebraic numbers is called the sum of the numbers. 
 
 From sections 33 and 34 we infer that: 
 
 1. The sum of two numbers with like signs is the sum of 
 their absolute values with their common sign prefixed to the 
 result. 
 
 2. The sum of two numbers with unlike signs is the dif- 
 ference of their absolute values with the sign of the number 
 which has the greater absolute value prefixed to the result. 
 
 Note 1. To add three or more algebraic numbers with like signs, 
 add the second to the first, to the result add the third, and so on. 
 
 Thus, +3+(+2)+(+4) + (+l) = +5 + (+4) + (+l) 
 
 = (+9) + (+l) = (+10). 
 
 Note 2. To add three or more algebraic numbers with unlike 
 signs, add the positive numbers and the negative numbers separately, 
 and then add the results. 
 
 Thus, +4+(+o) + (+6) + (-2)-f-(-7)=-|-15+(-9)= + 6. 
 
 Remark. When two numbers have the same absolute value and 
 unlike signs, their sum is zero. 
 
 Thus, (+2) + (-2) =0. 
 
 EXERCISE 9 
 
 
 
 
 Name at sight the sum : 
 
 
 
 
 1. +3 2.-2 3. 
 
 + 5 
 
 4. 
 
 -4 
 
 + 2 -1 
 
 + 7 
 
 
 -9 
 
26 ELEMENTARY ALGEBRA 
 
 5. 
 
 + 2 
 
 6. +3 
 
 7.-8 
 
 8.-6 
 
 -1 
 
 -5 
 
 + 5 
 
 + 9 
 
 9.-4 10. -10 11. +7 12. +8 
 
 + 4 - 1 -6 -8 
 
 Add, as indicated, at sight : 
 
 13. +5 + (+2). 14. -7 + (-3). 
 
 15. -8 + (-6). 16. -8 + (-3). 
 
 17. +9 + (_9). 18. +10 + (-12). 
 
 19. +2+(+l) + (+4). 20. _2-f-(-3) + (-4). 
 
 21. -4 + (+l) + (+2). 22. +6 + (-2) + (-l). 
 
 23. _l + (_2) + (+l) + (+3). 
 
 24. +2 + (-3) + (4-4) + (-l). 
 
 25. _l+(+l) + (+5) + (-3). 
 
 26. +3 + (-3) + (-4) + (+4). 
 
 27. _2+(-3) + (+4) + (-3). 
 
 28. 6 + (-3) + (+2) + (-7). 
 
 29. 5+(-|.3) + (-4) + (-8). 
 
 30. _7+(-2) + (+8) + (+2). 
 
 Subtraction 
 
 38. In algebra, as in arithmetic, subtraction is the 
 inverse operation of addition ; that is, subtraction is the 
 undoing of an addition. 
 
 Thus, 5 + 3-3 = 5, which shows that the addition of 3 has been 
 undone by the subtraction of 3. 
 
 Also, a + 3 — 3 = a, in which a is any algebraic number. 
 
 In general, the result of subtracting a number from an 
 equal number, whether it be positive or negative, is zero. 
 Thus, a - a = 0, since + a - a = 0. 
 
INTRODUCTION 27 
 
 Since a — a= and a -h ( — «) = [section 37, remark], 
 the following principle may be inferred : 
 
 The result of subtracting a number is the same as the 
 result of adding the opposite number. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Subtract + 3 from + 5. 
 
 Solution. (+ 5)-(+ 3) = (+ 5) + (- 3) = + 2. 
 
 2. Subtract — 3 from + 5. 
 
 Solution. (+5)-(-3) = +5+(+3)= + 8. 
 
 3. Subtract + 7 from + 2. 
 
 Solution. (+2)-(-|-7) = + 2 +(-7) = - 5. 
 
 4. Subtract + 3 from — 5. 
 
 Solution. (-5)-(+3) = - 5+(-3)=-8. 
 
 5. Subtract — 2 from — 5. 
 
 Solution. (- 5)-(-2) = - 5+(+2) = -3. 
 
 Remark. In algebra the terms minuend, subtrahend, and difference 
 have the same meaning as in arithmetic. 
 
 Thus, in (+ 6) — (+ 2) = + 4, the minuend is + 6, the subtrahend 
 is + 2, and the difference is + 4. 
 
 39. From the solutions of the illustrative examples of 
 section 38 the following rule may be derived: 
 
 Rule. The difference of two algebraic numbers is found by 
 adding to the minuend the subtrahend with its sign changed. 
 
 EXERCISE 10 
 
 Name at sight the difference : 
 
 1. + 5 2.-5 3. + 3 4.-4 
 
 + 2 -3 H-7 -9 
 
28 ELEMENTARY ALGEBRA 
 
 5. +9 6.-7 7. - 
 
 -9 
 
 8. 4-1 
 
 -2 4-3 
 
 -1 
 
 + 8 
 
 9.-6 10.-6 11. 
 
 
 
 12. 
 
 -6 +6 
 
 4-8^ 
 
 -5 
 
 Subtract, as indicated, at sight : 
 
 
 
 13. +6 -(4- 2). 14. -4 -(-3). 
 
 15. 
 
 4-7-(-l). 
 
 16. -9 -(4- 5). 17. -1-(4.1). 
 
 18. 
 
 -l-(-7). 
 
 19. 11 -(-4). 20. 7 -(-10). 
 
 21. 
 
 0-(4-5). 
 
 22. O-(-l). 23. 6 -(4- 5). 
 
 24. 
 
 5 -(-5). 
 
 40. In section 33 it is shown that 
 
 
 
 10 4-(4-6)=16 = 10 4-6, 
 
 (1) 
 
 also that 10 4- (- 6) = 4 = 10 - 
 
 -6. 
 
 (2) 
 
 Again, from section 39 we have 
 
 
 
 10-(4-6)=.10+(- 
 
 -6) = 
 
 10 - 6, (3) 
 
 also that 10 -(- 6)= 10 4-(4- 6) = 
 
 10 4-6. (4) 
 
 Examples (1), (2), (3), and (4) illustrate the following : 
 
 Rule of Signs 
 
 4- 4- or — — mat/ be replaced by 4-. 
 4- — or — 4- may be replaced by — . 
 
 From examples (1), (2), (3), and (4), it is evident that 
 in addition a number is written down with the sign before 
 it retained, while in subtraction the number is written 
 with its sign changed. 
 
 Thus, (+ 2) + (+ 3) + ( - 4) is written 2 + 3-4 = 1; 
 
 but, ( + 2) - (- 6) is written 2 + 6 = 8. 
 
 Also, - 4 +( - 3) is written - 4 - 3 = - 7 ; 
 
 but, - 4 - (- 3) is written - 4 + 3 = - 1. 
 
INTRODUCTION 29 
 
 Remark. An expression like 4 - 6 = — 2 does not occur in 
 arithmetic. In algebra, this expression means simply [see diagram, 
 § 29] that if we start at the point marked + 4 and count 6 units to 
 the left we shall end with the point marked — 2. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Simplify (4.2) + (-3) + (-f4). 
 
 Solution. (+2) + (-3)+(+4)=2-3 + 4=-l + 4 = 3. 
 
 2. Simplify (-}.2)4-(-3) + (-4). 
 
 Solution. (+2) + (-3) + (-4)=2-3-4 = -l-4=--5. 
 
 EXERCISE 11 
 
 Perform the indicated operations in the following ex- 
 amples : 
 
 1. (+2)-(+l). 2. 3 -(+2). 
 
 3. -5 + (-2). 4. (-2) + (H-5). 
 
 5. 2 -(-3). 6. -7 +(-2). 
 
 7. _2-(+l). 8. -3-h(-5). 
 
 9. -2 + (-l). 10. 3 + (-2). 
 
 11. 3 -(+5). 12. 3 + 5-(+2). 
 
 13. (_2) + (-3)-(-4> 14. (-3)+( + 2)-(-l). 
 15. -4'+(-4). 16. -2 + 2 + (-2). 
 
 17. -5 + 4+(-l). 18. -l-2-(+3). 
 
 19. l + 2 + (-3). 20. 1-1 + 2 + (-2). 
 
 21. -3 + 4 -(-5). 22. -l_2 + (-3). 
 
 23. _(_2)4-C-3)-(-l). 
 
 24. l4.(+9)-(+12)H-(-3) + (+17)-(+12). 
 
 25. 2 + (-9)-(-8) + (-l)-(+10) + (+20). 
 
 26. 8-(-7) + (-14)-(-5)-(4-12)4-(+7). 
 
 27. -6-(-10) + (-f 20)-(4-15) + (-25). 
 
30 ELEMENTARY ALGEBRA 
 
 Graphic Representation of Addition and Subtraction of 
 Algebraic Numbers 
 
 41. It is not possible in arithmetic to subtract a number 
 from a less number. The introduction of negative 
 numbers makes it possible in algebra to subtract in all 
 cases. The diagram of section 29 is here reproduced and 
 used in illustrating the addition and subtraction of alge- 
 braic numbers. 
 
 I I I I I I I I I I I 
 
 _5 _4 _3 _2 -1 +1 +2 +3 +4 +5 
 
 1. To add a positive number. 
 
 To add + 2 to +3 begin at the point marked + 3 and count two 
 spaces to the right, ending at the point marked + 5. In like manner, 
 to add + 2 to any positive or negative number begin with that 
 number and count two spaces to the right. In general, 
 
 In adding a positive number^ the counting is to the right, 
 
 2. To subtract a positive number. 
 
 Since subtraction is the undoing of an addition, to subtract + 2 
 from + 5, begin at the point marked + 5 and count two spaces to 
 the left, ending at the point marked + 3. In general. 
 
 In subtracting a positive number^ the counting is to the left, 
 
 3. To add a negative number. 
 
 Since the sum of — 2 and — 3 is equal to — 5 [§ 33], to add — 2 to 
 — 3, begin at the point marked - 3 and count two spaces to the left, 
 ending at the point marked — 5. In general. 
 
 In adding a negative number^ the counting is to the left. 
 
 4. To subtract a negative number. 
 
 Since subtraction is the undoing of an addition, to subtract — 2 
 from — 5, begin at the point marked — 5 and count two spaces to the 
 right, ending at the point marked — 3. In general, 
 
 In subtracting a negative number, the counting is to the 
 right. 
 
INTRODUCTION 31 
 
 Remark. Notice that in the two operations of adding a positive 
 and subtracting a negative number, the counting is done to the 
 right ; while in the two operations of adding a negative number and 
 subtracting a positive number, the counting is done to the left. Hence, 
 The subtraction of a number and the addition of its opposite number 
 lead to the same result. 
 
 EXERCISE 12 
 
 Using the diagram in section 41, verify the following by 
 counting : 
 
 1. (_2)-h(+2)=0. 2. (-5) + (+9)= + 4. 
 
 3. (_2)-(+2)=-4. 4. (-5)-(-l)=-4. 
 
 5. + (+3)=-f3. 6. 0-(+3)=-3. 
 
 7. 0+(-3)=-3. 8. 0-(-3)= + 3. 
 
 9. + = 0. 10. 0-0 = 0. 
 
 11. (-l) + (_2)=-3. 12. (_4)-C+l) = -5. 
 
 13. (-f-3)-H(-4)=-l. 14. (+5)-(+8)=-3. 
 
 Multiplication 
 
 42. In arithmetic, 3x2 means 2 + 2 + 2. In algebra, 
 
 1. (+3)x(+2) means +2+(+ 2) + (+2); that is, 
 multiplication by a positive integer means that another 
 number is to be repeated positively. 
 
 2. (-3)x( + 2) means -( + 2)-( + 2)-( + 2); that 
 is, multiplication by a negative integer means that another 
 number is to be repeated negatively. 
 
 Note. (+ 3) X (+ 2) and (- 3) x (+ 2) are usually written 
 (+ 3)(+ 2) and (- 3) ( + 2), respectively. 
 
 43. The number repeated is called the multiplicand; 
 the number which shows how many times the multiplicand 
 is repeated is called the multiplier; the result in multipli- 
 cation is called the product. 
 
32 ELEMENTARY ALGEBRA 
 
 44. The possible combinations of signs in algebraic 
 multiplication are given in the following : 
 
 1. (+3)(+2) = + (-f 2) + (+2) + (+2)=2 + 2 + 2=4-6. 
 
 2. (+3)(-2) = + (-2) + (-2) + (-2)=-2-2-2=-6. 
 
 3. (- 3)(+ 2)= -(+ 2)-(+ 2)- (+ 2) = - 2 - 2 - 2 = -6. 
 
 4. (-3)(-2) = -(-2)-(-2)-(-2) = +2 + 2 + 2=+6. 
 
 45. From 1, 2, 3, and 4 of section 44 we may infer the 
 
 Rule of Signs in Multiplication 
 
 The product of two numbers with like signs is positive; 
 the product of two numbers with unlike signs is negative. 
 
 EXERCISE 13 
 
 Name, at sight, the product : 
 
 1. ( + 3)( + 2). 2. ( + 4)( + 3). 3. (+3)(-2). 
 
 4. (-4)(+3). 5. (-3)(-2). 6. (-1X-2). 
 
 7. ( + 1)(-1). 8. (-1)(-1). 9. ( + 2)(-3). 
 
 10. (-3)(-3). 11. (-5)( + 2). 12. (+7)(-3). 
 
 13. (+2)(+3)(-4). 
 
 Suggestion. Multiply +3 by + 2, then find the product of this 
 result and — 4. 
 
 14. (+2)(+3)(-2). 15. (_2)(-3)(-4). 
 16. (_2)(+3)(-5). 17. (+5)(-7)(-2). 
 18. (_1)(+1X-1). 19. (+l)(_l)(+l). 
 
 20. In finding the product of three factors, what is the 
 sign of the product when only one of the factors is nega- 
 tive? When all of the factors are negative? When two 
 of the factors are negative ? 
 
Franciscus Vieta (Francois Viete) (1540-1603) was a French 
 lawyer who gave up most of his leisure to mathematics. He was 
 the author of the earliest work on symbolic algebra. He introduced 
 in this work the use of letters for known and unknown numbers. 
 His solution of the cubic equation continues in use at the present 
 time. 
 
INTRODUCTION 33 
 
 Division 
 
 46. In arithmetic, 12 (the dividend)-!- 3 (the divisor) 
 = 4 (the quotient), because 8 x 4 = 12. 
 
 That is, 2)/i;/5o;. X Quotient = Dividend. 
 
 This relation connecting divisor, quotient, and dividend 
 may be employed in establishing the rule of signs in 
 division, thus : 
 
 1. (+12)-f-(+4) = +3, since(+4)(+3)= + 12. 
 
 2. (+12)-(-4)=-3, since (- 4)(- 3)= + 12. 
 
 3. (-12)-(+4)=:-3, since (+4) (-3) = -12. 
 
 4. (-12)-(-4) = -f 3, since (-4)(+3) = -12. 
 
 From 1, 2, 3, and 4, we may infer the 
 
 Rule of Signs in Division 
 
 77ie quotient of two numbers with like signs is positive; 
 the quotient of two numbers with unlike signs is negative. 
 
 BXBRCISI 
 
 : 14 
 
 
 Name, at sight, the quotient : 
 
 
 
 1. (+6)^(+2). 
 
 2. 
 
 (+9K(+3). 
 
 3. (+6)^(-2). 
 
 4. 
 
 (-6) + (+2). 
 
 5. (-8)^(+l). 
 
 6. 
 
 (_3)^(-l). 
 
 7. (-24)^(-12). 
 
 8. 
 
 (-27)^ (-3). 
 
 9. (-14)^(-2). 
 
 10. 
 
 (+27)^(+9). 
 
 11. (+30)h-(-8). 
 
 12. 
 
 (+56)^(-8). 
 
 13. (+12)--(-t-3)^(+2). 
 
 Suggestion. Divide (+ 12) by (+ 3), then divide the resulting 
 quotient by (+ 2). 
 
 14. (+24) + (-2)H-(-3> 
 
 15. (_18)^(-3)-v(-3). 
 
CHAPTER II 
 
 FUNDAMENTAL PROCESSES 
 
 Addition 
 
 47. Addition of monomials which have no common factor. 
 
 1. The sum of 3 a, 2 5, and 5c is -\- Sa -\- 2b -\- 5c. 
 
 2. The sum of 3«,26,and -5cis +Sa -{-2b + (- 5 c), 
 which is equal to H-3a4-25 — 5e. 
 
 In general, 
 
 The sum of two or more monomials which have no common 
 factor is expressed by writing the monomials in turn, each 
 preceded by its own sign. 
 
 Remark. For convenience, a plus sign before the first term is 
 usually omitted. 
 
 Thus, +3a + 2& + 5cis written 3 a + 2 ft + 5 c. 
 
 48. Commutative law for addition. In arithmetic, it is 
 not necessary to call attention to the obvious fact that, 
 for instance, 2 -}- 3 = 3 + 2. In algebra, it is assumed 
 that the sum does not depend on the order in which the 
 terms are taken; this assumption is usually referred to 
 as the commutative law for addition. This law is ex- 
 pressed by the formula, 
 
 a^h=b-\- a. 
 
 EXERCISE 16 
 
 Name the sum in each of the following : 
 1. a and c. 2. x and 1. 
 
 3. b and <?. 4. 4 and a. 
 
 34 
 
FUNDAMENTAL PROCESSES 35 
 
 5. «, 5, and c. 6. 2 a, — 5, and — c. 
 
 7. 4 a;, 2^, and — 2. 8. — 4 m, — 5 ii, and p, 
 
 9. 3(a + 6)and -4(<?4-c?). 
 
 10. 2(2: + ^). and 3(m — n). 
 
 11. a, 5, <?, and — c?. 
 
 12. 3(a— 5) and 4(a + 6). 
 
 13. B^, r\ and Rr. 
 
 14. 4 r, - 2 y, 2/, and - 3 i. 
 
 49. Addition of monomials which have a common factor. 
 
 1. 2a + 3a = (2 + 3)a = 5a. [See problem 9, Exer- 
 cise 2, page 8.] 
 
 2. 2aH-3a-4«=(2-f3-4)a = a. 
 
 3. 2a + 3a-4a-5a = (2 + 3-4-5)a=-4a. 
 In general, 
 
 The sum of two or more monomials which have a common 
 factor is equal to the sum of the coefficients of the common 
 factor multiplied by the common factor . 
 
 EXERCISE 16 
 
 (Solve as many as possible at sight.) 
 
 Add: 
 
 
 
 
 1. 
 
 2. 
 
 3. 
 
 4. 
 
 4:X 
 
 — 4m 
 
 — m 
 
 Sab 
 
 3rr . 
 
 ' — 5m 
 
 — 5m 
 
 4ab 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 -2a 
 
 3mri 
 
 9a% 
 
 — 4a6c? 
 
 5a 
 
 — 8mw 
 
 -Sa% 
 11. 
 
 9abc 
 
 9. 
 
 10. 
 
 12. 
 
 -%a^ 
 
 irBP' 
 
 2(a + b) 
 
 ab(m-\-ny 
 
 ^7^ 
 
 AirB^ 
 
 3(a + 5) 
 
 -3a6(m+7i)2 
 
36 ELEMENTARY ALGEBRA 
 
 13. 
 
 
 14. 
 
 
 15. 
 
 
 16. 
 
 3a 
 
 
 -6m2 
 
 
 2m7i2 
 
 
 - xyz 
 
 4a 
 
 
 -8m2 
 
 
 3mn2 
 
 
 -4:xyz 
 
 5a 
 
 
 -5m2 
 
 
 5 mw2 
 19. 
 
 
 -2xyz 
 
 17. 
 
 18. 
 
 20. 
 
 -4m 
 
 
 -5a 
 
 
 — mw 
 
 
 - 18 xyz 
 
 5 m 
 
 
 7a 
 
 
 — 3mw 
 
 
 12 xyz 
 
 3m 
 
 
 -3a 
 22. 
 
 
 — bmn 
 
 
 xyz 
 
 21. 
 
 23. 
 
 24. 
 
 - :r2 
 
 
 - 10 pq 
 
 
 - Its 
 
 
 aW 
 
 3a:2 
 
 
 pq 
 
 
 9^8 
 
 
 - 8 ah^c^ 
 
 2^2 
 
 
 - ^^P^ 
 
 
 - ^« 
 
 
 - 6 a62tf2 
 
 -5a^ 
 
 
 - l^pq 
 
 
 - 11 ts 
 
 
 9a62c2 
 
 25 
 
 
 26. 
 
 
 27. 
 
 
 28. 
 
 2(a 
 
 + *) 
 
 - (m -f 
 
 71) 
 
 (^ + 
 
 1) 
 
 3(rr + ;5^)2 
 
 -4(a 
 
 + h} 
 
 -3(m + 
 
 n) 
 
 - 2(r + 1) 
 
 -4(a:H-y)2 
 
 5(a 
 
 + *) 
 
 5(m + 
 
 n) 
 
 -8(r4- 
 
 1) 
 
 - 8(0. + yY 
 
 - 6(a 
 
 + ^) 
 
 4(m + 
 
 n) 
 
 6(r + 1) 
 
 ii(^ + yy 
 
 29. 2 a;, 3 a;, and 5 x. 
 
 30. — mn, — 2 mw, and — 5 mn, 
 
 31. — 4 w, 5 w, and — n. 
 
 32. ^2^, _ 5 52^^ and 9 62^. 
 
 33. - R, -6R, and 8 7?. 
 
 34. 4 /,e5 _ 8 5^^ and - 3 brA 
 
 35. m2n, — 3 m2n, — m2w, and 4 m2n. 
 
 36. 2 7ri22, _ 4 ^i22^ 7ri22, and 3 wB^. 
 
 37. — 6 a;y2^ 10 xy^^ — xy^^ and — 4 xy^. 
 
 38. — J iB2, :| a^, — x^^ and | a:2. 
 
FUNDAMENTAL PROCESSES 37 
 
 39. (w + w), — 3(m + 7i), and — ^(m -f n). 
 
 40. a(2:. + y)\ - <o a{x + y^, 8 a(x + 2^)^. 
 
 Just as the sum of 3 m and 5 m may be written (3 + 5) m, so the 
 sum of am and 2 m is written (a + 2) ?w. Similarly, am + 6m = (a + 5)/w. 
 aw + bm — m =(a + 6 — l)/w, and c(m + n) + d (m-^ n) = (c -\- d)(m-^ n). 
 
 Add by combining the coefficients : 
 41. cy and by. 42. aa; and — x. 
 
 43. a5^ and (?6y. 44. mxb and — cyJ. 
 
 45. ^ aB Siud^ ab, 46. ^ J?LB and — J iTJ. 
 
 47. am^ — m, and <?w. 48. ttB^, tt/^, and 7ri2r. 
 
 49. i HttB^ -\- i Rirr^ -h i RirRr, 
 
 50. a(2: + y) and ^ b(^x+ y^. 
 
 51. ^(?/ + z) and r(y + z). 52. a(a: — y) and (a; — y). 
 
 50. Addition of monomials. The sura of 2 a, 3 5, — 3 a, 
 
 5 c, 2 6, — 6 a, and 7 b may be found as follows : 
 
 2a + 36-H<-3a)+5c + 26+(-6a)+7 6 
 
 = 2a + 3&-3a + 5c + 26-6a + 76 [§40] 
 
 = (2a-3a-6a)+(3 6 + 2& + 7&)+5c [§48] 
 
 = (2 - 3 - 6)a +(3 + 2 + 7)6 + 5c [§ 49] 
 
 = -7a + 12 6 + 5c. 
 In general, 
 
 TAg s-WTw of two or more monomials is obtained by finding 
 the sum of those that have a common factor and then adding 
 these sums. 
 
 51. Associative law for addition. In arithmetic, it is not 
 necessary to call attention to the obvious fact that, for 
 instance, 2 + 3 + 1 + 6= (2 +3) + (1 + 6) = 5 + 7 = 12. 
 In algebra, it is assumed that the sum does not depend on 
 the grouping of terms ; this assumption is usually referred 
 to as the associative law for addition. This law is ex- 
 pressed by the formula, 
 
 fl + (&+c) = (a+6) + c. 
 
38 ELEMENTARY ALGEBRA 
 
 EXERCISE 17 
 
 Add: 
 1. a and h. 2. a and —h. 3. a and 3 h, 
 
 4. 2 a and 3 5. 5. 3 a and —^x. 6. a, h, and c. 
 7. a:, ^, and— z. 8. m, — ti, andp. 9. 2 a, 3 5, 4 a, and 55. 
 
 10. ^x, 4a^, - 5a;, and -%x^. 
 
 11. — 3 a, 4 5, 5 a, — c, 4:c^ and — a. 
 
 12. — a;^, 2 rrz, 3 yz, 4 xt/^ 5 ?/z, and — 6 a:2. 
 
 13. 3 m^, — mn, 2 ^2^ 3 mn, —2n\ 4 w^, and 5 mn. 
 
 14. - 3^2^ 22:?/, 5^2^ 22^, - 4a:«^, and 62^2, 
 
 15. a\ - 2 a5, iX -a^ -2 ah, and - 52. 
 
 16. 4a25, 2ab, - bah\ - 6a% 2a% -ah^, -iah, and 
 8 a25. 
 
 17. 2(w + n), 3(m — w), — 5(m + n), and —(m — n). 
 
 18. 4(2^ + 2^), _4(a;-y), -2(a:-y), -(a: + y), 
 - 3(a; + ?/), and - (a; - 3/). 
 
 19. 2(a;-3^), 52, -4, -Cx-y\ -7, -452, and (a:-y). 
 
 20. a25, a52, 3 ab, 4 ^25, _ 6 ^25^ _ 5 ^52^ and - ab. 
 
 21. a2, a, (a H- 6), — 3 a, — 4 a2, — 5(a + 5), and 
 4(a + 5).. 
 
 22. x^i/ + 3 a;y — 2 a;y2 4. 5 a^t/ — xy^ — \xy-\-2 x^y. 
 
 52. Addition of polynomials. Since a polynomial is the 
 sum of two or more monomials, no additional rules are 
 necessary for finding the sum of two or more polynomials. 
 However, it is convenient to arrange the terms so that 
 like terms stand in the same column, then to add the 
 columns separately and combine the sums by section 50. 
 
FUNDAMENTAL PROCESSES 39 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Find the sum of 2a;+3?/ — 4 2, 'Sx — i/ -\-2z^ and 
 4:x-{-'2y — 5z, and check the work by letting a; = 1, y = 1, 
 and 2 = 1. 
 
 Solution Check 
 
 2a: + 3?/-43 2 + 3-4 = 1 
 
 3x- y-\-2z 3-1 + 2 = 4 
 
 4a: + 2y-5z 4 + 2-5 = 1 
 
 9a; + 4y-72: 9 + 4-7 = 6 
 
 2. Fin4 the sum of Ta;— 4(?/ + 2), 6 a; + 2(?/ -f- 2), 
 2 a: +(^4- 2), and a: — 3(«/ +- 2), and check the work by 
 letting a; = 2, y = 1, and 2 = 1. 
 
 Solution Check 
 
 7 a: - 4(?/ + z) 14 - 8 = 6 
 
 6x + 2(2/ + 3) 12 + 4= 16 
 
 2x+ {y -{-z) 4 + 2= 6 
 
 a: - 3(y + z) 2-6=- 4 
 
 16 a: - 4(y + 3) 32 - 8 = 24 
 
 Remark. Note that the coefficient of (y + s) in the third expres- 
 sion is 1. 
 
 3. Find the sum of ax +- hy^ hx — ady^ and cdx + ay^ 
 and check the work by letting a=2, i = — 1, <7 = 3, c? = l, 
 a; = 1, ^ = 1, and 2 = 1. 
 
 Solution Check 
 
 ax -\-hy 2-1=1 
 
 hx-ady - 1 - 2 = - 3 
 
 cdx + ay 3 + 2 = 5 
 
 (a + 6 + cd)x +(b - ad -{■ a)y 4 -1 = 3 
 
 EXERCISE 18 
 
 Add the following polynomials and check the results ; 
 1. a + h 2. a-\-b — c 
 
 a — h a -\- c 
 
40 ELEMENTARY ALGEBRA 
 
 3. a — b + 2c 4. a-\-h — c — d 
 
 a-\-b — 2c *2a — h+c—'2d 
 
 5. rr — ^ 4- 2 
 -x+y-z 
 
 7. 3 m — 2 w 
 
 4 771 H- 3 71 
 
 9. 
 
 - ab-\-S(^d 
 
 11. 
 
 a— b + c 
 -2a+46-6tf 
 
 6. 
 
 x-2y 
 
 
 X -2 
 
 8. 
 
 4:X-\-l y 
 
 
 -5x-Sy 
 
 10. 
 
 3 m^n — 8 mn^ 
 
 
 4 rrv^n — 3 mn^ 
 
 12. 
 
 ^7nn-2m^+ bn^ 
 
 
 -2 7WW+3m2-ll7l2 
 
 13. Ja + J5 14. |a + ^ 
 
 15. |aH-|5 16. _12:4-J«/-J2 
 
 i a-^-^b — x-\- y- g 
 
 17. i^ + ilZ-i^ 18- .5a;+.3y 
 
 19. .3 a + .2 a; 20. 1,2m -{-l.bn 
 
 5a— .Sx .37/1— .571 
 
 21. 2w + 4:v 22. S7rI^-27rRI£ 
 
 6w-7v -47ri22+ tt/?^ 
 
 8M;4-2t; - ttJ^^ -|- ttEH 
 
 23. 2^-f- J?- O 24. 2a:2- 2:i/+ y2 
 
 3^.-4^4-6(7 -42:2 4.4a;^_2^2 
 
 -5^1 + 4^-6(7 3r»-5a:y + 3.y2 
 
 25. 4 7w-2w + 3 26. 3a25- a5 + 4a^2 
 
 677i + 5w-l . - a% + 5ab-Sab'^ 
 
 -2m-4n-4: 2a%-6ab 
 
FUNDAMENTAL PROCESSES 41 
 
 27. m^ -\-6n^ 28. a^ + 3 ab 
 
 -Sm^-4:mn+ n^ - 3 a^ + ab -\- b^ 
 — 2 mn — 6n^ 6 a^ — 4 J^ 
 
 29. 3(w + n)— 3(7w — Ti) 30. 4(2^ — 3/)+ («^'--v) 
 
 6(m + ri) + (tw — n) — 6(2; — y)— '^{w — v) 
 
 — 2(m4-^) — 4(y/i — n) — \{x — y) + 8(e^ — v) 
 
 31. 2 a: H- 3 ^ - 2, a; - 4 y + 4, and 3 2; - 4 ^ — 8. 
 
 32. 3^2 — 2 m^i — w^, 4 w^ — 2 wn + 4 w^, and 
 
 — W^ — 4 WW + 2 7l2. 
 
 33. 3 2;2 + v^ - 2 ^2^ ^2 _ 2 v^ + t;2, and 4 v^ - 8 ^2 _ ^2. 
 
 34. ^_s2_^2^ 4r2H-2^^ 6 82_2r2, and4r24.2^2_3^. 
 
 35. ^Jrx^^-x-\, 3a:2-42;4-4, "io^-l^-^, and 
 2a:3_^_l. 
 
 36. a;3_^3^^^^2^ 1^-Zxy'^-1y\ 4 x^ -{- d 2^y 
 + 3 a;^2 _ ^2^ and 3 a^y — 2 a;^2 _j_ ^3^ 
 
 37. m^ — 3 7?i2n, 4 7ww2 — 2 w'^, w^ — w^, 2 7wri2 — ^2/1, and 
 n^ 4- 3 m7i2. 
 
 38. 3(w + w) + 2(2:-y), -2(m + w)-3(a;-y), (w + w) 
 + (re — ^), and 4(m + n)— 2(x — y). 
 
 39. am + 5w, 6m + en, and ciw + gw. 
 
 40. ax^ + 6^^ ^^^ — t?^2, and 2^2 _ yi 
 
 41. a6a; + w^, ex + wjt?^, and ^2; — ry. 
 
 42. 4 r + 3 r2, Trr H- 7rr2, and cr + 
 
 ar 
 
 43. ^22; + 52^, — 7i2^ — c2^, and jt?22; — y, 
 
 44. a(m H- w) + b{m — n), c(m 4- w) — e(m — w), and 
 d(^m-\- ri)— hQm — n). 
 
 45. a2; + 52:?/ — cz, 2 x -\- S xy -{■ z, and cx — xy-{- az. 
 
 46. 3 2:3 _^ 2 2:2^, 3 2^2?/ - 4 ^^2^ 5 2:?/2 _ 4 ^3^ and 3 ^3 _ 2 a^, 
 
 47. a2:3 ^ 52;2 4- C2; H- (^ and 3 2^3 _ 2 2^2 ^ 3 ^ _ 5^ 
 
42 ELEMENTARY ALGEBRA 
 
 Subtraction 
 
 53. Subtraction of monomials. Since monomials are 
 numbers, what is stated in sections 38 and 39 is applicable 
 to monomials. Hence, for subtracting one monomial 
 from another we have, from section 39, the following : 
 
 Rule. Change the sign of the subtrahend and add the 
 result to the minuend. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. From 5 m take 2 m. 
 
 Solution. 5m-(+2w)=5m-2m 
 
 [§40] 
 
 = 3m. 
 
 
 2. From 6 a; take -2x. 
 
 
 Solution. Qx-{-2x) = Qx + 2x 
 
 [§40] 
 
 = d>x. 
 
 
 3. From — 8 y take 3 y. 
 
 
 Solution. - 83/ - (+ 3 .v) = - 8;/ - 3 ?/ 
 
 [§40] 
 
 = -ll.y. 
 
 
 4. From —Qxy take — 2 xy. 
 
 
 Solution. - Q xy - {- 2 xy)= -Q xy -{- 2 xy 
 
 [§40] 
 
 = — 4:xy. 
 
 
 5. From X take y. 
 
 
 Solution. X -(+ y)= X — y. 
 
 [§40] 
 
 6. From — 3 xy take 2 yz. 
 
 
 Solution. -Sxy-(+2yz)=-S xy - 2 yz. 
 
 [§ 40] 
 
 7. From —2mn take - 4 vs. 
 
 
 Solution. — 2 wn - ( - 4 rs) = - 2 wm + 4 rs 
 
 [§40] 
 
 = 4 r.s — 2 mn. 
 
 [§48] 
 
 8. From amn take bmn. 
 
 
 Solution. mnn -(+ l)tnn)= amn - b7Jin 
 
 [§ 40] 
 
 = (a — b)mn. 
 
 
FUNDAMENTAL PROCESSES 43 
 
 EXERCISE 19 
 
 (Solve as many as possible at sight.) 
 
 In the first three examples subtract the lower mono- 
 mial from the upper monomial. 
 
 1.5a; — 7a 5w — 5y 1 z 
 
 2x —4a 7m -\-2y -4z 
 
 2. 
 
 -5E 
 -IB 
 
 -%^y 
 
 — a^ 3r^ mn 
 -5a2 7^3 4mn 
 
 — xyz 4(m + w) 2(x-\-y) 
 xyz 3(w + w) 7{x-\-y^ 
 
 -2r8 
 -6r8 
 
 3. 
 
 -5(r + «) 
 
 4. From 3 x take 9 x. 5. From 4 y^ take — 7 y^. 
 
 6. From — 7 tww take 2 mn. 7. From 4 Tri?^ take irB^. 
 
 8. From - 6 7-2 take - r2. 9. From TriJ^ take jTri^^. 
 
 10. From 6{m + n} take —2(m + 7i), 
 
 11. From (r - 1) take - 2 (r - 1). 
 
 12. From 2a(x + y^ take 3 a (rr + y) . 
 
 13. From 52 (a; _ ^) take -2Iy^Qx-y). 
 
 14. From subtract x. 15. From 1 subtract — a. 
 16. From a subtract — b. 17. From — a take — 6. 
 
 Subtract as indicated : 
 
 18. 
 
 m- (-11). 
 
 19. 
 
 2^-( + y). 
 
 20. 
 
 ax-C-6y 
 
 21. 
 
 a2_(_ ww). 
 
 22. 
 
 c2-(-(?0. 
 
 23. 
 
 _^2_^_^2). 
 
 24. 
 
 — 4a:y — (+ Suv). 
 
 25. 
 
 1^^2_(+^2). 
 
 26. 
 
 — cm^ — (^— dm^). 
 
 27. 
 
 ax— (— 6a;). 
 
 28. 
 
 3(^-y)-«(^-^). 
 
 29. 
 
 w(a;-|-y)-w(a;-f-2^). 
 
44 ELEMENTARY ALGEBRA 
 
 54. Subtraction of polynomials. In adding 2 a — Sb to 
 3 a 4- 5 5 — c, each term of 2 a— 3 J is added to 3a + o5 — <?; 
 the resulting sum is 5a-{-2h— c. Therefore, in the in- 
 verse operation of subtracting 2a— Sb from 5a-\- 2b — c, 
 each term of 2a — Sb is subtracted from ba+ 2b — c; 
 the resulting difference is Sa-\- 5b — c. 
 
 The work may be arranged thus : 
 
 6a -{■2b - c 
 -2a + 36 
 Sa + 5b - c 
 
 In accordance with section 39, the sign of each term of 
 the subtrahend, 2 a — Sb^ has been changed and the result 
 added to the minuend, 5 a -\- 2 b — c. In general. 
 
 To subtract one polynomial from another, change the sign 
 of each term of the subtrahend and proceed as in addit/ion, 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Subtract ^x—2y from 5 a; + 3 ?/, and check the work 
 4)y letting x— 1 and y — 1. 
 
 Solution 
 
 (5a: + 3y)-(3a:-23/) = (5:r + 3 3/) + (-3a: + 2y) 
 = 5 a: - 3 a: + 3 2/ + 2 y 
 = 2x + 5y. 
 
 Another Solution 
 
 5 a; + 3 ly 
 
 -3a: + 2y 
 
 2x + 5y. 
 
 Remark. In practice, the change of the signs in the subtrahend 
 and the addition should be performed mentally. Thus, in the above 
 example the work should be arranged as follows : 
 
 Solution Check 
 
 bx+^y 5 + 3 = 8 
 
 ^x-2y 3-2 = 1 
 
 2x+ 5y 2 + 5 = 7 
 
FUNDAMENTAL PROCESSES 45 
 
 2. Subtract ^a— 2xi/-{-5b from 5a — 35-}-2c, and 
 check the work by letting a = 1, 5 = 1, c = 1, a; = 1, and 
 
 Solution Check 
 
 5a-'db +2c 5-3 +2 = 4 
 
 3a + 5&-2xy 3 + 5-2 =6 
 
 2a - 86 + 2a:y + 2c 2-8 + 2 + 2=-2 
 
 3. Subtract bxy -{- 2bz— et from axy — 2bz -{-St^ and 
 check the work by letting a = 1, 6 = 1, c = 1, ^ = 1, 2: = 1, 
 and y = 1. 
 
 Solution Check 
 
 aary-2&2 + 3< 1-2 + 3 = 2 
 
 bxy + 2bz- ct 1 + 2-1 = 2 
 
 (a - b)xy - 4 62 + (3 + c)< 0-4+4 = 
 
 EXEBCISE 20 
 
 Subtract and check: 
 
 1. 
 
 2x-\-y 
 x-y 
 
 2. 
 
 3m4-4n 
 2m + 2n 
 
 3. 
 
 5a + 35 
 2a- b 
 
 4. 
 
 4m4-3w 
 
 2m-\-Sn 
 
 5. 
 
 3x-Sy 
 Sx-9y 
 
 6. 
 
 4a-35 
 ba-^b 
 
 7. 
 
 5mn—l 
 2mn-\-l 
 
 8. 
 
 5a;2- X 
 la^-h2x 
 
 9. 
 
 27n%2+4 
 -3^27^2+5 
 
 10. 
 
 2x^-ly^ 
 62:2-8^2 
 
 11. 
 
 -ia-ib 
 
 12. 
 
 4^2-. 4^2 
 - m2-.6/i2 
 
 13. 
 
 bx + my 
 cx-\-my 
 
 14. 
 
 am^ — bn^ 
 
 15. 
 
 2irRH+'TrR'^ 
 irRH-irR'^ 
 
 16. 
 
 4:x+^y-2 
 x-Sy+2 
 
 2m + Sn — 4p 
 Sm^-in— 5p 
 
 17. 
 19. 
 
 a + 
 — a — 
 
 b + e 
 b-e 
 
 18. 
 
 4r-h 
 
 \8-6t 
 8-\- 5t 
 
46 ELEMENTARY ALGEBRA 
 
 20. -3a;2 + 2a;-4 
 
 22. 
 
 4 77^2 4. 2 m + 5 
 m2 -1 
 
 
 24. 
 
 2r + « 
 -5*+ r-8 
 
 
 26. 
 
 x^- y 
 4:x-\-2y-bz 
 
 -3 
 
 28. 
 
 3 a + 3(m + w) 
 a — 2(m 4- w) 
 
 
 21. 
 
 23. 
 
 ^+ y + 2 
 
 5a:-4^/ 
 
 52^ -3 
 2a^ + 4a:-5 
 
 
 25. 
 
 am -{- S bn -\- 4: dx 
 cm -\- hn — ^dx 
 
 
 27. 
 
 — m — b n — p 
 
 — Sm —2p- 
 
 -3 
 
 29. 
 
 -2(a + i)-2 
 
 
 30. 2(a; + ^)+3(m+w) 31. _3(a+ ^)2-4(c-(i)2 
 4(a; + y)+2(7/^ + ^) 2(a+^)^-4(g-(^)2 
 
 32. a + 5 - c - (^ 33. (a + hy -z-^S 
 
 Sa-b-c-^d 2(a -|- 5)^ + g - 2 
 
 34. 5(10)2 + 6(10) +7 
 2(10)2 + 4(10) +8 
 
 35. 7.68-3-62+2.6 + 1 
 6.63-8-62-3.6 + 4 
 
 36. From a + b take 3 a + 4 5 — c?. 
 
 37. From Sm^ -\-4:m — n take 4 9^2 — ^2 + w. 
 
 38. From ax^ -{-a^x-h2 a^aP take Sa^x^-S aH + 2 ax^, 
 
 39. From a - 1 take ^2 + a2 + a + 2. 
 
 40. From x-^-7?' take 5 rr^ _ 3 ^ 9 a:. 
 
 41. From w + 2(a + 6) take 3(a + J) - 2 w + 4. 
 
 42. Subtract 4 a;^2 __ 3 ^y _|_ ^.^ from xy. 
 
 43. Subtract 3 r^ - a; + 4 from 0. 
 
 44. Subtract 2 a» - 3 a + 4 from 3 a" - 2 a + 4. 
 
FUNDAMENTAL PROCESSES 47 
 
 45. Subtract la^-^a-^-b-^ from 4 + -i- 5 + a2 _,. i ^, 
 
 46. From ax^ — bi/^ -\- cz^ take ma^ — my^ + 2 s^. 
 
 47. From rr^—^T^ — 'p^ take a/^i^ _ 5^2 _ ^^^2 
 
 48. Take 3 m^ — ti — 4 from the sum of vr^ —Zm-\-\n 
 and 3 m^ — m — 6. 
 
 49. Take 1 -^ r^ — ^ -\- r% from the sum of 3 r^ 4- r« and 
 
 50. Take the sum of a^ — ^a-\- 2 and — a^ — ^ a^ -\- a 
 from 1. 
 
 51. Take a^ -\- x^ — x -\- 1 from the sum of .r^ — x^i/ H- a:^ 
 and — a:^ — xt/^ + 1. 
 
 52. From the sum of wB^ and lirRH+'lirB? take 
 4 7ri22 _ ^RH. 
 
 53. From the sum of a + b — c and 2« — 35+2c take 
 the sum of — a-^ 2h and 2a — 36 + 2 c. 
 
 54. Take the sum of 32: — 4^ + 5 and 3 y — 4 a: — 4 
 from 2. 
 
 55. From x-\- 1/ -[-z subtract the sum of a: — 2 y — z and 
 2x-y-\-2z. 
 
 If A = Sx-\-2y-5z% B = 2x-Zy^4:z\ and (7 = 
 
 — X -{■ y + z\ find the value of : 
 
 56. A^B-0, 57. A-B+C. 58. B-^C-A. 
 59. A- B-0, 60. B-C- A. 
 
 61. From 3 a^hc - 2 ab^ -\- 5 b^c take 2 a2^c + 2 ^^2 
 
 — 3 62c + 2 flic. 
 
 62. From 4 xyz^ — 2 a;!/22 -|- 7 a^?/3 subtract — 5 xyz^ 
 
 — 13 2:2/^2 + a^yz. 
 
 63. From 22(a + 6)2 + 5(« + 6)- 7 take (a + 6)2 
 -5(a + 6)4 3. 
 
 64. From 5(2; -^ y)+ 13(a + 6) - 2 take 7(a + 6) 
 -(2: + ^)+2. 
 
48 ELEMENTARY ALGEBRA 
 
 65. From x + (^ + zy + t^ take 6x + t/-^l, 
 
 66. How much greater than x-\-l is x^? 
 
 67. How much less than a + 35 — 4<?is 2 a-{- b -\- c^l? 
 
 68. What must be subtracted from a so that the re- 
 mainder is h? 
 
 69. What must be subtracted from x'^ — xy -\- y^ so that 
 the remainder is 2 a:^ _}_ 3 v 
 
 70. From the sum of W xy^ — \% :)iP'yz -\- VI xyz and 
 3 xyz — 2 Tp'yz -{- b x^ — S y take the sum of 5 xyz,— 13 xyz^ 
 + 12y -19x^ and 7 x^yz -lSx^-\- y. 
 
 71. From ax^ + bxy + cy^ take J aa^ _ 1. 5^^ _|. 2 cy^. 
 
 72. Take the sum of Sh^-2hk + Sk^-^2h-^k-\-l 
 and 4A;2-3^^+57i + 3^ + 2 from 2h^ + 2hJc-nJ(^ 
 4- 11 A -16 A; +7. 
 
 Parentheses 
 
 55. 2 a 4- (3 6 -H c) means that the number (3 6 + c) is 
 to be added to 2 a. 
 
 2 a + (3 6 — ^) means that the number (3 6 — c?) is to be 
 added to 2 a. 
 
 By addition, 
 
 2a-f(3 5-hO=2« + 3J + c; 
 and 2a-}-(3 6-0=2a+35-c. 
 
 Therefore, 
 
 1. If an expression in parentheses is preceded by the plus 
 sign^ the parentheses may he omitted. 
 
 2. An expression may be inclosed within parentheses pre- 
 ceded by the plus sign. 
 
 Thus, 
 
 1. 2 a + (3 6 - c + ^) may be written 2a + 3A-c + (f. 
 
 2. 2 a + 3 ?> - c + rf may be written 2a+(3 6-c + d), or 
 2a + 36+(- c + c?). 
 
FUNDAMENTAL PROCESSES 49 
 
 2 a — (3 6 H- e) meaus that the number (3 6 + <?) is to be 
 subtracted from 2a. 2a — (— 5 — <?) means that the 
 number ( — 5 — c) is to be subtracted from 2 a. 
 
 Since the result of subtracting a number is the same as 
 the result of adding the opposite number, 
 
 2a-(3 6H-(?)=2a + (-3 6-c) = 2a-36-c; 
 and 2a-(-6-c?)=2a4-(6-|-c)=2a+6 + c. 
 
 Therefore, 
 
 1. If an expression in parentheses is preceded hy the 
 minus sign^ the sign of each term within the parentheses must 
 he changed when the parentheses are removed. 
 
 2. An expression may he inclosed within parentheses pre- 
 ceded hy the minus sign, provided that the sign of each term 
 be changed. 
 
 Thus, 
 
 1. 2 a — (3 6 + c) may be written 2a — Sb - c. 
 
 2. 2a - 3b + c - d may be written 2a-(Sb-c + d) or 2 a 
 -Sh-(-c + d). 
 
 Remark. Observe that when the parentheses preceded by the 
 minus sign are removed, the minus sign before the parentheses 
 is omitted. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Simplify 2 + [3 -(5 -2)]. 
 
 Solution. 2+[3 -(5-2)] = 2+[3- 5 + 2] 
 
 = 2 + 3 - 5 + 2, or 2. 
 
 2. Simplify 2a -[3 a- (2 a- 5)]. 
 
 Solution. 2a-[3a-(2a-Z>)] = 2a-[3a-2a + 6] 
 
 = 2a — 3a + 2a — 6, ora — 6. 
 
 3. Simplify 1 - [2 a: + J3 y - (4 « + 5) j] + [5 a: 
 -S4^4-(32-2)S]. 
 
50 ELEMENTARY ALGEBRA 
 
 Solution. l-l2x + {3y -(4:z + 5)}] + [5a: - {4y +(32 - 2)}] 
 = 1- [2a: + {3 2/ -42- 5}] + [5x - {4:y + Sz - 2]'] 
 = l-[2a: + 3y-42-5] + [5x-42/-3z+2] 
 = l-2a;-3y + 4z + 5 + 5a:-43/-32 + 2 
 = S + 3x-7y + z, 
 
 Remark. It will be observed in the solutions of illustrative 
 examples 1, 2, and 3, that the innermost parentheses have been re- 
 moved first. Although it is not essential to do so, yet the beginner 
 is advised to proceed in this manner. 
 
 4. Indicate that from the sum of 2a-\-Sb — c and 
 Sa + b-\-2c the sum of a — h-\-c and 4a+-26 — 3(? is to 
 be subtracted ; then perform the indicated operations. 
 
 Solution. 
 
 [(2a + 3&-c) + (3a + & + 2c)]-[(a-& + c) + (4a + 26-3c-)] 
 = [2a + 36-c + 3a + & + 2c]-[a-ft + c + 4a + 26-3c] 
 = [5a + 46 + c]-[5a+ />-2c] 
 = 5a + 46 + c - 5a - 6 + 2c 
 = 3 & + 3 c. 
 
 EXERCISE 21 
 
 (Solve as many as possible at sight.) 
 Remove the parentheses and simplify when possible : 
 
 2. a:+-(y-3). 
 
 4. m—{n—p). 
 
 6. x-\-(2x-y). 
 
 8. 5 + (7 -2). 
 
 10. 8 -(5 -4). 
 
 12. ^x — (2x — x). 
 
 14. r— (— 3r-fr). 
 
 16. 2k — [m — w] . 
 
 18. 9-7"=^. 
 
 20. 7 -(8 -5)- 2. 
 
 22. 6-(5-l)-(4-3). 
 
 1. 
 
 a-\-(h + c). 
 
 3. 
 
 x-iy-^z). 
 
 5. 
 
 r-(B-ty 
 
 7. 
 
 a-(a-\-b). 
 
 9. 
 
 6 -(3 + 2). 
 
 Il- 
 
 3-(-2-l). 
 
 ia. 
 
 2m-(-3m-m). 
 
 15. 
 
 7rH-!-2«-«S. 
 
 17. 
 
 8-5+2. 
 
 19. 
 
 9-(64-2) + 3. 
 
 21. 
 
 8-(6 + l)-(8-7) 
 
FUNDAMENTAL PROCESSES 51 
 
 23. a-(a-^) + (a- 25). 
 
 24. 2x-(x — i/} — (2x + 2i/), 
 
 25. 2^-(32^-6i/)-(z^-2y + 3y). 
 
 26. 4a: + [3a: +(3 2: -2)]. 
 
 27. 7a:- [3a:-(6«/ + 22)]. 
 
 28. 2m + 37i + fm — (m— n)}. 
 
 29. a:-2^-52a:-(a:-3^)S. 
 
 30. 6r+ J37--(2r-0-3«i. 
 
 31. 4m— [2w — (3m — n)-|-2 7i]. 
 
 32. (3r + s)-{r-(2r-«) + 3r|. 
 
 33. (4^-^)-[-;>-(3^ + ^)-4^]-(;> + ^). 
 
 34. a:- [a:-(«/ + 2)-Ja:-(^-2)i]. 
 
 35. ll_J10-[9-(8-7-6a:)]i. 
 
 36. 2-S-2-[2-(-2-2T2-2)-2]i. 
 
 37. r — \_— r — I — r — (r -\- r — r — r — r} — r \ — r^. 
 
 In examples 38-44, remove only the inner parentheses 
 and simplify when possible. 
 Thus, [a - (6 - c)] = [a - & + c]. 
 
 38. [a:-(^/ + 2)]. 39. [a: + (y + 2)]. 
 
 40. [m — (w — 1)]. 41. [{rn + n) — (p — q)], 
 
 42. [(2a:+3^)-(2a:-3y)]. 
 
 43. [(a:2_^2)_(^+2./2)]. 
 
 44. 5 (m + n) — (m — 7i) j . 
 
 In examples 45-50, indicate the operations before per- 
 forming them. 
 
 45. To3a2_ladda2_3a-Hl. 
 
 46. From 3 a take a — 2. 
 
 47. Add 7^-2o?y, 3 xy^ - y\ and a:^^ - 2 xy^. 
 
52 ELEMENTARY ALGEBRA 
 
 48. From 3 y take ^ — 5. 
 
 49. Add 1 to the sum of 3 a — 3 and 2 a + 2. 
 
 50. Take the sum of 3a — 26 +5 and 6 — 5a — 2 from 2. 
 
 Inclose the last three terms of the following polyno- 
 mials within parentheses, preceded by a plus sign. 
 
 51. 
 
 a-\-h-\-c -\-d. 
 
 52. 
 
 — a-^h — c-{- d. 
 
 53. 
 
 a-\-h-\-c—d. 
 
 54. 
 
 a-{-b — c — d. 
 
 55. 
 
 m — w+jp + l. 
 
 56. 
 
 m — n—p — 1. 
 
 57. 
 
 2^2 + ^2 _,.22/ + l. 
 
 58. 
 
 m^ — n^ — 2 n — 1, 
 
 59. 
 
 r2_82^2«-l. 
 
 60. 
 
 x^ — y^ — 2 yz — z^. 
 
 61-70. Inclose the last two terms of the polynomials 
 in examples 51-60 within parentheses, preceded by a plus 
 sign when the sign of the third term is plus, and by a 
 minus sign when it is minus. 
 
 71-80. Inclose the last three terms of the polynomials in 
 examples 51-60 within parentheses, preceded by a minus 
 sign. 
 
 EXERCISE 22. — GENERAL REVIEW 
 
 1. Simplify by collecting like terms : 
 2x—Sy-\-z — i/-^Sz-{-2x — Sx — y-^4:y — i2, 
 
 2. Evaluate a[^—2x^ — x-\-l when x = 1. 
 
 3. Evaluate a^ — 4: x^ -\- x -\- 1 when x=2. 
 
 Given 7r= 3.1416, find to three places of decimals : 
 
 4. 2 7rr, when r = 1. 5. 2 7rr, when r = 1.5. 
 6. 7rr, when r = 2. 7. 7rr, when r = 2.5. 
 8. 4 Trr^, when r = 4. 9. ^ Trr^ when r = 3. 
 
 10. Express the sum of the squares of a and h; the 
 difference of the squares of c and d. 
 
FUNDAMENTAL PROCESSES 63 
 
 11. If m and n represent two numbers, what does 
 m-\-n represent? What does m — n represent? m^ -\-n^'^ 
 TT^ — r^^. (771 + w)2 ? (m — 7i)2 ? 
 
 12. Express the sum of the cubes of x and y\ the 
 difference. 
 
 13. Simplify by collecting like terms : 
 
 3 a26 + 2 aJ - a^^- 2 a% - 3 ab^ - 2 a% -\- 2 ab -^ 4: ab\ 
 
 14. Add a; + 2y4-3z, 2x — y — 2z^ y — x— z^ and 
 
 15. Add a4-2a3_^3a2, a + a^ -\- a\ 2a^4-3a3, a2+5a 
 - 2, and - 2 r- 3 a - 4 a2. 
 
 16. Add -2(x-y), 6(x-y), -4(a:-2/), and 7(a:-y). 
 
 17. Simplify by collecting like terms : 
 
 ^x- 5(y - 2) — 2 a; + (^ - z) + 2; - 2(y - 0) 4- 2 2. 
 
 18. From 5 a take 2 a — 3. 
 
 19. From 3 take x^— x — 1. 
 
 20. From a^ -\- 2 ab -\- b^ take a^-2ab-\- b^. 
 
 21. From 2 take the sum of 2a— 3^ — 4 and 2 — 2b-\- a. 
 
 22. From 3 a6c - 2 a%x -\-l ay-2 take - 3 a6c - 3 a25a; 
 + 7 ay + 2. 
 
 23. What must be added to m — w + jt? to make 2 jp ? 
 
 24. What must be added to a: + y — z to make ? 
 
 25. What must be subtracted from a-\-b + c to make 
 a-b-\-cl 
 
 26. If 2 « = a + 6 + c, what expression is equal to 
 aH-6-c? 
 
 27. Given « = (a + 5) ^ ; for what number does « stand 
 when a = 1, ft = 3, and w = 6 ? 
 
54 ELEMENTARY ALGEBRA 
 
 28. Simplify by combining the terms having the same 
 powers of m, so as to have the plus sign before each set of 
 parentheses : 
 
 am^ + Jw2 + <?w — 3 w^ — 4 m — 7 — 5 m^. 
 
 29. Simplify by combining the terms having the same 
 powers of x^ so as to have the minus sign before each set of 
 parentheses : 
 
 30. If ^ = 2a:2-3a:+l, B^x^-^x-'l, and (7 = 
 rr - 4 + 0^2^ find the value of ^ + ^ + 6\ 
 
 31. Evaluate Sa; — [4^ — (2^; — y — Z y^ —2x'] when 
 x = l and y=2. 
 
 32. Subtract 2(x - y^ -(x - y)-{-l from 4(a: - y)2 
 
 33. If A= ax-{-2by — cz and B = x— Zby — kz, find 
 the value of ^ — J5. 
 
 34. If A = m(ix — y^ -^ r(^x + y^ and B = n(x — y) 
 — (a; H- y), find the value of ^ — ^. 
 
 Multiplication 
 
 56. Commutative law for multiplication. The product 
 of a and b is expressed by ab. Similarly, the product of 
 a, 6, and c is expressed by abc. 
 
 In arithmetic it is obvious that 
 
 3x5 = 5x3. 
 
 In algebra it is assumed that : 
 
 The product does not depend on the order in which the 
 factors are taken. 
 
 Thus, it is assumed that 
 
 axb=h X a. 
 
 This important principle is referred to as the commuta- 
 tive law for multiplication. 
 
FUNDAMENTAL PROCESSES 55 
 
 57. Associative law for multiplication. In arithmetic it 
 is obvious that 
 
 2x(3x5) = (2x3)x5. 
 That is, 2 X 15 = 6x5. 
 
 In algebra it is assumed that: 
 
 The product does not depend on the way in which the 
 factors are grouped. 
 
 Thus, it is assumed that 
 
 a X (6 X c) = (a X 6) X c. 
 
 This important principle is referred to as the associative 
 law for multiplication. 
 
 58. Index law. From section 14, we have 
 
 23 = 2 X 2 X 2 ; 
 and 2* =2x2x2x2. 
 
 Therefore, 2^ x 2* = (2 x 2 x 2) x (2 x 2 x 2 x 2) 
 =2x2x2x2x2x2x2 
 = 27, or 23+^ 
 Similarly, a^ ^ a* = (a x a X a) X (a x a x a x a) 
 =axaxaxaxaxaxa 
 = a^, or a3+4. 
 
 In general, when m and n are positive integers, 
 
 2%e exponent of the product of two powers of the same base 
 is equal to the sum of the exponents of the factors. 
 
 Remark. Observe that any power of a positive number is positive 
 and that all even powers of negative numbers are positive, while all 
 odd powers of negative numbers are negative. 
 
 For example, 
 
 (- a)2 =(- «)(- «)= a2 and (- «)» =(- a)(- a)(- a) = - a*. 
 
56 ELEMENTARY ALGEBRA 
 
 EXERCISE 23 
 
 Name the product in the following examples : 
 
 1. 
 
 x'a^. 
 
 
 2. 
 
 m^m. 
 
 
 3. 1/2^6. 
 
 4. 
 
 r^rK 
 
 
 5. 
 
 sh. 
 
 
 6. tK'. 
 
 7. 
 
 - aHK 
 
 
 8. 
 
 a(-a6). 
 
 
 9. w2( — 7W^). 
 
 10. 
 
 22 . 23. 
 
 
 11. 
 
 3.32. 
 
 
 12. TT X 7r2. 
 
 13. 
 
 Rm. 
 
 
 14. 
 
 a'^a. 
 
 
 15. a'^a*. 
 
 16. 
 
 x-^x\ 
 
 
 17. 
 
 Cmf^m^ 
 
 
 18. a;2a^. 
 
 19. 
 
 -x'^x. 
 
 
 20. 
 
 - x'xK 
 
 
 21. a:(-a:*»). 
 
 22. 
 
 a?"^a^'^. 
 
 
 23. 
 
 — x'^o?"^. 
 
 
 24. r(-y)- 
 
 25. 
 
 c-an- 
 
 a3). 
 
 
 26. 
 
 (- 
 
 -«)2(-a)3. 
 
 27. 
 
 (-«')(- 
 
 a8). 
 
 
 28. 
 
 (^ 
 
 ^+y)'(^+^). 
 
 29. 
 
 (x~\-i/y(ix-{-yy. 
 
 30. 
 
 ix 
 
 + y-2)(a;-|-y-2; 
 
 31. 
 
 a^ ' a^ - a^. 
 
 
 
 
 
 
 Suggestion, a?- • a^ ' a^ — {a^a^^a!^ — a^a* = a^. 
 
 32. (iab)\ahy(aby. 33. (x-^i/yix+i/y(x-\-yy. 
 
 34. (a + ^ - c)(« + ^> - cy(a + b- cy. 
 
 35. (m_ 1)2(^-1)3(^-1)7. 
 
 36. (2x- ^ yyC2x - S 2/y(2x - S yyC2x - S I/). 
 
 59. The product of two monomials. When written in 
 
 full, 
 
 2abxSab^='2xaxbxSxaxbxb 
 
 z=2xSxaxaxbxbxb [§ 5^] 
 
 = (2 X 3) X (a X a) X (6 X 6 X ^>) [§ 57] 
 = 6 a253. 
 
 In like manner it may be shown that 
 
 ( - 2 a^2^)( __ 4 ab^c) = 8 a%^(^. 
 Also, that 
 
 (_ 2a62)(- 3a262)(- 4a5) = - 24 aW 
 
Euclid (330 275 bc) was a successful teacher of mathematics 
 in Alexandria. His Elements has been the recognized textbook in 
 elementary geometry for 2000 years. In it are to be found geo- 
 metrical proofs of the commutative and distributive laws. 
 
FUNDAMENTAL PROCESSES 57 
 
 From the foregoing illustrations it is evident that : 
 The product of two or more monomials is equal to the prod- 
 uct of their numerical coefficients and all the different literal 
 factors that occur in the monomial factors^ each letter having 
 as exponent the sum of the exponents of that letter in the 
 monomial factors. 
 
 EXERCISE 24 
 
 Multiply : 
 
 1. 2 a by a. 2. 2 jK by ^. 3. - 3 a; by 4 x. 
 
 4. -Qyhy -2y. 5. 3a^ by 2a:3. g. ^Rhy^R. 
 
 7. a%'^ by a%^. 8. 6 a by - 3 6. 9. 2 i2 by J i. 
 
 10. - 4 xyz^ by - 2 A. ii. 4 22 by ^ i2. 
 
 12. I a% by - 3 he. 13. -2ah'^hy - 3 ax^. 
 
 14. - ?>a%c by 4 he^x, 15. (a + hy by 3(a -h 6)3. 
 
 16. 2(x + yyhy -^(x-^yy. 
 
 17. - 4 <a + 5)2 by 3 2:2(^^^)8. 
 
 18. —^a(x-\- yY by - 2 a2(2: -f yy, 
 
 19. 3(2: + y)3by (a: + 3/). 
 
 20. - 3(a; + yy by - (2; + y)^. 
 Perform the indicated multiplications: 
 
 21. (Za%'^y. 22. {-2abe^y. 23. (fa;)^ 
 
 24. (-fa3)2. 25. (-|a26)2. 26. (-3a2^,3^)2. 
 
 27. (a62)(3a252)(2a25). 
 
 28. (3mw2)(_2m2n3)(-4 7W7i). 
 
 29. (-2:3/)(32^./3)(_5^4^). 
 
 30. 
 
 (2^25)(_|«J)(2a358). 
 
 
 31. 
 
 (-y)(-/)(-^). 
 
 32. (2a26)3. 
 
 33. 
 
 (-3^)3. 
 
 34. C-^axy. 
 
 35. 
 
 (-.4a263c2)8. 
 
 36. (-Ja262^)8. 
 
68 ELEMENTARY ALGEBRA 
 
 37. b(x + yX^ + yy(x + yy. 
 
 38. 2(a-hl)2(a + l)3(a+lj)4. 
 
 39. -2>(x + y-{- z)\x ^y + z)\x -\- y -\- zy>. 
 
 40. 3a(a;-h «/) • 2a5(ajH-^y) . 3 5c(a7H-y). 
 
 41. ( - 3 a%c) (f a52) ( - 4 o^c^) ( - ^253^) . 
 
 42. (1 0^33^2254) ( _ 2 2:2^23) ( _ 3 ^^3^2) ( _ ^^2^5) . 
 
 43. 3 a;"*. 4 a;". 44. 5 ^''.4^. 
 
 45. 2 a"» . 3 a2*". 46. 5 a;"'^! • 3 rc*""^. 
 
 47. 3 af +^ . 2 a;«-*. 48. - 3 a^^^C - Ta:^). 
 
 49. _7^2o+l(_3^2-2a), 50 (_3a;".)2. 
 
 Find the product of : 
 
 51. a;, 3 a;2^ 2 x", and 4 a;^ 
 
 52. 2 a26, 6 62^, and 5 c^a. 
 
 53. - 3 hV, 5 H2, and - 7 ^3^2. 
 
 54. 28, - 32^, - 4, and 52^3. 
 
 55. ^x^yz^ — ^xy\ ^xyz^^ and — 2xyz. 
 
 56. |(a:+ y^{y+zy, - J(y + 2!)(25 + a;)2, and 
 -K2 + a:)(a; + ^)2. 
 
 60. Multiplication of a polynomial by a monomial. It 
 is obvious that 
 
 3x(4 + 5)=3x4 + 3x5. 
 Also, that 
 
 3 X (4 + 5- 2)= 3 x4 + 3x5-3x2. 
 
 In like manner, it is assumed that 
 
 a(h-\-c)-=ah-\-ac. (1) 
 
 Also, a(J>-\-c-d) = ah + ac-ad. (2) 
 
 Equations (1) and (2) express the fact that multiplying 
 every term of a polynomial hy a monomial multiplies the 
 polynomial hy that monomial. This principle is referred 
 to as the distributive law for multiplication. 
 
FUNDAMENTAL PROCESSES 59 
 
 From equations (1) and (2) we have the following rule 
 for multiplying a polynomial by a monomial : 
 
 Rule. Multiply each term of the polynomial hy the mono- 
 mial and write in succession the resulting products^ each with 
 its proper sign. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Multiply (h-\- c— d^ by 2 a and verify the result 
 when a = 2, 5 = 3, (? = 2, and d = l. 
 
 Solution. 2a(h + c - d)=2ah + 2ac -2ad. 
 Check. 2x 2(3 + 2-1)= 2x2x3 + 2x2x2-2x2x1 
 4x4 =12+8-4. 
 
 16 = 16. 
 
 2. Multiply (-2ar* + 3y-222) by ^x^yz and verify 
 when x= — 2, y = — 1, and 2 = — 3. 
 
 Solution Check 
 
 -2a:2 + 3y-222 _ 8 - 3 - 18 = -29 
 
 3 x'^yz 36 = 36 
 
 - 6 x^yz + 9 x'^yH - 6 x^y^ - 288 - 108 - 648 = - 1044 
 
 Remark. The form of solution used in example 2 is preferable 
 when the multiplication cannot readily be performed at sight. 
 
 3. Simplify 2(a - 26)+ 3(2 a - 5). 
 
 Solution. 2(a - 2 fe)+ 3(2 a - ft) = 2 a - 4 6 + 6 a - 3 6. 
 
 = Sa-7b, 
 
 EXERCISE 25 
 
 (Solve as many as possible at sight.) 
 1. a(x+-y). 2. a;(a — 6). 
 
 3. 2'irR{H-\-R). 4. y(a-6 + c). 
 
 5. x(x — y — z), 6. — a(— a — 6 — c). 
 
 7. ah{a — h — V). 8. Q[^{xy—xz-\-c). 
 
 9. 7r(^ + r -\- Rr). 10. m\mhi — m^p + pn). 
 
60 ELEMENTARY ALGEBRA 
 
 11. _2(3a-45 + c). 12. 6(2^-22/3-322). 
 
 13. 2ax(ax-l + Sa^3^), 14. - 2a2«/(12a3 + 3y). 
 
 15. Say^-x-^lhy ^x. 16. 2a: + «/+ 1 by -a;. 
 
 17. 3a:2 - a: + 2 by 3 x\ 18. 2aft2 -ah+h'^ by aJ. 
 
 19. a2_3^5 4.262by_2a5. 
 
 20. ^x^y — 2xy^ — ^y^ by xy^, 
 
 21. ah -\-hc— ea by — aftc. 
 
 22. ?;2 _ ^3 _ 4 ^4 |3y ^^^ 
 
 23. a(a; + «/) + (wi + 7i) by ab. 
 
 24. a(m + ?2) + ^(jo + 5')+ <?(a^ + ^) by aic. 
 
 25. (a:4-y)-(a: + ^)2 + (2J + «/)3by (rr + 3/)2. 
 
 26. 2 a(w + 7i) — 3 5(m + /i)^ + 4 <?(m + n)* by (m -\- w)2. 
 
 27. 4(a; H- «/)2— 3 ^(2; + ?/) — 5 c(rr + i/)^ by 2 a6c(2; + «/)*. 
 
 28. a^(^x + ^) — ^^(^ + «/)^ — abc(x + ^)3 by abc(^x + ^)2. 
 
 Simplify : 
 
 29. 3(a + 26)+2(a-6). 30. (2 - a:)y- a;(l + y). 
 
 31. (22 — 3^)2:2 + (3y — 42)^2 — (22 — 3a:)2y. 
 
 32. a[l +(^ +<?)]• 
 
 33. [a — 2(a;— a)]^; — [a;+2(a — a;)]a. 
 
 34. 5a3_2a(2a2_a + l)-3a(a2-2a + l). 
 
 35. 3(1 + 20 + 300), 36. 5(2 + 3a:-2^ + |2). 
 37. a»(l + a + a2). 38. x^'^x'^y'^-^- ^x^y"" - ^}, 
 
 39. a'"a;«( - 2 aa;2 + 3 a"»+ V+2 _ 7 a;"+3) . 
 
 40. -3m27lP(-fm«7l2P4.|t7^9+lw3P+l-|). 
 
 41. (2: - yyiCx - yy - 3(2: -yy + 2(x-y)'\-il 
 
 *'• "L"34 51- + -68-] 
 
 43. 7a» - 2a(a2 + 3a - 2) - SaCa^ - 2a + 5). 
 
FUNDAMENTAL PROCESSES 61 
 
 44. To the product of 3 a -|- 1 and 5 a add 3 a — 2, and 
 multiply the sum by a. 
 
 45. From the product of 3m2-j-3mH-2 and bm sub- 
 tract the product of 5 m^ — 2m — 1 and 3m. 
 
 61. Multiplication of a polynomial by a polynomial. 
 
 The product of two polynomials can be obtained by suc- 
 cessive applications of the principle given in section 60 ; 
 
 thus : 
 
 (3 a - 4 6) (a; - 2/) = (3 a - 4 6) a: - (3 a - 4 &)3/ 
 = (3 aa: - 4 hx) - (day - ^hy) 
 = 3 ax — 4: bx — S ay + 4: by. 
 
 In like manner : 
 
 (3 a _ 4 J + 2 c)(2 a - 3 6) = (3 a-4 & + 2 c)(2 a) - (3 a-4 b + 2 c)(3 b) 
 = Qa^-Sab-\-4ac-9ab +12b^-Qbc 
 = 6 a2-17 a6 + 4rtc + 12 b^-Q be. 
 
 Remark. From section 60, z(x — y)= zx — zy^ in which expression 
 the letter z represents any number. Substituting (3 a — 4 6) for z, 
 we have 
 
 (3 a - 4 6) (a: - y) = (3 a - 4 6)x - (3 a - 4 b)y, 
 
 which is the expression obtained in the first of the preceding 
 examples. 
 
 From the foregoing illustrations we have the following 
 rule for multiplying a polynomial by a polynomial : 
 
 Rule. Multiply each term of the multiplicand hy each 
 term of the multiplier aiid add the partial products. 
 
 Note. Before multiplying one polynomial by another, both of 
 them should, if possible, be arranged according to the ascendinr/ or 
 descending powers of a certain letter ; that is, in such a manner that 
 the exponents of a certain letter in successive terms decrease or 
 increase from left to right. 
 
 For example, 3 ar^ — a:" -f 4 a: — 3 is arranged according to the 
 descending powers of x ; and a — ay -\- 2by^ — y* is arranged accord- 
 ing to the ascending powers of y. 
 
62 ELEMENTARY ALGEBRA 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Multiply 3a^-f2a;— 1 by 2a;— 3 and verify when 
 
 a;= 2. 
 
 Solution Check 
 
 3a:2 + 2a:-l =15 
 
 2x -3 =1 
 
 6 a;8 + 4 a:2 - 2 a; 
 
 - 9 ar^^ - 6 ar + 3 __ 
 
 6a:«-5a:2-8ar + 3 =15 
 
 2. Multiply a— 5-f<?bya-f6 and verify when a = 5, 
 5=3, and c = 2. 
 
 Solution Check 
 
 a — ft + c =4 
 
 g + & = 8 
 
 + gfe - fc'' + ^>g 
 
 a2 + ac - fe2 ^. 2,c =32 
 
 3. Multiply 3 a:^ — 2 xy^ + 3 x^y — y^ by — y^ -\- xy •\' a^ 
 and verify when a; = 2 and^ = 1. (See note, section 61.) 
 
 Solution Check 
 
 3 a;8 + 3 x2y - 2 xy^ - f =31 
 
 x^ -\- xy — y^ = 5 
 
 3 a:« + 3 a;*y - 2 T^y^ - x^y^ 
 
 + 3 a:*3/ 4- 3 x^y"^ - 2 xh^ - xy^ 
 
 - 3 x^y^ - 3 x^y^ + 2 xy* + y^ 
 
 3 a:* + 6 xV - 2 a;8y2 _ 6 ara^s + ^,^4 _^ yi = 155 
 
 EXERCISE 26 
 
 Multiply, and check results : 
 
 1. a; + 3 by a: + 2. 2. 2wH-lbyw + 3. 
 
 3. a + 5by2aH-4. 4. 3 a; + 5 by a; -4. 
 
 5. 5 a + 2 by 2 a + 3. 6. 2 m - 4 by 3 w - 3. 
 
 7. 6-4a;by 5-2a;. 8. 3a -5 by 4 + 6a. 
 
FUNDAMENTAL PROCESSES 63 
 
 9. 5 + 2 ?w by 3 w - 2. lO. 8 a + 2 by 3 a - 2. 
 
 11. 4 — 2 ^ by 4 4- 2 y. 12. m -\- n hy m -\- n. 
 
 13. a — b by a — b. 14. r + » by r — 8. 
 
 15. 3 7W 4- w by 2 m + w. 16. 3 r + 2 « by r — 3 «. 
 
 17. 5a: + 4y by 3a; + 2i/. 18. 8m— 2n by 2 m-h3ri. 
 
 19. 3a;2H-2yby4a;-8«/2. 20. 3^24.4 ^3 by 2^2-4 w2 
 
 21. 2 a:^ -f- 3 2 by 5 x^t/ — 4:Z. 
 
 22. 5 3^t/^-2z^ by Sa^f-^z\ 
 
 23. 6a:3_i by 22^3+1. 
 
 24. 5 a^^2 _ 1 by 6 a:2^2 _ 1^ 
 
 25. m2 — 7WW + 7l2 by w + 71. 
 
 26. 2x^-\-^xi/ + 4:2/^hy x+1/, 
 
 27. 0:2 H- a:^ + i/2 by ic + y. 
 
 28. 2^2 _|. ^2- + a2 by 2 a: + 3 a. 
 
 29. 4r2 + 6r« + 9«2 by 2r4-3«. 
 
 30. 3ic3^ 2a:-l by 4a;2+2. 
 
 31. 4 a* - 3 a2 + a by 2 a3 _ 3 a\ 
 
 32. mV + 2 WW + 1 by ww H- 1. 
 
 33. 2:2^2 -f 4 2;y + 4 22 by a;y 4- 2 0. 
 
 34. 2a^ + Sx^-4x-Shy^a^-h2x. 
 
 35. 2a:34.3a;2_^^4by 22^-82: + 2. 
 
 36. 2^2 ^ 2:^ ^ ^2 by 2^2 _ 2;^ _|_ y2^ 
 
 37. 3 m2 - 2 m + 1 by m2 4- 3 m + 2. 
 
 38. 2^ + 2a;+2by2:2_2rr + 2. 
 
 39. 4 2^2 ^12^^ + 9 ^^2 by 2 2;+ 3^. 
 
 40. 2^ + a^^ + 2:?/2 ^ y3 by 2; — y. 
 
 41. 82:34. 122^^4. 182;2/2 + 27^by 22:-32/. 
 
 42. 6 TwV — 2 7w2w2 4- 4 7WW — 1 by 3 mV — 2 mn — 3. 
 
64 ELEMENTARY ALGEBRA 
 
 Expand : 
 
 43. (4a2+ 2a + l)(4a2_2a-|-l). 
 
 44. (9^2- 6mn-f-4w2)(9w2+6wn + 2w2). 
 
 45. (a^-x-l}(2^-2a^-x-{-l). 
 
 46. (a;-l)(a;-2)(2;-3). 47. (x^-x + iy. ' 
 48. (3^-^2x^-\-Sx-j-iy. 49. (2m4-47i + J)2. 
 
 50. (2r-«)(r + 2«)(2r + 8«)- 
 
 51. (m + w + ^)^. 52. (a — 6 — c)8. 
 53. (a;»4-2)(2:« + 3). 54. (a;» - 4)2. 
 55. (2 a +36)3. .56. (3 m- 2/1)8. 
 
 57. (a>4-4)(ir2n_4). 53. (2 a + 3 6 - c-H 4 (?)2. 
 
 59. {T^-^-T^y-^-x^y^-^-xy^^-y^^ix-y), 
 
 60. (3a;-2y + 42-l)2. 
 
 61. What is the area of a rectangle that is 2 a + 8 units 
 long and 3 a -f 1 units wide ? 
 
 62. By how much is the area of a rectangle of base h 
 and altitude a changed by increasing the base by 2 and 
 decreasing the altitude by 1 ? 
 
 Division 
 
 62. The quotient of two powers of the same base. 
 
 Since a2 x a^ = a^, it follows from the definition of divi- 
 sion that 
 
 a^ -J- a2 _ ^3^ Qp ^6-2 
 
 In general, when w and w are positive integers and n is 
 
 less than w, 
 
 0^ -5- a" = a"~**. 
 
 The exponent of the quotient of two power% of the same 
 base is equal to the exponerit of that base in the dividend 
 minus its exponent in the divisor. 
 
FUNDAMENTAL PROCESSES 65 
 
 EXERCISE 27 
 
 Name the quotients in the following examples : 
 
 1. c? -h a. 2. a^ -J- a^. 3. y^ -j- 1/^. 
 
 4. a^^2^. 5. R^-i-E. 6. m^-^m\ 
 
 7. (_^)^(4.a:4). 8. (a^)^(i-a^). 
 
 9. (_a;i2)^(_^2). 10. 58-^52. 
 
 11. 45-43. 12. (25) -(-23). 
 
 13. (ia-^by-T-(a-^by. 14. (m + ti)^— (m 4- ri)*- 
 
 15. (x-ht/y-^Cx-ht/y. 16. (rH-l)6-(r+l)8. 
 
 17. (l + a)7-(l + a)2. 18. (jt? + ^)io-(^ 4-^)6. 
 
 19. (2 2:+«/)i<^^(2 2;4-?/)^. 20. (a + ^-0^-^(« + ^-<?)- 
 
 21. (2a + a; — i/)7-;-(2a + 2:— «/)2^ 
 
 In examples 22-30, the literal exponents are assumed 
 to be integers and the exponent in the divisor in each case 
 is assumed to be less than the exponent in the dividend. 
 
 22. x° -i- x^. 23. x'^-^x. 
 24. a^^ H- a"". 25. ty^^ -r- t/"". 
 26. — a*" -r- a^. 27. a** — (— a»). 
 28. (-w°)-5-(- w*). 29. r3^^(-r^). 
 30. a^+1 -^ a^-i. 
 
 63. Meaning of zero exponent. The definition of expo- 
 nent given in section 15, page 11, does not apply to zero 
 used as an exponent. When a is any number other than 
 zero we shall assume that 
 
 Remark. It has been shown in section 62 that a"" -^ a^ = a*"-", 
 when m and n are positive integers and n is less than w. If n is 
 equal to m, this equation becomes a^ -^ a"* = a"*"*" = a^ ; but a"* -f- a"* 
 = 1. Hence, if a"" -^ a"* = a"*"**, in which a is difPerent from zero, is 
 
66 ELEMENTARY ALGEBRA 
 
 an identity when n = m, it is necessary to make the assumption given 
 in section 63 ; namely, that a9 = 1. 
 
 EXERCISE 28 
 
 1. Explain why aP = h^. 
 
 2. Explain why 20=50 = ( J)0. 
 
 Express in the simplest form : 
 
 3. 6^aO; c^a%^; a%^c^ -h c\ 
 
 4. a'^b^-i-a^; a%^-^h^(P; m^n^^n^r^. 
 
 5. a%^-^(P; ^ah^^a%\ a^m^bHm'^. 
 
 64. Division of monomials. Since division is the inverse 
 of multiplication, and since 
 
 2a52^x 3a2J3^ = 6a356^, 
 it follows that 6 a%^(^ -5- 2 ah^c = 3 ^253^ 
 
 and 6 a^JV ^ 3 ^253^ ^ 2 a52c. 
 
 Also, since — 8 o^yz x 2 a:y = — 16 a^y^z^ 
 
 it follows that — 16 o^yH -. — 8 x^yz = 2xy 
 and — 16 2:^^225 -i-2xy = — ^ x^yz. 
 
 From these examples we infer the following rule for 
 finding the quotient of two monomials : 
 
 Rule. Divide the numerical coefficient of the dividend hy 
 that of the divisor (^observing the laws of signs for division) 
 and write after the quotient each letter of the dividend^ giving 
 it an exponent equal to its exponent in the dividend minus 
 its exponent in the divisor. 
 
 Note 1. If there occurs in the dividend any letter which is not 
 found in the divisor, it may be understood to occur in the divisor 
 with an exponent 0. If the same "power of a letter occurs in both 
 dividend and divisor, the difference of its exponents being 0, the 
 letter is omitted from the quotient ; for, a^ = 1. 
 
 Note 2, The division of one monomial by another may be 
 
FUNDAMENTAL PROCESSES 67 
 
 expressed in various ways. Thus, 10 x^y^ -=- 5 xy^ = 2 x^y may also be 
 expressed : 
 
 l?^£^ = 2:.3y or 5:ry ^)10:rV , 
 5x2/2 ^ 2x^y ■ 
 
 Note 3. When the numerical coefficient in the dividend is not 
 exactly divisible by that in the divisor, the numerical coefficient in 
 the quotient is an arithmetical fraction and should be simplified as 
 in arithmetic. 
 
 Thus, 9 a%^ ^ 6 a62 = I ah, or | ad. 
 
 EXERCISE 29 
 
 (Solve as many as 
 
 possible at sight.) 
 
 Divide : 
 
 
 1. ^7^hj2x. 
 
 2. - 6 a6 by 3 a2. 
 
 3. 10 56 by -2 63. 
 
 4. 12a352by3a. 
 
 5. - 15 a%^ by - 3 ah\ 
 
 6. Ux'h^yhj4.h^y. 
 
 7. - 21 a%^c by 7 a%^. 
 
 8. — 10 2:^225 by — 2 xyz. 
 
 9. lIx^yH^hy -2xy^z. 
 
 10. 81a35cby9A. 
 
 11. '^'^x^y^z^hyl^xyz. 
 
 12. 96 2^3^225 by -^x^yH. 
 
 13. 8(a + 6)3 by 2(a -f 6). 
 
 14. -12(a:H-2^)*by -(a:H-y). 
 
 15. - 1 2 a553(^ + ^)2 by - 3 a^js^^ 4. ^). 
 
 16. 15 a:S«/225(^ 4. ^-)6 by _ 5 a:Sy2(^ _|_ 2;)4. 
 
 17. I Trigs by 7ri22, 18. | tt^s by i B. 
 
 In examples 19-26 the literal exponents are assumed to 
 be integers, and the exponent in the divisor in each case 
 is assumed to be less than that in the dividend. 
 
 19. 6 a* by 2 a?i. 20. 9 a^ by — 3 a. 
 
 21. - 12 X"' by 4 X». 22. - 15 X^ by 15 X"». 
 
 23. 12 a^+i by — 4 a. 24. 16 or by 4 a"»-2. 
 
 25. - 18 a"»+i by 9 a"»-i. 26. 20 a^+i by - 5 a*. 
 
68 ELEMENTARY ALGEBRA 
 
 65. Division of a polynomial by a monomial. Since 
 2 a(3 b — 4:c-\-6 d) =6 ab — Sac + 10 ad^ and since divi- 
 sion is the inverse of multiplication, it follows that 
 
 (6 a6 - 8 ac + 10 a<f)^ 2 a = 3 6 - 4 C+ 5 <? 
 
 _ ^ab 8 gg 10 at? 
 ""2a 2a 2a ' 
 
 From the foregoing illustration we infer the following 
 rule : 
 
 Rule. To divide a polynomial by a monomial, divide each 
 term, of the polynomial by the monomial and add the quotients. 
 
 Note. (Qab — 8ac + 10 ad) -r- 2 a may be written 
 
 Qab-Sac + lOad 
 2a 
 or 2 a )Q ah - 8 ac -f 10 ad. 
 
 EXERCISE 30 
 
 (Solve as many as possible at sight.) 
 Divide : 
 
 1. 3 a2 - 6 a by 3. 2. a^ + ab by a. 
 
 3. 6a^-9xhySx. 4. 2a2-2a6by2a. 
 
 5. 6 x^y — 9 xy^ by 3 xy. 6. x^yh'^ — xyz by xyz. 
 
 7. 4a2-652_8c2by 2. 
 
 8. 6 m^ — 8 mn 4- 4 mn^ by 2 m. 
 
 9. 3 a^bx — 2 ab^x — 3 a^ex by aa;. 
 
 10. 15 x^y — 20 a;^2 _ 5 g.^ by 5 a;y. 
 
 11. a^ — 2 a by — a. 
 
 12. 2 aSJ - 3 a52 + 2 aft by - aft. 
 
 13. 3 a^yz — 5 2:^%^ _j_ 2 xyz^ by ~ xyz. 
 
FUNDAMENTAL PROCESSES 69 
 
 Perform the indicated divisions : 
 
 ,^ irRL^irm „ 27r^+7rig3 
 
 14- -• -I-*** :;:: * 
 
 7 rn^n — 14 n ^ + 21 mm 
 16. = • 
 
 7 mn 
 
 ab(m + ny + ah^jm + ^) 
 a6 
 
 18. 
 
 19. 
 
 20. 
 
 {m-\-Yi) 
 
 — 18 mhfir — 12 m?tV + 6 mnr 
 — 6mnr 
 
 .5r 
 
 66. Division of a polynomial by a polynomial. Since 
 division is the inverse operation of multiplication, it 
 follows that the product of two polynomials divided by 
 one of them is equal to the other. Some of the steps 
 involved in dividing one polynomial by another are 
 suggested by a careful inspection of the following multi- 
 plication : 
 
 2 a;2 - 3 a: + 4 
 x^- X - 1 
 
 2 a:* - 3 a:8 + 4 a;2 
 
 -2a:8 + 3a:2-4a: 
 
 -2a:2 + 3a;-4 
 2x4-5x8 + 5x2- a: -4 
 
 Let it be required to find a^ — x — 1^ given 2 a^ — 3 a; + 4 
 and 2x:^ — 5a^-{-5a^ — X — 4; that is, to find the quotient 
 oi2a^-5a^-^5x^-x-4: divided by 2a^^ - 3 a: + 4. The 
 quotient, which is the other factor, a^ -- ir — 1, may be 
 found as follows : 
 
70 ELEMENTARY ALGEBRA 
 
 X^- X -1 
 
 1. 2x*-h2x^ = x^ 2x^-Sx-h4:)2x*-5x» + 5x^-x-i 
 
 2. Subtract (2 a;2 - 3 ar + 4) X a:2 = 2x*-dx^ + ^x^ 
 
 5. -2x»-h2x^ = -x -2x»+x^-x-4: 
 4. Subtract (2 ^2 - 3 a; + 4) x (- x) = -2x» + dx^-ix 
 
 6. -2x2-2x2 = -! -2x2 + 3x-4 
 6. Subtract (2x2- 3x + 4) (-1)= -2x2 + 3x-4 
 
 
 
 Explanation. The explanation of the process may be given thus : 
 
 The terms of both dividend and divisor are arranged according to 
 the descending powers of x. 
 
 The term of highest power in the dividend being the product of 
 the term of highest power in the quotient and that of the highest 
 power in the divisor, the term of highest power in the quotient is 
 obtained by dividing the term of highest power in the dividend, 2 x*, 
 by the term of highest power in the divisor, 2 x2. This gives x2, the 
 term of highest power in the quotient. 
 
 The dividend is formed by multiplying the divisor, 2 x2 — 3 x + 4, by 
 each term of the quotient and adding the resulting products. Hence, 
 reversing the process, 2 x2 — 3 x + 4 is multiplied by x2, and the result, 
 2 X* — 3 X* + 4 x2, subtracted from the dividend, 2 x* — 5 x* + 5 x* 
 — X — 4. 
 
 The remainder, — 2 x* + x2 — x — 4, must be the product of the 
 divisor and the part of the quotient to be found. Hence, the term 
 of highest power in the remainder must be the product of the term 
 of highest power in the divisor and the term of the quotient of next 
 higher power to that already found. Therefore, the term of next 
 higher power in the quotient is obtained by dividing the term 
 of highest power in the remainder, — 2 x^, by 2 x2. This gives — x, 
 the term of next higher power in the quotient. 
 
 Multiply the entire divisor by the second term of the quotient, — x, 
 and subtract the product, — 2 x* + 3 x2 — 4 x, from the remainder. 
 This leaves — 2 x2 + 3 x — 4, the second remainder. 
 
 The third term of the quotient (— 1) is obtained from the second 
 remainder just as the second term is obtained from the first re- 
 mainder. The entire divisor is then multiplied by ( — 1) and the 
 product, — 2 x2 + 3 X — 4, subtracted from the second remainder. 
 
 The final remainder being zero, the division is said to be exact. 
 
 The quotient is x2 — x — 1. 
 
FUNDAMENTAL PROCESSES 71 
 
 From the foregoing explanation we may derive the 
 following rule for dividing one polynomial by another : 
 
 Rule. 1 . Arrange the dividend and the divisor according 
 to the descending or ascending powers of the same letter. 
 
 2. Divide the first term of the dividend by the first term 
 of the divisor ; this gives the first term of the quotient. 
 
 3. Multiply the entire divisor by this first term of the 
 quotient^ and subtract the result from the dividend ; this gives 
 the first remainder. 
 
 4. Divide the first term of this remainder by the first term 
 of the divisor ; this gives the second term of the quotient, 
 
 5. Multiply the entire divisor by this second term of the 
 quotient., and subtract the result from this remainder ; this 
 gives the second remainder ; and so on. 
 
 Remark. Divisor, dividend, and successive remainders should 
 always be arranged in the same order of powers of the same letter. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Divide 2a3_4«_5a2 + 3by2a-l. 
 
 Arrange the terms according to the descending powers 
 
 of a. 
 
 a2 _ 2 a - 3 
 2a- 1)2 «» - 5 «2 _ 4 a + 3 
 2 a8 - a^ 
 
 -4:a^-]-2a 
 
 -6a + 3 
 
 -6a + 3 
 
 
 
 Check. Let a = 1. 
 
 Divisor^ 2 « — 1 = 2 — 1 = 1. 
 
 Dividend, 2a3-5a2-4a + 3 = 2-5-4 + 3=-4. 
 
 Dividend -^ Divisor =(— 4)-=-!= — 4. 
 
 Quotienty a^-2a-3 = l-2-3=-L 
 
72 ELEMENTARY ALGEBRA 
 
 Remark 1. No number should be used in checking which when 
 substituted makes the divisor zero. 
 
 Remark 2. Observe in the solution of example 1 that only the 
 part of the remainder about to be used is brought down at each stage. 
 
 2. Divide a^ -\- y^ hj x -\- y . 
 
 x^ — x^y -f xy^ — y^ 
 
 X + y)x^ 
 
 + t 
 
 x^ + x^y 
 
 
 -x^y 
 
 
 -^y 
 
 -xY 
 
 
 x^ 
 
 
 xY^xf 
 
 
 -xy^^y^ 
 
 
 - ^f - f 
 
 
 2v* 
 
 Remark. The quotient is 3^ — x^y + xy^ — y^ ; the remainder is 
 2 y\ The complete quotient may be expressed in the same way as in 
 arithmetic. In this instance the complete quotient may be written, 
 
 2 u* 
 X -^y 
 
 Check. Let a: = 1, and y = 1. 
 
 Dividend f a;* + 2/*=l + l=2. 
 
 Divisor, x-\-y = l-\-l=2. 
 
 Dividend -i- Divisor =2-^2 = 1. 
 
 Quotient, a;» - x'^y + xy^ - y^ + -^^ =1~1 + 1-1+| 
 
 x-\- y 2 
 
 = 1. 
 
 Another Check. As in arithmetic, if the division is correct, 
 Divisor x Quotient + Remainder = Dividend. 
 
 Hence, if we evaluate both members of the equation for the same 
 numerical values of x and i/, and the results do not agree, we shall 
 know that there is an error. 
 
FUNDAMENTAL PROCESSES 73 
 
 3. Divide abx^ -\- (a^ -f- h'^)x H- ah by ax + h. 
 Solution 
 
 hx -\- a 
 ax + h)ahx'^ + a^x + 5% + ah 
 
 abx^ + h'^x 
 
 a^x + db 
 
 a^x + db 
 
 Check 
 
 Let X = 1, a = 2, and 5 = 3. 
 
 Then, a&a:2 -\-a^x + 6% + a6 = 6 + 4 + 9 + 6 = 25. 
 ax + 6 = 5, and 6x + a = 5. 
 ^ = 5. 
 
 EXERCISE 31 
 
 Divide : 
 1. x'^-^^x-\-2hy x-\-l, 2. a:2_^5^^g by ^^3^ 
 
 3. x^-\-lx-\-12hyx + ^. 4. a2^3^_,_2 by aH-2. 
 5. 2;2- 9 a: 4- 20 by 2: -4. 6. ^2_ 7^ ^ 6 by ^ _ 1^ 
 7. a;2 — a: — 20 by a: — 5. 8. m^ + 2 w — 15 by ?7i + 5. 
 
 9. r2 + 6r- 7 byr+ 7. 10. t;2 - 4t; - 21 by v- 7. 
 
 11. x^ + x-^Ohy x + Q. 12. ^2 _ ^ _ 2 by ^ + 1. 
 
 13. 2:4-62:24.6 by 0^2 _ 2. 14. 2:4 + 2:2 _ 12 by 2:24.4. 
 
 15. 9 4-10^4-^2 by 1 4- w. 16. 4 - 3 A: - A:2 by 4 4- )fc. 
 
 17. 12-8r4-/^by 2-r. 18. 42:2 - 1 by 22:4- 1. 
 
 19. 9 w2 _ 4 by 3 w 4- 2. 20. rri^ — n^ hy m -\- n. 
 
 21. 2:2 — 2 2:y 4- y^ by 2: — y. 22. m2 4. 2 77m 4- ^2 by w + w. 
 
 23. ^2 4. 2^^-1-1 by^z + l. 
 
 24. 2a2+ii«5_f.l252by 2a4-36. 
 
 25. 6 ^2 — WW — 12 n2 by 3 w 4- 4 w. 
 
 26. 10r24-16r»-882by 5r-2«. 
 
74 ELEMENTARY ALGEBRA 
 
 27. 36 ?/2 _j. 24 a«/ — 5 a2 by 6 3^ -h 5 a. 
 
 28. 2a:2^2_^a:i/-6by 2a;y-3. 
 
 29. m^— 2m/i — 15^2 by m — 5?i. 
 
 30. 6 a:* -11 2:2 -35 by 82:24.5. 
 
 31. a:^ _j. 3 2;2 _|_ 4 a; _|. 4 by a:2 ^ a; + 2. 
 
 32. 4a:3^i0a^+4a:-2 by 2rr2+3a:-l. 
 
 33. 10^3_37^2^13^_21 by 5^2_^^3^ 
 
 34. 2«3 + 5a2_f.4^_,_l by 2a + l. 
 
 35. 8m3+7m2-10 + 9mby 3w-2. 
 
 36. 8y3_l9y_lo^2_i5by 2^-5. 
 
 37. a;^ — ?/3 by a; — ?/. 
 
 38. a^*" + 5^ by a"* + h^. 
 
 39. a:* 4- a:2y2 _j_ ^^ by 2:2 + a:^ -j- y%^ 
 
 40. a3_^ 1 by a4-l. 
 
 41. 8 w3 4. 27 by 2 w 4- 3. 
 
 42. 8«/3_64 by 23/ -4. 
 
 43. m^ + 8 7712/1 4- 3 mr^ 4- ti^ by w 4- w. 
 
 44. m* + w2 4- 1 by w2 — w 4- 1 • 
 
 45. 20 r^ - 36 r^s 4- 59 tH^ - 57 r^s^ 4. 29 r*^ - 15 «6 by 
 10r2-3r«4-5«2. 
 
 46. 7!^ — y^hy X — y. 47. w^ H- ^i^ by ?w + w. 
 48. w2 4-w^byw — w. 49. w*4-lbyw4-l. 
 
 50. 2:8_|_ 5^+ 102,^10 by 2:4-2. 
 
 51. 6 w^ — mhi — 14 mw2 4- 3 w^ by 3 m2 4- 4 mn — n^. 
 
 52. 2 a^ 4- 8 a* - 12 a3 4- 15 ^2 _ 11 a 4- 3 by - 8 a 4- 1 
 4- 2 a2. 
 
 53. 3/-16/4-1 by 3jp2-jt?4-l. 
 
 6 1 
 
 54. Find the quotient of ^ ~ ; check the result by 
 letting a = 2. 
 
FUNDAMENTAL PROCESSES 75 
 
 a^ — 1 
 
 55. Find the quotient of — ; check the result by 
 
 letting a = — 2. ~ 
 
 56. Find the quotient of -— ; check the result by 
 
 letting a = 1. 
 
 57. Find the quotient of — -4- ; check the result by 
 letting a = 1. ^ 
 
 58. What is the remainder when a* — a^ + 2 a + 1 is 
 divided by a2_2a + 3? 
 
 59. For what value oi m is a* — a^ — 8 a^-\- ma — 3 ex- 
 actly divisible by a^ -f 2 a — 3 ? 
 
 60. For what value of m is a^ -\- a^ -\- 2 a^ -\- a^ -\- ma — 2 
 exactly divisible by a^ — a^ + 2 a — 1 ? 
 
 61. What are the values of Q (quotient) and E 
 (remainder) in the following expression: 
 
 a + 1 ^^a+1' 
 
 62. Show that -^ = l+aH-a2 4.^8 + ^4^ ^ 
 
 a 1— a 
 
 63. If a = — , what error is made in assuming that 
 
 1-a 
 
 EXERCISE 32. — GENERAL REVIEW 
 
 1. Find the algebraic sum of a:, 4 y, and — 32. 
 
 2. Find the algebraic sum of 3(a;— ^), — 4(a;— y), 
 2(x - y\ and - b(x - y). 
 
 3. Simplify ^x — 2a + ^h — 2c—^a^2x—h-\-c 
 - X -2h + ^a-{-^c. 
 
 4. Simplify ah + 2 a% - ab^ -h 2 ab^ ^ S a% - 4 a^I^ 
 + 5 aJ- 3 ab^ •{. a% - 6 ab + aW + 5 ab^ -f 3 ^252. 
 
76 ELEMENTARY ALGEBRA 
 
 5. Add - 3(a - b), 4(<i + b), - 5(aH-5), and 6(a-6). 
 
 6. Add ^ x'i/ + ^ x^i/-\-l xf- .1 3^1/^ ^xy^-lxy-^x^y 
 + .7 x^y\ \xy — .3 a^y^ and x^y — \ xy^. 
 
 7. What must be added to a^—2ab-{- 5^ to give 
 
 8. What must be added to x^ -{- S xy — 2 y^ -\- 4: to give 
 -Sx^ + xy-y^-2? 
 
 9. What must be added to x^-^ 2xy + y^ to give 
 x^ - 2xy + y^? 
 
 10. r2-2r-7-(-2r2+r-3)=? 
 
 11. Simplify l-{_l-[-l_(-l)_l]-l} + l. 
 
 12. Inclose the last three terms oix — y — z + 1 within 
 parentheses preceded by the sign — . 
 
 13. Name at sight the products : 
 
 ab ' ab; a% • ab'^ ; a"5" • ab ; a"5" • a'^b^ ; 2 ax"" ■ 3 a^. 
 
 14. Expand (a+ l)(a + l)(a - l)(a— 1). 
 
 15. Multiply 2 a2 _,. 6 a6 + 3 52 by 2 a2 - 6 a6 H- 3 b\ 
 
 16. Simplify(2a: + 3«/)(2a;+3t/)-(2a:-3y)(2 2:-3y). 
 
 17. Multiply (2m-\-lfhy (2m- 1)2. 
 
 18. Evaluate 2 irRCH + i2) in terms of ir when i2 = 2 
 and ir= 3. 
 
 19. Evaluate irR^L -f i2) in terms of ir when i2 = 3 
 and X = 5. 
 
 20. Evaluate ^ irH^B^ -\- Rr-\- r2) in terms of tt when 
 ir=:6, i2 = 4, r = l. 
 
 21. Evaluate J irH^E^ H- i2r + r2) in terms of tt when 
 5^=9, i2 = 3, r=0. 
 
 22. Evaluate (mn — r8)(n8 — mr)(nr — ms) when 7?i = 2, 
 71 = — 1, r = 0, and « = 1. 
 
FUNDAMENTAL PROCESSES 77 
 
 23. Name the quotients in the following : 
 
 a^^a?] w>^w>\ a^'^a'"', 26-5-23. 
 
 24. Name the quotients in the following : 
 
 a%^^ aV) ',a%-^ah\ - 6 a^J*-^ 2 ^352 . _ (3 ^^)3^ _ (3 ^5)2. 
 
 25. Name the quotients in the following : 
 
 a**6" -r- ah ; x^^^y^^'^ -?- x^y^ ; ic^"^ -^ a;"y. 
 
 26. Divide irW' by 1 R. 
 
 21, Divide 2 irRH-^ 2 7ri22 by 27ri2. 
 
 28. Divide irRL + irR^ by ttR. 
 
 29. Divide a25 _ aja -|- 2 aW by - a6. 
 
 30. Divide (a - 6)2; + (a - h^x^- (a - 6)(a + 6)2:8 by 
 (a — h')x, 
 
 31. Divide m^ — n^ hy m — n. 
 
 32. Divide 1 -|- 2 a; by 1 -|- 2:, carrying the quotient to 
 four terms. 
 
 33. Divide a^+b^ + ^ aH -\- ^ aW' by a + 6. 
 
 34. Divide a* -^ 5* - 4 aft^ _ 4 a% -i- 6 a252 by a - 6. 
 
 35. Express m— n— 'p-\-q — r— ar^h—x— y in- tri- 
 nomial terms having the last two terms of each trinomial 
 inclosed within parentheses preceded by the sign — . 
 
 36. Show that (m — n)(^p— q) = {n — m)(^q — p), 
 31. Show that (a - 6)2= (h - ay. 
 
 38. Show that gi:^'=^'-^'. 
 
 a — h — a 
 
 39. Multiply W2:3 _ ^^ j^ ^x —c by moi^ + 2:^ -|- 4 2: + d 
 and arrange the result according to powers of x. 
 
 40. In the formula V= ^ttR^^ find the value of V 
 when 7r= 3.1416 and R = 8.' 
 
 41. Divide 3 a^ + a* - 12 a^ -|- 5 a2 - 15 a - 6 by a^ 
 4-2a2 + 3. 
 
78 ELEMENTARY ALGEBRA 
 
 42. Divide 3 aa^ - U aV _ 7 ^z^ ^ 5 ^i^ _ q ^6^2 by 
 6 xS - 3 aa^ + 2 A. 
 
 43. Multiply ^2 +(a + ft)a; + a^ by 2; — a, arranging the 
 result according to powers of x. 
 
 44. Show that Cax + bi/y+(ibx-ayy=(a^-\-b^){2^-^i/^). 
 
 45. If (x + a)(3 a^ - 2 5a; + 3) is the same as 3 a^s _ ^^ 
 — a; + 3, find a and b. 
 
 46. The expression aa^i/^ has the value 32 when a; = 2 
 and t/ = 1 ; find its value when a; = — 3 and y = — 2. 
 
 47. Find the value of a^-S a«+i + 2(a + 1)« when 
 a=2. 
 
 48. Find the value of 2a;— 3^+z when a; = a + ft — c, 
 ?/=a— 5+2c, and z=Sa-\-2b + c. 
 
 49. From the formula s = at -\- ^ft^^ find : 
 
 (1) the value of « when a = 12, f = 2,/= 32. 
 
 (2) the value of/ when « = 64, a = 64, e = 2. 
 
 50. To - -h - add ^ — -and take -a — ^b from the sum. 
 
 51. Simplify 3(a;-y)-2(y-2)-4(t-a;)-7(a;+y+0- 
 
 52. Find the value of n which satisfies the equation, 
 7(58 - w) = 3(n - 14) - 14(n - 25). 
 
 53. Subtract 2a + 36 — 4c from 5a —Sb + 2c and add 
 the remainder to — 3 a. 
 
 54. Simplify : 
 4}3(|a-f6)+2(f6-i.)+6(fc-|a){. 
 
 55. Find the value of [p^ + (a + l)jt> + a] -*- (/> + a). 
 
 56. A man has to walk to a place m miles away. How 
 far will he be from his destination in a given number of 
 hours (f) if his rate of walking is r miles an hour ? 
 
 57. Evaluate a""^, also a" — 1, when a = 3 and n sa 5. 
 
FUNDAMENTAL PROCESSES 79 
 
 58. Find the product oi x — 2y, x-\-2y^ and x — ^y. 
 
 59. In the year 1900 a man on his birthday found that 
 the number of months he had lived was half of the date 
 of the year of his birth ; how old was he ? 
 
 60. If (a + l)x -{-l=x-\-2a, express x in terms of a. 
 
 61. A man has four sons whose combined ages are equal 
 to his own. In 20 years their combined ages will be 
 double the age of their father ; what is his present age ? 
 
 62. Show that S a^ -\- b^ -\- (^ — 6 abc is exactly divisible 
 by 2 a H- 6 + c. 
 
 63. Show that 
 
 64. The volume of a circular cone is given by the 
 expression ^ ttt^A where h and r represent, respectively, the 
 measures of the height and the radius of the base in 
 terms of the same unit. Find to two decimal places 
 the volume of a cone 5 ft. high and 12 in. in diameter. 
 (Take w = 3.14.) 
 
 65. Find the value of Z, when 
 
 a^-a-l = (^a-\- 2)(a - 3). 
 
 66. Find the value of m, when 
 
 a^ - ma — 35 = (a + 5)(a - 7). 
 
 67. Find the value of w, when 
 
 ^a2 + 3 a - 6 = 3(3 a + 2)(2 a - 1). 
 
 68. First indicate and then perform the following 
 series of operations : To the product of (a + 2 5) and 
 (2 a — 6) add the product of (a — 2 b) and (2 a -f 6), and 
 divide the sum by 4. 
 
 69. If a gallon of diluted milk contains p pints of 
 water, how much pure milk is there in n gallons of 
 the mixture ? 
 
80 ELEMENTARY ALGEBRA 
 
 70. Representing the sum of two numbers by «, their 
 difference by d^ the larger number by iV, and the smaller 
 by n, express by formulae the following statements : 
 "Half the sum of any two numbers plus half their 
 difference is equal to the greater number, and half their 
 sum less half their difference is equal to the smaller 
 number." 
 
 71. If n denotes any positive integer, what kind of 
 integers are completely represented by2w? By 2/1 — 1? 
 
 72. If eggs are sold according to quality for 50 cents, 
 42 cents, and 38 cents per dozen, write a formula for the 
 total cost in dollars of m dozen of the highest grade eggs, 
 n dozen of the medium grade, and p dozen of the lowest 
 grade. 
 
 73. What number is represented hy St^-\-2t-\- 1 when 
 t denotes the number 10 ? Can any integer of three 
 digits be expressed in the form at^ -\-bt + c (read from 
 left to right) where a, 5, and e denote the digits used to 
 express the number and t denotes the number 10 ? 
 
 74. Using the notation of problem 73, write the six 
 numbers which can be represented by using the digits a, 
 5, and c. 
 
 75. a and b are two digits of which a is the larger. 
 Find a formula for the difference between the two num- 
 bers which can be represented by them. 
 
 76. If a represents the sum of the ages of n persons b 
 years ago, what expression represents the sum of their 
 present ages ? 
 
 77. Divide 2 a;"» + 3 rr'^+i + 14 x^+^ -|- 32 x"'+^ - 98 ic«+4 
 -|-39a:'«+5by 2 + E> x - '6 x^. 
 
CHAPTER III 
 SIMPLE EQUATIONS 
 
 67. Identity. If the two members of an equation are 
 such that the one can be transformed into the other, the 
 equation is an identical equation and is called an identity. 
 
 Thus, 3a + 4a = 7aisan identity. 
 
 Note. Since the two sides of an identity represent ways of ex- 
 pressing the same number, the statement that they are equal is 
 always true, whatever values may be assigned to the letters involved. 
 
 Thus, ^x — 2x = X is true whatever value be given to x. 
 
 68. Conditional equation. An equation which is true 
 only when the letter or letters involved are restricted to 
 certain values or sets of values is called a conditional 
 equation. 
 
 Thus, the equation x — 1 = is true only when x has the value 1. 
 The equation x + y — 2 is true for many sets of values of x and y, 
 but not for all values of x and y. 
 
 Remark. When in algebra the word equation is used, a condi- 
 tional equation is usually meant. 
 
 69. Notation. In algebra unknown numbers are usually 
 represented by the last letters of the alphabet, as, w, v, a?, 
 y, z ; and known numbers by the first letters, as, a, 6, c. 
 By this convention it is easy to distinguish at a glance 
 between the known and the unknown numbers which 
 occur in the equation. 
 
 81 
 
82 ELEMENTARY ALGEBRA 
 
 70. Satisfying an equation. Any set of values of the 
 letters of an equation which reduces the equation to an 
 identity is said to satisfy the equation. 
 
 Thus, X = 2 satisfies the equation 3 x = 6 ; x = 2 and y = 1 is a 
 set of values of x and y which satisfies the equation 2x -\-dy = 7. 
 
 71. Simple equation in one unknown number. Any 
 equation which can be put in the form ax -{-h = is called 
 a simple equation in one unknown number. 
 
 Thus, 5x-\-2 = 3x + 4: is a simple equation ; it may be written 
 2x-2 = [§25], 
 
 Remark. The simple equations considered in this chapter con- 
 tain only one unknown number. Observe that no higher power of 
 the unknown than the first occurs ; and that the unknown number 
 does not occur in the denominator of any fraction. 
 
 Note. A simple equation is called an equation of the first degree, 
 or a linear equation. 
 
 72. Solution of an equation. To solve an equation which 
 contains only one unknown number is to find all values 
 of the unknown which satisfy the equation. 
 
 73. Root of an equation. A root of an equation which 
 contains only one unknown is any value of the unknown 
 which satisfies the equation. 
 
 Thus, 2 is .a root, and the only root, of the simple equation 
 3a; -6 = 0. 
 
 74. Equivalent equations. Two equations in one un- 
 known are said to be equivalent when they are satisfied by 
 the same value or values of the unknowp ; that is, when 
 they have the same roots. 
 
 Thus, iix-\-S = 2x-{-5, and j^ + 3 = 5 are two equivalent equa- 
 tions ; each one has the single root 2. 
 
SIMPLE EQUATIONS 83 
 
 Two equations are therefore equivalent when: 
 
 1. Every solution of the first equation is a solution of the 
 second. 
 
 2. Every solution of the second equation is a solution of 
 
 the first. 
 
 Thus, the two equations a; - 1 = and a:^ - x = are not equiva- 
 lent; for although the one solution of the first equation, namely, 
 a; = 1, is a solution of the second, yet the second equation has a solu- 
 tion, namely, a: = 0, which is not a solution of the first. 
 
 Note. A simple equation is solved by transforming it into an 
 equivalent equation which shall contain the unknown number alone 
 in one member and its value in the other. 
 
 Thus, the equation 3a:-f-3 = 2a: + 5 
 
 is equivalent to x -}- 3 = 5, 
 
 which is equivalent to a: = 2. 
 
 75. Transposition. An important principle follows from 
 assumptions 1 and 2, section 25. 
 
 Thus: 
 
 Let ax — b = c. (1) 
 
 Adding b to each member of equation (1), 
 
 ax- b + h = c + b. (2) 
 
 Combining, ax = c -\- b. (3) 
 
 Observe that equation (3) differs from equation (1) in that the 
 term containing b is in the second member of (3) but in the first 
 member of (1), and that the signs of the terms containing b are 
 different in the two equations. 
 
 Again, let ax -\- b = c. (1) 
 
 Subtracting b from each member of equation (1), 
 
 ax + b - b = c - b. ■ (2) 
 
 Combining, ax = c — b. (3) 
 
 Compare equations (1) and (3) and observe as before that the 
 term containing b is in the second member of (3) but in the first mem- 
 ber of (1), and that the signs of the terms containing b are different 
 iu the two equations. 
 
84 ELEMENTARY ALGEBRA 
 
 It follows from the foregoing that : 
 Any term may he transposed from one member of an equa- 
 tion to the other, provided that its sign is changed. 
 
 76. Cancellation of terms in an equation. Cancellation 
 of terms in an equation may be illustrated by solving the 
 equation, 
 
 x + b = c-\-b. (1) 
 
 Transposing, x = c -\- b — b. (2) 
 
 Combining, x = c. (3) 
 
 Comparing (1) with (3) we may infer that : 
 When the same terms preceded by like signs occur in both 
 members of an equation^ these terms may be omitted. 
 
 77. Change of signs in an equation. It is sometimes 
 convenient to change the signs of all the terms of an 
 equation. This may be done by multiplying both mem- 
 bers by - 1 [§ 25]. 
 
 For example, let 2 — a: = — 5. (1) 
 
 Multiplying both members of equation (1) by — 1, 
 
 (-l)(2-:r) = (-l)(-5), (2) 
 
 or, _ 2 + x = 5. (3) 
 
 Observe that equation (3) is equation (1) with the sign of each 
 term of equation (1) changed. 
 
 Remark. It is evident that the members of an equation may be 
 interchanged. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Solve the equation Sx — 2 = 2x + 5. 
 
 Solution. 3 x - 2 = 2 X + 5. (1) 
 
 Transposing, 3 x — 2 a: = 5 + 2. (2) 
 
 Combining, x = 7. (3) 
 
 Check. 3 a: - 2 = 2 a: + 5. (4) 
 
 Substituting 7 ior x, 3x7-2 = 2x7 + 5. (5) 
 
 Simplifying, X9 = 19. (6) 
 
SIMPLE EQUATIONS 85- 
 
 2. Solve the equation x -p-^ = -(^x— 2). 
 
 Solution. X - 5^jt_? = hx-2). (1) 
 
 Multiplying each member of the equation by the least common 
 multiple of the denominators, 
 
 12.-lg(3^ + 2)^|(.-2). (2) 
 
 Performing the indicated divisions, 
 
 12ar - 3(3 a: + 2)=i(x - 2). (3) 
 Performing the indicated multiplications, 
 
 12x-9x-Q=4x-S. (4) 
 
 Combining, 3 a: — 6 = 4 a; — 8. (5) 
 
 Transposing, 8 — 6 = 4 a: — 3 a:. (6) 
 
 Combining, 2 = a:. (7) 
 
 Interchanging members, a: = 2. (8) 
 
 Check. x- ^^^^ = -(x-2). (9) 
 
 Substituting 2 fot x, 
 
 2_?Jil±-? = ?:(2-2). (10) 
 
 Simplifying, 2-2 = ^x0, (11) 
 or, = 0. 
 
 3. Solve the equation 2.5a; — 3 = .82: + 2.1. 
 
 Solution. 2.5 a: - 3 = .8 a; + 2.1. (1) 
 Expressing the decimals in (1) as common fractions, 
 
 i^-3 = i^+fi. (2) 
 
 Transposing, | a: - | ar = -|1 + 3. (3) 
 
 Combining, i^a: = ^. (4) 
 Multiplying both members of (4) by -J-^, 
 
 a: = 3. (5) 
 
 Check. 2.5 a: - 3 = .8 a: + 2.1. (6) 
 
 Substituting 3 f or ar, 2.5 x 3 - 3 = .8 x 3 + 2.1. (7) 
 
 Simplifying, 7.5 - 3 = 2.4 + 2.1. (8) 
 
 Combining, 4.5 = 4.5. (9) 
 
86 ELEMENTARY ALGEBRA 
 
 Remark. Various methods of procedure may be resorted to in 
 solving example 3. Thus, each term of equation (1) may be multi- 
 plied by 10; also, .Sx maybe transposed and combined with 2.5a: 
 and — 3 transposed and combined with 2.1. Again, both members 
 of equation (2) may be multiplied by 10, the least common multiple 
 of the denominators; then equation (3) would be replaced by 
 25a:-30 = 8x + 21. 
 
 EXERCISE 33 
 
 Solve the following simple equations, and check each 
 solution ; 
 
 1. 13x-h7 = 5a;-4. 
 
 2. 5u + 2=2u-4. 
 
 3. 13-6a = 13a-6. 
 
 4. 25c?-13 = -6^ + lll. 
 
 5. 3m + 2 = llw-J^. 
 
 6. 5jt? + 12 = 17 - 5j9. 
 
 7. 13r-ll = 2r-ll. 
 
 8. 15 -6f= 3^-12. 
 Suggestion. Divide each term by 3. 
 
 9. -3^ + 17 = 125^-58. 
 
 10. 13-lli/ = 133/ + 253. 
 
 11. 2 + 3(a:-5)=5 + 4(a:-6). 
 
 12. 3 -2(3^-4) =5(2^^ + 3) -84. 
 
 13. 3(jo 4- 2) - 2(2jt? - 3) = 11 (3 - 7 jo) + 72p - 1. 
 
 14. 11(1 -a:) +3(2 -a:) -5(3 -a:) =11. 
 
 15. 12«-5(3f-2) = 3-2«. 
 
 16. 3(2a;-3)=8-5(2a;-3). 
 
 17. 5(z-3)+2(z-3)-4(2-3)=0. 
 
 18. 2(3y-5)-7(23^ + 3)=5(3y-5)-8(2y + 3). 
 .19. a;(a:+3)=r^ + 6. 
 
SIMPLE EQUATIONS 87 
 
 20. (a:-hl)(a;H-3) = ar2-82:+27. 
 
 21. (a;4-3)(22;-5)=2a:(2;-2). 
 
 22. (^^-Hl)(y4-2)-(^ + 3)(y4■4)4-30 = 0. 
 
 24. 3a!-Ka^+2)=8. 
 
 2 5 
 
 26. i (3a; -1)= 11(2^ -7). 
 
 27. £l^-£ll2 £-_5^Q 
 
 10 15 20 
 
 28. 7-^^-(7x + l) = 0. 
 
 55 ^ 77 
 
 31. .3a; + 4=. 9a: -2. 
 
 32. .8a;-l = .la; + 2.5. 
 
 33. 1.5a;- .5 = . 7a; +.6. 
 
 34. .9a; -2.1 = 3.9 -.la;. 
 
 78. Solution of problems. In solving a particular prob- 
 lem which leads to a simple equation in one unknown 
 number it is necessary to : 
 
 1. Restate the problem in algebraic language in the form 
 of an equation. 
 
 2. Solve the resulting equation for the unknown number, 
 
 3. Verify the solution. 
 
88 . ELEMENTARY ALGEBRA 
 
 Remark. Many different kinds of problems occur which lead to 
 simple equations each in one unknown number, and only the foregoing 
 very general directions can be given for their solution. However, 
 when any such problem admits of a definite solution, it will be found 
 that there are in it as many distinct statements as unknown numbers. 
 These distinct statements enable us to express all the unknown num- 
 bers in terms of one of them. The algebraic form of the final state- 
 ment is an equation in this one unknown. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. What number is as much greater than 10 as it is less 
 than 54? 
 
 Solution. 
 
 Let X = the required number. 
 
 Then, a; — 10 = the difference between the required number and 10, 
 and 54 — a; = the difference between 54 and the required number. 
 
 Since the two differences are equal, 
 
 X - 10 = 54 - x. (1) 
 
 Transposing, a: -h a: = 54 -H 10. (2) 
 
 Combining, 2 a: = 64. (3) 
 
 Therefore, x = 32. (4) 
 
 Hence, the required number is 32. 
 Check. 32 - 10 = 54 - 32. 
 
 2. A dealer bought 500 oranges in two lots ; the first 
 lot at the rate of 2J cents apiece, and the second at the 
 rate of 2 cents apiece. He sold them all at the rate of 
 30 cents a dozen and gained $2.25. How many did he 
 buy at each price? 
 
 Solution. 
 
 Let X = the number of oranges in the first lot. 
 
 Then, 500 — x = the number of oranges in the second lot. 
 Then, f a? = the number of cents in the cost of the first lot. 
 
 and 2(500 — x)= the number of cents in the cost of the second lot. 
 .*. I" a; -f 2(500 — x)— the number of cents in the cost of both lots. 
 ^-^^ X 30 = the number of cents in the selling price of both lots. 
 
SIMPLE EQUATIONS 89 
 
 Then, I a: -f- 2(500 - x)+ 225 = ^^- x 30. (1) 
 
 Simplifying, | a: + 1000 - 2 x + 225 = 1250. (2) 
 
 Transposing and combining, J a: = 25. (3) 
 
 Whence, x = 50, 
 
 and 500 -a: = 450. (4) 
 
 Therefore, the dealer bought 50 oranges at 2-| cents apiece and 450 at 
 2 cents apiece. 
 
 Check. I X 50+2(500 - 50) + 225 = -^^^- x 30. 
 
 That is, 1250 = 1250. 
 
 3. A man traveled 30 mi. in 6 hr. 40 min., walking 
 part of the distance at the rate of 3 mi. an hour and rid- 
 ing the remaining distance at the rate of 6 mi. an hour. 
 How far did he walk? 
 
 Solution. Let x = the number of miles he walked. 
 
 Then, 30 — a; = the number of miles he rode. 
 
 Also, - = the number of hours he walked, 
 
 and = the number of hours he rode. 
 
 6 
 
 rvx. t a; , 30- a: 20 ,,v 
 
 Therefore, - + —^— = —• (1) 
 
 Multiplying both members of (1) by 6, 
 
 2 a: + 30 - a: = 40. (2) 
 
 Combining, x + 30 = 40. (3) 
 
 Whence, x = 10. (4) 
 
 Cheok. 10 3_0-ip^20. ^ 
 
 3 6 3 ^ ^ 
 
 That is, \^ = ^^- (6) 
 
 4. A number is composed of two digits ; the digit in 
 the tens' place is one more than twice that in the units' 
 place, and if 36 is subtracted from the number the result- 
 ing number is expressed by the same two digits taken in 
 the reverse order. Find the number. 
 
90 ELEMENTARY ALGEBRA 
 
 Solution. Let x = the units' digit. 
 
 Then, (2 a: -h 1) = the tens' digit. 
 
 Also, 10(2 X + l) + a: = the number, 
 and 10 a: + (2 0? + 1) = the number obtained by writing the digits 
 
 in the reverse order. 
 
 By the conditions of the problem, 
 
 10(2a:+ l)+a:-36 = 10a:+(2a:+ 1). (1) 
 
 Simplifying (1), 9 a: = 27. ' (2) 
 
 Dividing by 9, a; = 3, (3) 
 
 and 2 x + 1 = 7. (4) 
 
 Therefore, the required number is 73. 
 
 EXERCISE 34 
 
 1. If X represents a certain number, what represents 
 the number increased by 3? 
 
 2. What number increased by 3 is equal to 15? 
 
 3. If X represents a certain number, what represents 
 the number diminished by 2? 
 
 4. What number diminished by 2 is equal to 10 ? 
 
 5. If X represents a certain number, what represents 
 four times the number ? 
 
 6. If four times a certain number is 30, what is the 
 number ? 
 
 7. If X represents a certain number, what represents \ 
 of the number? What represents | of the number? 
 
 8. If I of a certain number is 2J, what is the number ? 
 
 9. A certain can filled with lard weighs 42 lb. ; if the 
 can weighs 4 lb., what is the weight of the lard? 
 
 10. Two boys have 36 cents ; if one of them has three 
 times as much as the other, how much has each? 
 
 11. Two men bought 100 fruit trees; if one of them 
 bought 10 more than the other, how many did each buy? 
 
SIMPLE EQUATIONS 91 
 
 12. If the sum of two angles is 90° and one of them is 
 20°, what is the other? 
 
 13. If the sum of two consecutive integers is 21, what 
 are the numbers? 
 
 14. The sum of two numbers is 276. One of them is 
 five times the other ; what are the numbers ? 
 
 15. The number of pupils in a certain school is 227 and 
 the number of girls exceeds the number of boys by 21. 
 How many boys are there? 
 
 16. I paid 8150 for two cows, one costing §30 more 
 than the other. What was the price of each? 
 
 17. The difference between two numbers is 7 and their 
 sum is 31. What are the numbers ? 
 
 18. If from five times a number 21 is subtracted, the 
 
 remainder is 9. What is the number? 
 
 19. Separate 24 into two parts so that one part may be 
 equal to three fifths of the other. 
 
 20. A woman bought a certain number of yards of dress 
 goods and one half as many yards of lining. If she bought 
 24 yards of cloth, how many yards of each did she buy? 
 
 21. If one half of a number added to one fourth of the 
 number is 7|^, what is the number ? 
 
 22. Find a number which when 100 is added to it will 
 give a result equal to five times the number. 
 
 23. Two dealers together bought 15,000 bushels of 
 wheat, one of them buying three times as many bushels 
 as the other. How many bushels did each buy ? 
 
 24. A wagon loaded with wheat weighed 6390 lb. If 
 the wagon weighed one half as much as the wheat, what 
 was the weight of the wheat ? 
 
92 ELEMENTARY ALGEBRA 
 
 25. Three times a certain number is 24 more than J of 
 the number. What is the number ? 
 
 26. The result of subtracting 96 from a certain number 
 is the same as the result of dividing the same number by 
 13. What is the number ? 
 
 27. The difference of two numbers is 24 and the smaller 
 is I of the larger. What are the numbers ? 
 
 28. A and B together own 466 acres of woodland. If 
 22 times A's share is 6 acres less than B's share, how much 
 does each own? 
 
 29. A dealer sold an article for il2, which was at a 
 gain of ^ of the cost. What was the cost? 
 
 30. A dealer sold an article for $12, which was at a loss 
 of ^ of the cost. What was the cost? 
 
 31. The wages of a man and his son for one month were 
 il20. If the son's wages were | of the father's, what 
 were the wages of each? 
 
 32. Find three consecutive numbers whose sum is 42. 
 
 33. The sum of three angles, A, B, C, is 180°. If B is 
 two times and A three times (7, how many degrees are 
 there in each? 
 
 34. The sum of the angles of any plane triangle is 180°. 
 If in a triangle ABC^ angle A is twice angle B and angle 
 (7 is J of angle -B, how many degrees are there in each 
 angle? 
 
 35. A storekeeper found that he had #6.50 in dimes 
 and quarters. How many had he of each if the number 
 of coins of both kinds that he had was 35? 
 
 36. A man wishes to divide a straight line 40 ft. long 
 into three parts so that the first part may be 4 ft. less 
 than the second and the second 7 ft. more than the third. 
 Required the length of each part. 
 
SIMPLE EQUATIONS 93 
 
 37. If ^ of a pole is in mud, ^ of it in water, and the 
 remainder, 15 ft. of it, above water, what is the length of 
 the pole ? 
 
 38. A baseball team won 63 games, which were | of the 
 games that it played. How many games did it play? 
 
 39. In sorting melons 27 less than | of them were 
 found to be defective. If 45 of the melons were found to 
 be in good condition, how many of them were defective? 
 
 40. A man sold 3 acres more than ^ of his lot and had 
 2 acres less than half of it left. Find the number of 
 acres in the lot. 
 
 41. If a; represents the number of dollars in the cost of 
 an article, what represents the number of dollars in the 
 gain if the rate of gain is 
 
 50%? 25%? 20%? 100%? 121%? 62i%? 
 
 50 1 X 
 
 Suggestion. 50% of a: = — — x = -x, or -■ 
 
 42. If X represents the number of dollars in the cost of 
 an article, what represents the number of dollars in the 
 loss if the rate of loss is 
 
 5%? 4%? 75%? 371%? 331%? 6i%? 
 
 43. If X represents the number of dollars in the cost of 
 an article, what represents the number of dollars in the 
 selling price if the rate of gain is 
 
 25%? 30%? 80%? 621%? 200%? 
 
 44. If X represents the number of dollars in the cost of 
 an article, what represents the number of dollars in the 
 selling price if the rate of loss is 
 
 20%? 10%? 121%? 16f%? 8|%? 
 
 45. A dealer gained 25 % by selling a coat at a profit of 
 $51. Find the cost of the coat. 
 
94 ELEMENTARY ALGEBRA 
 
 46. Some lemons were sold at a loss of 6 cents a dozen. 
 If the rate of loss was 16|%, what was the cost ? 
 
 47. A farmer sold a horse for §220, which was at a 
 gain of 10 %. What was the cost ? 
 
 48. A used automobile was sold for §600, which was 
 20 % less than cost. What was the cost ? 
 
 49. The difference between two numbers is 328, and the 
 larger is 42 times the smaller. Find one of the numbers. 
 
 50. A tennis court for doubles is 42 ft. longer than its 
 breadth. The distance around the court is 228 ft. Find 
 the length and breadth of the court. 
 
 51. An acre of wheat yielded 25,000 lb. more straw 
 than grain. The weight of the grain was f of the 
 weight of the straw. What was the weight of tlie grain? 
 
 52. A man bequeathed §45,000 to his wife, daughter, 
 and son. The daughter received §5000 more than the 
 son, and the wife received three times as much as the son. 
 How much did each receive ? 
 
 53. A man is 27 years older than his son ; 12 years 
 hence he will be twice as old as his son will be then. 
 How old is his son ? 
 
 54. Divide §30 among three persons so that the first 
 person shall receive three times as much as the second, 
 and the third person §5 more than the second. 
 
 55. Part of a sum of §3000 was invested at 4% and 
 the remainder at 4| % ; the total income from these invest- 
 ments was §126.25. How much was invested at each rate? 
 
 56. Divide 68 into two parts so that one third of one 
 part may equal one fourth of the other. 
 
 57. A man is 60 years old and his son is 30 ; how many 
 years ago was the man just three times as old as his son ? 
 
SIMPLE EQUATIONS 95 
 
 58. What number diminished by ^ of itself equals 1 less 
 than I of itself ? 
 
 59. All school buildings should have the total light 
 space equal to at least 20 % of the floor space ; what, then," 
 is the greatest amount of floor space that a schoolroom 
 should have whose light space is 180 sq. ft. ? 
 
 60. A straight line is divided into two parts, one of 
 which is 30 in. longer than the other. Seven times the 
 shorter piece equals two times the longer. How long is 
 the line ? 
 
 61. A has $3 more than B, and B has §6 more than C ; 
 together they have #111. How much has each? 
 
 62. On a farm there are twice as many turkeys as there 
 are dugks, and five times as many chickens as there are 
 turkeys. There are 260 of the three kinds in all. How 
 many are there of each ? 
 
 63. How many pounds of tea at 40 ct. a pound must be 
 mixed with 20 lb. at 75 ct., in order that the mixture may 
 be worth 50 ct. a pound ? 
 
 64. The perimeter of a rectangle is 1000 yd. and its 
 altitude is four times its base. What is the length of 
 the base ? 
 
 65.. Eight men hired a yacht, but by taking in four 
 more the expense of each was diminished f 1 ; how much 
 did they pay ? 
 
 66. A man bought 2-cent stamps, 5-cent stamps, and 
 11-cent stamps, of each the same number ; if he paid 72 ^ 
 for the lot, how many of each did he buy ? 
 
 67. A man saved §1350 in three years. He saved twice 
 as much the second year as the first, and three times as 
 much the third as the second. How much did he save 
 the first year ? 
 
96 ELEMENTARY ALGEBRA 
 
 68. A man spends ^ of his yearly income for board and 
 lodging, I of the remainder for clothes and other expenses, 
 and saves ^500 a year. What is his income ? 
 
 69. What number increased by | of itself equals the 
 sum of I of the number and 9 ? 
 
 70. A man invests f of his capital at 5%, and the re- 
 mainder of it at 4| % ; his annual income from both in- 
 vestments is i 240. Find his capital. 
 
 71. " Eight years ago," said a man to his son, " I was 
 thirteen times as old as you were, and four years hence, 
 I shall, if I live, be four times as old as you will be then.'* 
 What is the man's age ? 
 
 72. A merchant mixes 30 lb. of tea which cost 40 ct. a 
 pound, and 20 lb. which cost 60 ct. a pound. What is 
 the mixture worth per pound ? 
 
 73. If I spend $70 for rugs and $S6 for chairs, and then 
 have left one fourth of what I had at first, how much have 
 I remaining? 
 
 74. A dealer has coffee, some at 20 ct. and some at 35 ct. 
 per pound ; he wishes to make a mixture of 100 pounds 
 which shall be worth 30 ct. a pound. How many pounds 
 of each must he use ? 
 
 75. A train ran from Pittsburgh to Philadelphia in 7^ 
 hours ; if it had traveled 10 miles an hour slower, it would 
 have taken 10 hours. Find the distance from Pittsburgh 
 to Philadelphia. 
 
 76. It is required to find a number such that if it be 
 multiplied by 3 and the product increased by 7, the result 
 shall be the same as if it were increased by 8, and the sura 
 multiplied by 2. 
 
SIMPLE EQUATIONS 97 
 
 77. 10 lb. of tea and 12 lb. of coffee together cost 19.60. 
 If a pound of tea cost 30 ct. more than a pound of coffee, 
 find the cost per pound of each. 
 
 78. A packer, engaged to pack 500 tumblers, received 
 8 ct. for every one that arrived at its destination in 
 good condition, and forfeited 15 ct. for every one broken. 
 He received il7.34. How many were broken ? 
 
 79. A cask contains a mixture of 25 gallons of vinegar 
 and 5 gallons of water ; a certain quantity is drawn out 
 and replaced by water and then the mixture consists of 
 10 gallons of vinegar with 20 gallons of water. How 
 many gallons were drawn out? 
 
 Suggestion. If x represents the number of gallons drawn out ; 
 then I X represents the number of gallons of vinegar drawn out, and 
 25 — I a; represents the number of gallons of vinegar left in. 
 
 80. A bottle contains a mixture of 1 pint of cream and 
 3 pints of milk ; a certain quantity is removed and re- 
 placed by milk, and then the mixture contains J of a pint 
 of cream. How much was removed ? 
 
 81. I traveled 22 mi. in 8 hr., walking part of the 
 way at 4 mi. per hour, and riding the rest of it at 10 mi. 
 per hour. How far did I walk ? 
 
 82. A person wishing to give 50 cents apiece to some 
 boys, finds that he has not money enough by 25 cents ; but 
 if he gives them 40 cents apiece he will have 35 cents re- 
 maining. Required the number of boys. 
 
 83. A workman received ^2.50 and his board for each 
 day that he worked, and paid 60 ct. for board for each 
 day that he did not work. For 90 da. he received $132 ; 
 how many of these days did he work ? 
 
98 ELEMENTARY ALGEBRA 
 
 84. It is required to find two numbers whose sum is 
 12, such that if ^ the less be added to J the greater, the sum 
 shall be equal to ^ the greater diminished by ^ the less. 
 
 85. How many pounds of water must be added to 40 lb. 
 of a 5 % solution of salt to obtain a 4 % solution ? 
 
 86. How many pounds of salt must be added to 80 lb. 
 of a 10% solution of salt to obtain an 11| % solution ? 
 
 87. A certain number consists of two digits, the one in 
 the units' place being twice that in the tens' place. If 18 
 be added to the number, the resulting number is repre- 
 sented by the same digits reversed. What is the original 
 number ? 
 
 88. A certain number consists of two digits, the one in 
 the units' place being three times the one in the tens' 
 place. If the order of the digits be reversed and 16 be 
 added to the resulting number, the new number will be 
 three times the original number. What is the original 
 number ? 
 
 89. Into what two sums can $2700 be divided so that 
 the income from one at 5 % shall equal the income from 
 the other at 4 % ? 
 
 90. M's income is $500 a year more then N's and each 
 saves ^ of his income. At the end of 10 years M has saved 
 1| times as much as N. What is the yearly income of each ? 
 
 91. A certain medicine contains 80% alcohol. How 
 much water must be added to 1 quart of it so that the 
 mixture shall contain only 10 % alcohol ? 
 
 92. During one year a traction company carried 
 3,000,000 fewer passengers than in the preceding year ; 
 but, as the average fare had been raised from 4.1 ct. to 
 5.2 ct., the receipts were $394,000 more. How many 
 were carried in each year ? 
 
SIMPLE EQUATIONS 99 
 
 93. If one machine can grind 10 bu. of grain in 2| hr. 
 and another can grind 10 bu. in IJ hr., how long will it 
 take both machines to grind 100 bu. of grain ? 
 
 94. The digit in the units' place of a number composed 
 of two digits is 4 less than 3 times that in the tens' place; 
 the sum of the digits plus 27 is equal to the number. 
 Find the number. 
 
 95. Twice the digit in the tens' place of a number 
 composed of two digits is 7 greater than that in the units' 
 place ; if 7 is subtracted from the number, the remainder 
 is 5 times the sum of the two digits. Find the number. 
 
 96. If one machine can skim 75 gal. of milk in 
 1| hr. and another 60 gal. per hour, how long must both 
 run to skim 300 gal. of milk ? 
 
 97. A man invested $ 5500 in two business enterprises. 
 On the first investment he lost 6 % and on the second he 
 gained 5%. His net gain was $55. How many dollars 
 did he invest in each enterprise ? 
 
 98. There is a reservoir which can be supplied with 
 water from three different inlets ; from the first it can be 
 filled in 12 hr., from the second in 18 hr., and from the 
 third in 36 hr. In what time will it be filled if it is 
 being supplied from all inlets at the same time ? 
 
 99. A certain principal will earn 120 more interest in 
 8 yr. at 6% than it will in ^ yr. at 5%. What is the 
 principal ? 
 
 100. A certain principal will in 4 yr. at 5 % amount to 
 $10 less than the same principal will amount to in 5 yr. 
 at 4|-%. What is the principal? 
 
 101. The sum of two numbers is 30 and one of them is 
 6 less than the other. Find the numbers. 
 
CHAPTER IV 
 
 TYPE PRODUCTS AND FACTORS 
 
 79. Rational operations. Addition, subtraction, multi- 
 plication, and division are called the rational operations of 
 algebra. 
 
 80. Rational expression. An expression which in- 
 volves only rational operations is called a rational ex- 
 pression. 
 
 81. Integral expression with respect to any letter. An 
 expression is said to be integral with respect to any letter 
 when it does not involve a division either by that letter 
 or by a polynomial containing that letter. 
 
 Thus, a^ — 2a -h - is integral, but - and '^—^ are not integral, 
 
 .., ,. 3 a 2a— 1 
 
 with respect to a. 
 
 82. Integral expression. An integral expression is an 
 expression that is integral with respect to each one of the 
 letters which it contains. 
 
 Thus, a^h — Xxy -\- 2 is an integi-al expression. 
 
 83. Degree of a monomial. The degree of an integral 
 monomial is equal to the number of its literal factors. 
 
 Thus, 3 x^y is a monomial of the third degree. 
 
 84. Degree of an expression. The degree of an integral 
 algebraic expression is the same as the degree of the term 
 or terms of the expression which are of the highest 
 degree. 
 
 100 
 
TYPE PRODUCTS AND FACIO^RS I'^i 
 
 Thus, Sa% + 2 abc — 5 a is an algebraic expression of the third 
 degree, and a:^ + 5 a: + 6 is one of the second degree. 
 
 Note. When all the terms of an expression are of the same 
 degree, the expression is called homogeneous. 
 
 Thu^, 3 aH) + 2 h^c — 5 c% is a homogeneous expression. 
 
 85. Degree of an expression with respect to a particular 
 letter. The degree of an integral expression with respect 
 to a particular letter is the same as the index of the 
 highest power of that letter in the expression. 
 
 Remark. It is convenient to classify expressions according to 
 their degree with respect to a given letter. 
 
 Thus, with respect to x : 
 
 ax -\- b is a linear expression, or an expression of the Jirst degree; 
 
 ax^ 4- &x + c is a quadratic expression, or an expression of the 
 second degree ; 
 
 ax^ + bx^ -]- ex + d is a. cubic expression, or an expression of the 
 third degree; 
 
 ax* + bx^ + cx^ 4- dx -\- e is an expression of the fourth degree. 
 
 EXERCISE 36 
 
 1. State the degree of each of the following monomials: 
 
 3 X1/Z ; — 2 0^1/ ; 1 a^; — 3 ci^i/h. 
 
 2. State the degrees with respect to x of each of the 
 monomials in example 1. 
 
 3. State the degree of the following expressions : 
 2x-\-S; ax — by; aa^-^-bx+e; a^ — y^; x^yH'^ -^2xyz — Z. 
 
 4. State the degrees with respect to x of each of the 
 expressions in the preceding exercise. 
 
 5. Which of the following expressions are integral with 
 respect to each of the letters contained? 
 
 o^25 3^2 _ 5^ 2a:2 1 a^^hy^ 
 
 3 3 i + 1 a 2 ab 
 
102 STLEMENTARY ALGEBRA 
 
 6. State the degree of each of the following products : 
 
 7. State the degree of each of the following quotients : 
 3a6 -5-2^2. a253^2a52; Sx'-^'Sx; y^yH^~xy^^, 
 
 8. Which of the following expressions are homogeneous ? 
 a^^1a-\-W'\ a + 254-3c; 2^2^33,^. ^_^_f_22. 
 
 x + y -\-l; x^ -\- ip' -\- z^ — xyz ; he + ca + ah. 
 
 9. Write a homogeneous polynomial of five terms using 
 the letters a, 6, and c. 
 
 10. Write two homogeneous polynomials of three terms 
 each, find their product, and state whether or not it is 
 homogeneous. 
 
 86. The square of a monomial. From section 58, we 
 have, 
 
 That is, (ic»»)2 = a:2m^ 
 
 Hence, 
 
 Rule. To square a power of a number^ multiply its ex- 
 ponent hy '2. 
 
 By the commutative law of multiplication, section 56, 
 
 That is, (arb'^y = (a'»)2(5»)2 = a2m52n. 
 
 Hence, 
 
 Rule. To square a monomial^ multiply the exponent of 
 each of its factors by 2. 
 
 Remark. When a monomial has a numerical coefficient, it is 
 usually preferable actually to square the coefficient rather than to 
 indicate its square. 
 
 Thus, (3 x^yzY = 3\x^)Y^^ = » ^Y^^- 
 
TYPE PRODUCTS AND FACTORS 103 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Square a^. 
 
 Solution. (a^y = a\ 
 
 2. Square 6^. 
 
 Solution. (68'«)2 = 66«. 
 
 3. Square — e5 a^h^c. 
 
 Solution. ( - 5 a^b^cy = (- 5ya%^c^ = 2ba%*c^. 
 
 4. Square (a + by(c — dy. 
 
 Solution, [(a + b)\c - dy^ = (« + ^)* (^ " ^Y- 
 
 EXERCISE 36 
 
 (Solve as many as possible at sight.) 
 Find the square of each of the following, as indicated : 
 
 1. (-3)2; (-2a)2; (5a)2; (-2x 32^)2. 
 
 2. [(a + J)P; [2(« + 5)]2; [_3(a + 6)P. 
 
 3. ( - 5 2^y ^5)2 ; (-2.3 a63c2)2. 
 
 4. (-2a-)2; (3a3-)2. 
 
 5. [2a(5 + (?)P; [-3a3(5 + c?)4]2. 
 
 6. [|a]2; [-|a]2; [| .a^6]2. 
 
 7. [-fa2(6 + ^)2(2; + ^)3]2. 
 
 8. Why is the square of all numbers which we have 
 considered necessarily positive ? 
 
 9. Why are the exponents of the literal factors which 
 occur in the square of a monomial necessarily even 
 numbers ? 
 
 10. If the numerical coefficient of a monomial is a per- 
 fect square, and the exponents of the literal factors are all 
 even numbers, what can be said of the monomial ? 
 
104 ELEMENTARY ALGEBRA 
 
 87. The cube of a monomiaL From section 58, we have 
 
 That is, (af^y = a^. 
 
 Hence, 
 
 Rule. To cube a power of a number multiply its exponent 
 byZ. 
 
 By the commutative law of multiplication, section 56, 
 
 That is, (iaH^y = (a^)3(5»)3 = c^H^-. 
 
 Hence, 
 
 Rule. To cube a monomial multiply the exponent of each 
 of its factors by Z. 
 
 Remark. When a monomial has a^ numerical coefficient, it is 
 usually preferable to cube the coefficient rather than to indicate its 
 cube. 
 
 Thus, (3 x^yzY = S^x^yf^^ = 27 xy^- 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Cubea2. 
 
 Solution. (a2)8 = a^. 
 
 2. Cube a^"*. 
 
 Solution. (x^y = a;9«. 
 
 3. Cube - 5 flS^c?. 
 
 Solution. ( - 5 a%^cy = ( - 5y(a»y(byc» = - 125 a^'^cK 
 
 4. Cube (a 4- hy(c - dy. 
 
 Solution, [(a + by(c - dyf = (o + by(c - dy. 
 
 EXERCISE 37 
 
 (Solve as many as possible, at sight.) 
 Find the cube of each of the following, as indicated : 
 
 1. (2)8; (-3)8; (-2a)8; (5 ay. 
 
 2. (abc^y; (2ab^cyi {-^a^yhy. 
 
TYPE PRODUCTS AND FACTORS 105 
 
 3. [(a + b^Y; [2(a + 6)]3; [_3(«+^)]=^. 
 
 4. (-2a^y; (3a3'")3. 
 
 5. l2a(b-\-c)Y; [-Sa^b + cyy, 
 
 6. [faP; [-|aP; [2a26]3. 
 
 8. When is the cube of a number positive? When 
 negative ? 
 
 9. Why are the exponents of the literal factors which 
 occur in the cube of a monomial necessarily divisible by 3? 
 
 10. If the numerical coefficient of a monomial is a 
 perfect cube and the exponents of the literal factors are all 
 multiples of 3, what can be said of the monomial ? 
 
 88. Square root and cube root of a monomial. When 
 the square of a number a is equal to a given number 
 A^ the number a is called a square root of A. Also, 
 when the cube of a number a is equal to a given number 
 A^ the number a is called a cube root of A. 
 
 Thus, 3 is a square root of 9 since (3)^ is equal to 9. Also, 3 is the 
 cube root of 27, since (3)^ is equal to 27. 
 
 There are always two square roots of a number, the one 
 being positive and the other negative. 
 
 Thus, since (+2)2 = 4 and (-2)2 = 4, both + 2 and - 2 are 
 square roots of 4. 
 
 Since the cube of a positive number is positive, and the 
 cube of a negative number is negative, it follows that the 
 cube root of a positive number is positive and the cube 
 root of a negative number is negative. 
 
 Thus, (+2a)8 = + 8a3 and (-2a)8 = -8a8; therefore, +2a 
 is the cube root of + 8 a* and - 2 a is the cube root of — 8 a*. 
 
 Remark. For the present, a number will be considered as having 
 but one cube root. Later it will be shown that there are three dif- 
 ferent expressions which when cubed give the same number. 
 
106 ELEMENTARY ALGEBRA 
 
 89. Notation. The radical sign V is used in algebra to 
 indicate a square root ; similarly, V is used to indicate a 
 cube root. 
 
 Note 1. In such expressions as Vah fhe sign V does not include 
 h. The square root of the whole expression is indicated either by 
 y/(ab) or Vab, the vinculum over ah serving the purpose of paren- 
 theses. 
 
 Note 2. The sign ± is read plus or minus. 
 
 Thus, y/9 = ± 3 is read the square roof of 9 is equal to plus or minus 3. 
 It is agreed, however, that + VO shall mean + 3 and — V9 shall mean 
 -3. 
 
 EXERCISE 38 
 
 (Solve as many as possible at sight.) 
 
 1. Why is it not possible to find a numerical value of 
 the square root of a negative number ? 
 
 2. In finding a square root of a\ by what number 
 must the exponent be divided ? 
 
 3. In finding the square root of aP^^ by what number 
 must the exponent be divided ? 
 
 4. Give a rule for extracting a square root of a num- 
 ber such as ic^m [§ 86]. 
 
 5. Give a rule for extracting a square root of such an 
 expression as a^b^^ [§ 86]. 
 
 6. How can the result obtained by taking a square 
 root of a number be checked ? 
 
 7. In finding the cube root of a^ by what number must 
 the exponent be divided ? 
 
 8. In finding the cube root of a^"*, by what number 
 must the exponent be divided ? 
 
 9. Give a rule for extracting the cube root of a num- 
 ber such as a^"" [§ 87]. 
 
TYPE PRODUCTS AND FACTORS 107 
 
 10. Give a rule for extracting the cube root of such an 
 expression as a^b^"' [§ 87]. 
 
 11. How can the cube root of a number be checked ? 
 Find the following roots, as indicated : 
 
 12. V^; </'^; </S^; Vf^^p. ^8^356. VT6^*P. 
 
 13. V9"^^; V36^y^. 
 
 14. </21a%^; V-27 
 3 
 
 15. V25(a + 5)2; V-125(aH-5)3. 
 
 16. Vf dXb + ey ; ^2/«3(5 + ^)3. 
 
 18. >/>-1000(a + 2^)9(2 J -3)5*. 
 
 Type Products 
 
 90. Certain algebraic identities which occur in multi- 
 plication are specially important owing to their frequent 
 occurrence. They serve as models for other multiplica- 
 tions, and for this reason should be memorized. 
 
 91. The distributive law [§ 60]. 
 
 a(b-\-c} = ab-{-ac. (1) 
 
 a(b-c)=ab-ac. (2) 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. x(^y — z) = xy — xz. 
 
 2. 12(3a;^-^/)=36a; + 12^/. 
 
 3. — 3 x^y (a: — 2 y) = — 3 oi^y -f 6 a^y . 
 
 4. 2y{x-y^z) = 2yx — '2y{y + z) = 2yx — 2y'^-2yz. 
 
 EXERCISE 39 
 
 Multiply : 
 
 1. _2(a-4). 2. Zx(2x-'^y). 
 
 3. |a(J6-c)- *• 2wv(2w-3v). 
 
108 ELEMENTARY ALGEBRA 
 
 5. 3a6(3a-2J). 6. |(3a:-6y). 
 
 7. — xy {x^ — y)' 8. (5 — a) d^, 
 
 9. (- 2a;+ 3)(-2a;). 10. a{h-\-c + d). 
 
 11. f(52-|-J52). 12. (3a:2^-52)(-3a;?/«2). 
 
 €> 
 
 13. lm{l + m-\-l). 14. 2a;(i/-f2-0- 
 
 15. (2x-'6y^-bz){-2x). 16. 3a(5 - <? + ^ -/). 
 
 17. (-2a5)(-2a + 35). 18. (2 abc - 1 bcd^- hd), 
 
 19. - 3^2^(11 t;2- 7 «3). 20. (a-5 + 2c)(-3 5(?). 
 
 21. ^Iffi^fq-b^f), 22. \'KH(W'^}P'-^Bh) 
 
 92. The square of a binomial. By multiplication we 
 
 find that 
 
 (a4-&)2 = a2 + 2a&+62. (1) 
 
 This identity may be expressed in words as follows : 
 
 The square, of the sum of two numbers is the square of the 
 first plus twice their product^ plus the square of the second. 
 
 Replacing ^ by —b in identity (1), we have [§ 67 
 note] : 
 
 [a +(- 5)]2= a2 + 2 a(- 5)4-(- 6)2. Performing the 
 
 indicated operations, 
 
 (a-6)2 = a2_2a&+fi2, (2) 
 
 This identity may be expressed in words as follows : 
 The square of the difference of two numbers is the square of 
 
 the first minus twice their product^ plus the square of the 
 
 second, 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Square 35 by use of identity (1.). 
 
 Solution. 35^ = (30 + 5)2 
 
 = 30^ + 2 X 5 X 30 4 5« 
 = 900 + 300 + 2;-) ^- 1225. 
 
Leonard Euler (1707-1783) was born at Bale and died at 
 St. Petersburg. He wrote on almost all the branches of mathe- 
 matics then known, revising almost all those of pure mathematics. 
 In 1770 he published an algebra at St. Petersburg which was 
 translated into French in 1794 by the celebrated mathematician 
 Lagrange. 
 
TYPE PRODUCTS AND FACTORS 109 
 
 2. Square 19 by use of identity (2). 
 Solution. 19''=(20-1)2 
 
 = 20^ - 2 X 20 X 1 + 1* 
 = 400 - 40 + 1 = 361. 
 
 3. Square 2 a + 3 6. 
 
 Solution. (2 a + 3 6)2 = (2 a)^ + 2(2 a) (3 b) + (3 by 
 
 = 4 a2 + 12 a6 + 9 &2. 
 
 4. Square x^ — 2. 
 
 Solution. (xy - 2)2 = (xyY - 2(xy) (2) + (2)« 
 = a;2y2 - 4 a:y + 4. 
 
 5. Square | a; — f y. 
 
 Solution, (f.: -^yy=(ixy _2(|a:)(f i/) + (f y)2 
 = Aa;2- 4a,y+ _9^y2. 
 
 
 
 BXEBCISE 40 
 
 
 
 Write the squares 
 
 of the following 
 
 binomials as 
 
 indicated : 
 
 
 
 
 
 1. (2:4-1)2. 
 
 2. 
 
 (2 0.-^)2. 
 
 3. 
 
 (a -3 5)2. 
 
 4. (2(? + 3(^)2 
 
 5. 
 
 (7 a -2)2. 
 
 6. 
 
 (3 5 + 2)2. 
 
 7. (5«-3 6)2. 
 
 8. 
 
 (5cd-S ay. 
 
 9. 
 
 (a + 1)2. 
 
 10. (2 a; -1)2. 
 
 11. 
 
 (Sax-h2bi/y. 
 
 12. 
 
 (2 a; -9)2. 
 
 13. (a + i)2. 
 
 14. 
 
 (3 m - 2 71)2. 
 
 15. 
 
 Cal + sy. 
 
 16. (3 a: + 4)2. 
 
 17. 
 
 (5 a: -3)2. 
 
 18. 
 
 (2mn-S w)2. 
 
 19. (abc-^iy. 
 
 20. 
 
 (xi/z-iy. 
 
 21. 
 
 3P. 
 
 22. nl 
 
 23. 
 
 m\ 
 
 24. 
 
 78l 
 
 25. 29'. 
 
 26. 
 
 99^ 
 
 27. 
 
 (15:^ + 2^)2. 
 
 28. (13a- 3^)2. 
 
 29. 
 
 a^-i^)'. 
 
 30. 
 
 Cix-izy. 
 
 31. (fa; -1^)2. 
 
 32. 
 
 (la -1)2. 
 
 33. 
 
 (1 7/1 + 2)2. 
 
 34. Is 9 x^ -\- 4: y^ -^ 6 XT/ a perfect square ; that is, the 
 square of a binomial? 
 
no ELEMENTARY ALGEBRA 
 
 35. Give a rule for determining whether or not a 
 trinomial is a perfect square. 
 
 Which of the following trinomials are perfect squares ? 
 
 36. ar^ + 2a: + 4. 37. jt?2 + 6 jo -f 9. 
 
 38. m^ + n^-2 mn, 39. ^x^ + l + ^x. 
 
 40. 9 2^ + 4 + 6 a;. 41. a;^ + 4 y2 _ 4 ^^^ 
 
 42. a^-xi-l. 43. 4m2 + 25-20m. 
 
 44. 4:a^-\-4:x-l, 45. r2-2r8-«2. 
 
 46. l-2jp+jp2. 47. 4-12w + 3m2. 
 
 93. The square of a trinomiaL 
 
 Since a -^ b -\- e = a -{- (^b -\- c}, 
 
 (^a + b-^cy = la^(b-{-c}y 
 
 = a^^ 2 aQb + c)-h(b + cy 
 =:a^+2ab + 2ac-\-b^-h2bc+c\ 
 
 Hence, 
 
 (ia + b+cy = (f-{-b^+c^-\-2ab + 2ac-{-2bc. 
 
 This identity may be expressed in words as follows : 
 The square of a trinomial is equal to the sum of the squares 
 
 of its terms plus twice the sum of the products of all pairs 
 
 of the terms. 
 
 Note. In a trinomial there are three pairs of terms. 
 Thus, in the trinomial 2 x —3 y -\- dz the three pairs of terms are 
 2 X and — 3 y,2 x and 6z, — S y and 5 z. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Square x-\-y — z. 
 
 Solution. {x-{-y-zy = x'^-Vy^-\-{-zy + 2xy-\-2x{-z) + 2y{-z^ 
 = x^+ y^ -\- z^-{-2xy -2xz-2 yz. 
 
 2. Square 2 a; — 3 ^ — 1 and check ; let x = 2 and 
 
 y = i. 
 
TYPE PRODUCTS AND FACTORS 111 
 
 Solution. (2x -Sy -ly 
 = (2xy+(-Syy+(-iy + 2(2x)(-Sy)-]-2(2xX-l)+2(-^yX-l) 
 = 4: x^ + Q y^ + 1 - 12 xy -4:X+Qij. 
 
 Check. *(2x-3y-iy = 4:X^+9y^ + l-12xy-4:X-\-6y 
 (4 _ 3 - 1)2= 16 + 9 + 1-24-8+6 
 = 0. 
 
 EXERCISE 41 
 
 Square the following trinomials, as indicated, and check 
 by substituting a = 1, ^ = 2, <? = 3, a: = 1, «/ = — 1, ^ = 2. 
 1. (a; + ^ 4- 2)2. 2. (x-y - zy. 
 
 3. (a -+5 + 1)2.' 4. (a + 5+2;)2. 
 
 5. (x-\-2y + zy. 6. (ia-x-\-^e)\ 
 
 7. (2 a - 3 ^ - 1)2. a (3 a: - 2 1/ - 2)2. 
 
 9. (a + 2 6-3)2. 10. {yz -\- zx ^- xyy. 
 
 11. (a2 + « + i)2, 12. (x^-xy^y'^y. 
 
 94. The square of a polynomial. When a polynomial 
 contains more than three terms, it may be expressed as a 
 binomial or a trinomial by the use of parentheses. 
 
 Thus, the polynomial 2 a + 3 /> — c + 5 rZ may be written, 
 as a binomial, (2a + 3^) + (— c + 5rf); 
 
 as a trinomial, (2 a + 3 &) — c + 5 c?. 
 
 By repeated application of the rules for finding the 
 square of a binomial or a trinomial, it will be found in 
 every case that the square of a polynomial is expressed by 
 the principle employed to find the square of a trinomial ; 
 namely. 
 
 The square of a polynomial is equal to the sum of the 
 squares of its terms plus twice the sum of the products of all 
 pairs of the terms. 
 
 Note. A systematic way of naming the pairs of terms in a poly- 
 nomial is as follows : Take the first term with each of the terms that 
 
112 ELEMENTARY ALGEBRA 
 
 follows it; take the second term with each that follows it; take the 
 third term with each term that follows it ; contirme this process until 
 next to the last term is taken with the last. 
 
 Thus, the sum of the algebraic products of all pairs of terms of 
 the polynomial (a + b -\- c — d + e) is 
 
 ab -\- ac -h a(- r/) + o.e + be -\- b(- d)+ he + c(- rl) -\. (^e -\- (- d)e. 
 
 EXEBCISE 42 
 
 (Solve as many as possible at sight.) 
 Square the following polynomials as indicated : 
 1. (a + 6 + c?4-c?)2. 2. {x-\-y + z-uy. 
 
 3. (m-\-n—p-\- qy. 4. (r — 8 + ^ -h w)^. 
 
 5. (a — b — c-\- dy. 6. (m — n—p — q^. 
 
 7. {x-\-y^-z-\-iy, 8. (mH-372-jo + 2)2. 
 
 9. (a -h 5 + <? + (f + e)2. 10. (m—n—p — q — ty. 
 
 95. The product of the sum and difference of two 
 numbers. By multiplication we find that 
 (a + &)(a-&) = a2_62. 
 
 This identity may be expressed in words as follows : 
 The product of the sum and difference of two numbers is 
 equal to the square of the first minus the square of the second. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Find(a + l)(a-l). 
 
 Solution. (a + l)(a - 1)= a^ - V = a^ - 1. 
 
 2. Find (2 7w4-5y)(2m-5^/). 
 
 Solution. (2 m + 5 y) (2 m - 5 y) = (2 my - (5 y)^ = 4 m* - 25 y^. 
 
 3. Find (ax - 6) (ax + b) 
 
 Solution. (ax - b)(ax + 6) = (ax^ - (by = a^x^ - b^. 
 
 4. Find 101 X 99. 
 
 Solution. 101 X 99 = (100 + 1)(100 - 1) = (100)2 _ i 
 
 = 10000 - 1 = 9999. 
 
TYPE PRODUCTS AND FACTORS 113 
 
 5. Find (a + 6 + c)(a + ^ - 0- 
 
 Solution, (a + 6 + c)(a + 6 - c) = (a + 6 + c)(a + 6 - c) 
 
 = a2 + 2 a6 + 62 _ c\ 
 6. Find (x ■\- y — z)(x — y -\- z). 
 
 Solution, (x -\- y - z) {x - y -\- z) = {x + y - z){x - y - z) 
 
 = x^-{y-zy 
 = a;2 _ y2 _,. 2 y2 - 22. 
 
 EXERCISE 43 
 
 (Solve as many as possible at sight.) 
 Multiply as indicated : 
 1. (x-{-V)(x-l). 2. (a + 2 6)(a-2 6). 
 
 3. (m-n)(m-\-n). 4. (2 a + 5 5)(2 a- 5 6). 
 
 5. (^ah-c){ah + c). 6. (2 ah + c)(2ah - c). 
 
 7. (3a:^-2 2)(3a:z/H-2z). 8. (4 a:- 7 z/)(4a: + 7 ?/). 
 9. (_3a:+5^)(3a;4-5y). 10. (- 1 + a:)(rc + l). 
 11. ih-^yX\ + ^y)- 12. (a2 + 52)(a2-52). 
 
 13. (a2 + 62)(52_^2). 14. (10a:?/z+3)(10a:«/3-3). 
 
 15. 63 X 57. Suggestion. 63 x 57 = (60 + 3) (60 - 3). 
 
 16. 22 X 18. 17. 81 X 79. 
 18. 37 X 43. 19. 201 X 199. 
 
 20. 202x198. 21. (x^y^z)(x-\-y-z). 
 
 22. {a-h-{-c)(a-\-h-{-c). 23. (- a -h 6 + c)(a + 64-c). 
 24. (a-\-h — c){a — h + c). 
 
 Suggestion, (a + & - c)(a - 6 + c) = [a + (6 - c)] [a - (6 - c)]. 
 
 25. ( — aH-6 + c)(a— 5 + <?). 
 
 26. (a-hl-^)(a-l + J). 
 
 27. (2a + 35-c?)(2a-36 + (?). 
 
 28. (2a:-3 2/ + 2)(2a: + 32/-2). 
 
 29. [a2 + (5 + c)2] [a2 - (6 + O^J • 
 
114 ELEMENTARY ALGEBRA 
 
 30. (x3H-?/3)(2^-?/3). 
 
 31. (m* — n^) (w* + w*). 
 
 32. (r6+l)(rS-l). 
 
 33. State a rule for telling whether or not a binomial 
 is the product of the sum and difference of the same two 
 numbers. 
 
 34. Which of the following may be expressed as the 
 product of a sum and difference ? 
 
 a4 _ ^ . ^2 - 2 aft + 52 + 1 ; a^ + f. 
 
 35. Find in two different ways the value of 
 
 (a + J)(a — 5) when a = x+l and b = x—1. 
 
 36. Find in two different ways the value of 
 
 (x -{-y^(x — y)-, when a; = 2 a + 3 and y = '2a — b. 
 
 37. Find in two different ways the value of 
 
 {x -\- y^(x — y)^ when x = a+ h-{- c and y z^a + b — c. 
 
 96. The product of two binomials having a common 
 term. By actual multiplication we have : 
 
 X -\- a 
 x-\-b 
 x^+ ax 
 -\- bx -\- ab 
 
 a;2 + (a + b)x + ab 
 Hence, 
 
 (x+a)(jc + &) = x^ + (a-h6)jc + fl&. 
 
 This identity may be expressed in words as follows : 
 The product of two binomials having a common term is 
 equal to the square of the common term plus the product of 
 the algebraic sum of the unlike terms and the common term^ 
 plus the product of the unlike terms. 
 
TYPE PRODUCTS AND FACTORS 115 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Find the product oi x + a and x — b. 
 
 Solution, (x + a)(x — b)= x^-\-(a - h)x +(«)(- V) 
 = x2 + (a - h)x - ah. 
 
 2. Find the product oi x— a and x—b. 
 
 Solution, (x - a)(x -b)= x^ +(- a- b)x + (- a)(- b) 
 = x^ —(a + b)x + ab. 
 
 EXERCISE 44 
 
 (Solve as many as possible at sight.) 
 Multiply as indicated : 
 
 1. (ia + b)(a-\-c). 2. (a + 2)(a+l). 
 
 3. (2aH-l)(2aH-3). 4. (3a + 2)(3a -4). 
 
 5. (7H-2)(w + 5). 6. (a: + 3)(2:H-2). 
 
 7. (a;H-2)(a;-3). 8. (a; 4- 4) (a: - 5) . 
 
 9. (a:-3)(a:-2). 10. (x-5)(a;-7). 
 
 11. X^b-hc)(ab-\-d). 12. (xi/ + z^(xi/-\-l), 
 
 13. (2a5c + l)(2a6c+3). 14. (i a + 1)(J a + 3). 
 
 15. (ia:+2)(Ja:-5). 16. (3 a - l)(3a - 2). 
 
 17. (a: + y + l)(2: + y + 2). 18. (a + 6 + 2)(a + ^ + 3). 
 
 97. The cube of a binomial. 
 
 By actual multiplication we have, 
 
 a^-\-2ah + b^ 
 
 a -\-b 
 
 a» + 2 a% + ab^ 
 
 + a% + 2ab^ + b^ 
 a^ -\- 3 a% -\- 'S ab^ + b* 
 
 Hence, 
 
 (a+&)3 = a3_j_ft3^3fl2ft + 3afi2. 
 
116 ELEMENTARY ALGEBRA 
 
 By the use of this identity the cube of any binomial can 
 at once be written as in the following : 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Find the cube of a — h. 
 
 Solution, (a - &)8 = a» + ( - ft)8 + 3 a2( - 6) + 3 a( - 6)2 
 = a8-&8-3a26 + 3a62. 
 
 2. Find the cube of 2 a + 3 6. 
 
 Solution. (2 a + 3 &)8 = (2 ay + (3 &)»+ 3(2 a)2(3 6) + 3(2 a)(3 hy 
 = 8a8 + 27 68 + 36a26 + ^ah\ 
 
 3. Cube 98 by the use of the identity of section 97. 
 Solution. (98)8 = (100 - 2)8 
 
 = (100)8 -f(- 2)8 + 3(100)2(- 2)4- 3(100)(-2)« 
 = 1000000-8-60000 + 1200 
 = 941192. 
 
 EXERCISE 45 
 
 (Solve as many as possible at sight.) 
 
 Cube the following, as indicated : 
 
 1. (x + yy. 2. (x-yy, 3. (a + 1)8. 
 
 4. (a -1)8. 5. (re -2)8. 6. (a: + 2)3. 
 
 7. (2 a + 5)3. 8. (2a; + 3y)3. 9. (3-2ic)8. 
 
 10. (4iir-33/)3. 11. (99)3. 12. (101)8. 
 
 13. Is 8 a3 + 12 a% + 6 a62 + 63 a perfect cube ? 
 
 EXERCISE 48. REVIEW 
 
 (Solve as many as possible at sight.) 
 
 By use of type identities perform the indicated multi- 
 plications in examples 1-33. 
 
 1. a(2-3 6). 2. 2 6(3a; + 52/). 3. (x+\y. 
 
 4. (a; -7)2. 5. (5 a; +2)2. 6. (3 a; -4)2. 
 
 7. (6a; + 32)2. 8. (2w-7w)2. 
 
10. 
 
 (^_9)(a:-l). 
 
 12. 
 
 (a:-ll)(a:+3). 
 
 14. 
 
 (2 2:-ll)(2a; + ll). 
 
 16. 
 
 (x-hl)(x-\-a}. 
 
 18. 
 
 (2r + w)(2r-7i). 
 
 20. 
 
 (2r + l)3. 
 
 22. 
 
 {S-^a-by. 
 
 24. 
 
 (xy-lX^^-^^y 
 
 26. 
 
 (2-32/)3. 
 
 28. 
 
 (l-h2:-^^)(H-2: + y). 
 
 TYPE PRODUCTS AND FACTORS 117 
 
 9. (a: + 5)(2:4-3). 
 11. (a:4-10)(2:-2). 
 13. (a:-9)(rc+9). 
 
 15. (4/71 — 7l)(4 7W + W). 
 
 17. (a:- 6)(a;H-4). 
 
 19. (2 2:+?/ + 5)2. 
 
 21. (l + 3c)(l + 4(?). 
 
 23. (a:-y + l)(a:-y-|-2). 
 
 25. (a: + 3)3. 
 
 27. (2a— 5c)(2a + 5<?). 
 
 29. (xy — yz + zx)(xy -\- yz — zx). 
 
 30. (l + a + 2a2)2. 31. («+3)(^_i). 
 
 32. (a + f)(a-2). 33. (ia + l5)(Ja-^6). 
 
 34. Verify the identity : 
 
 (a2 + 62)(2:2 + y2) ^ (^^ ^ 5^)2^ (^y __ 5^)2. 
 
 35. Verify the identity : 
 
 (a2 ^ 52 4. c2)(^2^ y2^ 22) = (aa: + % + C2)2+ (hz - cy^ 
 
 + (ca; — azy + (ay — hxy. 
 
 Simplify the following by performing the indicated 
 operations and when possible uniting terms : 
 
 36. (l + a;)2-(a;-l)(a: + l). 
 
 37. (a + 5)2(a - 6)2. 
 
 Suggestion, (a + 6)2(a - hy =[(a + h){a - 6)]2 
 
 38. (2x+3«/)2(2a:-3 2/)2. 
 
 39. (rr2 4.a.4.i)(2^2-ir + l). 
 
 40. (a^^x-k-lX^-x^-l^ia^-x^^V). 
 
 41. (a + 5 + c)(a + 6- c)(a- 5 + <?)(- a + 6 + <?). 
 Suggestion, (a -\- b + c)(a -f 6 - c)(a - b + c)(- a + b -\- c) 
 
 = [(a + 6 + c)(a + 6 - c)][(a - 6 + c)(- a + 6 + c)]. 
 
118 ELEMENTARY ALGEBRA 
 
 42. (a-^h + e + d)(a-^h + c- d). 
 
 43. a(b — c) + b(^c — a)+ c(a — h). 
 
 44. 2a;(3y-42)+3?/(42-2a:) + 4 2(2a;-3y). 
 
 45. (a -f ^ + <?)(«^ + ^^ + ^'^ - 5(? - <?a - aft). 
 
 46. (a2 -h a6 -h ^2)(a - b'). 47. (a2 - aft + b^)(a + ft). 
 
 Factors 
 
 98. Integral algebraic factors. In this chapter we 
 shall consider only the integral factors of integral expres- 
 sions of the more common form. 
 
 99. Factors of integral expressions. By factors of an 
 integral expression are meant those integral expressions 
 which when multiplied together will produce the given 
 expression. 
 
 Note 1. The word integral refers only to the literal part of the 
 expression. Thus, | a and ^ are integral expressions. 
 
 Note 2. A factor of each of two or more expressions is called a 
 common factor of the expressions. 
 
 100. Prime number and prime expression. In arith- 
 metic a prime number is defined as an integer which is 
 divisible by itself and 1 and by no other integer ; as, 3, 5, 7. 
 In elementary algebra an integral expression with integral 
 coefficients is said to be prime when it is divisible by itself 
 and 1, and by no other integral expression or integer. 
 
 Thus, (x^ + y^) is a prime expression. 
 
 Note. An integer or an integral expression with integral coeffi- 
 cients may often be expressed in more than one way as a product of 
 factors, but it can be expressed in only one way as the product of its 
 prime factors. 
 
 For example, 24 = 12x2 = 8x3 = 2x2x2x3. 
 
 When an expression is to be factored it is understood that, in 
 general, its prime factors are required. 
 
TYPE PRODUCTS AND FACTORS 119 
 
 101. Use of type forms. The type forms in multiplica- 
 tion are of great service in giving the factors of an inte- 
 gral expression. In many cases they furnish the clue as 
 to the kind of factors to expect. 
 
 102. Factors of monomials. The literal factors of a 
 monomial can always be seen at a glance. 
 
 Thus, the factors of — 2 a%'^c are — 2, a, a, a, b, b, and c. 
 Note. Here, as elsewhere, the sign before the monomial is to be 
 regarded as belonging to the numerical coefficient. 
 Thus, -a%=(-l)a^b. 
 
 103. The converse of the distributive law. We learned, 
 section 60, that 
 
 w(a -{• b -{- € -\- d) = ma -\- mb -\- mc -\- md ; 
 hence, conversely, 
 
 ma -\- mb -{- mc -\- md = m(a + 6 + c -h (f). 
 
 From this identity it follows that: 
 
 A monomial which is a factor of every term of an expres- 
 sion is a factor of the whole expression. 
 
 Note. The first step in factoring an integral expression is to 
 remove all monomial factors. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Factor ^x + ^y. 
 
 Solution. 3 2; + 6 2/ = 3(a: + 2 y). 
 
 2. Factor mx -\- my — m. 
 
 Solution. mx + my — m = m(x -f y — 1). 
 
 3. Factor a(l — m) 4- a(m -\- n). 
 
 Solution. a(l - w) + a(m + n) = a[(l - m) + (m + n)] 
 
 = a[/ - m + m -\- n^= a{l -\- n). 
 
 4. Factor (a + 5) (2: + ?/) - (a + 5) {y+z). 
 
 Solution. {a^b){x + y)-{a + b){y + z) = (a-{-b)l{x^y)-{y + z)-\ 
 
 = (a + b)\_x + y - y - z'] 
 = ia-\-h){x-z). 
 
120 ELEMENTARY ALGEBRA 
 
 5. Factor (a + i)(6-c) + (a-|-6)(<?-a)H-(a-f-J)(a--6). 
 Solution, (a -\- b)(b - c)-\-(a -\- b)(c - a) + (a + 6) (a - b) 
 
 = (a + b)[b -c-^c-a + a-b]. 
 
 = (a + b)0 = 0. 
 
 EXERCISE 47 
 
 (Solve as many as possible at sight.) 
 Factor the following expressions by removing the mo- 
 nomial factors : 
 
 1. 4iX-\-12y. 2. am — an. 
 
 3. 5 a; 4- 10 ^ — 15 2. ^. px — py -\- pz. 
 
 5. ax -{■ ay -\- az, 6. | a 4- 1 J — f c. 
 
 1. 2 ax— 2 ay. 8. 3 aw — 6 aw 4- 9 ar, 
 
 9. 12x-\-^0xy. 10. x^y — xy'^. 
 
 11. \ab+^ac, 12. -3a^*H-12a:. 
 
 13. ax + a. 14. ax— ay -\- a. 
 
 15. 4ca; + 6cy-2c. 16. a% -2 a^b^ + ?> ah^. 
 
 17. a:2y25 + 2 aj?/% — 3 a:^^^^ is. — abc —hc — h. 
 
 19. 5 a^ + 10. 20. a:?/ — x. 
 
 21. xy ■\- xz — X. 22. a(ft + <?) 4- «(aJ 4- y)» 
 
 23. (^x + y)(J>^c) + (iy^z^{h-{-c), 
 
 24. (a 4- h)x — (a 4- ^)«/. 
 
 25. (a4-^>-(a4-i)«/H-(a4- J>. 
 
 26. (a4-^)<?4-(«4-*)c2-(a4-^)e. • 
 
 27. (a;4-y)(2a+ft)4-(a^4-?/)(^»-a). 
 
 28. (3 a + 2 5)a 4- (3 a 4- 2 J)5 - (3 a 4- 2 6)(a - 6). 
 
 29. axyz — ayux -\- axut. 
 
 30. 2(a 4- ^)<? 4- 2(a 4- ^)c?. 
 
 31. 2{a-h)cd-2{a + h)ia-1)), 
 
 32. 2(a: + ^)« - 4(a: 4- ,y)e. 
 
TYPE PRODUCTS AND FACTORS 121 
 
 33. SQa + h)c+6(^a-^b')d, 
 
 34. 2(3 a - 2 a:)(3 « + 2 or) + 3(3 a - 2 2;)(2 2: - a). 
 
 35. a(6 + l)+<6+l). 
 
 36. x(2/-l) + ?/(y-l) + («/-!). 
 
 37. a;(2-l)4-«/(2 -l) + 2- 1. 
 
 38. x(a—l)-\-i/{a-l^-a + l. 
 
 The expression may be written x(a — 1) + y(a — 1) — (a — 1). 
 
 39. a(^x — y^ — x-\- y. 
 
 40. a(^y —l)-\-by — h. 
 
 41. a(h + c)-\-2b-\-2c. 
 
 42. 2(x — y)-\-^x — 4:y. 
 
 43. 3 a(?/ -\-z)-^hy — Zhz. 
 
 44. (a + 5)(a; + y^ - (a + 5)2(a; + ^)-f-(a + 6)(a; + y). 
 
 45. (a + hy 4- (a + J)c. 
 
 46. 2(a + 6)^2 - 2(a + ^)26-. 
 
 47. (w+w)Cjt? + ^)24-(m + w)(p + 9) — (wH-w)2(jo-f-5'). 
 
 104. Factors found by grouping terms. Examples 37 
 to 43 inclusive of exercise 47 furnish simple instances of 
 the grouping of terms. In each of these examples the 
 grouping required merely the insertion of a single set of 
 parentheses. In general, in factoring an integral expres- 
 sion of four or more terms by the aid of rearrangement 
 and grouping of terms, those terms should be grouped 
 together which have a common monomial factor. Two or 
 more ways of grouping may be possible, and some of these 
 ways may lead to a common factor and others may not. 
 It is not possible, however, to give any simple rule for 
 proper grouping when several ways of grouping exist, but 
 a careful study of the following illustrative examples will 
 prove helpful. 
 
122 ELEMENTARY ALGEBRA 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Factor ax -^ bx -\- a^ -\- hy. 
 
 Solution, ax -\- hx -{■ ay -\- hy = (ax + hx) + (ay + hy) 
 
 = x(a -ir b) Vy(a + h) 
 = (a-\-h)(x-{-y). 
 
 2. Factor ax—hx — ay — h-\-a-\-hy. 
 
 Solution. ax — hx — ay — h-\-a-\-hy = {ax — ay + a)-\-{ — hx — h-\-hy). 
 
 — a{x - y -\- 1)— h{x - y -{■ V) 
 = {x-y + l){a-h). 
 Another Solution. 
 
 ax — hx — ay — h -{■ a -^ hy = (ax — hx) + (— ay -\- hy) -\- (— h -^ a) 
 = x(a — h)— y(a — h)-\-(a — h) 
 = (a-h)(x-y-^l). 
 
 3. Factor a^ -\- a% + ah^ -\- b^. 
 
 Solution. a8 + a% + ah^ + h» = a^a + h)+ h\a + h) 
 
 EXERCISE 48 
 
 Factor : 
 
 1. 2a-\-2b -\- ac + hc. 2. ax-^ x + a-\'l, 
 
 3. am -^ an — 4: m — 4:71, 4. by -{- y — 2b — 2, 
 
 5, Ip + qm -\- mp H- ql. 6. am — cm -{- cu — au, 
 
 7. xy-\-2y-2x-4. 8. 6 - 9 a + 46 - 6a6. 
 
 9. 6 m - 3 - 2 am + «. 10. 6 rs - 9 /• - 2 «2 4. 3 «. 
 
 11. ^x^+\bxy-\-QyK 
 
 Suggestion. 9 a:^ + 15 2:^ + 6 ^2 = 3(3 x^ + 5 ary + 2 y^) 
 
 = 3(3 x^ + ^xy -^2xy + 2y^) 
 = 3[(3x2 + 3x3^) + (2x3/-l-2y2)] 
 
 = ^lZx(x -\- y)-\-2y(x + y)-\. 
 
 12. a^ -\- ah -\- a — ca — cb — e + a -\- b + \, 
 
 13. 2^2 — a:^ -I- a;— rrz + 22/ — z 4-a: — y -h 1. 
 
TYPE PRODUCTS AND FACTORS 123 
 
 14. a{x-y^-^h{y -x). 15. 2(a - h)- x(h - a), 
 
 16. a^-x'^ + x-l. 17. T^-\-x^+x-\-\. 
 
 18. 82:3_4^2^2a:-l. 
 
 Suggestion. 8 ar^ - 4 a;^ + 2 a; - 1 = (2 a:)8 - (2 xy+ (2 x) - 1. 
 
 19. %a^ + ^x^+2x+\. 
 
 20. aa;^ — ha^ -\- ax — cx^ — bx— ex. 
 
 21. a:2-3 3: + 2. 
 
 Suggestion. x^-Sx + 2 = x^-x-2x + 2. 
 
 22. flw: + a^ + ^2 4- ^a: + ^^ -I- ^2 H- ca: 4- CI/ + cz. 
 
 23. a^ + a52 -|- a% -\- b^. 24. x^ 4- a:?/^ — a:^^ — y^. 
 25. a^^; _ 2 — ^2 _|_ 2 a:. 26. 15 — 6 a: — 10 ^ + 4 xy. 
 27. 1 — abed -\- ae — bd. 28. ??i2n2-f m2p2-)_^7^2_|_^2^2 
 29. (2:2 _ ^2^ 2 — (^2 _ ^2) 2-^ 
 
 Suggestion. First remove the given parentheses, then group 
 terms. 
 
 20. \ — abx^ — (^a — b)x, 31. x^ •\-{a-{-b')x-\- ab. 
 
 32. (ia^^-y^)c+(b^+(P'')a. 33. aijy^^- (^)-b{<^+ a^). 
 
 105. Trinomial squares. We learned, § 92, that 
 
 (a + 5)2= ^2 ^ 2 a5 + 2>2 and (a - 6)2= «2 _ 2 a5 + ^2 ; 
 
 conversely 
 
 (1) fl2 + 2a& + 62 = (a+6)2. 
 
 (2) fl2-2a&+2r^ = (a-6)2. 
 
 From identities (1) and (2) it is obvious that 
 
 If two terms of a trinomial are perfect squares and the 
 
 third term is equal to plus or minus twice the product of the 
 
 square roots of the other two terms, then the trinomial is 
 
 the square of a binomial. 
 
 When two terms of a trinomial are perfect squares and 
 
 the third term is equal to plus or minus twice the product 
 
124 ELEMENTARY ALGEBRA 
 
 of the square roots of the other two terms, the binomial 
 square root — that is, the binonlial which, when squared 
 is equal to the given trinomial — may be found by taking 
 the positive square roots of the two terms of the trinomial 
 which are perfect squares, and adding one of these square 
 roots to the other, or subtracting it from the other, accord- 
 ing as the third term of the trinomial is positive or 
 negative. 
 
 Remark, (a — by=(b — a)^ since a — b and b — a differ only in 
 sign, and (+ a)2=(— 0)2 whatever expression may be represented by 
 a. However, it is customary to write a^ — 2ab -\- b^ equal to either 
 (a ~ by or (b — ay at pleasure and not to write it equal to 
 [±(a - 6)]2. Similarly, a^ -\-2ab + b^ is written (a + by and not 
 [±(a + 6)p. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Factor a2+ 2a + 1. 
 
 Solution. a2 + 2 rt + 1 = (a)2+ 2(a) (1) -f (1)2, which satisfies the 
 conditions for a perfect square. 
 
 ,'.a^-\-2a + l=(a + iy. [§105,1] 
 
 2. Factor x^ — 4: x^ -\- 4. 
 
 Solution, x* - 4 x2 4- 4 = (x^y - 2(2) (x^) + (2)2, which satisfies 
 the conditions for a perfect square. 
 
 .-. a:* - 4 a;2 + 4 = (^2 - 2)2. [§ 105, 2] 
 
 Remark, (x^ - 2)2 may be written (2 - x2)2. [§ 105, Remark] 
 
 3. Factor 4 r*- 12 2:2^ + 9/. 
 
 Solution. 4:x^ - 12xy + 9y^ =(2xy- 2(2x) (3 y) + (3 yy 
 
 = (2x-dyy. 
 
 4. Factor (a -f 6)2- 6 (a + 5)(c - c?) + 9((7 - dy. 
 Solution, (a + 6)2- 6(a + b)(c - rf)+ 9(c - dy 
 
 = (a + by - 2(a + 6)[3(c. - d>] + [3(c - d)Y 
 = [(a + &)-3(c-rf)]2 
 = (a + 6_3c + 3rf)2 
 
TYPE PRODUCTS AND FACTORS 125 
 
 5. Factor x^ — x -\- \. 
 
 Solution. x^-x + l = x^- 2(i) x + Qy. 
 
 Remark. Although not all of the numerical coefficients of 
 x^ — X -\- i are integral, yet the terms of the trinomial satisfy the 
 conditions for a perfect square. 
 
 EXERCISE 49 
 
 (Solve as many as possible at sight.) 
 Factor the following trinomials : 
 
 1. m2 -f- 2 mw 4- n^. 2. r^ — 2 rs -f- «2. 
 
 3. aP'j-2x-{-l. 4. a2-h4a + 4. 
 
 5. 7w2-6m + 9. 6. 4:a^-\-4:x-{-l. 
 
 7. x^-6xi/-\-9y^, 8. x^ + x-\-\. 
 
 9. a2-a+i. 10. 9^:2+32: + ^ 
 
 11. a^+2rc2 + l. 12. 4a*-h20a2+25. 
 
 13. 25a4-20a2 4.4. 14. x^i/^ -\- 2x1/ -{■ 1^ 
 
 15. 4a252-4a5 + l. ' 16. x^ -\- 2xf -\- 1/^. 
 
 17. a254_2a52-|_l. is. 2^/ - 2 a;?/V + 2*. 
 
 19. x^-Qxf-\-9f. 20. 4 a2 _ 20 a52 + 25 5*. 
 
 21. 9 x^t/^z^ -\- 6 axi/z -{■ a^. 22. a2 - 12 a -f- 36. 
 
 23. 25m24-70ww+49n2. 24. (p^qy+2(p + q}-\-l. 
 
 25. a2« + 2 a"»5" + ^". 
 
 Suggestion, a^^ + 2 a'^b'' + J^n _ (am)2_^ 2(a'") (&»*) + (b'^y. 
 
 26. a:2p — 2 a:^?/' + ^2«. 27. a^ + 4 aH"" + 4 62n. 
 28. 4 a63 - 4 ^252 _^ ^5, 29. 7?y -2 :^y + a;y. 
 
 30. a25^ + 4 aj^ + 4 5^2. 31. 2:2^22 _ 14^^22 + 49^2, 
 
 32. o?-V\%^-\-^\x. 33. w2wjt?4-20mwp + 100rip. 
 
 34. (a:+^)2_6(2: + y)(a+i)+9(a+5)2. 
 
 35. 2:2_iOx(?/ + 2)+25(?/ + z)2. 
 
126 ELEMENTARY ALGEBRA 
 
 36. a^-^b^+c^ + 2hc-\-2ca-\-2 ah. 
 Suggestion, a^ + b^ + c^ + 2bc + 2 ca -\- 2 ab 
 
 = a2 + 2(b + c)a + (62 + 26c + c«). 
 
 37. a^ + h^-^(^-\.2hc^2ca-2ah, 
 
 38. 4a2-28a(6 + c)4-49(5+c)2. 
 
 39. 49a:V+28a:«/2(3^4.2)+4i/2(y + 2)2. 
 
 40. l-6(a-5)+9(a-6)2. 
 
 106. The difference of two squares. We learned, 
 section 95, that 
 
 (a + 6)(a — 5) = a2 _ 52 . hence, conversely, 
 a2-63 = (a4.6)(a_5). 
 
 From this identity we infer the following rule for 
 factoring the difference of two squares : 
 
 Rule. Find the positive square root of each of the two 
 squares and form the sum and the difference of these square 
 roots in the order in which their squares occur in the expres- 
 sion. The sum and the difference of the square roots are the 
 two factors, 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Factor 9a^-25. 
 Solution. 9 x2 - 25 = (3 x)^ - (5)2. 
 
 The positive square roots of the squares are 3 x and 5. The sum 
 of the square roots is 3 x + 5 and the difference is 3 a: — 5. 
 
 .-. 9ar2 - 25 =(3x 4- 5)(3ar - 5). 
 
 2. Factor a2 - (5 - c)2. 
 
 Solution. The positive square roots of the squares are a and b — c. 
 The sum of the square roots is a + (b — c), and the difference is 
 a — (6 — c) ; that is, a + 6 — c and a — b + c. 
 
 .-. a2 -(6 - c')2 =(a + b- c)(a - 6 + c). 
 
TYPE PRODUCTS AND FACTORS 127 
 
 In practice, the work of factoring a^ —(b — cy may be arranged 
 thus: 
 
 a^-(b- c)2 = [a +(b - c)] [a-(b- c)] 
 = (a -{- b — c)(a — b + c). 
 
 3. Factor a2 _ 1 52. 
 Solution. a2 _ A 52 = ^2 _ (2 5)2 
 
 4. Factor a* — b^. 
 
 Solution. • a* - 6* = (a^y - (b^y 
 
 = (a^ + b^)(a^-b^) 
 
 = la^ + b^X^ + b)(a - b). 
 
 EXERCISE 50 
 
 Factor : 
 
 1. m^ — n^. 2. 2^2 _ y2 Z, W — r^. 
 
 4. a^-\. 5. 1-32. , 6. m2-4. 
 
 7. 9-^2. 8. 42:2_^2, 9. ^2_1622. 
 
 10. 2^2 _ 100. 11. 252;2-64?/2. 12. ^m^-\^n'^, 
 
 13. w*-w2. 14. a;6_^2, 15. 36a:4-49«/8. 
 
 16. 25 a254 _ 36 52^4, 17, ^ - x. 
 
 18. a^-4ir2, 19. 2^-?/*. 
 
 20. ^ — {y-\- 25)2. 21. ( w H- w)2 - jt?2, 
 
 22. {a -f 5)2 - 9. 23. 2^2 - I y2. 
 
 24. (a+6)2-((?+(^)2. 25. l_(a:-y)2. 
 
 26. 4(a-6)2-9(a + 5)2. 27. 25(2: -^)2- 36(2^4-^)2. 
 
 28. (&-Q>-cy. 29. 9c2_(a-}-6 + (?)2. 
 
 30. a2+2a(? + c2-52. 31. a;2_l_2^_^2. 
 
 32. 2^2 ^10 a; 4. 25- 25 «/2. 33. \-m^ -"Imn-n^. 
 
 107. Trinomials of the form x2 4- ex + {/. From sec- 
 tion 96 we have, 
 
 {x 4- a)(2; + 5) = 2;2 + (a + h^x + ah ; conversely, 
 Jt2 4- (a 4- &)jir + a6 = (jr + a)(jr + 6). 
 
128 ELEMENTARY ALGEBRA 
 
 From this identity we see that : 
 
 Any trinomial of the form a^-\- cx-{- d can be factored 
 when c, the coefficient of x, is the sum of two expressions, 
 and d, the last term, is the product of the same two expressions. 
 
 Remark. When c and d are given integers and not too large, it is 
 possible to determine by inspection whether two other integers a and h 
 exist such that a + b = c and ab = d. When t^o such integers are 
 found the factors oi x^ -\- ex -\- d are {x + a) and (x + b). 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Factor x^ -{- 1 x -\- 12. 
 
 Solution. The two integers whose sum is + 7 and whose product 
 is + 12 are evidently -f 3 and + 4. 
 
 . ... a:2-<-7a; + 12=(x + 3)(x + 4). [§107] 
 
 2. Factor q^-{-x — 12. 
 
 Solution. The two integers whose sum is + 1 and whose product 
 is — 12 are evidently + 4 and — 3. 
 
 .-. a:2 + a: - 12 = (x - 3)(a: + 4). [§ 107] 
 
 3. Factor a;^ - 9 ic + 20. 
 
 Solution. The two integers whose sum is — 9 and whose product 
 is + 20 are evidently — 4 and — 5. 
 
 .-. a;2 - 9 a: + 20 = (x - 4)(a: - 5). [§ 107] 
 
 4. Factor x^ -\- a(h — c)x — c^hc. 
 
 Solution. The two expressions whose sum is ab — ac and whose 
 product is — aV)c are evidently ab and — ac. 
 
 .'. x^ + a(b - c)x - a%c = (a: + ab){x - ac). [§ 107] 
 
 5. Factor x^-2 x^y^ - 15 y^. 
 
 Solution. The two monomials whose sum is - 2 y* and whose 
 product is — 15 y^ are — 5 y^ and 3 y^. 
 
 ... x^-2 xhf -\by^= {x^ - 5 y'^ix^ + 3 ^2). [§ 107] 
 
TYPE PRODUCTS AND FACTORS 129 
 
 BXBROISB 51 
 
 Factor the following : 
 
 1. a2 + 3a + 2. 2. y2 ^ 2 ^ - 3. 
 
 3. 22_2_6. ^. h^-Qh + b. 
 
 5. jo2 + 6jD + 8. 6. ^-7c? + 12. 
 
 7. x^-bx-1^. 8. «2_4^_45. 
 
 9. 52-12 5 + 32. 10. 22 + 13^ + 30. 
 
 11. m2 + 4m-221. 12. .c2 + 18 a: 4- 72. 
 
 13. jp2_i0jp_ll. 14. 2^ -10 a; + 24. 
 
 15. ;r2-27a; + 50. 16. a2-18a + 80. 
 
 17. x^-l^x-\-^\. 18. «24.8a_209. 
 
 19. a2+9a-36. 20. 2^2 -2; -2. 
 
 21. 62 + 27 5 + 152. 22. w2-16m + 55. 
 
 23. 22.^52-24. 24. 2:2 ^_ 10 a: - 39. 
 
 25. a2_i2a_l33. 26. w2- 28 m + 171. 
 
 27. 2:2 _ 78 2; + 365. 28. 2^2 _^ 10 a: - 119. 
 
 29. 2:2 + 3 2; -154. 30. 22_|.62-91. 
 
 31. a^ + a-mO. 32. Z2_j.24Z+23. 
 
 33. 2:2 _ 26 a; - 155. 34. a2_i8^_i9. 
 
 35. a^y^ + 4 2:2^/2 _ 5 xy. 36. x^-{-(a + S)x + 3 a. 
 
 37. a2_(2- 5)a- 2 5. 38. «/2 -(5 + ^)^ + 5 2. 
 
 39. ^2 _j_ 2;(?/ _ ;2)^ _ x^yz, 40. 2:2 + 3 2: — a2: — 3 a. 
 
 41. 2:2 _ ^2; + 3 ^^ _ 3 ^^ 42. a^ _|_ 2 52: — C2: — 2 5(7. 
 
 43. 6/2 4- 3 a5 + 2 52. 44. 2:2 _ 6 ^^2 + 5 ?/*. 
 
 45. a* + 4 «2J - 221 52. 46. X^-{-S 2:2/ _^ 15 2^. 
 
 47. 2:2- 2a2:-2 52: + a2_|_2a5. 
 
 48. 2:2^^2:— 52: + 6 — 3 a. 
 
130 ELEMENTARY ALGEBRA 
 
 108. The general quadratic trinomial ax^ -\- bx -h c. By 
 
 actual multiplication we have 
 
 (^px -f q) (rx + «) = pro^ + (ps -f qr^x + qs; 
 conversely, 
 
 prx^ -h (ps + qr}x -^qs = (px + q)(rx + s). 
 
 We observe in the first member of this identity that 
 the coefficient of x is the sum of two terms + ps and -h qr 
 whose product is -i-pqrs and that the product of the coeffi- 
 cient of x^, namely pr^ and the last term, namely qs, is also 
 +pqr8. These facts furnish a clew which is of assistance 
 in factoring a trinomial of the form aa^ +bx'^c whenever 
 it is possible to express b as the sum of two numbers whose 
 product is equal to ac. 
 
 Note. In the foregoing, we have, 
 
 a = pr, h = ps ■{■ qr, c = qs. 
 
 Therefore, V^ = ph^ + 2 pqrs + qh-^ ] ^ 
 
 and, 4 ac = ^pqrs ; 
 
 hence, h^ — ^ac = p^s^ — 2pqrs + q^r^ = (ps — qry. 
 
 If, therefore, in aaj-^ -i- bx -\- c, the square of the coefl&cient of x 
 minus four times the product of the coeflBcient of x^ by the last term 
 is not a perfect square, it is not possible to express the quadratic 
 ax^ + bx -\- c a,s the product of two rational factors. 
 
 The converse of this statement (namely, that when b^ — 4: ac is a 
 perfect square, integral values of p, q, r, and s exist such that a = />r, 
 ^ =z ps -\- qr, c = qs) will be proved in a later chapter. (The letters a, 
 b, c are assumed here to represent positive or negative integers, or 
 integral expressions.) 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Factor 6 2^2 .,.19 ^^10, 
 
 Solution. If possible, we must find two integers whose sum is 19 
 and whose product is 6 x 10, or 60. These integers are evidently 15 
 and 4. 
 
TYPE PRODUCTS AND FACTORS 131 
 
 Therefore, 6 a:^ + 19 x + 10 = 6 a:^ + ISx + 4 x + 10 
 
 = S x(2 X + 5) + 2(2 X -{■ 5) 
 = (2x+ 5)(3a; + 2). 
 
 That is, 6x2 + 19a: + 10 =(2x + 5)(3x + 2). 
 
 2. Factor 10 2:2 _ 7 a- _ 12. 
 
 Solution. If possible, we must find two integers whose sum is 
 
 — 7 and whose product is 10(— 12), or — 120. Since their product 
 is negative, one of these integers is positive and the other negative. 
 Moreover, the larger integer is negative, since the sum of the two 
 integers is negative, namely —7. The required integers are evidently 
 
 - 15 and + 8, since - 15 + 8 = - 7, and (- 15)(8) = - 120. 
 
 .-. 10x2 - 7x - 12 = 10x2 _ I5x + 8x - 12 
 = 5x(2x-3)+4(2x-3) 
 = (2x-3)(5x + 4). 
 That is, 10x2 -7x- 12 =(2x - 3)(5x + 4). 
 
 (In this example a = 10, 6 = - 7, and c = - 12. ^>2 _ 4 q^. = 629 = 23*. 
 Since b^ — 4:ac is a perfect square, the given expression can be ex- 
 pressed as the product of two rational factors.) 
 
 3. Factor acx^ + (^bc — a^x — b. 
 
 Solution. If possible, we must find two expressions whose sum 
 ia be — a and whose product is — abc. The required expressions are 
 evidently be and — a. Hence, 
 
 acx2 + (be — a)x — b = aex^ + bex — ax — b 
 = ex(ax -\- b) — (ax + b) 
 = (ax+ 6)(cx- 1). 
 Note. See problem 29, exercise 48. 
 
 EXEBCISB 52 
 
 Factor : 
 
 1. 
 
 Qa^-x-1. 
 
 3. 
 
 6x^-\-x-5. 
 
 5. 
 
 102:2- 13a: -3. 
 
 7. 
 
 3p^-7p-Q, 
 
 9. 
 
 622 + 113 + 3. 
 
 11. 
 
 262+116-21. 
 
 2. 
 
 Ux^^^x-1, 
 
 4. 
 
 14arJ + 2:-3. 
 
 6. 
 
 9a2_9^_4. 
 
 8. 
 
 6m2-5wi-4. 
 
 10. 
 
 862_i45_i5. 
 
 12. 
 
 142:2_4i^ + 15 
 
132 ELEMENTARY ALGEBRA 
 
 13.. 6x^-x-2. 14. 4y2 + it3^ + 15. 
 
 15. 6a^-{-Ux-\-6. 16. 20^2 + 13 m -15. 
 
 17. 4a2H-8a + 3. 18. 6^2^ii^_10. 
 
 19. 1522 + 162 + 4. 20. 8^>2_266-45. 
 
 21. 10^:2+9^+2. 22. 10j92 + 29jo + 10. 
 
 23. 9 a2 + 18 a + 8. 24. 28 a2 + 51 « + 20. 
 
 25. 6^2+112 + 4. 26. 6a;2 + 23a: + 20. 
 
 27. 21a2+8a-4. 28. 12rr2+ 59a; + 55. 
 
 29. 16^2 + 2a- 3. 30. 16a:2 + 34a:-15. 
 
 31. '24:X^+7x-6. 32. 20a^ + 53a:+35. 
 
 33. 22a2 + 27a-9. 34. 6ic2 + (9 + 2a> + 3a. 
 
 35. 3rr2 + (^+6>)^ + 2a. 36. 6^2 + (2a-9)y-3,a. 
 
 37. a<?rr2 + (56?+2a>+2 5. 38. aba^ + (a^-lr^^x—€d>. 
 
 39. aJa^ + (a^ + 52)a; + ^5. 40. 3 <?a;2 + (6 - 2 <?> - 4. 
 
 41. 6 aV — 5 ax — 6. 
 
 42. 6 wwp2 + (3 7^2 + 2 7i2^jt> H- mw. 
 
 109. Sum and difference of two cubes. By actual mul- 
 tiplication we have, 
 
 {a + h)(a^-ab + h^}=a^-^h^ 
 and (a-6)(a2 + a6 + ^>2^=a3-63; 
 
 conversely, 
 
 (1) a8 + 63 = (a+6)(a2-a&H-62). 
 
 (2) (^-i^ = (a-b)(i(^ + ab-hh^). 
 
 By use of identities (1) and (2), the factors of any expres- 
 sion which has the form of the sum or the difference of 
 two cubes may be found. It is obvious that one factor of 
 the sum of the cubes of two numbers is the sum of the num- 
 bers and that the other factor is a trinomial which is the sum 
 of the squares of the two numbers minus their product ; also, 
 
TYPE PRODUCTS AND FACTORS 133 
 
 that one factor of the difference of the cubes of two numbers 
 is the difference of the numbers and that the other factor is 
 a trinomial which is the sum of the squares of the two num- 
 bers plus their product, 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Factor a^ -}- 8. 
 
 Solution. a3 -H 8 = a3 + (2)8 
 
 = (a + 2)(a2-2a + 4). [§109,1] 
 
 2. Factor a^ - 216 R 
 Solution. o8 _ 216 &8 = a3 _ (6 &)« 
 
 = (a - 6 6)(o2 + 6 ab + 36 fe^). [§ 109, 2] 
 
 3. Factor x^ + y^. 
 
 Solution. a-« + y^ - {jp-y^r (y^Y 
 
 = (:r2 + 2/8)[(x2)2 - (x2)(|,8) + (3^«)2] 
 
 That is, a:» + y^ = (x'^ + y^){x^ - -^V + 2/®)- 
 
 4. Factor a^ — y^. 
 
 Solution. a:« - yS = (a:« - y^){x^ + /) [§ 106] 
 
 = (a: - .y)(a:2 + xy + 2^2) (^ _,. y^(^x2_xy + y^). 
 
 That is, a:® — y^ = (a: — y)(x + .v)(a:2 _j. -^y _|. y2)(-y2 — a;y+y2y^ 
 
 Factor : 
 
 
 EXERCISE 53 
 
 
 
 1. m^ + n^. 
 
 2. 
 
 m^ — ?l3. 
 
 3. 
 
 63 + ^. 
 
 4. 2:»-l. 
 
 5. 
 
 r3+l. 
 
 6. 
 
 a3-8. 
 
 7. 1-./3. 
 
 8. 
 
 2/3-0^. 
 
 9. 
 
 2:3+27. 
 
 10. a3H-216R 
 
 11. 
 
 8 a3 4- 27 63. 
 
 12. 
 
 27a3_l. 
 
 13. 8 53-1-1. 
 
 14. 
 
 ^-h 
 
 15. 
 
 27^3 + 64713. 
 
 16. a^-\-l. 
 
 17. 
 
 a^-1. 
 
 18. 
 
 m^-{-m. 
 
 19, a^-fa:^. 
 
 20. 
 
 ma^ 4- mab^. 
 
 21. 
 
 16r3« + 2«*. 
 
 22. a;6^y6. 
 
 23. 
 
 x^-Sx. 
 
 24. 
 
 a363 _ c^. 
 
134 ELEMENTARY ALGEBRA 
 
 25. rn^ + S, 26. w«-8. 27. a6+27. 
 
 28. mV-a%^. 29. r«-27. 30. 2 7w8w + 128w. 
 
 31. (a; -2)3 + 1. 32. l-(^x + ^y. 
 
 33. (a;4-y)3+(a;-y)3. 
 
 34. Give, at sight, one factor of 125 — (a -|- 4)^. 
 
 35. Give, at sight, one factor of (a + by — (a — by. 
 
 36. Give, at sight, one factor of (2m — ny-{-(m + 2 w)^. 
 
 37. By use of the factors of a:^ — y^ [See Illustrative 
 Example 4, p. 133], find four factors of 999,999. 
 
 Suggestion. 999,999 = lO^ - 1. 
 
 Factor : 
 
 38. l + ^\f. ' 39. (a-6) + Ca3-63). 
 
 40. a-\-b + a^-\-b^. 41. 2(jt?+ ^)+jt>3-f- ^. 
 
 42. (1+ a;) +3(1 + 2^3). 43. (^a-\-b-cy-(a-b + cy, 
 
 44. m%* + m^n. 45. 1 + (jt? + 5' — 1/. 
 
 46. m^ — mn(m + w) + w^. 
 
 Suggestion, tw* — mn(m + w)+ n* = (w?* + w*) — mn(in -f n). 
 
 47. m^ — 2 m^w + 2 Tww^ — w,8, 43. 2m3 — 12m2+24w — 16. 
 49. a8 + fa25 4.|a62+i63. 50. a^ ^ a^ Jf.\a-{-^j. 
 
 110. Special methods of factoring. There are certain 
 integral expressions whose factors may be found, and 
 which are not classed under any of the preceding general 
 cases of factoring. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Factor a* +a2 + l. 
 
 Solution, a* + a2 + 1 = a* + 2 a2 + 1 - a2 
 
 That is, a* + a* + 1 = (a2 + rt + l)(a« - a + 1). 
 
TYPE PRODUCTS AND FACTORS 135 
 
 Remark. The solution of example 1 illustrates the method of 
 factoring by the aid of adding and subtracting the same number. 
 
 2. Factor a^-{-b^-^ 4c^- 4:be-^4:ca-2ab. 
 
 Solution, a^ + b^ + 4: c^ - 4:bc + 4: ca - 2 ab 
 
 = a2 + 2a(2c - &) + (4c2 - 46c + f^) 
 
 = a^ + 2a(2c - b) + (2c - by 
 
 = (a-h2c-by. [§105] 
 
 3. Factor hc(h — <?) -f ca(^c — a)-\- ahQa — 6). 
 
 Solution. In order to arrange this expression according to the 
 powers of a, it is necessary to perform the indicated multiplications in 
 the last two terms. Then we have, 
 
 bc{b — c) -\- ca(c - a) + ab(a — b) = bc(b — c) + c% — ca^ + a^b — ab^ 
 
 = (P- c)a2 - (62 - c^)a + bc(b - c) 
 = (6 - c) [a2 -(b + c)a + be] 
 = (b - c)(a - b)(a - c). 
 
 Remark. Many expressions when arranged according to the 
 powers of some letter are seen to be factorable. The solutions of 
 examples 2 and 3 illustrate the method of factoring such expressions. 
 
 . EXERCISE 54 
 
 Factor tlie following : 
 
 1. w*+m2+l. 2. w^ + mV-hw*. 
 
 3. l-|-9a^^ + 81a^, 4. m^ + m^^ + n^. 
 
 5. ;?8+Jt>V + ?*- 6. 3^+a^i/^-\-y^, 
 
 7. a^-\-a^-\-l. 8. 16/ + 4^2^!^ 
 
 9. x^-lla^t/^-^^^ 10. a4_27a2^ + 5*. 
 
 11. w*-123m2H- 1. 12. a:4 + /_7a;2/. 
 
 13. a* - 34 a2 -f- 1. 14. p^ _ 14 jo2^2 ^ ^4. 
 
 15. 2:2 + 4 y2 _j_ ^2 _|_ 4 ^2 _ 2 2a; _ 4 ^^ 
 
 16. yz(jf — z)-\'Zx(z — x)-^xy{x — y). 
 
 17. x^ + y^+l-\-2y + 2x-\-2xt/. 
 
 18. a^(b - c) + bXc - a) + (^(a - 5) . 
 
136 ELEMENTARY ALGEBRA 
 
 19. a\b-o)+b\c-a}+c\a-b). 
 
 Suggestion. Arrange according to powers of «, remove a factor, 
 then arrange the remaining factor according to powers of b, remove 
 a second factor, finally arrange the remaining factor according to 
 powers of c. 
 
 20. 9a*-37a262+4 6*. 
 
 • 21. 4a^-21a^f-^9i/^, 
 
 22. 16a^ + SQa%^ + Slb^. 
 
 23. bc(b^ - c2) + ca((^ - a2) ^ ab(a^ - 52). 
 
 24. a(b^-c^) + b((^-a^') + c(a^-b^). 
 
 25. 62<?2(62 - (?2) ^ cH\c^- a2) + ^2^2(^2 _ ^2)^ 
 
 111. Summary of factoring. The identities, rules, and 
 solutions of illustrative examples as given in this chapter 
 are sufficient to cover the simple cases of factoring which 
 occur in elementary algebra. The following summary will 
 be of assistance in the work of factoring. 
 
 I. A% the fir %t step in factoring an integral expression re- 
 move all numerical and monomial literal factors. 
 
 II. Iw factoring a binomial use one of these identities : 
 1. a2_j2=^(a_&)(a + 6). 
 
 3. a3 + 68 = (a+6)(a2-a6+62). 
 
 III. In factoring a trinomial use one of these identities: 
 
 1. ff^ + 2a&-f &2 = (a+&)2. 
 
 2. a2-2a&-|-&2 = (a-&)2. 
 
 3. jc2 + (a4-&)Jc+a&=(x+a)(x+ft). 
 
 4. ax^-\-hx-\-c = ipx + q)<irx-¥s). 
 6. (^^(^y^-^l^^ia^+h^y-iaby. 
 
TYPE PRODUCTS AND FACTORS 137 
 
 IV. In factoring a polynomial he guided hy one or more 
 of the following directions : 
 
 1. Rearrange and group terms. 
 
 2. Consider the polynomial the difference of two squares. 
 
 3. Arrange the polynomial according to the powers of 
 some one letter. 
 
 4. Consider the polynomial the square of a polynomial. 
 
 EXERCISE 55 — REVIEW 
 
 Find the factors of: 
 
 1. 2a;+2. 2. a?-\'bx^. 
 
 3. ax 4- ay — hx — hy. 4. ah -f- he. 
 
 5. 3 2;(a~5)-2^(6-a). 6. p'^-QA:. 
 
 7. mn\x — y) + 2 m(x — y^n -\- m(x — y). 
 
 8. a^^-2x'^-x-2. 9. a^+^a^-x-^. 
 10. a^-2a;-15. 11. a^ - a\ 
 
 12. 64a2 + 144«6 + 81J2. 13. ^x^-yK 
 
 14. a2» - 52. 15. 1 -j- ac? - &(i - ahcd. 
 
 16. (a:2 + «^2)2 4-(^2 + 2>. 
 
 17. 7i(7i4-l)(w4-2) + (7i4-l)(w+ 2)(w + 3). 
 
 18. 2x^ + 2ax-[-2ac-\-2cx. 19. 2ax-^ay—2hx+Sby 
 20. awi — 2 5w + «n — 2 5m. 21. (a + l)2— 3(a + l). 
 
 22. a2 - 2(a -h)- h\ 23. a^ - 2(a -h)- h\ 
 
 24. a^+aV- 25. (m + w)2— p2, 
 
 26. 49^262^- 64 a2c?2. 27. 49a252^-64a262^. 
 
 28. 25a2+10a+l. 29. 3a2-7a?>4-452 
 
 30. 2^2 -12 a: + 36. 31. jo22:4+23prc2_,_130. 
 
 32. a;2+2a;-8. 33. \2m^ -1 pm^ ^-p^. 
 
 34. 8a;3+27. 35. 4m2w2-(w2 + w2-;?2)2. 
 
138 ELEMENTARY ALGEBRA 
 
 36. Sa^-h6Sa^-S. 37. a^^ - a^ 
 
 38. a6 H- 62 4. a _ 1. 39. ab-b^-a + 1. 
 
 40. a^ + 6aP + 5x+l. 
 
 41. a2_4^(^^^) + 3(a;4.t^)2. 
 
 42. (a2_2a)2 + 2(a2-2a) + l. 
 
 43. a2(^ + l)2-(a-|-l)2(a + 2)2. 
 
 44. (a_6)2_(6_a)(a + <?). 
 
 45. (a- 6)(a + <?)-(^-«)(^ + ^)- 
 
 8 68. 
 
 46. 
 
 7i8 + W* + 1. 
 
 47. 
 
 a8_3^25_6a62 + ^ 
 
 48. 
 
 2^ +68 a; +1092. 
 
 49. 
 
 12a2-a-6. 
 
 50. 
 
 4a^-20a%^-hhK 
 
 51. 
 
 2^ - 19 :c2/ + 9 ^. 
 
 52. 
 
 l-Ux-Bla^. 
 
 53. 
 
 a^« + 7 a:?^ - 30 ^2. 
 
 54. 
 
 lx^-^lxy + 12y\ 
 
 55. 
 
 10a2_29a6 + 2162. 
 
 56. 
 
 (^2x-yy-Cx + i/y, 
 
 57. 
 
 a*+46^ 
 
 58. 
 
 216-(:c+3)3. 
 
 59. 
 
 a^ + 2ae-b^-2bc. 
 
 60. c2 4.2a6-a2_52, 61. a6 4-64c6. 
 
 62. m^ - p^m^ - nhn^ ■\- phi\ 63. 14 2^2 + n^;^ _ 15^2. 
 
 64. x^y^ — x^—u^y^ + l. 65. wi^ + w^-fl. 
 
 66. 2(a;-3)2+3(a;-3)-2. 67. a^x- y^- c(y - x^. 
 
 68. (2a-36)2+ll(2a-36)+30. 
 
 69. a6" — <?6»+2 + c?6«+i. 
 
 70. aa^ -f- aa;y +xz-j- bxy + 6^2 -1-^2. 
 
 71. a^J^-a^c^-b^c^ + d^. 
 
 72. w(w -f Jo)— w(7l+Jt?). 
 
 73. 4a252_(^2 + 52__^>)2. 
 
 74. a2(a + 6 + <? H- (^) + (6<?c? + cda H- c?a6 + a6c). 
 
 75. a2 + 62+i + 2a + 26 + 2a6. 
 
 76. ^a2 + i62+^c2-Ja6~^6c + ^^a. 
 
TYPE PRODUCTS AND FACTORS 139 
 
 77. a%^ + 4 ^>V + 16 c2a2 - 4 ahH - 16 h(?a + 8 ca%, 
 
 78. xyH — y^ + 2 (^ — 2). 
 
 79. 2aa:2_(3^^2> + (a + 2). 
 
 80. 3aa;2+(2a-5>-5(a-l). 
 
 112. The remainder theorem. When a polynomial in x 
 is divided by a binomial ic — a, the remainder may be 
 found by substituting a for x in the polynomial. 
 
 Thus, when ^7? — Z x"^ -\-2x — f)\& divided by a; — 2, the remainder 
 is the value of 1:x? — Zx'^-\-1x — h when 2 is substituted for x ; 
 namely, 
 
 2 . 23 - 3 . 22 + 2 . 2 - 5 or 16 - 12 + 4 - 5, which is 3. 
 
 Similarly, when 3 x* + 2 a: — 5 is divided by a: — 1, the remainder 
 is 3 • 1* 4- 2 • 1 — 5, which is 0. Hence, 3a:*4-2a:— 5 is exactly 
 divisible by a; — 1. 
 
 The proof in the case of the first of the foregoing ex- 
 amples is as follows : 
 
 We know that 2a;8-3a;2 + 2a:-5=(a:-2)Q + /2, where d and 
 R represent, respectively, the quotient and remainder when Ix^ -Ztc^ 
 + 2 a: — 5 is divided by (a: — 2). Since one member of this equation 
 is the same polynomial as the other, the two members are equal for 
 all values of x. Substituting 2 for x, we have 2 • 2^ - 3 • 2^ + 2 . 2 
 
 - 5 = (2 - 2)Q + /? or, since (2 - 2)Q = • Q = 0, 2^ - 3 • 2^ + 2 • 2 
 
 — 5 = i?. A precisely similar proof holds when any polynomial- in x 
 is divided by a: — a. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Find the remainder when 7p'-\-Zx-\-b is divided by 
 a:- 3. 
 
 Solution. i2 = 32 + 3.3 + 5 = 23 
 
 2. Find the remainder when ^ -\-^^ — bx-\-Z\B divided 
 by a; + 2. 
 
 Solution. Since a: + 2 = a; -(- 2), we substitute - 2 for a: in 
 the polynomial ; hence 
 
 i? =(_ 2)8 + 3(- 2)2 - 5(- 2)+ 3 = 17. 
 
140 ELEMENTARY ALGEBRA 
 
 3. Find the remainder when 2a^— 3a;-h7 is divided 
 by JT. 
 
 Solution. Since the divisor may be written a; — 0, we have 
 i2 = 20-3.0 + 7 = 7. 
 
 4. Find the remainder when 
 
 a^-(a-{-2}a^ + (2a — S')x-\-4:a is divided by x-a. 
 
 Solution. i2 = aS - (a + 2)a^ + (2 a - 3)a + 4 a 
 
 = a8-a8-2a2 +2a2-3a + 4a = a. 
 
 5. Find the remainder when 
 
 a\h - (?) + ^K^ - «) + ^(« - ^) is divided by a-b. 
 Solution. The given expression may be regarded as a polynomial 
 in a. Hence, the remainder is the value of the polynomial when b is 
 substituted for a. Thus, 
 
 R = b^(b - c) + b^c -b)+ c^(b - b) 
 = 53 _ 52c 4. 52^ _ fts + 
 = 0. 
 Since the remainder is zero, the polynomial is exactly divisible by 
 a-b. 
 
 113, The remainder theorem in factoring. The linear 
 factors of a polynomial can often be found by an applica- 
 tion of the remainder theorem. In order that a poly- 
 nomial in X should have a: — a as one of its factors, it is 
 sufficient that the polynomial should vanish, that is, 
 should become equal to zero, when a is substituted for x. 
 This is evident since, by the remainder theorem, when 
 the polynomial is divided by a; — a, the remainder is zero, 
 if the polynomial vanishes when a is substituted for x. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Find the factors oi a^-\-2x^— 5x — 6. 
 
 Solution. If re - a, where a is an integer, is a factor of ar^ + 2 x- 
 - 5 X - 6, it is necessary that the last term - 6 should be divisible 
 
TYPE PRODUCTS AND FACTORS 141 
 
 by a. We therefore substitute in turn the different factors of — 6, 
 namely, 1, — 1, 2, — 2, 3, — 3, 6, — 6, in the polynomial. 
 
 When a; = l, a:8 + 2a:2-5a;-6 becomes 1 + 2-5-6=- 8. 
 
 When x = -l,7? + 2x^-5x-Q becomes -1 + 2 + 5-6 = 0. 
 
 When x = 2,3^-\-2x^-5x-Q becomes 2^ + 2 • 2'^ - 5 • 2 - 6 = 0. 
 
 When x = -2, 3?-\-2x^-bx-Q becomes (- 2)3 + 2(- 2)'2 
 _5(_2)-6=4. 
 
 When a: = 3, a:3 + 2ar2-5a:-6 becomes 38 + 2 • 3^-5 • 3-6 = 24. 
 
 Whena: = -3, a;« + 2x2-5a:-6 becomes (-3)3+2(-3)2-5(-3) 
 -6 = 0. 
 
 Hence, when 3^ + 2 x^ - bx — Q is divided by a: — (- 1), by a: — 2 
 and by a: — (— 3), the remainder is 0, and therefore the factors of 
 ar3 + 2 a:2 - 5 a; - 6 are a: + 1, a; - 2, and a: + 3. 
 
 Question. Why is it unnecessary to substitute 6 and — 6 for x 
 in the given expression ? 
 
 2. Show that a;" — ^", where n is a positive integer, is 
 exactly divisible hj x — y. 
 
 Solution. When a:** — y« is divided by a: — y, the remainder, JR, 
 is the value of a:" — y^ when y is substituted for x. 
 Hence, R = y^ — y^ = 0. 
 
 3. When is a:" +- ^" exactly divisible by a; + ^? 
 
 Solution. R =(— yY + ^, which is or 2 y", according as n is 
 an odd or an even positive integer. Therefore, when n is an odd 
 positive integer, a:** + y" is exactly divisible hjx-\-y and when n is 
 an even integer it is not exactly divisible hj x -\- y. 
 
 4. Show that a^ -\- h^ + (^ — 2> ahc is exactly divisible by 
 a-{-h-\- c. 
 
 Solution. Since a -^ h -^ c = a -{- h - c) the remainder, /?, of 
 the division is the value of a^ + 6^ + c^ — 3 abc when — 6 — c is sub- 
 stituted for a. 
 
 Hence, R=-(h-\- c)' + 6^ + c^ + 3(^> + c)hc 
 
 = -b^-3I^c-d bc^ _ c3 + 63 + c8 + 3 b'^c + 3 bc^ 
 = 0. 
 
 Therefore, the division is exact. 
 
142 ELEMENTARY ALGEBRA 
 
 BXEBOISB 66 
 
 Factor : 
 
 1. a^-x^-x + 1. 
 
 2. 2^-Sz'\-2, 
 
 3. 3^-i-X^-X-l, 
 
 4. a:8_3a;-2. 
 
 5. a^-6x^-{-llx-6. 
 
 6. a^ + 5x^-x-5. 
 
 7. 2a^-x^-6x-2. 
 
 
 Suggestion. When two factors have been found, the third factor 
 
 may be obtained by dividing the given expression by the product of 
 the two known factors. 
 
 8. Sa^-2a^-19x-e. 9. 2a^ -\-x^-lSx-\-6. 
 
 10. a^4-5aj3 + 5a:2_5a._6, 
 
 11. Show that a;2» — ^2n^ where w is a positive integer, is 
 exactly divisible by a; + y. 
 
 Equations Solved by Factoring 
 
 114. Rational and integral equations in one unknown 
 number. By transposition, all the terms of an equation 
 can be brought to one member of the equation. The other 
 member then is zero. An equation which contains one 
 and only one unknown number is said to be a rational and 
 integral equation in one unknown, provided that, when 
 all the terms are written in one member, the polynomial 
 which occurs in that member is rational and integral with 
 respect to the unknown. 
 
 Thus, 3 a:» - I a;2 + 1 X - ^ = is a rational integral equation. 
 Remark. When the word equation is used in this chapter, it will 
 be understood to refer to a rational integral equation in one unknown. 
 
 115. Degree of an equation. Any equation may be put 
 in the form ^ = 0, in which A represents a rational and 
 integral expression with respect to the unknown. Equa- 
 tions are classified according to the degree of the expres- 
 
TYPE PRODUCTS AND PACTOPB 143 
 
 sion, represented by A^ with respect to the unknown 
 number. For example, 
 
 aa; + & = is a linear equation^ or an equation of the first degree. 
 
 ax^ + 6a: + c = is a quadratic equation^ or an equation of the 
 second degree. 
 
 ax^ + 6a~^ + car + c? = is a cubic equation^ or an equation of the 
 third degree. 
 
 ax^ + 6a;* + ca;2 4- rfa: 4- e = is an equation of the fourth degree. 
 [§ 85, Remark.] 
 
 116. Solution of an equation. An equation in one 
 unknown is said to be solved when all of its roots have 
 been found. 
 
 117. Roots of an equation found by factoring. If one 
 
 member of an equation is zero, the roots of the equation 
 may be found easily, provided that the polynomial in the 
 other member can be expressed as a product of factors, 
 each one of which is of the first degree in the unknown 
 number. This important method of solving an equation 
 is applied and explained in the illustrative examples which 
 follow. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Solve the equation ^x^-\- ^x=2x^ - Zx'\-^^. 
 
 Solution. 3 a;2 + 5 a: = 2 a;2 - 3 a; + 33. (1) 
 
 Transposing, 3 a;2 + 5 a; - 2 x2 + 3 a: - 33 = 0. (2) 
 
 Combining, a;2 -|- 8 a; - 33 = 0. (3) 
 
 Factoring, {x - 3)(a: + 11) = 0. (4) 
 
 Notice here, that the first member of equation (4) is a product of 
 factors and that a product cannot be equal to zero unless at least one 
 of its factors is zero. Hence, any value of x which satisfies equation 
 (4) must cause at least one of the factors of the first member to vanish. 
 Moreover, any value of x which causes either of the factors to vanish 
 will satisfy the equation. Therefore, the required roots are found by 
 equating each of the factors to zero. 
 
144 ELEMENTARY ALGEBRA 
 
 Therefore, a; - 3 = 0. (5) 
 
 Whence, x = d. (6) 
 
 Also, a; + 11 = 0. (7) 
 
 Whence, x = - 11. (8) 
 
 That is, the roots of 3 a;^ + 5 a; = 2 a;^ - 3 a; + 33 are 3 and -11 
 
 2. Solve the equation ^a^ -2a^- ^ x-\- 2 = 0. 
 
 Solution. 3 x8 - 2 a:2 - 3 a: + 2 = 0. (1) 
 
 Grouping terms, (3 a:8 - 2 a:2) - (3 a; - 2) = 0. (2) 
 
 Factoring first term of (2), a;2(3 a; - 2) - (3 a; - 2) = 0. (3) 
 
 Factoring, (3 a; - 2) (a?^ - 1) = 0. (4) 
 
 Factoring completely, (3 a; - 2) (a; - l)(a: + 1) = 0. (5) 
 
 Equating (3 x - 2) to 0, 3 a; - 2 = 0. (6) 
 
 Equating (a: - 1) to 0, a; - 1 = 0. (7) 
 
 Equating (a; + 1) to 0, a; + 1 = 0. (8) 
 
 Solving (6), a; = |. (9) 
 
 Solving (7), a; = L (10) 
 
 Solving (8), a: = - 1. (11) 
 That is, the roots of 3 x^ - 2 a;^ - 3 a: + 2 are 1, — 1, and ^. 
 
 3. Solve the equation a^ — Sx^-{-16 = 0. 
 
 Solution. a;4 - 8 a:2 + 16 = 0. (1) 
 
 That is, (a:2 - 4)2 = 0. (2) 
 
 Factoring, [(a: - 2)(a: + 2)]2 = 0. (3) 
 
 That is, (x - 2)(x - 2)(x + 2)(ar + 2) = 0. (4) 
 
 Equating each factor to 0, a: — 2 = 0, a: — 2 = 0, a; + 2 = 0, x + 2 = 0. 
 Solving simple equations, a:= 2, a: = 2, a: = — 2, a: = — 2. 
 That is, the roots of a;^ - 8 a:2 + 16 = are 2, 2, - 2, and - 2. 
 Note. The equation has four roots, two pairs of equal roots ; that 
 
 is, as many roots as there are linear factors. 
 
 Remark. The student should carefully check all roots obtained 
 
 from the solutions of illustrative examples 1, 2, and 3. 
 
 4. Solve Qa^-llx^-S5x=0. 
 
 Solution. 6 a;8 - 11 a;2 - 35 a; = 0. 
 
 Factoring, a:(3 a: + 5) (2 a: - 7) = 0. 
 
 Equating each factor to 0, a: = 0, 3 a: + 5 = 0, 2 a: — 7 = 0. 
 
 Solving simple equations, a: = 0, a: = — J, a: = |. 
 
 That is, the roots of 6 a:8 - 11 a:2 - 35 x = are 0, - J, and J. 
 
TYPE PRODUCTS AND FACTORS 145 
 
 From the solutions of the foregoing illustrative ex- 
 amples, the following rule for solving an equation by 
 factoring may be inferred : 
 
 Rule. Transpose all the terms to one member of the equa- 
 tion^ factor the resulting expression into its linear factors^ 
 equate each factor to zero^ and solve the resulting simple 
 equations. 
 
 Note. In solving equations by factoring, care should be exer- 
 cised to bring all terms to one member of the equation. The following 
 is an example of an error which is the direct result of disregarding 
 this practice. 
 
 Solve the equation (2 a; + 3) (a: - 1) = (a: + 2)(a: - 1). 
 
 Incorrect Solution. (2 a: + 3)(x - 1) = (ar + 2) (a: - 1). (1) 
 
 Dividing both members of (1) by (x — 1), 
 
 2 a: + 3 = X + 2. (2) 
 
 Transposing and combining, a: + 1 = 0. (3) 
 
 Solving, X = — 1. (4) 
 
 Correct Solution. (2 a: + 3)(a; - 1) = (a: + 2) (a: - 1). 
 
 Transposing, 
 
 (2a; + 3)(a: - l)-(x + 2)(a: - 1)= 0. (1) 
 
 Factoring, (x - 1)[(2 a: + 3) - (a: + 2)] = 0. (2) 
 
 SimpUfying, (x - l)(a: + 1) = 0. (3) 
 
 Equating each factor to 0, a: — 1 = 0, a: + 1 = 0. 
 
 Solving simple equations, a; = 1, a: = — 1. 
 
 That is, the roots of (2 a: + 3) (a: - 1) = (a: + 2) (a; - 1) are + 1 and 
 -1. 
 
 The error in the incorrect solution arises from dividing both mem- 
 bers of equation (1) by a factor which contains the unknown num- 
 ber, and which vanishes when x has the value 1. 
 
 The equations (2 a: -f- 3) (a: - 1) = (x + 2) (a: - 1) and 2x +3 = x-\-2 
 are not equivalent, the first equation having a root which is not a 
 root of the second. In solving equations, every transformed equation 
 or set of equations wMch occurs in the solution must be equivalent to 
 the original equation [§74]. 
 
146 ELEMENTARY ALGEBRA 
 
 EXERCISE 57 
 
 Solve the following equations and check the roots : 
 
 1. a?(a:-2)=0. 2. r» (3 a; -h 2) = 0. 
 
 3. (2a:+l)(3a;-l) = 0. 4. a^-2x-^l = 0, 
 
 5. ar2-3a;H-2=0. 6. a^-9x + 20 = 0, 
 
 7. 2^2 + 2a; -3 = 0. 8. rrZ + 3^; _ IQ = 0. 
 
 9. a;2-13a; + 42 = 0. 10. a^*- 6a: - 55 = 0. 
 
 11. a^-5a;+6 = 0. 12. a^^- 4a:- 21 = 0. 
 
 13. a^-\-x-SO = 0, 14. a:2_7^^io^O. 
 
 15. a^-Ux-15=:0. 16. 2a:2-3a:-2 = 0. 
 
 17. 3a;2-h2a:-8 = 0. 18. 4a:2 - 3 a: - 85 = 0. 
 
 19. 3a^2-5a:-12 = 0. 20. 4a:2 _ 3^ _ 45 ^ 0. 
 
 21. 5a^ + a:-6=0. 22. 4x^^lx-Ul = 0. 
 
 23. 7a:2_5a:-78 = 0. 24. 11 a^^- 13a; 4- 2 = 0. 
 
 25. 15a^i + 2a;-56 = 0. 26. 13ar2- 9a; - 414 = 0. 
 
 27. 3a;2^2a: + 5 = 5a;2_3^_2. 
 
 28. a;(a;+l) = (2a; + l)(a; + l). 
 
 29. a;(a;H-l)(a; + 2)=a;(2a; + 3)(a;-t-l). 
 
 30. (2a;2_3^_,.i)2_(^_ 1)2 = 0. 
 
 Highest Common Factor— Lowest Common Multiple 
 
 118. Highest common factor. The highest common 
 factor (H. C. F.) of two or more integral expressions is the 
 integral expression of the highest degree, with greatest 
 numerical coefficient, which exactly divides each of them. 
 
 Thus, the H. C. F. of 4 a%» and 6 a%^ is evidently 2 a%^. 
 
 119. Greatest common diyisor (G. CD) in arithmetic. 
 In arithmetic, the greatest common ^ivisor of two or 
 more numbers may be found by expressing each of them 
 
TYPE PRODUCTS AND FACTORS 147 
 
 as the product of powers of its different prime factors, 
 and then taking the product of the common prime factors 
 of the numbers, giving to each common prime factor the 
 least exponent which it has in any of the numbers. 
 
 ILLUSTRATIVE EXAMPLE 
 
 Find the greatest common divisor of 180, 252, and 270. 
 Solution. 180 = 22 x 3^ x 5 
 
 252 = 32 X 7 X 22 
 
 270 = 2 X 38 X 5 
 G. C. D. of 180, 252, and 270 = 2 x 32, or 18. 
 
 120. Highest common factor of monomials. The high- 
 est common factor of two or more literal monomials can, 
 obviously, be found by inspection. 
 
 ILLUSTRATIVE EXAMPLE 
 
 Find the highest common factor of 12 a^^c^, 18 cfil^(^^ 
 and 24 a%'^c. 
 
 The greatest number which will exactly divide 12, 18, and 24 is 6. 
 The highest power of a which will exactly divide a, a^, and a* is a. 
 The highest power of h which will exactly divide 62, &», and 6* is h^. 
 The highest power of c which will exactly divide c* and c is c. 
 Evidently, the required H. C. F. is 6 ab^c. 
 
 From the above illustration, we have the following: 
 
 Rule. To find the highest common factor of two or more 
 monomials, multiply the product of the lowest powers of 
 their common literal factors hy the greatest common divisor 
 of their numerical coefficients. 
 
 Note. The numerical coefficient in the highest common factor 
 is taken as positive. 
 
 Thus, the H. C. F. of - 4 a^h and 6 a/>2 jg regarded as 2 ah and 
 not — 2 ah. 
 
 Remark. When the greatest common divisor of the numerical 
 coefficients of two or more monomials cannot be readily seen, it may 
 be found as in section 119. 
 
148 ELEMENTARY ALGEBRA 
 
 EXERCISE 58 
 
 Find by inspection the highest common factor of : 
 
 1. a%^ and a¥. 2. 3 x^yz and 12 s^y. 
 
 3. 6 a'^h^c and 4 a^h\ 4. a!^c, a^c^, and a%<?. 
 
 5. 2 a^b'^c^ 4 a^ftc^ and 6 aJ^. 6. (a: + y)z and (a; — y)2. 
 
 7. — 2(a + 6)a:^ and — 2(a + 6):r2. 
 
 8. (a + ^) V and (a + ft) V. 
 
 9. 2a25(c + e^)2, 4a52(c + (^), and 10ah{c^-dy, 
 
 10. a;(a;-l)(a;-2), a;(2:4-l)(a:-l), and 3a:(a;-l)(a;-2). 
 
 121. Highest common factor of polynomials by factoring. 
 
 Expressions which are completely factored; i.e. each of 
 which is expressed as a product of powers of its prime fac- 
 tors, are in the form of monomials, and their highest com- 
 mon factor may be found by inspection, as in exercise 58. 
 
 ILLUSTRATIVE EXAMPLE 
 
 Find the highest common factor of 18 a;^ -|- 15 a; — 18 
 and 36 a:2 + 78 a: + 36. 
 
 Solution. 18 a;2 + 15 a: - 18 = 3(2 a; + 3)(3 a: - 2) 
 36 x2 + 78 X + 36 = 6(2 x + 3)(3 x + 2). 
 
 Therefore, the required highest common factor is the H. C. F. of 
 3(2 a:+3)(3 x- 2) and 6(2 a: + 3)(3 a; + 2), which is evidently 3(2 a:+3). 
 
 Therefore, to find the highest common factor of two or 
 more polynomials which can he readily factored., the method 
 of procedure is to factor each polynomial completely., thus 
 changing each into the form of a monomial, and then to find 
 the highest common factor of the monomials by inspection. 
 
 EXERCISE 59 
 
 Find the highest common factor of : 
 
 1. (2a:4-4)(a;+2) and (3a: + 6)(a; + 2). 
 
 2. x^-^'^x and 2 a; + 6, 
 
TYPE PRODUCTS AND FACTORS 149 
 
 3. x^+2x-{-landa^-Sx-4:. 
 
 4. 2^2 + 3 2; _ 10 and a^ - 5 2; + 6. 
 
 5. a% + ab^2inda^+a%. 
 
 6. 3^ — x^ and 2^ -\-a^. 
 
 7. aHSft^and 2a^-\-^ab. 
 
 8. a^ — y^ amd x!^ -{- 3^ -\- 1. 
 
 9. a3 ^ 53 and (2 a + 3 5)2 - (a + 2 5)2. 
 
 10. 3 (a + 5), 6 a2 + 6 a5, and 2 a^ + 2 a^. 
 
 11. ax — ay -\-hx — by and aa: + 5a; + 5y + ay. 
 
 12. a:2_9^ ^_6a;_^9^ and 2a^2_5a,_3^ 
 
 13. 1155 a%, 910 a52, and 595 ah. 
 
 14. a2-52_c2+25cand a2_52_^^^2ac. 
 
 15. 4 a;2 + 12 a; + 9, 4 a:2 - 9, and 6 a:3 _^ 13 2,2 ^ 5 ^, 
 
 16. a2+52 + c2_25c-2ca + 2a5and a^-\-b'^-(^ + 2ab. 
 
 17. iK3-2a^ + a;-2auda:3_ 3a:2 + ^_3. 
 
 18. a;2-3a;H-2 anda;3_^5^2_3a._3^ 
 
 Suggestion. The first expression is the product of two factors. 
 Find whether the second expression is exactly divisible by one or both 
 of these factors. 
 
 19. x^ + ix-5 and a;^ + 3 a;2 - 9 a; + 5. 
 
 20. x^ — X— 6 and a:^ + a;2 — 9 rr — 9. 
 
 122. Lowest common multiple. The lowest common 
 multiple (L. C. M.) of two or more integral expressions, 
 the numerical coefficients of which are integers, is the in- 
 tegral expression of lowest degree with least numerical 
 coefficient which is exactly divisible by each of them. 
 
 Thus, the L. C. M. of 2 a^b* and 4 a^b^ is evidently 4 aSJ*. 
 
 123. Least common multiple (L. C M.) in arithmetic. 
 
 In arithmetic the least common multiple of two or more 
 numbers may be found by expressing each of them as the 
 
150 ELEMENTARY ALGEBRA 
 
 product of powers of its different prime factors and taking 
 the product of all the different prime factors of the num- 
 bers, giving to each different prime factor the greatest 
 exponent which it has in any of the numbers. 
 
 ILLUSTRATIVE EXAMPLE 
 
 Find the least common multiple of 90, 189, and 300. 
 
 Solution. 90 = 2 X 32 X 5 
 
 189 = 38 X 7 
 300 = 22 X 3 X 52 
 L. C. M. of 90, 189, and 300 = 22 x 38 x 5^ x 7, or 18,900. 
 
 124. Lowest common multiple of monomials. The low- 
 est common multiple of two or more literal monomials 
 can, obviously, be found by inspection. 
 
 ILLUSTRATIVE EXAMPLE 
 
 Find the lowest common multiple of 6 a^h(^^ 8 ahc^^ and 
 12 aW(^, 
 
 Solution. 
 
 By inspection it is readily seen that : 
 The least number which will contain 6, 8, and 12 is 24. 
 The lowest power of a which will contain a^, a, and a' is a*. 
 The lowest power of b which will contain h and b^ is ft*. 
 The lowest power of c which will contain c^ and c^ is c*. 
 Evidently, the required L. C. M. is 24 amc\ 
 
 From the above illustration, we have the following: 
 
 Rule. To find the lowest common multiple of two or more 
 monomiaU^ multiply the product of the highest powers of their 
 different literal factors hy the least common multiple of their 
 numerical coefficients. 
 
 Remark. When the lowest common multiple of the numerical 
 coefficients of two or more monomials cannot be readily seen, it may 
 be found as in section 123. 
 
TYPE PRODUCTS AND FACTORS. 151 
 
 EXERCISE 60 
 
 Find, at sight, the lowest common multiple of: 
 1. 2 a^h and 3 ac^, 2. ahc^ 3 a^c^ and 5 aft^. 
 
 3. 5 a;, 3 y, and 2z, 4. 21 :^y, %\yh, and 22%. 
 
 5. Qx^yz^ ISa^z, and \%xyz^, 
 
 6. 2 mVp\ 3 a^TTjp^ and 6 a6<?. 
 
 7. a;(a; — 1) and y(x — 1). 
 
 8. a;2(a; — 1) and xy(x — l)^. 
 
 9. x^(x - iy(x + 3) and y\y - r)\x + 3). 
 10. 17 x^y\x + yy and 10 a:8«^(2J + yy. 
 
 125. Lowest common multiple of polynomials by factor- 
 ing. Expressions which are completely factored, that is, 
 each of which is expressed as a product of powers of its 
 prime factors, are in the form of monomials, and their 
 lowest common multiple may be found by the rule of sec- 
 tion 124. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Find the lowest common multiple of x^'\'4iX and 
 3a;+12. 
 
 Solution. a^* H- 4 X = x{x + 4). 
 
 3a: + 12 = 3(a; + 4). 
 Therefore, the required L. C. M. = the L. C. M. of x(x + 4) and 
 3(a; + 4), or 3 x{x + 4). 
 
 2. Find the lowest common multiple of 3 a:^ __ 27 a; + 60, 
 a;3— 5a^4-a: — 5, and a^ — 4:3^ + x — i. 
 
 Solution. S x^ - 27 X +Q0 = S(x - ^)(x - 5). 
 
 x^-5xi-\-x-6=(x- 5)(x^ + 1). 
 
 x»-ix^ + x-^ = (x- 4)(a^ + 1). 
 Therefore, the required L. C. M. = the L. C. M. of 3(x - 4)(x - 5), 
 (x - 5)(a:2 + 1), and (x - 4)(a^^ + 1), or 3(a; - i)(x - o)(x^ + 1). 
 
152 ELEMENTARY ALGEBRA 
 
 3. Find the lowest common multiple of a:^ -{-x^ — 6x 
 and a^ — 6 x^ -^ 11 X — Q, 
 
 Solution. x» -\- x^ - Q X = x(x + S)(x - 2). 
 
 It is now necessary to find whether or not a:* — 6 x^ -f 11 a; — 6 is 
 exactly divisible by any of the factors oi t^ -\- x^ — Q x. By actual 
 division a:^ — 6 x^ + 11 a; — 6 is found to have a: — 2 as a factor ; hence, 
 x9-Qx^ + nx-Q = (x-l)(x-2)(x- 3). 
 
 Therefore, the required L. C. M. = the L. C. M. of x(x + 3)(a; - 2) 
 and (x - l)(x - 2)(x - 3), or a;(a; - l)(a: - 2)(a: - 3)(a; + 3). 
 
 EXERCISE 61 
 
 Find by factoring the lowest common multiple of : 
 
 1. a% + ab^aLnda^-a%. 
 
 2. a* — a^ and a^ — a. 
 
 3. a^+6a;and4a;4-24. 
 
 4. (2a; + 4)(a;4-2) and (3rr + 6)(a; + 2). 
 
 5. a2 _ 52 and a3 _ 53. 
 
 6. x^ — 1 and s^—1. 
 
 7. a3 4- h^ and a* + a%^ + 6*. 
 
 8. a2-62and (a-i)(a2 + 62). 
 
 9. x^ + 5x-68Lnda^-dx + 2. 
 
 10. x^-\- bx — 14 and x^ — 5x-{-6. 
 
 11. x^-f,x^-}-xy-2fsindx^-\-Sxt/-\-2f. 
 
 12. 122:2-18a:y + 453/2 and 18a:2-33a;y-30^2. 
 
 13. 3^—2x^-\-xsinda^ + x^—x — l. 
 
 14. a^ — ^y, y^ — xy, and a^—Zx^y-\-xy^ + y^. 
 
 15. (x + y)2 - xy, 7^ - y^, and a:^ 4. a^?^ + xy^. 
 
 16. (a + 6)2 - c2, (a+c)2- J2, and (6 + 0^ - «^- 
 
CHAPTER V 
 FRACTIONS 
 
 126. Definition. A fraction in algebra is the quotient 
 of two numbers or expressions. The expression - means 
 
 
 
 a-*-6 (§ 6, Remark). A fraction, however, is usually re- 
 garded ^s an indicated division. The dividend, a, is called 
 the numerator, and the divisor, b, is called the denominator. 
 The numerator and denominator are called the terms of 
 
 the fraction. ^ ^^ ^^^^ ^ ^^^^ 6 or a divided by h. 
 
 
 
 127. Laws governing algebraic fractions. Algebraic 
 fractions are subject to the same laws as arithmetical frac- 
 tions. This is, in part, a direct consequence of the fact 
 that both kinds of fractions remain unchanged in value 
 when both numerator and denominator are multiplied or 
 divided by the same number (excepting zero). For arith- 
 metical fractions this important principle is established in 
 arithmetic. It may be proved in algebra as follows : 
 
 Let 7 denote any fraction, and m the number by which its terms 
 
 
 
 are to be multipHed. Representing the quotient of a divided by h by 
 q we have, 
 
 Since the dividend is equal to the product of the divisor and the 
 quotient, a = bq. (2) 
 
 163 
 
154 ELEMENTARY ALGEBRA 
 
 Multiplying both members of identity (2) by m, 
 
 am = bmq. (3) 
 
 Dividing both members of identity (3) by bm, 
 
 "£=*• (*) 
 
 From identities (1) and (4), we have, 
 
 b bm 
 
 Again, ^ may be obtained from — by dividing both terms of — 
 bm bm 
 
 by m, and from identity (5) the value of the fraction remains un- 
 changed ; that is, we may write 
 
 ^ = ?. (6) 
 
 bm b 
 
 From identities (5) and (6) we have the following 
 principle : 
 
 Multiplying or dividing both terms of a fraction by the 
 same number (zero excepted) does not change its value. 
 
 Note. The denominator of a fraction cannot be zero. The expres- 
 sion - has no meaning, and, therefore, does not represent a 'number. 
 
 In other words, division by zero is excluded. Care should be exer- 
 cised in assigning numerical values to letters to see that the values 
 assigned do not cause the denominator of a fraction to vanish. 
 
 128. Signs affecting a fraction. The sign of a fraction 
 
 is the plus or minus sign before the fraction. 
 
 — 2 
 
 Thus, in the expression -i , the sign which stands first is the 
 
 + 3 
 sign of the fraction. 
 
 There are, therefore, three signs which affect a fraction ; 
 namely, the sign of the fraction, the sign of the numerator, 
 and the sign of the denominator. The signs of the nu- 
 merator and denominator combine according to the rule 
 for signs in division. 
 
FRACTIONS 155 
 
 Thus, 
 
 ±| = + |. (1) ^ = + |. (2) 
 
 -8=-i- («) fl=-i- (*> 
 
 In algebraic symbols, 
 
 ±^ = -l- (3) • =^ = -r (*) 
 
 — 00 -\- 
 
 By comparing equation (1) with equation (2) and equa- 
 tion (3) with equation (4), it is evident that 
 
 I. The signs of both terms of a fraction may he changed 
 mthout altering the value of the fraction. 
 
 Thus, ±^ = :::^and ^=^. 
 
 + b -b -b + b 
 
 II. Any two of the three signs affecting a fraction may be 
 changed without altering the value of the fraction. 
 
 The foregoing statement is evident from (I) and from 
 the following : 
 
 Changing the sign of the fraction and the sigti of the 
 numerator in (1), we have 
 
 Changing the sign of the fraction and the sign of the 
 denominator in (1), we have 
 
 ^51=+' + 
 
 !)=+!• c^) 
 
 Comparing identities (1), (2), (3), we have 
 
 — ^^—r = + ^^ = -\- ^— , since each is equal to -|- ^ 
 4-0+6—6 6 
 
156 ELEMENTARY ALGEBRA 
 
 From I and II it is evident that 
 
 in. A fraction may he written in at least four ways with- 
 out changing its value. 
 
 Thus, +±i^ = + Zl^ = _±^ = _ZL^. 
 
 -fft -h - b +6 
 
 In like manner, 
 
 I ^ ~ -' = \ - ^ + y - - ^ + .y ^ X- y 
 
 X — y — z — X + y + z x — y — z — x -\- y + z 
 
 Remark. When no sign is written before a fraction, + is under- 
 stood. 
 
 Thus, - means +-• 
 b b 
 
 129. Change of signs of factors in the terms of a frac- 
 tion. To change the sign of one factor of an expression 
 is equivalent to multiplying that expression by — 1. 
 Therefore, when either or both terms of a fraction are ex- 
 pressed as a product of factors, the signs of an even num- 
 ber of these factors may be changed without altering the 
 value of the fraction ; but if the signs of an odd number 
 of them are changed the sign of the fraction must be 
 changed in order that its value may not be changed. 
 
 Thus, (g - b)(c - d) ^ (b - aXd - c) ™, ^-. 
 
 (x - y)iz -w) {x - y){z - w) 
 
 = -(« -h){c-d) [Why?] 
 
 {X - y)(ia -z) I y ^ 
 
 ILLUSTRATIVE EXAMPLE 
 
 Without altering the value of the fraction — ^~^ , ex- 
 press it in a form in which each of the three signs affecting 
 it is plus. 
 
 Solution. Changing the sign of the fraction and the sign of the 
 denominator, we have 
 
 y - X _ y - X 
 -2a 2a 
 
FRACTIONS 157 
 
 EXEBCISE 62 
 
 Tell, at sight, which of the statements in examples 1-15 
 are true. 
 
 1. +£ = +^^. 
 
 y -y 
 
 3. +E = -I1^. 
 
 y -y 
 
 5. --^ = -z:£. 
 -y y 
 
 ' X X 
 
 2. 
 
 y -y 
 
 4. 
 
 X —X 
 
 y y 
 
 6. 
 
 X — X 
 
 -y -y 
 
 8. 
 
 X — X 
 
 9. 
 
 10. 
 
 11. 
 
 12. 
 
 13. 
 
 X-y y-x x-y y-x 
 
 X _ — a; 
 
 x-y x-y 
 
 a a 
 
 (a - 6)(e - df) (h - a){d - <?) 
 
 a — a 
 
 (a _ h)(c -d)~ (h-a)(c-d) 
 a a 
 
 (a _ hXe -d) (5 - a)(c - d) 
 a-\-b ^ a — h 
 c— d d — c 
 
 a-\- b _ — a — h a-\-h __ _ — a — h 
 
 c — d d — c . c — d c — d 
 
 Without altering the value of the fractions in examples 
 
 16-30, express each in a form in which the three signs 
 affecting it are plus. 
 
 16. =^. 17. '^^^. 
 
 — — c 
 
 18. . • 19. ^. 
 
 y —a 
 
158 ELEMENTARY ALGEBRA 
 
 a(h — g) 
 
 90 
 
 — a — h-\- c 
 
 
 ah 
 
 22. 
 
 h-\-c 
 
 
 c — a 
 
 24 
 
 ^(a^h)(x^y) 
 
 
 x-\- a 
 
 91*. 
 
 ~ a 
 
 
 y(a-5) 
 
 23. 
 
 y-x 
 
 25. 
 
 a-\-h 
 -c-d 
 
 on 
 
 « 
 
 5(c — a) (c — a)6 
 
 28. 
 
 
 29. _JL!lA_. 30. - -(^-^) . 
 
 — (m — w) — (r — «) 
 
 Lowest Terms 
 
 130. Numbers or algebraic expressions prime to each 
 other. Two numbers in arithmetic or two algebraic ex- 
 pressions are said to be prime to each other when their 
 only common factor is 1. 
 
 131. Reduction of fractions to lowest terms. A fraction 
 is said to be in lowest terms when its numerator and de- 
 nominator are prime to each other. 
 
 It was shown in section 127 that both terms of a 
 fraction may be divided by the same number without 
 changing the value of the fraction. Hence, we have the 
 following: 
 
 Rule. To reduce a fraction to lowest terms, cancel all 
 factors common to the numerator and denominator ; that is, 
 divide both terms of the fraction by their highest common 
 factor. 
 
FRACTIONS 169 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Reduce ^^ „, ^ , to lowest terms. 
 
 28 a%M 
 
 Solution. To reduce the given fraction to lowest terms, we 
 divide its numerator and its denominator by their H. C. F., which is 
 
 4 ahcH. 
 
 36 ah'^chJ^ _ 3 6 ah'^cM'^ -^ 4 ahc^d ^ 9 hcP 
 *'* 28 a^bcH ~ 28 a%cH -f- 4 ahcH ~ 7 a^^ ' 
 
 2. Reduce -r -— to lowest terms. 
 
 a^4-3a:- 18 
 
 Solution. ^^-9 ^(x-3)(. + 3)^(^j^, 
 
 a:2+3jr-18 (x-3)(x + 6) (j: + 6) 
 
 Note. In practice it is customary to separate the numerator and 
 the denominator into their prime factors and cancel the factors 
 common to both. 
 
 ^, 3 m^ -3m _ a>#»{77r=n) (m + 1) ^ m + 1 
 
 ^^' 6 7«5 + 6 m* _ 12 m8 ~ jS.**a(rrr^ri) (m + 2) ~ 2 y«2(m + 2) ' 
 2^2 
 
 Remark. Care should be exercised not to cancel a common term of 
 the numerator and denominator of a fraction when they are poly- 
 nomials. 
 
 EXERCISE 63 
 
 (Solve as many as possible at sight.) 
 Reduce the following fractions to lowest terms : 
 
 ^. 2. — . 3. ?^. 
 
 xz 4: be xyt 
 
 ^ -7^ . 12a8 
 
 1. 
 
 a" 
 
 c^ 7^ -16 a^ 
 
 ^^ - 12 mhiY 9p^qr UmV 
 
160 ELEMENTARY ALGEBRA 
 
 13. ^m.^ 
 
 2irRH UttB^ 
 
 xi/{^ + zy (a: H- z) ' x"" 
 
 17. — • 18. — ^— • 
 
 2 ^n—lym+2 jf^+lym—l 
 
 19. ^ • 20. i^^ 
 
 ^, -35a2«-352p ^2_52 
 
 21. • 22. • 
 
 23. xr_J — £_. 24. 
 
 25. ^- 4 26. ^ ^ 
 
 (a;-l)(a^+l) 46 + 4a 
 
 2^ a2 + 2«a; + 2^ ^ 28. ^'-y' 
 a^ -]-a^x ' iy — ^Y 
 
 29. • 30. ^- ^ • 
 
 53 _ ^8 (f—p^ 
 
 31. ^' + ^ . 32. ^'"^^ 
 
 2a5 + 26a: a:2_8^^16 
 
 a^-2xy^-y^ a^ + h^ 
 
 ^^ ax-^bx + ay-^ by ^^ xy 
 
 ' ax-^-ay — bx—by ^y + xy'^ 
 
 37. <^. 38. 8-3 
 
 a^ — 1 
 
 4-42^ 
 
 (a + l)(a + 2)(a + 3) ^ 2 2^+17^-19 
 • (a + 2)(a4-3)(a + 4) * Z:^-bx^-1 
 
 4^:2^3^,22 ^2 ^-(y-gy 
 
 " b^-Zx-X^ ' (x + yy-z^ 
 
43. 
 
 FRACTIONS 161 
 
 (x + zy-y^ ' * x^-a^ + yi^x-^-y) 
 
 ^g 7?-x^y-\-xy'^-7? ^g \-x-^y-xy 
 
 a? + o^y + xy"^ + ^ 1— x — z + xz 
 
 a^-\-2x^-\-x-{-2 4 52^ _ (52 ^ g2 _ ^2)2 
 
 a5 + 3a;24.a,^3* * 4c2a2_ (^.^ ^2. 52)2" 
 
 Multiplication of Fractions 
 132. Multiplication of a fraction by an integral expres- 
 sion. Let ^ denote any fraction and c any integral ex- 
 
 pression. Representing the quotient of - by 5^ we have, 
 
 i=.. (1) 
 
 Since the dividend is equal to the product of the divisor 
 
 and the quotient, 
 
 a = bq. (2) 
 
 Multiplying both members of identity (2) by c, 
 
 ac = bcq. (3) 
 
 Dividing both members of identity (3) by 6, 
 
 ac bcq a ^^ . 
 
 T= 6 =^^ = ^^ft (4) 
 
 That is, ^=^Xt- (5) 
 
 
 
 . . ac ac -^ c a ^ci\ 
 
 Again, — = = (6) 
 
 b 0-7- c b-i-c 
 
 Therefore, from identities (5) and (6), 
 
 = cxl (7) 
 
 b-i-c I? 
 
162 ELEMENTARY ALGEBRA 
 
 Identity (5) shows that multiplying the numerator of 
 - by <? multiplies - by c. and identity (7) shows that divid- 
 
 
 
 ing the denominator of - by <? multiplies r by <?; hence, 
 
 
 
 Rule. To multiply a fraction hy an integral expression^ 
 either multiply the numerator or divide the denominator hy 
 that expression. 
 
 Note. Divide the denominator when possible. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Find the product of — and a. 
 
 n 
 
 Solution. ax- = — • ' [§132] 
 
 n n 
 
 2. Find the product of ^ and h. 
 
 
 
 Solution. bx^= -^ = ? = a. [§ 132] 
 
 -^ 1 
 
 3. Multiply — hy 4 ah, 
 xy 
 
 Solution. — X 4 a6 = 
 
 xy xy 
 
 2 1 2 
 
 4. Multiply ^^_^ by a; - ?/. 
 
 (a;8 - y8) -=- (a: - y) x^ + xy -\- y^ 
 
 Solution, -^-^f^ ' (^ - y) = 
 
 5. Multiply ^ by a:"»+i. 
 
 Solution. — x »»+i = = 
 
FRACTIONS 163 
 
 EXERCISE 64 
 
 (Solve as many as possible at sight.) 
 Multiply as indicated : 
 
 1. 2x|. 2. 2xi. 3. CX^' 
 
 5 8 n 
 
 *.p.l. 5. 2ax|. e. *x^. 
 
 7. 2X ^^ . 8. <5<fx-g^. 9. baxx ^^^ 
 
 10. 9a;y2X^^^. 11. (a + i)xl. 
 
 12. (m-n)x—-^—-^ 13. (a + 6)x^. 
 
 6(7w — n) c — a 
 
 {x-yY x^ — y^ 
 
 Multiply : 
 
 - !a^^^v(a-H5). 17. 5±|by8(. + *). 
 
 IS- J^o^ ^ hy^-x^X, 19. -^, byaf*. 
 
 ^'^ I ''y *• "• |;r by ^• 
 
 23. 
 
 g?jg7^'>^<'-»)<«-.0. 
 
 24. _?-^bv(a^ + y)2. 25. r^>y5. 
 
164 ELEMENTARY ALGEBRA 
 
 Addition and Subtraction of Fractions 
 
 133. Adding and subtracting fractions with the same 
 denominator. By the distributive law, Section 60, 
 
 \m m mj \mj \mj \mj 
 
 = a + 6-|-c [§ 132, Rule]. (1) 
 
 Dividing both members of identity (1) by m, 
 
 «4.1 + £ = ^±M^. (2) 
 
 m m m tn 
 
 Again, mi— ]=m[—]—m[ — \ 
 
 \m mJ \mj \mj 
 
 = a-b. (3) 
 
 Dividing both members of identity (3) by w, 
 a b a— b 
 
 mm m 
 
 (4) 
 
 From identities (2) and (4), we have the following 
 Rule. To add or subtract fractions which have the same 
 
 denominator^ add or subtract their numerators and place the 
 
 result over their common denominator. 
 
 134. Lowest common denominator. Two or more frac- 
 tions whose denominators are not the same may be re- 
 placed by other fractions equivalent to them, respectively, 
 each of whose denominators is the lowest common multiple 
 of the denominators of the given fractions. 
 
 The lowest common multiple of the denominators of 
 two or more fractions is called their lowest common 
 denominator (L. C. D.). 
 
FRACTIONS 165 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Express as a single fraction --\-- • 
 
 a b e 
 
 Solution. The L. C. D. of the fractions is abc. 
 
 a abc^ 
 1 _ ac 
 b dbc^ 
 
 c abc 
 
 abc 
 
 2. Express as a single fraction in lowest terms : 
 
 x^ + x a^+Sx-t2 x^^-lx 
 Solution. The L. C. D. of the fractions is x{x + l)(x + 2). 
 
 1 a: + 2 
 
 x^ -\- X x(a:+ l)(a:+ 2) 
 1 X 
 
 a;2 + 3a;+2 x{x ^ l)(a: + 2) 
 1 2; + ! 
 
 a:2 + 2x a:(a: + l)(a: + 2) 
 
 Sum = (^ + ^)+^-(^ + ^> 
 a:(a:+l)(a: + 2) 
 
 _ a; + 2 + a:-a;-l 
 
 a;(a: + l)(a:4-2) 
 
 1 
 
 a:(x + 2) * 
 
 3. Express as a single fraction ^ ■\- xy -\- y^ ■{ — ^ — 
 
 x-y 
 
 Solution. :r2 + zy + y2 + _J^ = £iHl£y±l! + -J^ 
 a; -y 1 x-y 
 
 a:- y a: - y 
 a:8 
 
166 ELEMENTARY ALGEBRA 
 
 4. Simplify: 
 
 b -{- c , c-^ a a-\-h 
 
 (^a — b){a — c} (h — c)(h — a) (^c — a)(c — b^ 
 Solution. ^-±^ + i±i5 + « + ^ 
 
 (^a-b)(a-c) (b-c)(b-a) (r - a)(c - b) 
 b -\- c c + a a -\- b 
 
 (a-bXc-a) (a-b)(b-c) (b - c){c - a) 
 
 The L. C. D. of the fractions is (b — c){c — a) (a — b). 
 b + c _ (6 + c)(b - c) 
 
 (a — b)(a — c) (b — c)(c — a)(a — b) ' 
 
 c + a (c -\- a)(c — a) 
 
 (b - c){b -a)~ (b- c)(c - a){a - b)' 
 
 a + b _ (a-\- b)(a - b) 
 
 (c -a)(c - b) (b - c)(c - a)(a - b)' 
 
 (b — c)(c — a) (a — b) 
 _ 52 _f. c2 _ c2 + a2 - a2 _,. 52 
 
 (b — c)(c — a) (a — 6) 
 
 0. 
 
 From the solutions of illustrative examples 1, 2, 3, and 
 4, pages 165 and 166, we have the following: 
 Rule. For adding or subtracting fractions : 
 
 1 . Reduce^ when necessary^ the fractions to their lowest 
 common denominator. 
 
 2. Find the algebraic sum of the numerators of the re- 
 suiting fractions. 
 
 3. Write the algebraic sum of the numerators for the nu- 
 merator of the result and the lowest common multiple of the 
 denominators for its denominator. 
 
 4. Simplify the resulting fraction. 
 
 Note. In general, before operations on fractions are performed, 
 each fraction involved should be reduced to its lowest terms. 
 
FRACTIONS 167 
 
 EXERCISE 65 
 
 Express as a single fraction in lowest terms : 
 
 1. l + l- 2. 1+1. 3. ^ + ^. 4. 2x + ^ 
 
 a a zx Sx x 
 
 S. x+±-f. 6. a + *. 7. X-&. 8. 1-1 
 
 ZX OX c z he ca 
 
 4:a 6a a+la 
 
 13. -f+*. X4. 1 1 
 
 ha a-\- h a — h 
 
 15. -i- + l. 16. _2--A. 
 
 2H-a a a + 2 2a 
 
 17. -^ + . ^ ■ 18. ' 2 
 
 x + ^z 2a:+2y a: + 3 a;+2 
 19. — • 20. + 
 
 aH-2 3a-l a-1 a + 1 
 
 21. - + -. 22. ^-hf-' 
 z X by OX 
 
 x^ 1 1 
 
 23. 2: 24. 
 
 a; + ^ 2a + 6 3a + J 
 
 25. ^ + ?^. 26. 1 +-J_. 
 
 22; 2y a-\-h c + d 
 
 27. 1+1. 28. 9_t2a^9-25 
 
 a-1 3a 36 
 
168 ELEMENTARY ALGEBRA 
 
 31. —±-,^^-±^. 32. ^ 
 
 33. 
 
 (a;+y)5 {x+i/}a a^ — ab P — ab 
 
 la /a-\-5b b—2a\ 
 
 -( 
 
 b \ b ■ b J 
 
 34. B-^ + £. 35. f' + *i+4- 
 
 ax ox ex DC ca ab 
 
 36. ^±l-l±l^^±l. 37. ?_^4.i5 
 
 a:z/ 2/5J za: a 3a 9a 
 
 38. -• 39 
 
 2a(a;-a) 2a(a;+a) 6(2 a;- 3) 6(2 a: + 3) 
 
 40. — - — I 41. a H- 6 
 
 a^ a* a + 5 
 
 42. 1 • 43. 
 
 a a+1 • Sx^—Sxy '^iy'^ — ^yx 
 
 o^ _i_ ^ _ b^i^^-^y^) _ 3 a^^ 
 
 45. 1 +-J_+ 1 . 46. ^+ ^ 
 
 6 — c c— a a — 6 b — c c — a a — b 
 
 49- -^ T^T r + 
 
 50 
 
 (a - 6)(a 4- <?) (b — a)(b + <?) 
 1 3 
 
 (a - 6)8 (5 - a)3 
 
 gj^^ 3^-\-xy-\-y^ x^-xy-\-y^ 
 a^ — y^ a^ + y^ 
 
 52. 2a;y ^ 2x-{-y x-\-y 
 x^-y^ x->ty y — x 
 
 53. 1+^ + ^ ' ^ + ^ 
 
 a;a;2_9 x^-^x-\-^ 
 
FRACTIONS 169 
 
 54 flt + g . h4-c 
 
 55. .. il 4 2 . 8 
 
 56. 
 
 (3a:+2X2a;-3) (22:-3Xa:-2) (2-2:X3a:+2) 
 
 2 4 5_ 
 
 2^24.43.4.3 a,2_2a;-3 l-a:2* 
 
 57. Z § + I 
 
 2a^-bx--Z ^a^-lOx^Z Q:^^x-1 
 
 x-^\ 942:- 186 . 71a:-135 
 
 08. — — — — — — — — — — - — — — — 4" 
 
 Qx^-llx+12 152^-14a:-8 Qx^-Vlx+ll . 
 
 In examples 59-64, combine not more than two fractions 
 at a time. Thus, in example 59, combine the first two 
 fractions and then the result with the third. 
 
 59. ^+-^+ 2 
 
 1 — a 1 + a l + a^ 
 
 a h -2_A! 
 
 60. -^ + —^4 
 
 a + h a-h a^-^h^ 
 
 x — y x-^y a^ + y^ x^ -\- y^ 
 
 __ a a4-4, a a — 4 
 
 oz. \- • 
 
 a + 4 a a — 4 a 
 
 a^+x-h^ a^-1 2a;-8 
 
 63. 
 
 x-1 x-2 x^-Sx + 2 
 
 64. -^+ 1 1 
 
 x-S x-\-S x-2 x + 2 
 gg 3(22;2+l) 2a:+l 2a:-l 
 
 3^^x^+l x^-x-^1 x^^-x^-1 
 
 Reduction of Fractions to Integral or Mixed Expressions 
 
 135. Simple fraction. When both terms of a frac- 
 tion are integral, the fraction is called a simple fraction. 
 
170 ELEMENTARY ALGEBRA 
 
 136. Proper and improper fractions. Simple fractions 
 are classified as proper fractions and improper fractions. 
 A proper fraction is a simple fraction in which the degree 
 of the numerator is less than the degree of the denomi- 
 nator. An improper fraction is a simple fraction in which 
 the degree of the numerator is either equal to or greater 
 than the degree of the denominator. 
 
 Thus, — is a proper fraction ; — is an improper fraction. 
 
 a^ + o a -\- h^ 
 
 Remark. When the terms of a fraction contain more than one 
 
 literal number, the fraction may be a proper fraction with respect to 
 
 one of these numbers and an improper fraction with respect to 
 
 another. 
 
 Thus, — — — is a proper fraction with respect to 6, but it is an 
 a -\- h^ 
 
 improper fraction with respect to a. 
 
 137. Mixed expression. An expression, some of whose 
 terms are integral and some fractional, is called a mixed 
 expression. 
 
 Thus, a + -, X -\- y — , and 1 + - are mixed expressions. 
 
 c ^ x-\-y y 
 
 Note. An improper fraction can be reduced either to an integral 
 expression or to a mixed expression in which the fractional part is a 
 proper fraction. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Reduce r to a mixed expression. 
 
 Solution 1. 
 
 a2 _ 52 _ q _!■ 2 h a^-b'^-a + b + b 
 a — b 
 
 
 
 a-b 
 
 
 
 
 a2 
 
 -62 
 
 -b 
 
 — a 
 
 : + b 
 
 h 
 
 
 a 
 
 a - 
 
 -b -^a 
 
 — 
 
 b 
 
 a 
 
 -62 
 -b 
 
 a — 
 a — 
 
 b b 
 b^a- 
 
 1 
 
 
 a 
 
 + 6- 
 
 -1 + 
 
 h 
 I, 
 
 
 
FRACTIONS 171 
 
 Remark. Since = — + — — — it is evident that a f rac- 
 
 m m m m 
 
 tion whose numerator is a polynomial can always be written as the 
 
 algebraic sum of two or more fractions. As a step in reducing a 
 
 fraction to a mixed expression, it is desirable to express it in this 
 
 manner, whenever the terms can be grouped at sight in such a way 
 
 that each numerator with, in general, the exception of the last, is 
 
 exactly divisible by the denominator. 
 
 Solution 2. 
 
 a - 
 
 a-l + & 
 
 
 _ j)a2 _ a _ ^,2 ^ 2 6 
 a^ — ah 
 
 
 
 -a + ab-b^-{-2b 
 
 
 
 -a +6 
 
 ab-b^ + 
 ab-b^ 
 
 a^-a-b^ + 2b 
 
 = a — 1 + + 
 
 a-b 
 
 Remark. Solution 2 is preferable when it is not evident at 
 sight how the terms of the numerator should be grouped. 
 
 Check. Let « = 2, 6 = 1. 
 
 Dividend = Quotient x Divisor + Remainder. 
 ai-b^-a-\-2b= (a-{-b-l)(a-b) -{-b. 
 4_1 _2 + 2=(2+l-l)(2-l) + l 
 3=2x1+1 
 3 = 3. 
 
 2. Reduce 2a:« - 3a:V + 4r.y^ + 5y3 ^^ ^ ^.^^^ ^^ 
 S x^ -\- 2 XT/ — 4: y^ 
 
 sion. 
 
 X - -U 
 
 til. 
 
 Solution. Sx^ -\-2xy -4: y^)2 x^ - S x^y -\- 4: xy^ + 6 y^ 
 
 2x^ -^ ^x^y -fxyg 
 
 -^x^y + \^-xy^+5y^ 
 
172 ELEMENTARY ALGEBRA 
 
 Hence, 
 
 3 a:2 + 2 xy - 4 2/2 3 9 ^ 3 x^ + 2 xy - ^ y^ 
 
 = ^x-^y+ S6xy^-7y^ 
 
 3 9 "^ 9(3x^ + 2xy-iy2) 
 
 Check : Let x = 2, y = 1. 
 
 2aH»-3x2.y + 4xyH52/^==(|x-^3,)(3x2 + 2x3/-4y2) + ?^^^^^ 
 
 16 _ 12 4- 8 + 5 = (i-MVl2 + 4 - 4)+ ^^^~T . 
 \3 9 / ^ ^ 9 
 
 17=-f + ^^ = 17. 
 
 EXERCISE 66 
 
 (Solve as many as possible at sight.) 
 Reduce to either an integral or a mixed expression : 
 
 1. 
 3. 
 5. 
 7. 
 
 
 a 
 
 
 2a^ 
 
 - 3 rr2 + 2 a; - 
 
 1 
 
 
 X 
 
 
 4 w 
 
 -f- 3 mn — 2m 
 
 
 
 n 
 
 
 x^- 
 
 -a2 
 
 
 X — 
 
 ■a 
 
 
 x^- 
 
 -2ir + l 
 
 
 x-1 
 
 
 a2_ 
 
 b^ + a~b 
 
 
 
 a-b 
 
 
 2a2 
 
 -ft2 + l 
 
 
 mn 
 
 Sa^-2ab-\-5 ' 
 
 2a 
 l + 3a6-2a2j + 5j8 
 
 11. 
 
 a-\-b 
 15. ^-y' . 
 
 X+1/ 
 
 
 2ab 
 
 8. 
 
 ma-\- mb 
 a + b 
 
 10 
 
 2 'irRH+ 2 irm 
 
 
 H+R 
 
 12. 
 
 a^+b^ 
 a-b 
 
 14 
 
 a^-b^j^a-2b 
 
 
 a-b 
 
 16 
 
 2x'^-\-^xy + y^ 
 
 
 2x^Zy 
 
FRACTIONS 
 
 173 
 
 17. 
 
 19. 
 
 21. 
 
 3 a;2 - 2 rg,y + 2 
 3a;-2«^ 
 
 x^-\-2xy ■\-'^y'^ 
 Sx^-2x-\-2 
 2a^-^x-\-S' 
 
 18. 
 
 20. 
 
 22. 
 
 Sx^-\-Sxy + 2 
 x + y 
 
 2a^+3a;+l 
 
 x^+2x-,^ ' 
 
 Zx^-2xy + 4:y^ 
 2x^-^4xy-Sy^' 
 
 Multiplication of Fractions 
 
 138. Product of two fractions. 
 
 Let j=:q (1) and ^ = r 
 
 a 
 
 Then, a= bq (3) and c = dr 
 
 ac = hdqr. 
 
 Dividing both members of (5) by hd^ 
 
 ac 
 
 (2) 
 
 (4) 
 (5) 
 
 [§ 25, 3] 
 
 bd 
 
 = qr. 
 
 But 
 
 a c 
 
 H v£ — ^ 
 b d~bd' 
 
 (6) 
 
 a) 
 
 (8) 
 
 From identity (8) we have the following : 
 
 Rule. To find the product of two fractions, multiply the 
 
 numerators together for the numerator of the product and 
 
 the denominators for the denominator. 
 
 ILLUSTRATIVE EXAMPLES 
 
 Scd 
 
 1. Find the product of ^ — i:: and 
 
 Scd^ 
 
 ^ah^ 
 
 Solution. 
 
 2a% 
 3c(P 
 
 Scd 
 
 a 
 
 2hd 
 
 (I 
 2M 
 
 Remark. In practice it is customary to cancel as shown in solu 
 tion of example 2, which follows. 
 
''^'^ :,2 + 5^+6 
 
 "^ 4 
 
 a:-4 
 
 
 
 x^-hSx-4: 
 
 3a: 
 
 -k6_ 
 
 = (3>^)(^ + 
 
 4) 
 
 3(^^K2) 
 
 ' a;2+5a; + 6 
 
 4a; 
 
 -4 
 
 (2>K2)(a: + 
 3(a: + 4) 
 4(a: + 3) 
 
 3) 
 
 i(^^^) 
 
 Let x = 2. 
 
 
 
 
 
 
 a;2 + 3 a; _ 4 
 
 3a; 
 
 + 6 
 
 = 3(^ + 4). 
 
 
 
 a:2+5a; + 6 
 
 4a; 
 
 -4 
 
 4(a; + 3) 
 
 
 
 4 + 6-4 
 
 6 + 6. 
 
 _3(2 + 4). 
 
 
 
 174 ELEMENTARY ALGEBRA 
 
 2. Multiply V.,. • 
 
 Solution. 
 
 Check. 
 
 4 + 10 + 6 8-4 4(2 + 3) 
 
 Remark. It is evident that x — 1 and a; + 2 may each be can- 
 celled in the numerator of the one fraction and the denominator of 
 the other [Solution, example 2], for, if expressed in the product, each 
 would be a factor common to the numerator and denominator of the 
 product and hence could be cancelled. 
 
 3. S>mphfy 1 - ^, . ^^^/^^ • Vzfr- 
 Solution. 
 
 (C _ y)2 ' c* + CV ^ y4 c^ _ yi 
 
 ^l (c + y)(c^-cy + y^) ^ (c - y)(c^ -[- cy -{- y^) ^ (c + .y)2 
 
 (c - 2^) 2 (c2- cy + y^)(c2+ C3/ + y2) (c - 2/) (c + y) 
 
 ^1 (^ + yy 
 {c-yy 
 
 ^ (c-yy-(c + yy 
 (c - yy 
 
 ^ l(c-y) + (c + y)M(c - y) - (c + y)] 
 ^ (2c)(-2y) ^ -4:cy 
 
 {c - yy (c - yy 
 
FRACTIONS 175 
 
 EXERCISE 67 
 
 (Solve as many as possible at sight.) 
 Simplify : 
 
 2a 2^— si— 4^§f 
 
 3*5* ' Sh'b^' ' a^'a^' ' b ' d' 
 
 5. ^.?5.J_. e. 2a. ^. 
 
 5 21 - 2 c 
 
 „ 2 «^ o 2a 4 5 
 
 2xy^ 4x — 3 mn^p^ 4m 
 
 * 3a52*3z/' ' 2a5V *9a* 
 5a:yV -2a;y 2 a^^gS 4 a^^g^ 
 
 ' 3a5g * 3ag ' * Sde^^' 9 cPef^' 
 
 2(x + ^) (x + yy ^^ -3(a + 5) 5(a -f 5)3 
 
 3(2:-^)'(a;-^)3' ' 2(a-5) -15(a-5)2* 
 
 15. ^ + y ^-y 16. 3(^H-^^^-^^ 
 
 ' X - 1/ 2^ -\- 1/^ ^ — y^x-\-y 
 
 17 ^!zil (^ + ^)^ 18 ^^ 2(5+3) 
 
 • a2_52- ^4.1 '54-3 3a2 
 
 19. 
 
 21. 
 
 a52 6(^2 gf 4a5gc2 Ix^fz^ %ac 
 
 — . . -sL • 20. . ^ • • 
 
 cd^ e^f ah 1 xyz ZaVycH %xyz 
 
 a52/l_l\ _ 20252/02 _a\ 
 
 3U hj ' 5 W 5/ 
 
 23 a; l-a2 2^ a2^2a + l -352 
 
 a — 12x-\-xy 5^ o + l 
 
 27. 
 
 yv z \^ 
 
 35-6a: 2a*+2a35 6x-^Sx^ 
 
 2a + 2x dax-\-9x 2ab — ^ax 
 
176 ELEMENTARY ALGEBRA 
 
 28 <^^ - 2 aa; ab-\-SP 2abi/-Sxy^ 
 a^ + Sab' 2abx-'d2^'i/' ax-2a^ 
 
 2m^-2 5?n2 + 5 bp^ + bq ^ bp + bq 
 ^p^-Sq' ap^ + aq^' 3^2+8' cm-c' 
 
 (y + l)2V4-l' 
 
 b--a a^-b^ 6(x^-y^) (x + y)2 
 (a; + y)2' 10 * a-^-b ' x-y 
 
 29. 
 
 30. 
 
 31. 
 
 139. Powers of a fraction. An important case in the 
 multiplication of fractions is that in which the fractions 
 to be multiplied are equal; their product is, therefore, a 
 power of one of the given equal fractions. 
 
 -- ^-hhh%=k 
 
 
 
 m-^^^x=^>^^>^^-^- 
 
 -32 
 243 
 
 32 
 243 
 
 (^y = ^^.^... ton factors 
 
 
 
 _ a. a. a. ..ton factors ^a«j^^ 
 b'b'b"-ton factors 6'* 
 
 
 
 W IP 
 
 
 
 
 Identity (1) may be expressed in words as follows : 
 Any power of a fraction is equal to the same power of the 
 numerator divided by the same power of the denominator. 
 Conversely, we may write, 
 
 ?-©■• « 
 
FRACTIONS 177 
 
 ILLUSTRATIVE EXAMPLES 
 .2 
 
 ^'-"(ffDHfcf)' 
 
 1. 
 
 y 
 
 Solution. 
 
 \x — y } \ a — h I L X — y JL a — 6 J 
 
 2. Simplify L________L. 
 
 Solution. (2 ^^ + 3. + 1)3 ^/ 2.^ + 3. + 1 \» 
 
 (X + 1)8 V X + 1 I 
 
 EXERCISE 68 
 
 (Solve as many as possible at sight.) 
 Raise the following to the indicated powers : 
 
 ^- (2^) • '' [-^r) '' (32^) • 
 
 10 
 
 13 
 
 IV 
 
 3 xyH J \b xy 
 
 ■ {-IT '■ 6)"- 
 
 . (ff. „. (=^)\ „. (i+i) 
 
 \oJ V xyz ) \x yj 
 
 Suggestion. Add the fractions before squaring. 
 
 •e-p" "S-'T- -c--")" 
 
 16 fi-iY. 17 i^'+'^y 18 C^'+^^'^' 
 
 V a:y * (a2-a+-l)3' * (a3 + jsy 
 
 19^ i^ - ^yY 20. i^tuMt.. 21. (^^ - ^^^)^ . 
 
 (^ — a;)^ * (a; — ^)^ * (m^ — w^)^ 
 
178 ELEMENTARY ALGEBRA 
 
 Write each of the following as the square of a fraction : 
 
 • J2_45_^4 • a:2 + 2a;+l " * / 
 
 25 ^^ 26 ^^ ^'^'^' 27 4-^-12a:+9 
 
 * 9a456 * 64ar*2/V0* * 9a:2^_i2:i;+4- 
 
 Division of Fractions 
 
 140. Reciprocal of a number. The reciprocal of a 
 number is 1 divided by that number. Hence, whenever 
 the product of two numbers is equal to 1, either one of 
 these numbers is the reciprocal of the other. 
 
 Thus, since - x - = 1, the reciprocal of - is -, or - = 1 -f- - and 
 ha h a h a 
 
 - = 1 -^- -. The reciprocal of a fraction is obtained by interchanging 
 a b 
 
 the numerator and the denominator of the fraction ; that is, by invert- 
 ing the fraction. 
 
 141. Quotient of two fractions. The quotient of two 
 fractions may be obtained by use of the identity 
 
 Dividend = Divisor x Quotient (V) 
 
 Thus, let it be required to find the quotient of - h- -. 
 
 a 
 
 Multiplying both terms of - by cd, 
 
 a _ acd 
 b bed 
 
 Expressing the second member of (2) as the product of two 
 fractions, 
 
 ? = £ X — (3) 
 
 b d be 
 
 Observing that - in identity (3) corresponds to dividend in 
 b 
 
 identity (1), that - in identity (3) corresponds to divisor in identity 
 
 (1), and that — corresponds to quotient in identity (1), and since 
 be 
 
 (2) 
 
FRACTIONS 179 
 
 dividend divided by divisor is equal to quotient, we have, 
 
 a ^ c _ ad _ a d ft^. 
 
 b d be b c 
 
 From identity (4) we have the following: 
 Rule. To find the quotient of two fractions multiple/ the 
 dividend hy the reciprocal of the divisor, 
 
 142. Two special cases of division. 
 
 1. When the dividend is a fraction and the divisor is 
 an integral expression. 
 
 Any integral expression can be written in the fractional 
 form. 
 
 Thus, c may be written -. 
 
 Therefore, --c=-h-- = -x- = -. (1) 
 
 b b 1 b c be 
 
 That is, «^c=-^. (2) 
 
 b be 
 
 From identities (2) and (3) 
 
 From identities (2) and (4) we have the following: 
 Rule. To divide a fraction hy an integral expression^ 
 
 either divide the numerator or multiply the denominator of 
 
 the fraction hy the integral expression. 
 
 Note. In practice, divide the numerator rather than multiply the 
 denominator whenever the numerator is exactly divisible by the 
 integral expression. 
 
 2. When the dividend is an integral expression and 
 the divisor is a fraction. 
 
 a^^ = «^^ = «x^ = ^. (1) 
 
 c \ c 1 b b ^ ^ 
 
 That is, a-^ = ^. (2) 
 
 c b 
 
180 ELEMENTARY ALGEBRA 
 
 From identity (2) we have the following: 
 
 Rule. To divide an integral expression by a fraction^ find 
 
 the product of the reciprocal of the fraction and the integral 
 
 expression. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Divide — — by — —. 
 
 21 xy^^ 28 xy^ 
 
 2ab 4 
 
 Solution IQ ""^^^ ^ - 15 ab^c^ _ ^0^^^^ .2S-^vt^ 
 21 xy^z^ ' 28 xyz» -iirxy^z^ ^-l^^*3c»- 
 Sy -3c2 
 
 Sy 3c2 
 
 __ Sab 
 
 Solution. £^x3£%^^=^x-i:^x:i^*t' 
 
 « 2 J 
 
 a 2 ^ 
 
 2 ay' 
 
 q. ,.£ (rg H- y^^ x^ -\- xy . (a^ + y)^ 
 
 x^ — y^ (x -\- ^)2 — xy 7^ — y^' 
 Solution. 
 
 (a^ + y)^ j^ a:^ + xy ^ {x + yY 
 
 = 7^7^ — ^^^ 
 
 1 1 x + y 
 
 __ X 
 ~ x-\-y 
 
FRACTIONS 181 
 
 EXERCISE 69 
 
 (Solve as many as possible at sight.) 
 Simplify : 
 
 1. — '- m. 2. —-5-6. 3. --5-C. 
 
 n 
 
 1 - w ^ ^ 
 
 4. a-i--. 5. m-5 . 6. r-=--. 
 
 c 7i 8 
 
 c c 
 
 9. _^?_^(a;+i/). 10. '^^-^^ ^(m-n^ 
 
 X — y X 
 
 U. "'' + ^' ^(,» + n). 12. ^^(x-1). 
 
 13. (a:-3/)-5-^— . 14. (2^_^2)^£l^. 
 
 a^ + y a 
 
 a x-\- Z 
 
 a 
 
 19. -^ r-^ -5- «(6 -h c). 20. — 5- (w + ri). 
 
 — 1 m 
 
 «, 1 1 oo 3 4 o, 1 1 
 
 21. --5--. 22. --5--. 23. -^-. 
 
 X 2 a X y 
 
 «. « 2 ^^ 2a3 4a2 ^^3 5 
 24. — ; — . 25. ! . 26. ; . 
 
 2 a 3 63 9 63 2a 4a2 
 
 27. ^-^^. 28. ^-5-^1^. 
 
 63 3 62 I y 
 
 5£^_15^ 3^ (g-f ^>)^ . 2Ca-h6)8 
 
 Zmy^' 2m^f' * 3(a:-2/)S * 5(a:-^)2' 
 
182 ELEMENTARY ALGEBRA 
 
 31. 
 
 2ab,Zb 32 
 
 a 
 
 2d 
 Zcx 
 
 33. 
 
 2a . 8a 
 36 * 3 6* 
 
 34. 
 
 ^-^. 35. 
 cd ' bd 
 
 ax ^ X 
 
 % ■ y' 
 
 36. 
 
 2a6 2 6 
 Zed' Zc 
 
 37. 
 
 2 a5V 3 a^^c 
 9 xi/^^ 4 3^y^z 
 
 
 38. 
 
 H-- 
 
 
 39. 
 
 -■... 
 
 
 40. 
 
 
 
 41. 
 
 .-^.!|?. 
 
 
 42. 
 
 («+6)2h 
 
 2(a + 6)3 
 3a 
 
 ,^ -2a2: 2x _ -a362c4 2 a36c2 
 ^^^ _t_ ^ 44. ! • 
 
 3 y2 * _ cfiy' ' 2 a;2«/32 * 3 x^y'^z 
 
 a% b'^c abx ^^ . -,x x^-\-x-\-l 
 
 45. -^-^—7r~- • *^- (^ — 1)"^ T"^ • 
 
 x^y y^z yz 1 — x 
 
 47. (2;2_. y2)^^Zll. 48. (2: + a)2H--^±^. 
 
 ^!±i!^(«2 + ,j + 52). 50. 2(:r+y)^^3(.+ y)8 
 a-6 ^ ^ ^ . 3(:r-^)3 2(x-yy 
 
 (2a;+l)(3a;-2) . (3 rr- 2)(2 a; + 1)2 
 (3a:+2)(2a:-l) * (3 2: + 2)(2 2: - 1)2* 
 
 (2a;+3)(2 2;-3) . (2 a; + 5)(2 a;- 3) 
 (3a: + 2)(3a:-l) * 3(3a:-l)(2a:-l)* 
 
 6a;2H-5a;-6 . 2a; + 3 
 Qx'^-bx-Q^ Zx + 2' 
 
 9^:2-1 3 2:2 + 2 a;- 1 
 
 49. 
 
 51. 
 
 52. 
 
 54. 
 
 6 2;2-5a; + 6* 2a:2-2;-3' 
 
 f - x^y ^ f y - a^ Y 
 -\- yx-\-x^ \y ■\- x) 
 
 y^^-y) y'^ 
 
 a^ -\- ab — ae (a — c^ — b^ a 
 
57. 
 
 58. 
 
 59. 
 
 FRACTIONS 183 
 
 3 x^y^ + 3 + 6 a:!/ , "Ixy-^-l 
 4x^7/^ + 4: -8x1/ ' Sxy-S' 
 
 2^3 _ y3 x^ — xy + y^ _^ x^ -{- xy -^ xz-\- yz 
 
 x^ — xy -\- xz — yz x-\- y a^ -j- x^y^ + y^ 
 
 (x-^yy-z^ ^ z^-x'^-\-yiy-2z) , (y + z^-x ^ 
 ^x-yy-z^ z^-x^ + y(2x-y} * (y-z^-x^' 
 
 60. What is the reciprocal of — f ? Of — - ? 
 
 61. What is the reciprocal of the reciprocal of a frac- 
 tion ? Illustrate by taking ^. 
 
 62. When equal factors are cancelled from the numera- 
 tor and the denominator of a fraction, what operation are 
 we performing on the terms of the fraction ? 
 
 63. Why can we not cancel the like terms in the nu- 
 
 2 -\-x .2 + 2^ 2 + 1 
 
 merator and denominator of _ and obtain = - 
 
 o 1+x 1+/ 1+1 
 
 = -9 
 
 2' 
 
 X A- \ 
 
 64. For what value of x has no meaning? 
 
 Complex Fractions 
 
 143. Definition. A complex fraction is a fraction which 
 has one or more fractions in either or both of its terms. A 
 complex fraction is said to be simplified when it is reduced 
 to an equivalent simple fraction or integral expression. 
 
 In simplifying a complex fraction it is usually most convenient to 
 express each term of the fraction in its simplest form before attempt- 
 ing to perform the indicated division. Sometimes, however, labor 
 is saved by first multiplying both numerator and denominator by 
 the L. C. D. of aU the fractions contained in the terms of the given 
 fraction. . 
 
184 ELEMENTARY ALGEBRA 
 
 ILLUSTRATIVE EXAMPLES 
 
 1 + 1 
 
 1. Simplify =- - 
 
 Solution. Multiplying both terms of the given fraction by ar, 
 
 x_ X + 1 
 
 1 
 
 2. Simplify :j T-'^'i T"* 
 
 a b-\- c b a-\- c 
 
 Solution. Multiplying both terms of the first fraction by 
 a(h + c) and both terms of the second fraction by b(a 4- c), we have, 
 
 1 1_ 1 1_ 
 
 a b -\- c b a -\- c b + c — a a + c — h 
 
 i+ 
 
 a 
 
 1 
 
 6 + c 
 
 r 
 
 b 
 
 ^ 1 b + c -\-a 
 
 a + c 
 
 _b -\- c - a 
 
 a+ c 
 
 + b 
 + 6 
 
 
 
 
 b-\-c + a 
 
 a-\-c 
 
 -b 
 
 
 
 
 b + c - a 
 
 
 
 
 
 
 a -\- c — b 
 
 
 
 
 
 
 EXERCISE 70 
 
 
 
 Simplify : 
 
 
 
 
 
 
 '^- 
 
 
 
 -=t- 
 
 3. 
 
 -2.5 
 
 -^ 
 
 a 
 
 4.1 
 c 
 
 b 
 
 
 
 mn 
 5 ""^^ 
 ab 
 
 6. 
 
 m + Si 
 . 1 
 
1+^ 
 
 FRACTIONS 185 
 
 ^ — yi- 
 
 ^ _mr ^— y 
 
 n^ a 
 
 4__ X 
 
 X 
 
 10. 11. 
 
 a 
 
 1 
 
 is _0' X — a 
 
 X 
 
 X— y X — z 
 13. ^ 14, 
 
 V z 
 
 
 r-1 
 
 9. 
 
 r+l 
 r+1 
 
 
 r-1 
 
 
 X w 
 
 12. 
 
 y z 
 
 X w 
 
 y z 
 
 1 , 
 
 1 
 
 a + 1 ' a 
 
 ;-i 
 
 1 
 
 1 
 
 y—x z—x a—\ a+1 
 
 (j-')^')Hr')e-') 
 
 15. 
 
 1+^ 
 
 62 
 
 16. T— 70 17. j—-^ 
 
 ao + tr a— -\- c 
 
 ab—l^ a—h—c 
 
 X y fl . 1\2 1^1 1^1 
 
 X \x yj 
 
 y X \x yj x y -\- z X z — y 
 
 18. ! — = = — • 19. = = '- T — . 
 
 x y 1^_J^ 1 1 1 1 
 
 y X a^ y^ X y — z X y -\- z 
 22. 
 
 
186 ELEMENTARY ALGEBRA 
 
 (i — ^ — —V(— — —)' 
 
 \ i/-\-z x + yj \y + z x-Vyl 
 
 23 
 
 V \ 
 
 1 
 
 24. x-\- 
 
 .+1 
 
 X 
 
 Suggestion. Multiply both terms of the complex fraction by x 
 and then reduce the resulting mixed number to an improper fraction. 
 A complex fraction of the form given in this example is called 
 a continued fraction. 
 
 1 3 
 
 25. iH 7. 26 
 
 1+i* ' 24- ^ 
 
 
 oc^ — y^ 
 
 X 2-\-x 
 
 EXERCISE 71.— REVIEW 
 
 1. Name at sight the result in each of the following : 
 
 b X m^ f Tt 
 
 ax-; ax—; ^-m; -z=i-i- rt. 
 
 c ae n R 
 
 2. Name at sight the result in each of the following : 
 
 ^{x-y^, 
 
 z 
 
 3. Reduce — — — - to lowest terms. 
 
 arm + 3 a^ -f am + 3 a 
 
 4. Does("-y=-(^^l^% Why? 
 
 c — a d — c 
 
 ( 'YY\. — 92, I ^ 
 
 5. Reduce 1 — ^^— ^ to an improper fraction. 
 
 ^2 ^ ^2 ^ ^ 
 
 \ — ^ 
 
 6. Reduce to a mixed expression. 
 
 7. Add - to a. 
 
FRACTIONS 187 
 
 8. Subtract x from - 
 
 Si„pli„g.f).g-f). 
 
 Simplify : 
 1 
 
 10 
 
 m-\-n m— n 
 
 11. J-+-1-, 
 
 12. 1 
 
 2 X— y Z x — ly 
 
 13. ^^^ 
 
 I ^% 
 
 nhx -f ^i^y max -\- may 
 14. , ^ . + „ 1 . + 1-6^ 
 
 15. -JL+ 2 ^ J_ 3a^ 
 
 a — 5 a -f- 5 a^ — b^ (a — by 
 16. ^ + .-^„+ 1 
 
 a;— 1 1 — a;2 a:^ — 1 
 17. -1-+ 1 1 1 
 
 x-\-a x—a x — b x-\-b 
 
 ^^y y ~~ ^ z — x x — z y — x z — y 
 
 a + 36 a-36 a3_9^52 
 20. fl- /-g Y-±l. 
 
 \ 3?-X-Vx-\ 
 
 ■i-a l-a2 
 
188 ELEMENTARY ALGEBRA 
 
 .1.1 a3 + 3a2_l 
 
 22. aH-l+ \vo 
 
 a a^ -{-2 a 
 
 ix + 2yy ix-2yy 
 
 24. l-^_ + -J_ + ^+1 
 
 a a— 1 a — 2 a(a — l)(a — 2) 
 
 26. 
 
 1_1 ■ 1^1 
 
 X y X y 
 
 Suggestion. Factor the resulting numerator by arranging accord- 
 ing to powers of a. 
 
 (jd — h)(a— c) (h — c)(h — a) (^c — d)(^c — b) 
 28. , i, , 4- 1 1 
 
 a(a— 6)(a— (?) h(h — d)(h — c) e(^c — d)(^c— h) 
 
 29. a + a; 
 30. 
 
 a— a; a -\-x 
 1 1 
 
 (:r4-l)(^H-2) (a; + 2)(a: + 3) (a;4- 3)(2: + 1) 
 1 a^-1 0^4-1 
 
 34. - — 7 + 
 
 x^-l'x^ + 1 a^ + a^ + 1 2^-a^ + l 
 
FRACTIONS 189 
 
 • ^ x+1 ^ (a: + 3)2^ (a; + 3)2(2^+1) 
 2a^ 10 4 
 
 36. x + S-^ 
 
 (a; -1)2 2;2_i (a;+ l)(a;- 1)2 
 
 x-^1 
 
 37 (2xy + ^x-y 2xy-^x-y\ . a^y 
 
 V 32;-^ 32: + y J ' ^x^-f 
 
 a^ + 2a;+3 ^2^2^ + 3 
 X y 
 
 38 -^ 
 
 3a^ + 2a;+l 3^/2+2^ + 1 
 
 rc2 4- a: a^^+7a; + 10 x^ + 1 x-^12 
 
 a^ + 5a: + 4 a:2^2a; 2^ + 82; + 15 
 
 1 r^ I 2x-1 lfx-2 (a; + l)(a:-3) \n 
 * a; L 1 ^ 3U + 2 a^(a; + 2) jlj 
 
 41 
 
 X 
 
 3 8A2 + 7c?2 2^-6<? 
 
 42 
 
 43. 
 
 2A + (? 8^3+^3 4A2-2Ac + c2 
 
 3a;-l 2a: + 3 5a:2_2a;-l 
 a;+3 1-a: ar2 + 2a;-3 ' 
 
 44 <? + ^ ^2 _,_ ^5 ^ ^2 
 
 a-6 "*" a2 _ 52 • 
 
 a^ — y^ x^-\- y^ 
 
 a^-b^ '^^^^ a2_^ 
 45. X X 
 
 0^ -\- xy -\- y'^ x^ — xy-\- y'^ x^ — y^ 
 
 a — b a-{-b 
 
190 ELEMENTARY ALGEBRA 
 
 a^ + ab + 1^ a^-l^ a-b ' V^ a-bJV^ a) 
 
 X 
 
 ^ x + y X 
 
 a-\-h a^^h^ a-\- b 
 
 - «/ \x y) 
 
 x — y x-\- 
 x-^ y x — y 
 
 a 
 
 h ^ 
 
 49. Is the following a true statement ^ - = ^ ? What gen- 
 
 h 
 eral rule aan you derive from it ? Apply your rule to find 
 
 4 
 
 the value of %i- 
 
 A 
 
 a 
 
 50. Is the following a true statement : - = ? What gen- 
 - ^ b 
 
 eral rule can you derive from it? Apply the rule to find 
 the value of |. 
 
 51. Show that 
 
 a.„)(. + 3(,+|)(..i)=(.+i..+i 
 
 Reduce each of the following fractions to a mixed number 
 before performing the indicated additions or subtractions: 
 
 52. 
 
 x-1 x-{-l 
 
 OS. ■ -j • 
 
 x-^1 x—1 
 
CHAPTER VI 
 FRACTIONAL AND LITERAL EQUATIONS 
 
 Fractional Equations 
 
 144. Definition. A rational equation in one unknown 
 number which is not integral with respect to that number 
 must contain the unknown number in the denominator of 
 one or more terms. Such an equation is called a frac- 
 tional equation in one unknown number. 
 
 11 3 a: 2 5 
 
 Thus, - + - = 2 and -\ ^^r + - = are fractional equations. 
 
 X a a; — 2a: + 22 
 
 145. Clearing an equation of fractions. From a frac- 
 tional equation in which each fraction has been reduced 
 to lowest terms, an integral equation can be derived by 
 multiplying both members of the equation by the lowest 
 common denominator of all its fractions. This process is 
 called clearing the equation of fractions. 
 
 146. Solution of a fractional equation. The solution of 
 a fractional equation is found by solving the integral 
 equation obtained from it. Hence: 
 
 Rule. To solve a fractional equation, in which each frac- 
 tion is in its lowest terms : 
 
 1. Clear the equation of fractions hy multiplying both mem- 
 bers by the lowest common denominator of all the fractions. 
 
 2. Solve the resulting integral equation. 
 
 3. Test the roots by substituting them in the given equation. 
 
 191 
 
192 ELEMENTARY ALGEBRA 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Solve -^- — = 0. 
 
 a; + 3 3 a; 4-1 
 
 Solution. 2^+J:_|£jI^=:0. (1) 
 
 Clearing (1) of fractions by multiplying both members by 
 (a: + 3)(3x + l), 
 (2a; + l)(3a: + l)-(x + 3)(5a: - 2)= 0. (2) 
 
 Performing the indicated operations in (2), 
 
 a;2_8a: + 7 = 0. (3) 
 
 Factoring, (a: - 7)(a: - 1)= 0. (4) 
 
 Equating (x - 7) to 0, a; - 7 = 0. (5) 
 
 Equating (a; - 1) to 0, a: - 1 = 0. (6) 
 
 Solving (5), a; = 7. (7) 
 
 Solving (6), a: = L (8) 
 
 rtx. ^ 2 a: 4- 1 5 X — 2 rt ^^x 
 
 Check. ^ - = 0. (9) 
 
 a: + 3 3ar + l ^ ^ 
 
 Substituting 7 for x in (9), 
 
 2x7+1 5x7-2 
 
 0. (10) 
 
 7+3 3x7+1 
 
 Simplifying, I " I = «• (H) 
 
 Substituting 1 for x in (9), 
 
 2x1 + 1 5x 1-2 ^Q ..^ 
 
 1 + 3 3x1 + 1 • ^ ^ 
 
 Simplifying, f - f = 0- (13) 
 
 2 Solve ^-^^ + ^ 16(a:+l) _^ 
 
 Solution. 4zl|£±1_ 16{^_L11^=0. . (1) 
 
 ar2 + 3 a; - 4 5 a:^ + 46 a: + 41 ^ ^ 
 
 Factoring, _(^-^D! IH^^l) ^^ ^^^ 
 
 ^' (a: - 1) (a: + 4) (5 a: + 41) (a: + 1) ^ ^ 
 
 Reducing the fractions to lowest tferms, 
 
 a:-l 16 
 
 a: + 4 5 a: + 41 
 
 = 0. (3) 
 
FRACTIONAL AND LITERAL EQUATIONS 193 
 Clearing (3) of fractions, and dividing by 5, 
 
 
 x2 + 4a:- 21 = 0. 
 
 (4) 
 
 Factoring, 
 
 (a: + 7)(a'-3)=0. 
 
 (5) 
 
 Equating (x + 7) to 0, 
 
 X + 7 = 0. 
 
 (6) 
 
 Equating (x - 3) to 0, 
 
 a: - 3 = 0. 
 
 (7) 
 
 Solving (6), 
 
 x = -7. 
 
 (8) 
 
 Solving (7), 
 
 Check. I"'^-^! 
 x^ + 3x -4: 
 
 x = S. 
 16(. + 1) _o 
 5 x^ + 46 j; + 41 
 
 (9) 
 (10) 
 
 Substituting - 7 f or ar in (10), 
 
 
 
 (_ 7)2- 2(- 7)4-1 16(-7+l) _Q 
 
 (11) 
 (12) 
 
 (_7)2_,.3(_7)_4 5(_7)2+46(_ 7)+4i 
 
 Simr^lifvinp- 49 + 14 + 1 -96 
 
 Simplifying, 49 _ 21 - 4 
 
 245 - 322 + 41 
 
 or, 
 
 f-f-0- 
 
 (13) 
 
 Substituting 3 for x in (10), 
 
 
 
 (3)2 - 2 X 3 + 1 
 
 16(3 + 1) _ Q 
 
 
 (3)2 + 3x3-4 5x 
 
 32 + 46 X 3 + 41 
 
 
 0. 1-^ • 9-6 + 1 
 Simplifpng, g^g_^ 
 
 64 _Q 
 
 
 45 + 138 + 41 
 
 
 or 
 
 1-1=0. 
 
 
 Note. In certain exceptional cases the integral equation obtained 
 by clearing a given equation (in which each fraction is in its lowest 
 terms) of fractions by multiplying both members by the lowest com- 
 mon multiple of the denominators, is not equivalent to the given 
 equation. It is important, therefore, that each root of the integral 
 equation should be checked by substituting it for the unknown num- 
 ber in the given fractional equation. 
 
 The way in which these exceptional cases arise may be seen from 
 a consideration of a particular example. Thus, let it be required to 
 determine the value of x from 
 
 
 •r + 1 1 1 _ 1 
 
 
 X x(x — 1) X — 1 
 
 Clearing of fractions, 
 
 (a;2-l)+ l=x. 
 
 Simplifying, 
 
 x^ - X = 0. 
 
 Hence, 
 
 X = OoTh 
 
194 ELEMENTARY ALGEBRA 
 
 On substituting the value for x in the given equation, the first 
 fraction becomes -, which is without meaning, and, therefore, this 
 
 value of X does not satisfy the given equation. In like manner, the 
 second root of the integral equation does not satisfy the given 
 equation. 
 
 By writing the given equation in the form 
 
 
 X 
 
 + 1+ 
 
 X 
 
 1 
 
 x(x — 
 
 1) X 
 
 1 
 -1 
 
 = 
 
 and simplifying, we 
 Simplifying, 
 
 obtain 
 
 
 
 x^- 
 x(x- 
 
 ■ X 
 
 -1) 
 
 1 = 
 
 = 0. 
 = 0. 
 
 Now this is an impossible number relation, and the reason for the 
 non-existence of any value of x which will satisfy the given equation 
 is evident. In fact, although the given relation is expressed in the 
 form of an equation, the expression is not actually an equation. 
 
 EXERCISE 72 
 
 Solve the following equations and check each solution : 
 1. 1 = 2. 
 
 X 
 
 *■ - + - = !• 
 
 X X 6 
 
 _ 1 2 
 
 x — 2 X 
 
 9.5 + 
 
 X S — X 
 
 11. -^ ^— = 0. 
 
 X-\-4: 1X-\-l 
 
 15 " 5 
 
 2. 2=3. 
 
 
 3. 8 = 1 
 
 X 
 
 
 X 
 
 '■ i- 
 
 2x 
 3 
 
 6. 2=1 
 8x 4 
 
 8. 
 
 2 
 x + 2 
 
 .§ = 0. 
 
 X 
 
 10. 
 
 4 , 
 
 ^ -0 
 
 x-^ ' 
 
 x + 1-^- 
 
 12. 
 
 2 
 x+S 
 
 %=0. 
 x — 1 
 
 14. 
 
 5 
 x-^1 
 
 ^ -0 
 
 x-1-^- 
 
 1A 
 
 5 
 
 1 
 
 2a:-3 ^x-2 2x-]-b 2x+l 
 
FRACTIONAL AND LITERAL EQUATIONS 195 
 
 17. — - — -f — - — = 0. 18. — ?_+— i — = 0. 
 Sx-2 5x + 2 2x-h2 Sx-S 
 
 19. -^ L- = 0. 
 
 Sx-\-2 x-\-l 
 
 on ^ ^ 1 4 1 
 
 20. h 
 
 27. 
 
 28. 
 
 so: 
 
 2x Sx-9 Zx 
 21 l+i= ^ 
 
 3 a; 6 x{x-T) 
 
 22. ?+l = |- 23. 3^ + 2 = ^. 
 
 ^^ 3a;-4 , x-\ . o« 2a: 3a:-5 
 
 24. = 1. 25. = 
 
 2x^-\ a:+4 3a;-8 x-\-2 
 
 X , x-4: 2x^-1 
 
 26. - + 
 
 2x-l x+S 2a^-a;-21 
 
 3a:-2 3a: + l 29 a; + 19 ^^ 
 x-\-l 2a:-l 2a:2^.a;_l 
 
 4(a:4-2) 7(a:-l) 3(a:+l) ^n 
 
 2^_l a^-a^-2 ar2-3ar-h2 
 
 29. ^_4._1_4. ^ 6a.» + 30a: + ll 
 
 a;+l a:4-2 a; + 3 (a:H- l)(a: + 2)(a:4- 3) 
 a^ + 1 a; 4- 2 a; + 5 a; + 6 
 
 a: + 2 a;+3 x -i- 6 x + 1 
 Suggestion. Simplify each member before clearing of fractions. 
 
 31. A can perform a piece of work in two thirds of 
 the time in which B can ; B can perform it in one half of 
 the time in which C can ; they all can perform it in 22 
 days. Find the time in which each alone can do the 
 work. 
 
196 ELEMENTARY ALGEBRA 
 
 , 32. A and B working together can do a certain piece 
 oi work in 6 days. A working alone can do the same 
 work in 10 days. In what time can B alone do the work? 
 
 Suggestion. Let x = the number of days required. 
 
 Then, - = the part B can do in 1 day. 
 
 33. What number must be added to each term of the 
 fraction ^ so that the resulting fraction shall equal §? 
 
 34. What number must be subtracted from each term 
 of the fraction l| so that the resulting fraction shall equal 
 
 35. Find a proper fraction whose numerator and denomi- 
 nator differ by 1, and such that the result of adding i^ to 
 the fraction is double the result of subtracting the fraction 
 from 2. 
 
 36. A can do a piece of work in 10 days and B can 
 do it in 8 days. In what time can they do it working 
 together? 
 
 37. A certain pipe can fill a cistern in 10 hours and 
 another can fill it in 12 hours. In what time can they fill 
 it if both pipes run together? 
 
 38. A certain fraction is equal to |. When its numer- 
 ator is diminished by 5 and its denominator by 8, the 
 resulting fraction is equal to |. Find the fraction. 
 
 Suggestion. Let the fraction be denoted by ^ — 
 
 X 
 
 39. A can do a certain piece of work in 3 days, B can 
 do it in 4 days, and C can do it in 6 days. In what time 
 can they do it working together? 
 
 40. Three pipes are connected with a certain reservoir. 
 The first pipe can fill it in 2 hours, the second in 5 hours, 
 
FRACTIONAL AND LITERAL EQUATIONS 197 
 
 and the third can empty it in 10 hours. In what time 
 will the reservoir be filled if the three pipes are set to flow 
 at the same time? 
 
 Literal Equations 
 
 147. Numerical equation. A numerical equation is an 
 equation in which all the known numbers are expressed 
 by numerals. 
 
 Thus, 6a: — l=2x4-3isa numerical equation. 
 Remark. The equations which have been considered in the pre- 
 ceding pages are, in general, numerical equations. 
 
 148. Literal equation. A literal equation is an equation 
 which involves one or more known literal numbers. 
 
 Thus, ax -f 6a; + c = is a literal equation. 
 
 Remark. In section 69 it was stated that in algebra known num- 
 bers are usually represented by the first letters of the alphabet. 
 
 149. Solution of a literal equation. A literal equation 
 in one unknown number is solved in the same manner as 
 a numerical equation, but the roots of a literal equation 
 are algebraic expressions in the known numbers. 
 
 The known numbers in a literal equation are called such to dis- 
 tinguish them from the unknown numbers. In an equation it is 
 sometimes convenient to consider one of the literal numbers as the 
 unknown and at other times another. 
 
 Thus, from the formula A = ab, which expresses the area of a 
 rectangle in terms of its base and altitude, A, a, or 6 can be found 
 when the other two are known. 
 
 ILLUSTRATIVE EXAMPLES 
 
 X— b X— a a^-\-l^ 
 
 1. Solve the equation 
 
 a— h a-\- b cfi — J)^ 
 
 Solution. x^^_x_-a^a^ + h\ 
 
 a-ba + ba^-b^ ^ ^ 
 
198 ELEMENTARY ALGEBRA 
 
 Clearing (1) of fractions, 
 
 (a + b)(x-b)-(a- b)(x - a) = a^ 4- b\ (2) 
 
 Performing the indicated multiplications, 
 
 ax + bx - ab - b^ - ax + bx + a^ - ab = a^ -\- b^. (3) 
 
 Combining like terms, 
 
 2bx-^a^ -2ab -b^ = a^-\- b^. (4) 
 
 Transposing terms, 2bx = 2ab + 21^. (5) 
 
 Dividing by 2 6, x = a + b. (6) 
 
 2. Solve the equation 
 
 x-{-a x-\-b x—a-\-b. 
 
 Solution. -^ ^ = ""-^ . (1) 
 
 X -\- a X + b X — a -\- b 
 
 Combining terms in the first member, 
 
 (g —b)x _ a — b 
 (x + a)(x + b) X — a + b 
 
 Dividing both members of (2) by (a — 6), 
 
 X 1 
 
 Check : Substituting for x in (1), 
 
 a-b 
 
 !+« -|+* -!-«+* 
 
 (2) 
 (3) 
 
 (x -{■ a)(x -{■ b) X — a -{■ b 
 Clearing (3) of fractions, 
 
 x^ — ax -\- bx = x^ -{- ax + bx + ab. (4) 
 
 Transposing in (4) and uniting like terms, 
 
 2ax = - ab. (5) 
 
 Dividing both members of (5) by 2 a. 
 
 - = -|. (6) 
 
 (7) 
 
 Simplifying, f^Ta^ f^a <«> 
 
FRACTIONAL AND LITERAL EQUATIONS 199 
 
 3. What number must be added to both numerator and 
 denominator of the fraction — so that the resulting 
 
 fraction shall be equal to - ? 
 
 P 
 
 Solution. Let x denote the required number. 
 From th6 conditions of the problem, 
 
 I + X _n .y. 
 
 (2) 
 (3) 
 
 (4) 
 
 
 m-\- X p 
 
 
 Clearing (1) of fractions, 
 
 
 
 Ip -\- px = nm + nx. 
 
 
 Transposing in 
 
 (2) and uniting like terms, 
 (p — n)x = nm — Ip. 
 
 
 Dividing both members of (3) by {p - n), 
 
 
 
 ^_nm-lp 
 
 
 
 p-n 
 
 
 Check. 
 
 1 + X p-n 
 
 
 
 wi + ^ ^ ^ nm- Ip 
 
 
 
 p-n 
 
 
 
 _ pi — In + nm - 
 
 '¥ 
 
 
 pm — nm + nm 
 
 -Ip 
 
 
 _ n(m — I) _ n 
 
 
 
 p{m - 1) p 
 
 
 Application of formula. Substitute Z = 5, w = 6, n = 16, p = 17, 
 in formula (4), 
 
 nm — Ip 
 
 p- 
 
 - n 
 
 
 6 X 
 
 16- 
 
 5x17 
 
 
 17- 
 
 16 
 
 96- 
 
 -85, 
 
 or 11. 
 
200 ELEMENTARY ALGEBRA 
 
 EXERCISE 73 
 
 ' Solve the following and check all roots: 
 
 1. 
 
 X — h = a. 
 
 2. 
 
 ^ = b. 
 
 3. 
 
 1_1 
 X a 
 
 4. 
 
 a 
 ax= b. 
 
 5. 
 
 2x-\-ax = l. 
 
 6. 
 
 ax — x-^bx=sc. 
 
 ."■■' 7. 
 
 ax-}- bx — x = 0. 
 
 8. 
 
 X 
 
 9. 
 
 X a 
 
 10. 
 
 1 
 
 11. 
 
 ax-{-bx = 1. 
 
 12. 
 
 ax-l = 2-bx 
 
 13. 
 
 1 a 
 x-1 2 
 
 
 • 
 
 14. 
 
 ax—bx-^(c-^ d)x = a 
 
 -b-^c-\-d. 
 
 15. 
 
 ax — b = d — ex. 
 
 16. 
 
 ^-5 = ^ 
 
 a-{-b _ 1 aa: bx ex ^ 
 
 19. 2 aa: — 3 6a; = 2 5a; — 3 aa; -h <?. 
 
 20. (a + ^>-l>-(a-6+l) = (a-h6-l)-(a-6+2)a;. 
 
 21. a(a: - a) + 6(a; - 6) = 3 «a; - (a + 5)2. 
 
 22. a(^x + 5) + 5(a: + a) = 2 aa: + 52 - (a - 5)2. 
 
 23. (a: + a)2 = 4a2 4-(a;-a)2. 
 
 24. (a;-a)(a;-5)-(a;-(?)(a;-(^)=0. 
 
 25. a + -=e. 
 
 26. - + - = <? 
 
 XX X 
 
FRACTIONAL AND LITERAL EQUATIONS 201 
 
 27. «_* + £ = !. 
 
 XXX 
 
 28. fL±£+*±£+fL±* = o. 
 
 ax hx ah 
 
 29. £^^ + *±f + l + U0. 
 
 c^x y^x a h 
 
 ^^ 2 h a-2x 
 
 30. — — = 
 
 b X 2h^bx 
 
 31. £zif + £+* = 2. 
 x + a x^o 
 
 32. 1 = a^ 4- 6. 
 
 x — aax—o 
 
 -,-, ^ . ^ .1 
 
 33. + — = a H- 6. 
 
 1 — aa; 1 — 6a; 
 
 34. ^+ * ■ 
 
 x-\- b~ 2x-\-b-\-e 
 
 ac — x ah — X ah— x 
 35. 
 
 36. ah:(hx-l)+h^x(ax-l) = (a-^b')(ax-t)(ibx-'l}, 
 
 EXERCISE 74 
 
 1. A= ah. 
 
 (a) Solve for a. 
 (6) Solve for 6. 
 
 2. (7=2 7rr. Solve for r. 
 
 3. The volume of a prism is expressed by the formula 
 
 (a) Solve for B, 
 (5) Solve for ^. 
 
202 ELEMENTARY ALGEBRA 
 
 4. The volume of a rectangular parallelepiped is ex- 
 pressed by the formula V= ahc. 
 
 (a) Solve for a. 
 (6) Solve for 5. 
 
 (c) Solve for e. 
 
 (d) Find a when F= 100, b = 10, and c = 6. 
 
 5. The lateral area of a regular pyramid is expressed 
 by the formula S = ^ L x P. 
 
 (a) Solve for L, 
 (h) Solve for P. 
 (c) Find L when i^ = 40 and P = 8. 
 
 6. The volume of a pyramid is expressed by the 
 formula V=\B x H, 
 
 (a) Solve for B, 
 (h) Solve for ^. 
 (c) Find ^ when r= 63 and j5 = 15. 
 
 7. The lateral area of a cylinder of revolution is ex- 
 pressed by the formula S = 2 m-RH. 
 
 («) Solve for R, 
 (5) Solve for ^. 
 (c) Find R when aS' = 24 and ir= 4. 
 
 8. The lateral area of a cone of revolution is expressed 
 by the formula S = tt RL. 
 
 (a) Solve for R, 
 . (5) Solve for X. 
 
 (c) Find aS' when i? = 4 and i = 5. 
 
 (d) Find L when aS' = 15 and i2 = 3. 
 
 9. A = \(h H- B^a is the formula stating the area of a 
 trapezoid. 
 
 (a) Solve for a. 
 
 (h) Solve for (h + J?). 
 
FRACTIONAL AND LITERAL EQUATIONS 203 
 
 (6?) Solve for h. 
 
 (cT) Find a froAi the following data : ^ = 95, 
 ^ = 20, 5 = 18. 
 
 10. V = ^* is a formula from physics, 
 (a) Solve for t. 
 lb) Find t, when g = 32.16 and v = 80.4. 
 
 11. h = z— is a formula from engineering. 
 
 1.2b r 
 
 (a) Solve for w. 
 (h) Solve for r. 
 
 12. - = - + - is a formula from physics. 
 fab 
 
 (a) Solve for/. 
 
 (6) Solve for a. 
 
 Prt 
 
 13. ^ = P + — — is a formula from arithmetic. 
 
 (a) What do ^, P, r, and ^, respectively, repre- 
 sent in the preceding formula ? 
 (6) Solve for P. 
 (c) Solve for t. 
 
 id) Solve for ^. 
 
 14. /=— is a formula used in electrical work. 
 
 R 
 
 (a) Solve for ^. 
 (5) Solve for E. 
 
 (c) Find 7 when i: = 200 and i2 = 250. 
 
 (d) Find T; when 7= 1.5 and i2 = 200. 
 
 (e) Find R when 7= .4 and ^ = 120. 
 
 15. The formula for converting a temperature of F 
 degrees Fahrenheit into its equivalent of degrees 
 
204 ELEMENTARY ALGEBRA 
 
 Centigrade is C = - (F- 32). Express F in terms of C, 
 
 and compute F when : 
 (a) (7=20. 
 (6) (7=30. 
 (0 (^=27. 
 
 16. What must be added to a; + a to make y — h? 
 
 17. What is the cost of 3 oranges if a oranges cost e 
 cents ? 
 
 18. Divide the number n into four parts such that, if a 
 is added to the first, a subtracted from the second, the 
 third multiplied by a, and the fourth divided by a, the re- 
 sults will be equal. What are the results if 7i = 10, and 
 a = l? 
 
 19. If my age is such that in n years I shall be a times 
 as old as I was m years ago, what is my age ? 
 
 20. If each side of a square were n feet longer, its area 
 would be p^ square feet greater. Find the length of its 
 side. 
 
 21. How many pounds of coffee worth a cents a pound 
 must be mixed with b pounds worth e cents a pound so 
 that the mixture may be worth d cents a pound ? 
 
 22. Find the number the sum of whose ath and bth 
 parts is c. What is the number when a = 5, 6 = 7, and 
 c=12? 
 
 W 
 
 23. t = 
 
 •\ V J V 
 
 W-\- w' 
 
 (a) Solve for W, 
 
 (b) Solve for w. 
 ((?) Solve for V. 
 
 (c?) Calculate Tfwhen t = .0019, w = 2, V= -4, 
 V = 100, a = 0.1. 
 
Blaise Pascal v 1 623- 1 662) was born at Clermont and died at 
 Paris. Perhaps no man ever displayed greater natural genius 
 than Pascal. His contributions to mathematics were not extensive, 
 but his name will always be mentioned in connection with the 
 arithmetical triangle. Much of his mathematical work was done 
 when he was a boy. His essay on conic sections was written in 
 1639, when he was but sixteen years old. 
 
CHAPTER VII 
 SYSTEMS OF LINEAR EQUATIONS 
 
 150. An equation may involve more than one unknown 
 number. 
 
 Thus, the equation 2x + 3 y = 5 contains two unknown numbers, 
 and the equation ax -\- by + cz + d = contains three. 
 
 151. Degree of an equation. By the degree of a rational 
 integral equation in two or more unknowns is meant its 
 degree with respect to the unknown numbers. 
 
 Thus, 2 X + 3 y = 5 is an equation of the first degree in the two 
 unknown numbers x and y ; 3 x^ — 2 xy = 1 is an equation of the 
 second degree ; 2 x^y + 3 s;^ = 5 is an equation of the third degree. 
 
 152. Linear equation. An integral equation of the first 
 degree in any number of unknowns is called a linear equa- 
 tion, or a simple equation. 
 
 Thus, 2 a: = 6 and 2 x -\- 3 y = 12 are linear equations. 
 
 153. Solution of an equation. A solution of an equation 
 
 in more than one unknown is afi^ set of values of the un- 
 known numbers which satisfies the equation. 
 
 Thus, the linear equation 2 x -\- 3 y = 5 has among other solutions 
 the following sets : 
 
 x = l, y=l; x = 0, y = ^; x = ^, y = 0. 
 
 154. Number of solutions of an equation. The number 
 of solutions of an equation in more than one unknown is 
 not determinate. 
 
 205 
 
206 ELEMENTARY ALGEBRA 
 
 Thus, in the equation 2 r + 3 ^ = 5, we may assign any value to x 
 and solve the resulting linear equation for y ; for example : 
 
 Let . a: = 7 ; 
 
 then, 14 + 3 3/ = 5, 
 
 whence, y = — 3. 
 
 A solution of the equation 2 a: + 3 y = 5 is, therefore, a: = 7, y = — 3. 
 Remark. It is because the number of solutions of an equation in 
 more than one unknown is not determinate that such an equation is 
 called an indeterminate equation. 
 
 ILLUSTRATIVE EXAMPLES 
 
 Find four solutions of the indeterminate equation 
 2a;-|-3y=10, and check each solution found. 
 
 Solution. 2 X 4- 3 3/ = 10. (1) 
 
 Solving (1) for y in terms of x, 
 
 y = '^^- (2) 
 
 Assigning any four values to x, as 0, 1, - 1, 2, we have, 
 10 - 2 X 10 
 
 when a; = 0, 
 when a; = 1, 
 
 3 3 ' 
 
 10 - 2 X 1 8 
 
 3 3' 
 
 whenar = -l, ^ ^ 10 - 2(- 1) ^ ^. 
 
 3 
 
 when a: = 2, V = ^^ ~ ^ ^ ^ = 2. 
 
 ' ^ 3 
 
 Four solutions of 2 x + 3 y = 10 are, therefore, (0, ^2-), (1, |), 
 ( — 1, 4), (2, 2), in each of which the value of x stands first. 
 Check. 2 a; + 3 y = 10. 
 
 Substituting for x and ^^- for y in the given equation, 
 2x + 3(J^)=10. 
 
 Substituting 1 for x and J for y, 
 
 2 X 1 + 3 X I = 10. 
 
SYSTEMS OF LINEAR EQUATIONS 207 
 
 Substituting — 1 for x and 4 for y, 
 
 2(-l)+3 x4 = 10. 
 Substituting 2 for x and 2 for y, 
 
 2x2 + 3x2 = 10. 
 
 EXERCISE 76 
 
 Find four solutions of each of the following inde- 
 terminate equations and check each solution found. 
 1. y — 2x. • 2. x-\-y — 4:. 
 
 Z. x-\-1y = l. 4. 2x — Sy = 5, 
 
 5. 3 2: + 3«/ = 5. €. y = x + l. 
 
 7. y = mx. B. y = mx + h. 
 
 155. Independent equations. Two linear equations in 
 two or more unknowns are called independent equations if 
 each has solutions which are not solutions of the other. 
 
 Thus, X -\- y = ^ and j- — y = 2 are two independent linear equa- 
 tions, as a- = 8 and y = is a solution of the first equation which is 
 not a solution of the second ; and x = 2 and y = is a solution of 
 the second equation which is not a solution of the first. 
 
 156. Dependent equations. Two linear equations in two 
 or more unknowns are called dependent equations when 
 every solution of the one is also a solution of the other. 
 
 Thus, X -{- y = \ and 2 ar + 2 y = 2 are two dependent linear 
 equations. 
 
 157. Inconsistent equations. Two equations which have 
 no common solution are called inconsistent equations. 
 
 Thus, X -\- y = 2 and ar + y = 1 are evidently inconsistent, since the 
 sum of two numbers cannot be both 2 and 1 at the same time. 
 
 158. Simultaneous equations. Two or more equations 
 in more than one unknown number, when considered 
 together, are said to be simultaneous if they have at least 
 one solution in common. 
 
208 ELEMENTARY ALGEBRA 
 
 159. System of equations. Equations considered to- 
 gether are called a system of equations. 
 
 160. Number of solutions of two independent linear 
 equations in two unknown numbers. Two independent 
 and consistent linear equations in two unknown numbers 
 are satisfied by one and only one set of values of the 
 unknown numbers. Such a system of linear equations 
 has, therefore, only one solution. - 
 
 Note. The student in his later algebraic work will find that the 
 fact stated in section 160 covers a particular case which is included in 
 a more general statement. He can, however, satisfy himself that 
 two independent linear equations in two unknowns have no more 
 than one solution in common. For instance, to show that x -\- y = S 
 and X — y = 4: have one and only one solution, we may write 
 
 x + y = S, (1) 
 
 x-y = i; (2) 
 
 then by adding (1) and (2) we have 2 a: = 12. Therefore, any value of 
 X which satisfies both equations must be such that when multiplied 
 by 2 the result is 12, and there is but one such number. Moreover, by 
 subtracting (2) from (1), we have 2 y = i, and there is but one value of 
 y. It will be made evident in the course of this chapter that a similar 
 proof based on the theory of equivalent systems of equations holds 
 for the general equations of the form ax + by = c and mx + ny = p. 
 
 Remark. Two systems of equations in two or more unknown 
 numbers are said to be equivalent when every solution of the first 
 system is a solution of the second and every solution of the second 
 is a solution of the first. 
 
 161. Method of solution. Two independent linear equa- 
 tions in two unknowns are solved by combining these 
 equations in such a way that there results a simple equa- 
 tion in one unknown. One of the unknowns does not 
 appear in the resulting equation ; it has been eliminated. 
 The process by which the unknown is eliminated is called 
 elimination. 
 
SYSTEMS OF LINEAR EQUATIONS 209 
 
 162. Elimination by addition or subtraction. The solu- 
 tion of a system of two linear equations by employing the 
 method of elimination by addition or subtraction is ex- 
 plained in the following : 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Solve the system |!^ + ^^ = f' « 
 
 Solution. Multiplying (1) by 2 and (2) by 3 so that the co- 
 efficients of y in the two equations shall have the same absolute 
 value, 
 
 4 a: + 6 3/ = 10. • (3) 
 
 9x-6y = 3. (4) 
 
 Adding (3) and (4), 13 a: = 13. (5) 
 
 Dividing by 13, a: = 1. (6) 
 
 Substituting the value of x in (1), 
 
 2x1 + 33/ = 5. (7) 
 
 Transposing and uniting terms, 
 
 33^ = 3. (8) 
 
 Dividing by 3, 2/ = 1. (9) 
 
 Therefore, the solution of equations (1) and (2) is a: = 1, y = 1. 
 Check. Substituting 1 for x and 1 for y in equations (1) and (2), 
 we have, 
 
 2x1 + 3x1 = 5, 
 3x1-2x1 = 1. 
 
 Remark. In the solution of example 1, equations (1) and (2) 
 were replaced by the equivalent system (3) and (4) ; that is, the solu- 
 tion of (1) and (2) is the same as the solution of (3) and (4). The 
 actual operations of addition, subtraction, multiplication, and division, 
 here as elsewhere in solving equations, depend on the principles'stated 
 in section 25. 
 
 Solve the system 
 
 3a7+-2y _ Zx-ly ^ ^ _ 2^- 21 x 
 
 12 
 
 (1) 
 
 bx-Zy 52;+3y ^^ x-\-y .^^ 
 
 2 3 6 ' ^ ^ 
 
210 ELEMENTARY ALGEBRA 
 
 Solution. Clearing equations (1) and (2) of fractions, trans- 
 posing, and uniting terms, we have the equivalent system, 
 
 81 ar - 74 3^ = - 60. (3) 
 
 Qx- y = 9. (4) 
 
 Multiplying both members of (4) by 74, so that the coefficient of p 
 
 in the resulting equation shall be equal to the coefficient of y in (3), 
 
 444 ar - 74 y = 666. (5) 
 
 Subtracting each member of (3) from the corresponding member 
 
 of (5), 
 
 363 X = 726. (6) 
 
 Dividing by 363, x = 2. (7) 
 
 Substituting the value of x in (4), 
 
 12 - 2/ = 9. (8) 
 
 Solving (8) for y, y = S. (9) 
 
 Therefore, the solution of equations (1) and (2) is x = 2, y = 3. 
 
 Check. Substituting 2 for x and 3 for y in equations (1) and (2), 
 we have 
 
 3x2 + 2x3 3x2-2x3_i 2 x 3 - 21 x 2 ^^m 
 • 3 5 " 12 ' A^"^ 
 
 5x2-3x3 5x2 + 3x3 ^ g 2 + 3 ,, , . 
 
 2 3 6 ' ^ ^ 
 
 Simplifying (10), 4 = 4. (12) 
 
 Simplifying (11), ^^ = ^K (13) 
 
 ^ , 1 , \ax-\-by=:m^ (1) 
 
 3. Solve the system { , ;^. 
 
 •^ \bx — ay — n. (2) 
 
 Solution. Multiplying both members of (1) by a and both mem- 
 bers of (2) by b in order that in the two resulting equations the coeffi- 
 cients of y may have the same absolute value, 
 
 a^x + ahy = am. (3) 
 
 6% — ahy = hn. (4) 
 
 Adding (3) and (4), (a^ + h^)x = am -{■ hn, (5) 
 
 Dividing by a2 + 62, :, = ^_±^. (6) 
 
SYSTEMS OF LINEAR EQUATIONS 211 
 
 Substituting in (1) the value of x as found in (6), 
 
 ahn + abn , , . ^r,^ 
 
 ~^^-^ + by = m. (7) 
 
 Clearing (7) of fractions, 
 
 a^in -t- abn + (a^ + b^)bt/ = wm^ -f inb^. (8) 
 
 Transposing and collecting, 
 
 (a2 + b^)bt/ = mb^ - abn. (9) 
 
 Dividing both members of (9) by 6, 
 
 (o2 + b^)y = mb- an. (10) 
 
 Dividing (10) by a^ + b\ 
 
 mb — an x- - . 
 
 Therefore, the solution of equations (1) and (2) is 
 
 nm + bn mb — an 
 x= ■ , u = • 
 
 Remark. The value of i/ might also have been found as the value 
 of X was found ; namely, by making the coefficients of -r alike, com- 
 bining the resulting equations so as to eliminate x, and solving the 
 resulting equation for i/. 
 
 From the solutions of illustrative examples 1, 2, and 3 
 we may infer the following : 
 
 Rule. To eliminate an unknown number, as «/, from two 
 simultaneous linear equations hy addition or subtraction, mid- 
 tiply, if necessary, the members of one or both equations by 
 such a rmmber or such numbers as will make the absolute 
 values of the coefficients of y in the two equations the same ; 
 then add or subtract the corresponding members of the two 
 resulting equations according as the coefficients of y in the 
 equations have opposite signs or the same sign. 
 
212 
 
 ELEMENTARY ALGEBRA 
 
 1. 
 
 3. 
 
 7. 
 
 9. 
 
 11. 
 
 13. 
 
 15. 
 
 17. 
 
 19. 
 
 21- 1 - 13 a; + 11 «/ = 2. 
 
 23. 
 
 Sx , 2y . 
 
 X 1 
 
 ^ 2 
 
 2. 
 
 EXERCISE 
 
 Solve the following systems 
 the solution of each : 
 
 x-\- 1/ = 6, 
 2^=3. 
 
 f2:r+3y = 7, 
 \3x-S^=:U, 
 
 |3^ + 2y = 0. 
 ll2x + 4i/ = S, 
 
 Sx-^S^ = 1. 
 
 17«/ + 22 = 19, 
 
 15^-7;2=8. 
 
 3a; + 5?/ = 9, 
 
 2a; + 5?/=5. 
 
 4 a; -3?/ = 16, 
 
 14a;4-5^ = 25. 
 
 4m4-3jt? = 17, 
 
 5m — 4:p = 2, 
 
 2 w - 3 V = 4, 
 
 5v-3i^=-9. 
 
 ll:r + 12y = l, 
 
 17rr-l-19y = 2. 
 
 76 
 
 of equations, and check 
 
 8. 
 
 10. 
 
 12. 
 
 14. 
 
 16. 
 
 18. 
 
 20. 
 
 22. 
 
 24. 
 
 a; + ^ = 10, 
 x — ^ = 4. 
 5a:4-73/=17, 
 5a;+ 3^ = 13. 
 6a;4-3i/ = 4, 
 82^-9^=1. 
 3 a: - 5 i/ = 9; 
 y— 4^: = 5. 
 lla;-12 2/ = 13, 
 6y-13a; = l. 
 17ic + 15?/ = l, 
 8 a; + 6^ = 1. 
 3m + 2^1= 7, 
 4m — 57i= 6. 
 5?^;-3^ = 2, 
 7f-15i^ = 2. 
 6w + 5z^=-8, 
 4w+ 25«^ = — 1. 
 13a;4-17i/ = 4, 
 31 a;-f 37^ = 6. 
 
 2 3 2' 
 
 if + ^ = 2. 
 2 6 
 
 5^ 
 3 
 
 3^^19 
 5 30' 
 
 256^^19 
 7 5 35 
 
SYSTEMS OF LINEAR EQUATIONS 
 
 213 
 
 25. 
 
 27. 
 
 29. 
 
 31. 
 
 33. 
 
 35. 
 
 37. 
 
 39. 
 
 40. 
 
 41. 
 
 3m^5n^2-0, 
 
 26. 
 
 ^" 3=2' 
 
 6m + 57i-l = 0. 
 
 
 6i^+5^; + l = 0. 
 
 I + I+B.O, 
 
 28. 
 
 ^-^ = 3, 
 14 15 ' 
 
 ^-1 = 1. 
 10 9 
 
 Zx by^^ 
 
 2 17 
 9 a: Yly o 
 
 2 5 
 
 30. 
 
 Sx 2y_. 
 17 13" ' 
 
 5:.-6^/ = 7. 
 
 X — y = n. 
 
 32. 
 
 y = mx-\-e, 
 y = lx-hd. 
 
 x-\-ay=b. 
 
 34 < 
 
 - + !=!' 
 
 ax-\- y = c. 
 
 «3V* 1 
 
 6 a 
 
 - + ! = !' 
 ax—hy = c. 
 
 36. 
 
 ax -y = h 
 
 [Sx-\-y=e. 
 
 ^+f=i, • 
 
 a 
 x-y=\. 
 
 38. i 
 
 y-l = m(x-l), 
 m^y -h ma; + w^ H- 1 = 
 
 x-\-2a y-'2a_o 
 a+1 a—1 
 
 a, 
 
 
 x-y-4:a_ x-\-y 
 
 
 
 2a a^-{-l 
 
 
 H-y . ^— y _lQ _ a:+i/ + a x — y — h ^ 
 a 6 3 a 5 
 
 |a: + 2/=a + 6, 
 \ax—hy = a^— ft^. 
 
 42. 
 
 I aa; + by-\-c = 0, 
 joa: + ^-y + r = 0. 
 
214 ELEMENTARY ALGEBRA 
 
 \ ah; 4- h^i/ = 5 be. 
 
 44. 
 
 45. 
 
 46. 
 
 lb a \a bj 
 
 1 (a: + «/)(a2 + ^2)= ab(x + «/) -h 2(a8 + 63). 
 
 j aa; — 5^ = ^2 _ 2 a6 — 62^ 
 
 \ax -h bi/ = a^ -\- If^. 
 ax+bt/ = a-^ b, 
 (a + 5)a^ — (a-b)t/=2b. 
 
 ^^ Ua-b)xA-(a+b 
 \ax ^ by = a^ — 1)^. 
 
 ax by _ a^ — a% -{- ab -{- b^ 
 
 48. la-^-b a-b a^-U^ 
 
 I (« ^. h)x - (^2 + 52)^ = b(a - b). 
 
 163. Elimination by substitution. The solution of a 
 system of two linear equations by employing the method 
 of elimination by substitution is explained in the following : 
 
 Solve the system 
 
 ILLUSTRATIVE EXAMPLE 
 
 3a:+5^ = 21, (1) 
 
 2 a: -h 3 ^ = 13. (2) 
 Solution. Solving equation (1) for the value of x in terms of y, 
 
 . = 2JL^. (8) 
 
 Substituting in (2) the value of x as found in (3), 
 
 42_-10^^3^^^3 ^^^ 
 
 Solving (4), 3, = 3. (5) 
 
 Substituting in (3) the value of y as found in (5), 
 
 21-15 o /«^ 
 
 x= — = 2. (6) 
 
 Therefore, the solution of (1) and (2) is x = 2, y = 3, 
 
SYSTEMS OF LINEAR EQUATIONS 215 
 
 From the solution of the foregoing illustrative example 
 we may infer the following: 
 
 Rule. To eliminate an unknown number^ as x, from two 
 simultaneous linear equations hy substitution, solve one of the 
 equations for x and substitute the resulting value of x in the 
 other equation. 
 
 EXERCISE 77 
 
 Solve the following systems of equations, using the 
 method of elimination by substitution: 
 
 5a: + 3?/ = ll, ^ f3a:-22/ = 4, 
 
 y==2x. ' \Qx + by = ll. 
 
 3x-y==S, 
 
 ix—y=l, 
 
 5x+7y = 41. ^' \Sx-\-2y = 10. 
 l2y=Sx, jx + y = 5, 
 
 \5x-^y=2. *• \5x-ly=l 
 
 ax = by, 
 
 (jOL H- b)x -I- (a — b)y = a-\-b, 
 
 ■^"- ^(a-6H-2> + 3?/ = 4a + 26 + 2. 
 
 ^1 
 
 ax+by= c. 
 ^ ^ax+by = d, 
 mx -\-ny=c. 
 
 (For further practice in solving a system of two linear equations 
 by employing the method of elimination by substitution, one or more 
 of the examples in exercise 76 may be taken.) 
 
 164. Elimination by comparison. The solution of a 
 system of two linear equations by employing the method 
 of elimination by comparison is explained in the following : 
 
216 ELEMENTARY ALGEBRA 
 
 ILLUSTRATIVE EXAMPLE 
 
 Solve the system I ^^ ^ ~ ' ^^^ 
 
 ^ (10a;H-92/ = 10. (2) 
 
 Solution. Solving (1) for x, 
 
 (3) 
 
 4-3w 
 
 Solving (2) for x, x = ^^~^^ ' (4) 
 
 Equating the two values of x as found in (3) and (4), 
 
 4-3.y ^ lO-9y .g. 
 
 5 10 ■ > ^ 
 
 Simplifying (5), 3y = 2. (6) 
 
 Solving (6), 2/ = |. (7) 
 
 Substituting in (3) the value of y as found in (6), 
 
 x = f (8) 
 
 Therefore, the solution of (1) and (2) is x = ^, y = ^. 
 
 From the solution of the foregoing illustrative example 
 we may infer the following : 
 
 Rule. To eliminate hy comparison an unknown number, as 
 x^from two simultaneous linear equations, solve each of the 
 two equations for x and equate the two resulting values. 
 
 EXERCISE 78 
 
 Solve the following systems of equations, using the 
 method of elimination by comparison : 
 
 2y-\-'2>x = b, [3a; + 4y=-l. 
 
 11 x-l^y = 1, j2x-Sy = l, 
 
 Ux-\-14:y = 27. I5a; + 7y = 46. 
 
 5x-^2y=-l, \llx-2y = l, 
 
 Sx-\-5y = ll. \Ux-\-l y=:-65. 
 
 ■I 
 
SYSTEMS OF LINEAR EQUATIONS 217 
 
 1892:4-3^ = 3, jllx-2S^ = 2, 
 
 ^' \lSx-5y = -b. 123 2: + 71^ = 236. 
 
 (For further practice in solving a system of two linear equations 
 by employing the method of elimination by comparison, one or more 
 of the examples in exercise 76 may be taken.) 
 
 Remark. The three methods of elimination considered in sections 
 162, 163, and 164 are manifestly applicable to any system of two 
 simultaneous linear equations. Of these methods, that of elimination 
 by addition or subtraction is most frequently employed. However, 
 when one of the equations gives the value of one of the unknowns, 
 as X, in terms of the other, elimination by substitution may be used 
 to advantage. 
 
 165. Elimination by use of an undetermined multiplier. 
 The solution of a system of two linear equations by use of 
 an undetermined multiplier is explained in the following : 
 
 ILLUSTRATIVE EXAMPLE 
 
 Solve the system 1 ! ^ "^ ? ^ "" !'^' ^1} 
 
 ^ [3a:-5i/ = l. (2) 
 
 Solution. Multiplying (2) by m, 3 mx — 5 my = m. (3) 
 
 Adding (1) and (3), (7 + 3w)a: +(3 - 5m)y = 17 + m. (4) 
 
 Equating the coefficient of y in (4) to 0, 3 — 5 m = 0. (5) 
 
 Solving (5), ^ = I- (6) 
 
 Substituting the value of m in (4), (7 + ■|)a: + • y = 17 + |^. (7) 
 Solving (7), x = 2. (8) 
 
 Substituting the value of x in (1) and solving, y = I- 
 
 Remark. The number m in equation (4) of the foregoing solution 
 is undetermined ; that is, it may have any numerical value assigned 
 to it. We assign such a value to m that the coefficient of one of the 
 unknown numbers shall vanish. It is evident that instead of first 
 eliminating y, the coefficient of x might have been placed equal to 
 zero and the value of y determined. 
 
218 ELEMENTARY ALGEBRA 
 
 EXERCISE 79 
 
 Solve the following systems of equations, eliminating 
 one of the unknowns by use of an undetermined multiplier: 
 
 6m + 5jt? = — 2. 
 
 |4^-5y = 22, f 
 
 l32:+.23/ = 5. ^' \ 
 
 j7w-llv = 26, |a:-h2^ = 2, 
 
 I15w-f 5t; = -30. *• 1 
 
 2a:-3y = 54. 
 
 \(m -f n) -h J(^ - w) = 2, 
 I (m 4- w) + f (w - 7i) = 17. 
 
 15^ + 19y = 18, 
 ^' h9a; + 15^ = 50. 
 
 1*^ 
 
 IK 
 
 ■■! 
 
 10. 
 
 18a:+23?^==13, 
 23 a: + 18 ^ = 28. 
 
 2.5 a: + 3.7 3/ = 7.69, 
 
 3.6 a; -2.9^ = 1.20. 
 
 f. 05 a: +.03?^ =.011, 
 1 .72 a: + .93 2^ = .258. 
 f aa; + 6^ = a^ ^ 52^ 
 \ a^ — 52^ = a^ _ 58^ 
 
 166. Special systems of simultaneous equations. Cer- 
 tain systems of simultaneous fractional equations, which 
 are of frequent occurrence, should be solved by the 
 methods already employed in this chapter. In such 
 systems, the equations are not cleared of fractions, 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Solve the system 
 
 ?+^=i, (1) 
 
 ?+^ = 2. (2) 
 
 X y 
 
SYSTEMS OF LINEAR EQUATIONS 219 
 
 Solution. Writing (1) and (2) in another form, 
 
 Equations (3) and (4) are in the form of two simultaneous linear 
 
 equations in the unknown numbers - and -. Multiplying both 
 
 X y 
 
 members of (3) by 5 and those of (4) by 3, 
 
 Subtracting the members of (6) from the corresponding members 
 of (5), 
 
 - = -1. (7) 
 
 X 
 
 Solving (7) for a:, a: = - 1. (8) 
 
 Substituting in (1) the value of x as found in (8), 
 
 -2+? = l. (9) 
 
 y 
 
 Solving (9) for y, y = l. (10) 
 
 Therefore, the solution of (1) and (2) is ar = — 1, y = 1. 
 Check. Substituting — 1 for x and 1 for y in (1) and (2), we 
 have, respectively, 
 
 -2 + 3 = 1, ai) 
 
 -8 + 5 = 2. (12) 
 
 Note. The system (1) and (2) whose solution has just been 
 given is not equivalent to the system obtained by clearing (1) 
 
 and (2) of fractions ; namely the system j ^ x — xy ]^ ^ rjy^^^ 
 
 [3y + ox = 2xy} 
 new system is not composed of linear equations and it has other 
 solutions than the one obtained from (1) and (2). For example, 
 X = 0, y = is evidently a solution of this new system but is not a 
 solution of the system (1) and (2). When, however, two equations 
 
 of the form I ^^ + ^ — ^^V I are given, one solution of the 
 \mx-\- ny = pxy J 
 
220 
 
 ELEMENTARY ALGEBRA 
 
 system may, in general, be obtained by dividing each member of 
 both equations by xy and proceeding as in the solution of the fore- 
 going illustrative example. 
 
 + -^ = 12, 
 
 2. Solve the system 
 
 x-1 
 5 
 
 ^ + 2 
 3 
 
 1. 
 
 x-1 y+2 
 Solution. Writing (1) and (2) in another form, 
 
 (1) 
 
 (2) 
 
 (8) 
 (4) 
 
 In equations (3) and (4) we may regard the unknown numbers as 
 and „ . Multiplying both members of (3) by 3 and both 
 
 X -I y -\-2 
 
 members of (4) by 2, and adding 
 
 19 
 
 {-^^- 
 
 38. 
 
 Dividing by 19, 
 
 = 2. 
 
 x-1 
 
 Clearing (6) of fractions, 1 = 2(a; - 1). 
 
 Solving (7), ar = f . 
 
 Substituting in (1) the value of x as found in (8), 
 
 6 + 
 
 = 12. 
 
 y + 2 
 
 Solving (9), y = - f • 
 
 Therefore, the solution of (1) and (2) is x = \, y = - 
 Check. Substituting in (1) and (2), 
 
 (5) 
 
 (6) 
 
 (7) 
 (8) 
 
 (») 
 (10) 
 
 -1 -f+2 
 5 3 
 
 = 12, 
 = L 
 
 Simplifying, 
 
 1 -I+.2 
 
 I 6 + 6 = 12, 
 1 10 - 9 = L 
 
SYSTEMS OF LINEAR EQUATIONS 
 
 221 
 
 EXERCISE 80 
 
 Solve the following systems of equations, regarding each 
 as a system of simultaneous linear equations in two un- 
 knowns : 
 
 5. 
 
 3, 
 
 ?4-l 
 X y 
 
 1 + 1 = 2. 
 
 X y 
 
 X y 
 3_2 
 X y 
 
 7 
 
 = 2. 
 
 x—\ y+b 
 
 = 11, 
 
 x-1 y+B 
 
 17 
 
 ■ 2 
 
 2a 5a 3 
 5 a 2 a 17 
 
 
 2. 
 
 4. 
 
 6. 
 
 8. 
 
 '^+^=-1, 
 
 X y 
 
 ?-?=-4. 
 X y 
 
 + 
 
 x+l ?/ + l 
 
 1 
 
 = 5, 
 
 ah 
 x y 
 a-\- h a 
 
 X 
 
 2, 
 h 
 
 = 1. 
 
 a h 
 
 y ~h a 
 
 y 20 
 
 3a 5a^7 
 hx cy 4 ' 
 
 7a_3« 
 6a: c^ 
 
 23 
 
 20 
 
 10. 
 
 11. 
 
 f 2(a + 6) , 3(a-6) _ 3 
 X y 2 
 
 -3(^4-6) 5(a- 6) ^11 
 X y 12" 
 
 5 3a^ a-18 
 6x 5 y 6 
 2 3a^ 2a + 45 
 3a: 5^/ 15 ' 
 
 a _b 
 X y 
 a h 
 
 = m, 
 
 = n. 
 
 y 
 
222 
 
 ELEMENTARY ALGEBRA 
 
 12. 
 
 13. 
 
 14. 
 
 h a_ 
 
 4 ax 4:hy 
 
 2b 
 
 3 ax 
 
 a 
 hy 
 
 a— c 
 x^h 
 
 a + c 
 
 a — h 
 
 = 0, 
 
 x—h 
 1 
 
 y-c 
 a + h ^ 2a(c-b') 
 y—c (a— b){a — c} 
 1 2 
 
 x + b-\- c 
 2a-\-b-\-c 
 
 2b-\- c-\-a a—b 
 
 x+b-\-e y-\-c + a a-\-b + c 
 
 167. Fractional equations. In the case of systems of 
 simultaneous fractional equations which are not included 
 among those considered in section 166, it is usually best 
 to clear the equations of fractions. 
 
 Solve the system 
 
 ILLUSTRATIVE EXAMPLE 
 
 2x-\- 5y 
 x 
 
 4:X 
 
 = 1. 
 
 2 
 
 1^ = 2. 
 
 (1) 
 
 (2) 
 
 x + y 
 
 Solution. Clearing equations (1) and (2) of fractions and com- 
 bining like terms, 
 
 x^by = 2, (3) 
 
 2 ar - 5 y = 0. (4) 
 
 Adding (3) and (4), 3 a: = 2. (5) 
 
 Solving (5), X = 2 . (6) 
 
 Substituting in (3) the value of z as found in (5), 
 
 .V = y = T^- (7) 
 
 Therefore, the solution of (1) and (2) is j: = J, y = ^. These 
 values of x and y are found to satisfy the given equations (1) and (2). 
 
 Remark. Before clearing an equation of fractions, each fraction 
 should be expressed in its lowest terms (see also note, page 193). 
 
SYSTEMS OF LINEAR EQUATIONS 
 
 223 
 
 EXERCISE 81 
 
 Solve the following systems of equations and check the 
 results ; 
 
 1. i 
 
 3. 
 
 7. 
 
 11. 
 
 £±1=3, 
 
 X— 2 
 
 x-hy 
 
 y-2-y-r 
 
 x + 1 
 
 y-1 
 
 = 1. 
 
 2y + 3 ^ 3y + 2 
 2a:4-3 32^ + 2' 
 
 a: + ^ + 2 = 0. 
 
 1 . a;_2^-3a: 
 1 -l — — — , 
 
 y y 
 
 Zx^y=l. 
 
 x-\-2 y-3' 
 
 a b 
 
 a _h 
 y^ x' 
 2a 1 
 
 ax-\-by b 
 
 2. 
 
 4. 
 
 8. 
 
 10. 
 
 12. 
 
 = 3, 
 
 y 
 
 X 
 
 x + 
 
 = 4, 
 
 3y-2 
 
 ^y 2 
 
 2a;-3y ^2 
 3a:-2?/ 3' 
 3a;H-2y _3 
 2x-^y 2* 
 
 a; 4- ^ _ 53 13 a; 4- 5 
 3 2: "39 26 a; " 
 
 a:+l a;+l 
 
 a: -f 2 _ ,y — 6 
 a a— 6' 
 
 x— y = a — b^ 
 
 a H- c_ 6+ c 
 a: ^ 
 
 EXERCISE 82 
 
 1. The sum of two numbers is 15 and one of them is 
 one greater than the other. What are the numbers ? 
 
224 ELEMENTARY ALGEBRA 
 
 Suggestion. Let x = the larger number and y = the smaller. 
 Then, p + y = 15, (1) 
 
 1 ^ = ^ + 1. (2) 
 
 2. The sum of two numbers is 12, and their difference 
 is 6. What are the numbers ? 
 
 3. The sum of two numbers is 27, and five times the 
 first number is equal to four times the second. What are 
 the numbers ? 
 
 4. The difference between two numbers is 5, and the 
 sum of the numbers is twice their difference. Find the 
 numbers. 
 
 5. Twice a certain number is 4 greater than 5 times 
 a second number ; the sum of the two numbers is 80. 
 Find the numbers. 
 
 6. A bushel of corn and a bushel of oats together 
 weigh 88 lb., and the weight of a bushel of corn is 24 lb. 
 greater than the weight of a bushel of oats. What is the 
 weight of a bushel of each ? 
 
 7. The weight of 3 bu. of bran is equal to the weight 
 of 1 bu. of wheat, and the weight of 1 bu. of wheat 
 exceeds the weight of 1 bu. of bran by 40 lb. What is 
 the weight of a bushel of each ? 
 
 8. The sum of two numbers is 18, and 4 times the 
 larger is equal to 5 times the smaller. What are the 
 numbers ? 
 
 9. Three times a certain number is 7 greater than 
 four times the sum of 8 and a second number ; the sum 
 of three times the first number and four times the second 
 is 63. Find the numbers. 
 
 10. One half of one number is equal to two thirds of 
 a second; the sum of the first number and twice the 
 second is 20. What are the numbers ? 
 
SYSTEMS OF LINEAR EQUATIONS 225 
 
 11. A classroom has 54 desks, some of which are single 
 and some double ; the seating capacity of the room is 72. 
 How many desks of each kind are there ? 
 
 12. Two opposite numbers which differ by 8 have the 
 same absolute values. What are the numbers ? 
 
 13. 2 lb. of coffee and 6 lb. of sugar cost 11.18; 51b. 
 of coffee and 3 lb. of sugar cost f 1.99. Find the cost of 
 a pound of each. 
 
 14. If 12 gallons of milk will just fill either 152 bottles 
 and 5 jars, or 32 bottles and 20 jars, what are the separate 
 capacities of a bottle and a jar ? 
 
 15. A dealer bought 30 bu. of wheat and 10 bu. of rye 
 for $46. He also bought at the same time 50 bu. of wheat 
 and 30 bu. of rye for $87. Find the price of each per 
 bushel. 
 
 16. The sum of two numbers is equal to 5,5 diminished 
 by the second number ; three times the first number dimin- 
 ished by twice the second number is — 1.1. What are 
 the numbers? 
 
 17. " Give me five of your marbles," said a boy to his 
 brother, "and I shall have twice as many as you." His 
 brother replied, ^ Give me five of your marbles and then 
 I shall have as many as you." How many marbles had 
 each ? 
 
 18. Three years ago a boy was twice as old as his sister, 
 and fifteen years hence | of his age will equal J of his sis- 
 ter's age. How old is each? 
 
 19. A bill amounting to $8.70 was paid with 60 coins, 
 some of which were dimes and the rest quarters; how 
 many of each were there ? 
 
226 ELEMENTARY ALGEBRA 
 
 20. Divide $10 between A and B, so that the number 
 of half-dollars in A's share may be ten less than the num- 
 ber of quarter-dollars in B's share. 
 
 21. A certain number is equal to seven times the sum 
 of its two digits, and the left-hand digit exceeds the right- 
 hand digit by 2. Find the number. 
 
 Suggestion. Let x = the tens* digit, 
 and y = the units' digit. 
 
 Then, \^ x ■\- y = the number. 
 
 Whence, f 10 x + ^^ = 7(:r + y), (1) 
 
 \ x-y^% (2) 
 
 22. The length of a room is 25 % greater than the width, 
 and the perimeter is 35 ft. Find the dimensions. 
 
 23. A merchant has tea worth 50 cents per pound and 
 also tea worth 70 cents per pound ; how many pounds 
 of each must he use to make a mixture of 25 pounds worth 
 62 cents per pound ? 
 
 24. The cost of sending a day telegram of 17 words 
 from Philadelphia to Richmond, Indiana, is 71 cents, and 
 the cost of sending one of 23 words is 89 cents. What is 
 the rate for the first ten words in such a message and for 
 each additional word? 
 
 25. Five first-class fitters and 7 plain sewers earn $160 
 a week; 7 first-class fitters and 2 plain sewers earn il85 
 a week. Find the weekly wages of a first-class fitter and 
 those of a plain sewer. 
 
 26. The sum of two digits of a certain n.umber is 10 
 and if 18 be added to the number, its digits will change 
 places. Required the number. 
 
 Suggestion. Let x — the tens' digit, 
 and y = the units' digit. 
 
 Then, 10 a; + y = the number, 
 
 and 10 y + a: = the number with the digits interchanged. 
 
SYSTEMS OF LINEAR EQUATIONS 227 
 
 27. One digit is one greater than twice a second digit ; 
 the difference between the numbers which can be repre- 
 sented by the two digits is 45. P'ind the digits. 
 
 28. A man rode a certain distance, at a uniform rate, 
 in 7 hr. If the distance had been 4 miles less and his rate 
 per hour 1 mile more, the time required would have been 
 6 hr. Find the distance and his rate. 
 
 29. One man and three boys can do a piece of work in 
 2f working days of 10 hours each ; two men and one boy 
 could do it in the same time. How many hours would 
 one man alone require to do the work? 
 
 Suggestion. 
 
 Let X = the number of hours in which a man can do the work, 
 
 and y = the number of hours in which a boy can do the work. 
 
 Then, - = the part of the work the man does in 1 hr., 
 
 X 
 
 and - = the part of the work a boy does in 1 hr. 
 
 y 
 
 Whence, 
 
 l + ? = i- (1) 
 
 X y 24 ^^ 
 
 ? + i = i-. (2) 
 
 X y 2^ • ^ ^ ^ 
 
 30. One man and two boys can do a piece of work in 
 9 days; two men and five boys could do it in 4 days. 
 How long would one boy alone take to do the work ? 
 
 31. If 1 is added to the numerator of a fraction, the 
 value of the fraction becomes |- ; if 1 is added to the de- 
 nominator of the same fraction, the value becomes J. What 
 i^ the fraction ? 
 
 Suggestion. Let - = the fraction. 
 
228 ELEMENTARY ALGEBRA 
 
 32. If 1 be added to both terms of a fraction the re- 
 sulting fraction will be |, but if 1 be subtracted from both 
 terms, the resulting fraction will be ^, What is the 
 fraction ? 
 
 33. Separate 53 into two parts such that the greater 
 part divided by the less shall give both a quotient and a 
 remainder of 2. 
 
 34. A owes 1250 and B owes $375. A could pay all 
 his debts if in addition to his own money he had ^ of B's ; 
 and B could pay all of his debts and have $25 left if in 
 addition to his own money he had | of A's. How much 
 money has each? 
 
 35. The base of a rectangle is 10% greater than the 
 altitude, and the perimeter is 126 ft. Find the dimensions. 
 
 36. A part of $3000 is invested at 5|^% and the remain- 
 der at 4^%. The yearly income from the investments is 
 $147.25. Find the amount in each investment. 
 
 37. In a certain family each son has twice as many sis- 
 ters as brothers but each daughter has as many brothers 
 as sisters. How many children are in the family? 
 
 38. In a certain family each daughter has as many 
 brothers as sisters, but each son has three times as many 
 sisters as brothers. How many children are in the family? 
 
 39. A man has $7000 which he wishes to invest in two 
 enterprises so that his total income Avill be $ 330 ; if one 
 enterprise pays 5 % and the other 4| %, how much must he 
 invest in each? 
 
 40. A certain principal will amount to $260 if loaned 
 at simple interest for 5 yr. , and to $ 240 if loaned at the 
 same rate for 4 yr. Required the principal and the rate. 
 
SYSTEMS OP LINEAH EQUATIONS 229 
 
 41. A certain principal in a given time will amount to 
 $744 if loaned at simple interest at 6%, and to $708 if 
 loaned for the same time at 4| %. Required the principal 
 and the time. 
 
 42. In an athletic meet the winning team scored 42 
 points and the second team 35 points. The winning team 
 took first place in 6 events and second place in 4 ; the sec- 
 ond team took 4 first and 5 second places. How many 
 points does a first place count and how many does a second 
 place count? 
 
 168. Simultaneous linear equations in three unknown 
 numbers. Three consistent linear equations in three 
 unknown numbers have one and only one solution when- 
 ever by elimination two independent and consistent linear 
 equations in two unknowns can be derived from them. 
 
 ILLUSTRATIVE, EXAMPLES 
 
 2; 4-2^/4-3 2 = 4, (1) 
 
 2^ + 3^/4-42 = 7, (2) 
 
 [^x--ii/-5z = S. (3) 
 
 Solution. Multiplying (1) by 2, 2 x + 4 ?/ + 6 z = 8. (4) 
 
 Subtracting (2) from (4), y -h2z = l. (5) 
 
 Multiplying (1) by 3, 3 a; + 6 2^ + 9 2 = 12. (6) 
 
 Subtracting (3) from (6), 10y-hUz = i. (7) 
 
 Dividing (7) by 2, 5y-{-7z = 2. (8) 
 
 Equations (5) and (8) are two independent equations in two 
 
 unknowns and are solved by methods previously explained ; thus : 
 Multiplying (5) by 5, 5 3/ + 10 z = 5. (9) 
 
 Subtracting (8) from (9), ' 3 z = 3. (10) 
 
 Solving (10), z = l. (11) 
 
 Substituting in (5) the value of 2 as found in (11) and solving 
 
 resulting equation for y, ?/ = — !• (12) 
 
 Substituting in (1) the value of y from (12) and the value of z 
 
 from (11), and solving for x, x = 3. (13) 
 
 Therefore, the solution of the given system is x=3, y= —1, z = l. 
 
 Solve the system 
 
230 ELEMENTARY ALGEBRA 
 
 Check. Substituting 3 for x, — 1 for //, and 1 for z in equations 
 (1), (2), and (3), we have respectively, 
 
 r 3 - 2 + 3 = 4. 
 
 (14) 
 
 . 6 - 3 + 4 := 7. 
 
 (15) 
 
 9 4. 4 _ 5 = 8. 
 
 (16) 
 
 ?-? + l = i, 
 
 (1) 
 
 X y z 
 
 
 4,23 2 
 
 r - 5 
 
 (2) 
 
 X y z 3 
 
 
 2 5 2_1^ 
 
 (3) 
 
 2. Solve the system 
 
 Solution. Regard equations (1), (2), and (3) as linear in the 
 
 three unknowns -,-,-• 
 X y z 
 
 Multiplying (1) by 3, §_? + ?= 3. (4) 
 
 X y z 
 
 Adding (2) to (4), --- = ^ (5) 
 
 X y Z 
 
 Multiplying (1) by 2, ? _ i + ? = 2. (6) 
 
 X y z 
 
 Subtracting (3) from (6), ' ^ + 1 = 11. (7) 
 
 X y Q 
 
 Multiplying (7) by 4, 16_^4^^ ^g^ 
 
 X y 6 
 
 Adding (8) to (5), ??=f. (9) 
 
 X 6 
 
 Solving (9), x = 3. (10) 
 
 Substituting in (7) the value of x as found in (10), 
 
 .. h]-l ("> 
 
 Solving (11), y = 2. (12) 
 
 Substituting in (1) the value of x from (10) and the value of y 
 
 from (12), and simplifying, - = 1. (13) 
 
 z 
 
 Solving (13), z = \. (14) 
 
 Therefore, the solution of the given system is a; = 3, 3/ = 2, * = 1. 
 
SYSTEMS OF LINEAR EQUATIONS 
 
 231 
 
 -l + i 
 
 
 1 = - 
 1 
 
 Check. Substituting 3 for ar, 2 for y, and 1 for 2, in equations (1), 
 (2), and (3), we have, respectively, 
 
 - - ' 1, (15) 
 
 (16) 
 6- (17) 
 
 From the foregoing illustrative examples we may infer 
 the following : 
 
 Rule. To solve three linear equations in three unknown num- 
 bers^ eliminate any one of the unknowns^ as x^from any pair 
 of the equations^ and then eliminate the same unknown from 
 another pair ; solve the resulting two linear equations in two 
 unknowns for these unknowns^ substitute the values of the two 
 unknowns in one of the given equations^ and solve for the third 
 unknown number. 
 
 BXEBCISE 83 
 
 Solve the following systems of equations, and check the 
 results : 
 
 1. 
 
 3. 
 
 7. 
 
 2a?-3 «/-2=12, 2. 
 
 3a: + ^+ 2z = 5. 
 Sx+2y-lz=-14:, 
 Sx-2y + 5z = m, 4. 
 x-{-1 y-2z=-29. 
 llx+2y-{-Sz = 24:, 
 5a;H-3^-42=-18, 6. 
 2x-5y + l z = ^2. 
 5x-Sy-\-2z = U, 
 4a; + 4^-3 2 = 57, 8. 
 Sx-h2y-\-5z = 16. 
 x-\-y-\-z = 2, 
 2x-Sy-{-llz = S, 10. 
 [Sx-^ly-2z=5, 
 
 [2x-3^ + 5z = 15, 
 
 3a;+2y-4z=-7, 
 
 x-\- y-^z=:2, 
 \5x-\-2y-\-Sz=4:, 
 
 Sx-Sy-\-4z=-19, 
 
 2x-\-3y-l z = 41. 
 
 Sx-^2y+Sz = S, 
 
 22:4-3^ + 22 = 27, 
 
 1 x-5y-5z = 91. 
 
 5x-2y = S, 
 
 3a;4-2z = 5, 
 
 By-Sz=2, 
 
 x-\- y=0, 
 
 ?/ + 2 = -l, 
 
 z-^x = l. 
 
232 
 
 ELEMENTARY ALGEBRA 
 
 11. 
 
 13. 
 
 15. 
 
 [1 . 1 
 
 1_ 
 
 
 - + -- 
 
 
 6, 
 
 X y 
 
 z 
 
 
 1 1 
 
 \ _ 
 
 
 --- + 
 
 
 -2, 
 
 X y 
 
 z 
 
 
 1 1 
 
 1_ 
 
 
 -4-- + 
 
 
 0. 
 
 .^ y 
 
 z 
 
 
 W-= 
 
 5, 
 
 
 X y 
 
 
 
 1 1 
 
 
 
 -+- = 
 
 6, 
 
 
 y z 
 
 
 
 1+1= 
 
 7. 
 
 
 Z X 
 
 
 
 12. 
 
 14. 
 
 11 
 
 6"' 
 
 f2_^3_l 
 X y z 
 3 1 ,2_7 
 X y z Q 
 
 X y z S 
 
 1.1 o 
 
 - + - = 2 a, 
 
 1 + 1 = 25, 
 
 1^1 O 
 
 - + - = 2 (?. 
 
 2 X 
 
 { ax -\- hy + cz = a^ -\- h'^ -\- c^^ 
 
 I (5 + c)x + (<? 4- a)^/ + (a + 6)2 = 2 6c + 2 m 4- 2 ah, 
 
 \ hex H- m?/ + ahz = 3 a5<?. 
 
 EXERCISE 84 
 
 1. I paid 91 ct. for 2 lb. of sugar, 1 lb. of coffee, and 3 
 lb. of lard. If I had bought 3 lb. of sugar, 1 lb. of coffee, 
 and 21b. of lard, my bill would have been 84 ct.; but if 
 I had bought 1 lb. of sugar, 21b. of coffee, and 1 lb. of 
 lard, my bill would ha-ve been 83 ct. What did I pay 
 for a pound of each ? 
 
 2. For 16 1 can buy 3 lb. of tea, 8 lb. of coffee, and 
 30 lb. of sugar ; or 4 lb. of tea, 7 lb. of coffee, and 25 lb. 
 of sugar ; or 8 lb. of tea, 1 lb. of coffee, and 15 lb. of sugar. 
 What are the prices ? 
 
 3. There are three numbers whose sum is 162 ; the 
 second exceeds the first as much as the third exceeds the 
 second ; 18 times the first equals 5 times the third. 
 What are the numbers ? 
 
SYSTEMS OF LINEAR EQUATIONS 233 
 
 4. A dealer shipped 100 doz. of eggs on Monday and 
 Tuesday, 110 doz. on Tuesday and Wednesday, and 90 
 doz. on Monday and Wednesday. How many dozen did 
 he ship each day ? 
 
 5. A man has a triangular lot which he desires to 
 fence. 100 rods of fencing are required for the sides AC 
 and BC, 111 rods for the sides AB and BC, and 90 rods 
 for the sides AB and AC. Find the number of rods of 
 fencing required for each side. 
 
 6. The sides of a certain triangle are denoted by a, 6, 
 and c. What is the length of each side of the triangle if 
 the sum of the sides is 42, the sum of a and h is 27, and 
 the sum of h and c is 29 ? 
 
 7. The sum of the three angles of any plane triangle 
 is 180°. If these angles are denoted by A^ B, and (7, and 
 if O exceeds A by 50° and A exceeds B by 10°, find the 
 size of each angle. 
 
 8. The sum of three numbers is 24. The quotient of 
 the first divided by the second is | and the quotient of the 
 second divided by the third is |. Find the numbers. 
 
 9. The middle digit of a given three-digit number is 
 equal to the sum of the two remaining digits ; a second 
 number which is 594 less than the given number is 
 expressed by the same digits written in the reverse order ; 
 if 19 be added to the original number and 13 be added to 
 the second, one of the resulting numbers will be four 
 times the other. Find the original number. 
 
 10. I made three shipments of goods from Philadelphia 
 to Chicago. The first consisted of 300 lb. first-class 
 freight, 200 lb. second-class freight, and 200 lb. third- 
 class freight. My freight bill was $4.65. The second 
 shipment consisted of 1000 lb. first-class freight, 500 lb. 
 
234 ELEMENTARY ALGEBRA 
 
 of second-class, and 100 lb. of third-class freight. My 
 freight bill was 111.51. The third shipment consisted of 
 700 lb. of first-class freight, 800 lb. of second-class freight, 
 and 400 lb. of third. My freight bill was $12.71. Find 
 the rate of shipping 100 lb. of freight of each class from 
 Philadelphia to Chicago. 
 
 11. A and B together can do a piece of work in 20 
 days. After they have worked 12 days on it, they are 
 joined by C, who works twice as fast as B. The three 
 finish the work in 4 days. How long would it take each 
 man alone to do it ? 
 
 12. A number is composed of three digits, whose sum 
 is 12 ; the digit in the hundreds' place is one greater 
 than that in the tens' place ; if ten times the units' digit is 
 subtracted from the number the remainder is 257. Find 
 the number. 
 
 EXERCISE 85. — GENERAL REVIEW 
 
 (Solve as many as possible at sight.) 
 
 1. Adda;+ 2^-1-3 2, 1x—y — ^z^ y — x — z^ and z — x—y. 
 
 2. Add 4 m^ — 3 mn— 2 mhi — n^ -\- ^ mn^^ 2 mhi — 4 mn^ 
 
 — m^-\-Sn^^ 4 mh^ — 3 m^ -|- 4 mn — m'n?^ and 5 mn^ —2n^ 
 
 — 2 m^n. 
 
 3. What must be added to x-\- y -\- z that the sum may 
 be c — zl 
 
 4. What must be subtracted from x'^ — x-\-l that the 
 difference may be a^ — 1 ? 
 
 5. Subtract 2(r — «) -h 1 from the sum of 7(r — «) and 
 -4(r-«). 
 
 6. Simplify 
 2-(-2)-[-(-2)]-[-!-(-2)i-2]. 
 
SYSTEMS OF LINEAR EQUATIONS 235 
 
 7. Express m — n — p—q + r—s — t — u — x in trino- 
 mial terms having the last two terms of each inclosed in 
 parentheses. 
 
 If A = E^-Rr + r\ ^ = 2i2H- r-f- 1, and C=4R^ 
 — 2 Br — 2, find the value of : 
 
 8. A + B-^a 9. A-B+O. 10. A-B-C. 
 
 11. What is the product of mVp and 2 m^pq ? 
 
 12. What is the product of a:"-i«/"+%" and xy^~^z ? 
 
 13. What is the product of 2(a: + ^), Z(x-\-y)\ and 
 
 14. Divide 6 m^n^p^r by 2 m^n^pr. . 
 
 15. Divide 6a:;"+V^""^^ by 3 2:"y»-V+2. 
 
 16. Divide 6(m + 7i)2 — 4(m-|-^)^+.(w + 7i) by2(m-hw). 
 
 17. Multiply a:2 — 2 a;^ 4- «/^ -f ^2 by a;2 + 2 a:?/ + ^2 _ ^2. 
 
 18. Multiply a:*" + ^p — 2 2" by 2 a;"» — 3 «^. 
 
 19. Multiply f :r2 + f a^y 4- i ^2 by I a;2_ 1:^:^4- 1^2. 
 
 20. Multiply 3"» — 4" by 4^ -h 3". 
 
 21. Divide 
 
 24 mVpif — 36 m^n^pT^ + 48 mn^r^x by — 6 ww2. 
 
 22. Divide Qi?-^y^-\-^—^xyzh^x^^-y^-\-z^ — yz — xz — xy, 
 
 23. Divide 1 - 547 :r« + 546 a:^ by 1 + 2 a: - 3 2^2. 
 
 24. Divide 3-5:^-487:^5-1-489 2:6 by l-4a; + 32^ 
 and find the value of the quotient when rr = — 1. 
 
 1 7^ 
 
 25. Show by division that -—■ — = \-\-x-\'Q^-{-^-\- . 
 
 \—x Y—x 
 
 1 x^ 
 
 26. Show by division that = 1 — rr-ha;2— a^S-f- 
 
 \-\- X 1 — x 
 
 27. When the divisor is m^ + n^ and the quotient is 
 m^ — mV H- w%* — mV H- w^, what is the dividend ? 
 
236 ELEMENTARY ALGEBRA 
 
 28. When the divisor is m — n^ the quotient m^ + rn^n 
 _j_ y^2^2 ^ ^^3 ^ ^4^ an(j ^]^g remainder 2 7i^, what is the 
 dividend ? 
 
 29. Simplify 4w — {2 w— \2n(r-\- «)— 2 7i(r — «)] j, 
 
 30. Divide aP — y^ hy x^ + y^ -\- a^y -f- xy^ + a;^^^^ 
 
 31. Divide (1 — m^) by [(1 — m)(l + w)]. 
 
 32. Divide 1 + a^ 4. ^4 ]3y j ^ ^ ^ ^2^ 
 
 33. From (2m -\-n ■\-p)x take (m 4- 7i)x. 
 
 34. From (r + s)y + (« + 0^^ take (r — «)?/ — (« — ^)z. 
 
 35. Square as indicated : 
 
 (2«+5)2; (1-2^)2; (l-fm)2; (2 m - J)2. 
 
 36. Square as indicated : (m^n—p)^; (mn—ph'^')^, 
 
 37. (a + 5)(a-6)(«2+62)^? 
 
 38. (m-l)(m + J)=? 
 
 - (i-5)(>-i)=' 
 
 40. Express 26 x 24 in the form of (a + b)(a— h) and 
 state the product. 
 
 41. 21x19=? 51x49 = ? 53x47 = ? 101x99 = ? 
 
 42. Does(-^)(+2:)(-y)(+2)=<-2:)(+^)(-z)? 
 
 Why? 
 
 43. Find in the shortest way the value of 748 x 680 
 - 748 X 670. 
 
 44. Find in the shortest way the value of 2 TrMff-}- 2 ttR^ 
 when TT = 3.1416, i2 = 1, and iT = 9. 
 
 45. What must be added to x^ + 4:X that the sum may 
 be (x 4- 2)2 ? 
 
 46. What must be added to a:^ -far that the sum may be 
 
SYSTEMS OF LINEAR EQUATIONS 237 
 
 47. What must be subtracted from x^-^^xy-^y'^ that 
 the difference may be {x—yYl 
 
 Factor : 
 
 48. 
 
 m^ + m3. 
 
 
 49. 
 
 K^-^)-H^-s^) 
 
 50. 
 
 r(a-h)-8{h- 
 
 -a). 
 
 51. 
 
 a2-b^-2bc-c\ 
 
 52. 
 
 mx-\-ny — nx- 
 
 my. 
 
 53. 
 
 64 - m^ 
 
 54. 
 
 l^a-a^-haK 
 
 
 55. 
 
 l-x^-x + x\ 
 
 56. 
 
 m2 
 
 
 57. 
 
 X x^ 
 
 58. 
 
 m^ n^ 
 n^ m^ 
 
 
 59. 
 
 y.3 
 
 60. R^ -h Eh-^ -\- r^, 61. 4x^-^x-l. 
 
 62. Find all the factors of Sx^-\-a^^l x^- lOx- S, 
 being given that x^-\-x-\-l is one of them . 
 
 63. Find all the factors of x^ -\- x^ — 1 x^ — x + 6, being 
 given that two of them are x — 2 and x — 1. 
 
 64. Reduce to lowest terms ^ ~ ^ » 
 
 ix- yy 
 
 1 —(r — 8^^ 
 
 65. Reduce to lowest terms ^^ ^ • 
 
 8 — rs -\- 8^ 
 
 66. Simplify 2- ^"^ ~^ . 
 
 Z-^y-X 
 
 67. Simplify -1- + -J— 1-. 
 
 ^—y y —X ^-\- y 
 
 68. 
 
 69. Simplify — — ^ -^— — x , , < 
 
 TTjii — 77171 -\-n^ mP — w 
 
 (x + iy 
 
 7^-1 
 
238 
 
 70. Simplify 
 
 ELEMENTARY ALGEBRA 
 
 x^ — y"^ 
 
 {m + n)^ m^ 4- 3 n^m + 3 nm^ -f- n^ 
 
 71. Find the value of when x = 
 
 n— X m-\-n 
 
 72. Given a = jt? + 'prt ; find f in terms of a, r and t» 
 
 »o c 1 1,111 
 
 73. Solve — + - = 
 
 m X n X 
 
 74. Solve 0.3 a: -0.1 4- ^'^ '^"^•^ = 0.4 a: -0.05. 
 
 1.2 
 
 75. Solve 
 
 76. Solve 
 
 77. Solve . 
 
 1 1 
 
 = -1. 
 
 X y 
 3 
 
 x-{-y—\ x-^y-\'2 
 4 2 
 
 = 0, 
 
 = 0. 
 
 78. Solve 
 
 a;— 8 3/ — 4 
 
 x-{-y-\-z = S, 
 4:x-Sy-\-2z = -2y 
 6x-2y-Sz = l, 
 
 X y z 
 
CHAPTER VIII 
 
 RATIO, PROPORTION, AND VARIATION 
 
 Ratio 
 
 169. Definition. The ratio of a number a to a number b 
 
 is the quotient - obtained by dividing a by b. The ratio 
 
 a to 6 is sometimes written a : b. 
 
 Note. By the ratio of one quantity to a second quantity is meant 
 the number of times that the first contains the second ; as the ratio 
 of 4 quarts to 3 quarts is ^. Obviously, no ratio exists between 
 quantities which are not of the saine kindy and before the ratio of two 
 quantities which are of the same kind can be found, they must be ex- 
 pressed in terms of the same unit. 
 
 Thus, the ratio of one gallon to three quarts is the ratio of four 
 quarts to three quarts, which is ^. 
 
 170. Definitions. In the ratio f the dividend, or nu- 
 
 b 
 
 merator, a, is called the first term or antecedent, and the 
 divisor, or denominator, 6, is called the second term or 
 consequent of the ratio. 
 
 EXERCISE 86 
 
 In examples 1-24 express the ratios as fractions and 
 simplify when possible : 
 
 1. 2:4. 2. 6:8. 3. 9 : 3. 
 
 4. 15:. 10. 
 
 5. a2 . ^^ 
 
 6. aP:2^, 
 
 7. J:i. 
 
 8. i:|. 
 239 
 
 9. ab^ : b. 
 
240 ELEMENTARY ALGEBHA 
 
 10. 
 
 ax\ Q^. 
 
 
 11. 
 
 n : 5f . 
 
 12. 
 
 {x-y):i^- 
 
 f\ 
 
 13. 
 
 Cx-\-i/y:(x-^y). 
 
 14. 
 
 9 a2^ : 15 a2^J. 
 
 
 15. 
 
 3 pk. : 8 pk. 
 
 16. 
 
 2 rd. : 161 rd. 
 
 
 17. 
 
 9 in. : 1 ft. 
 
 18. 
 
 2 yd. : 1 rd. 
 
 
 19. 
 
 5 pk. : 2 bu. 
 
 20. 
 
 5 gal. : 3 qt. 
 
 
 21. 
 
 1 mill. : 1 hr. 
 
 22. 
 
 ahc : - . 
 c 
 
 
 23. 
 
 $x : y ct. 
 
 24 (a - 5) yd. : {a + 5) ft. 
 
 •25. Arrange the ratios, |, |, J, |, in ascending order of 
 magnitude. 
 
 26. What number must be added to both terms of the 
 ratio 1^ in order to convert it into the ratio ^? 
 
 27. Two numbers are in the ratio of 7 to 4 ; if 3 be 
 subtracted from each number, the differences are in the 
 ratio of 11 to 5. Find the numbers. 
 
 Proportion 
 
 171. Definitions. Four numbers or quantities are said 
 to be in proportion when the ratio of the first to the sec- 
 ond is equal to the ratio of the third to the fourth. 
 
 Remark. In what follows we shall use the expression, the ratio 
 of one number to another instead of, the ratio of one number or quantity 
 to another. 
 
 If Y = -, then a, 5, <?, d are in proportion. A proportion 
 a 
 
 is, therefore, an expressed equality of two ratios. In 
 
 proportion, as 7 = - , the n 
 a 
 
 the terms of the proportion. 
 
 proportion, as 7=^5 the numbers a, 6, c, d are called 
 d 
 
. RATIO, PROPORTION, AND VARIATION 241 
 
 Note. A proportion is often written a:b = c:d; read, the ratio 
 of a to b is equal to the ratio of c to d. An old form and one less 
 frequently used is a:b : :c :d. 
 
 172. Extremes and means. The first and fourth terms 
 of a proportion are called the extremes and the second 
 and third terms the means. 
 
 Thus, in the proportion - == -, a and d are the extremes and b and 
 b ' d 
 c the means. 
 
 173. Important identities. If the four numbers, a, b, c, 
 and c?, are the four terms of a proportion, they satisfy the 
 identity, ^^^ 
 
 Multiplying both members of identity (1^ by bd^ we 
 ^a^^' ad=bc. (I) 
 
 Identity (I) may be expressed in words as follows : 
 In any 'proportion the product of the extremes is equal to 
 the product of the means. 
 
 Again, if -t=-7 5 ^^^^ the reciprocal of - is equal to the 
 b d b 
 
 reciprocal of - ; that is, 
 d 
 
 - = -. (II) 
 
 a c 
 
 Since the reciprocal of a fraction is obtained by invert- 
 ing the fraction, identity (II) is sometimes expressed as 
 follows : 
 
 1^ four numbers are in proportion^ they are also in pro- 
 portion by inversion. 
 
 If 7 = ;^> WG have from identity (I), 
 
 ad = be. (2) 
 
242 ELEMENTARY ALGEBRA 
 
 Dividing both members of identity (2) by ah^ 
 
 {='- (HI) 
 
 o a 
 Again, dividing both members of identity (2) by dc^ 
 
 c a 
 
 Comparing identities (III) and (IV) with the propor- 
 tion ^ = - , we observe that if four numbers taken in a 
 h d 
 
 certain order are in proportion, they continue to be in 
 proportion when either extremes or means are inter- 
 changed. This fact is usually expressed as follows : 
 
 If four numbers are in proportion^ they are also in propor- 
 tion hy alternation. 
 
 If 1 be added to both members of the identity 
 
 (3) 
 
 (4) 
 
 (V) 
 
 
 a c 
 h^d' 
 
 3 have, 
 
 h d 
 
 Combining, 
 
 a + b c-h d 
 
 Subtracting 1 from both members of identity (3), 
 
 Combining, ~ = "" . (VI) 
 
 d 
 
 Dividing the members of (V) by the corresponding 
 members of (VI), 
 
 fl-t- fr ^ c + rf rvin 
 
 a-b c-d' ^ ■ 
 
RATIO, PROPORTION, AND VARIATION 243 
 
 Identities (V), (VI), and (VII) are usually expressed 
 in order as follows: 
 
 If four numbers are in proportion^ they are also in propor- 
 tion hy composition. 
 
 If four numbers are in proportion^ they are also in pro- 
 portion by division. 
 
 If four numbers are in proportion^ they are also in propor- 
 tion by composition and division. 
 
 EXERCISE 87 
 
 Test identities (I)- (VII) by the use of the propor- 
 tions of examples 1-4. 
 
 ^ « 31 210 
 
 a — b 
 5a _ 30 ab ax-\- ay — bx ^ by _a -\- b 
 
 3 5 18 52 ' ax — ay -\- bx — by x — y 
 
 x-\- y 
 Find the value of x in each of the proportions stated in 
 examples 5-12. 
 
 5. 1 = -. 6. 5: 2 = 2:: 10. 
 
 3 X 
 
 7. 3: 2:= 6: 14. 8. 2: : 10 = 5 : 2. 
 
 9. — i^=— ^t^. 10. a-\-x:b-\-x=:c-{-x:d+x. 
 
 S -\- X lb -\- X 
 
 11. 3-x: -2 = 3a:+4:32. 12. a:b::x:c. 
 
 13. Write by inversion ; 
 
 yd y 3 n q 
 
 14. Write (a), (6), and (c) of example 13 by alter- 
 nation, 
 
244 ELEMENTARY ALGEBRA 
 
 15. Write (a), (6), and (<?) of example 13 by com- 
 position. 
 
 16. Write (a), (6), and (c?) of example 13 by division. 
 
 17. Write (a), (6), and (c) of example 13 by compo- 
 sition and division. 
 
 18. If four numbers are proportionals (in proportion), 
 prove that either mean is equal to the product of the 
 extremes divided by the other mean. 
 
 19. If four numbers are proportionals, prove that either 
 extreme is equal to the product of the means divided by 
 the other extreme. 
 
 20. If the product of two numbers is equal to the 
 product of two other numbers, prove that a proportion 
 may be formed by taking one pair of numbers for the 
 extremes and the other pair for the means. 
 
 21. Using the statement of example 20, write the eight 
 proportions that may be expressed from the identity 
 mq — np. 
 
 22. It - = -, prove that — ' — = — ' — . 
 
 d a c 
 
 Suggestion. Write the given proportion by inversion, and then 
 apply identity (V). 
 
 23. It 7 = -, prove that = . 
 
 d a c 
 
 174. Continued proportion. Three or more numbers are 
 said to be in continued proportion when the ratio of the 
 first to the second is equal to the ratio of the second to 
 the third, and so on. 
 
 Thus, a, 6, c, d are in continued proportion if, 
 a:h = h :c = c :d. 
 
IIATIO, PR0I>0RTI0N, AND VARIATION 245 
 
 175. Mean proportional. When three numbers are in 
 continued proportion, the second is said to be a mean pro- 
 portional between the other two. 
 
 Thus, if - = -, the number & is a mean proportional between the 
 b c 
 
 extremes a and c. 
 
 176. Third proportional. When three numbers are in 
 continued proportion, the third is said to be a third pro- 
 portional to the other two. 
 
 Thus, if - = -, the number c is a third proportional to a and b. 
 b c 
 
 VJl. Fourth proportional. A fourth proportional to 
 three numbers a, 5, e, taken in the order given is the 
 fourth term of the proportion 
 
 a : h = c : X. 
 
 178. Composition of equal ratios. Let -, -, and - be 
 
 ha J 
 
 equal ratios and each equal to r ; that is. 
 
 
 ace 
 
 Then, 
 
 a — hr^ c= dr, e —fr. 
 
 Adding, 
 
 a + c+^ = (^ + c? -h/)r. 
 
 Dividing, 
 
 « + <?+«_ 
 
 Therefore, 
 
 a-fc + e a c e 
 h+d+f b d f 
 
 This identity may be expressed in words as follows : 
 In a number of equal ratios the sum of the antecedents is 
 
 to the sum of the consequents as any antecedent is to its 
 
 consequent. 
 
246 ELEMENTARY ALGEBRA 
 
 EXERCISE 88 
 
 1. If a, 6, and c are three numbers in continued pro- 
 portion, show that 6^ = ac. 
 
 2. If a, 6, and c are thre^ numbers in continued pro- 
 portion, show that c, the third proportional to a and 6, is 
 
 equal to — • 
 a 
 
 3. What positive integer is a mean proportional be- 
 tween 256 and 36 ? 
 
 4. Express by proportion the fact that 6 is a mean 
 proportional between 4 and 9. 
 
 5. Express by proportion the fact that 18 is the 
 third proportional to 2 and 6. 
 
 6. What is the third proportional to 10 and 5 ? 
 
 12a 
 
 7. If - = - = -, what are the values of a and h ? 
 
 2 a J' 
 
 8. If !^ = £ = !:, show that ^?^+^±^ = ^ = ^ = ^. 
 
 n q 8 n -\- q-\- s n q 8 
 
 9. From the proportion | = f , derive another propor- 
 tion by inversion ; derive two other proportions by alter- 
 nation ; derive another proportion by inversion and 
 composition ; derive another proportion by division ; de- 
 rive another proportion by inversion and division. 
 
 ,^ y£ a c 4.1 4. « ^ 3a-f- 2 c 
 
 10. If 7 = - , prove that - = - = —: —-_ • 
 
 Suggestion. Since - = - , then — = — ^ ; now see section 178. 
 h d 3 6 2r/ 
 
 11. If _^ = = — ^ , and a-\-h + c is not equal 
 
 h -\- e c + a a-\- 
 
 to zero, prove that each fraction is equal to J. When 
 a-\- b-\- c is equal to 0, what is the value of each of the 
 fractions ? 
 
RATIO, PROPORTION, AND VARIATION 247 
 
 Variation 
 
 179. Constant. A number which has always the same 
 value is a constant. 
 
 Thus, 2, — ^, and a, supposing the value of a to be known, or 
 given, are constants ; also the root of a simple numerical equation in 
 one unknown number is a constant ; as the root of 2 2- — 4 = 0. 
 
 180. Variable. Many of the numbers of algebra are 
 not constants. A number which is not a constant is 
 called a variable. 
 
 Thus, if the case of an express train running from New York to 
 Philadelphia be considered, and i denotes the number of minutes 
 which have elapsed since it left New York, s the number of feet that 
 it passed over in t minutes, and v the number of feet that it passed 
 over in each minute, v supposed to be constant, then the formula 
 connecting the time, the rate, and the distance passed over is, 
 
 s = vt. 
 
 In this formula v is constant, but t and s are both variables during 
 the whole time that the train is in motion. 
 
 Remark. The unknown numbers in an equation in two or more 
 unknowns are called variables. 
 
 Thus, in y = 2 r, any value whatsoever may be assigned to either 
 X or y. The fact that it is not necessary to assign one fixed, or con- 
 stant, value to X or y, as is the case in a simple equation in one un- 
 known, in order to obtain a solution of the equation, is sufficient 
 reason for calling x and y variables in the equation. 
 
 181. Application of terms. In general, a letter is said 
 to be a variable ; that is, it represents a variable number 
 if it may have a number of different numerical values in 
 a discussion or problem. It is a constant if it can have 
 only one numerical value. 
 
 182. Direct variation. If one variable, y, depends 
 upon another variable, x^ in such a way that the ratio of 
 ^ to a: is a constant, then y is said to vary as x, or vary 
 directly as x. 
 
248 ELEMENTARY ALGEBRA 
 
 If y varies as x^ then, 
 
 y.=ic (a constant). ^ ^ 
 
 X 
 
 X 1 
 Hence, - = - (a constant). (2) 
 
 y 
 
 From (1) and (2) it is evident that if y varies directly 
 as a:, then x varies directly as y, 
 
 183. Illustrations of direct variation. 1. The number 
 of cents (c) in the cost of cloth bought at the fixed price 
 of 60 cents per yard varies as the number of yards (n) 
 
 bought, since the ratio — is a constant, namely, the num- 
 
 n 
 
 ber of cents in the cost of one yard. 
 
 2. In the formula 8 = vt^ in which i; is a constant, « 
 
 varies directly as t ; that is, the space passed over varies 
 
 directly as the time. 
 
 Note. In the equation s = vt, as elsewhere, s, v, and t represent 
 numbers, and the statement, *^ space passed over varies directly as the 
 time" means that the number s varies as the number /. 
 
 184. Inverse variation. If the variable y varies as 
 the reciprocal of x; that is, if y is equal to a constant 
 times the reciprocal of x^ then y is said to vary inversely 
 as X. 
 
 Thus, \i y —-, where c is a constant, y varies inversely as x. 
 
 X 
 
 Also, from y =- we have x = -, and hence x also varies inversely 
 
 X y 
 
 as y ; moreover, from y z= - we derive xy = c ; hence, 
 
 X 
 
 If the product of two variables is a constant^ the one varies 
 inversely as the other. 
 
 185. Illustration of inverse variation. The number of 
 yards of cloth (n) that can be bought for a certain fixed 
 
RATIO, PROPORTION, AND VARIATION 249 
 
 price, as $8, varies inversely as the number of dollars (r) 
 that the cloth costs per yard, since ti = - ; that is, n is a 
 constant times the reciprocal of r. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. If y varies directly as x^ and y is 100 when x is 20, 
 what equation expresses the relation between y and a;? 
 
 Solution. Since y varies as x, then y is some constant, as c, 
 times X ; 
 
 Therefore, y = ex. (1) 
 
 Since y is equal to 100, when x is equal to 20, 
 
 100 = 20 c. (2) 
 
 Solving (2) for c, c = 5. (3) 
 
 Substituting the value of c in (1), 
 
 y = 5x. (4) 
 
 2. The number of feet that a body falls from rest under 
 the action of gravity is proportional to (varies directly as) 
 the square of the number of seconds during which it falls. 
 If a body falls 144 feet in three seconds, how far will it 
 fall in five seconds? How many seconds are required for 
 it to fall 192 yards? 
 
 Solution. Let s represent the number of feet that the body falls, 
 and t the number of seconds during which it falls ; also, let c repre- 
 sent a constant. 
 
 Since s varies as t^, s = cfi. (1) 
 
 Since s = 144, when < = 3, 144 = 9 c. (2) 
 
 Solving for c, c = 16. (3) 
 
 Substituting the value of c in (1), 
 
 s = 16 1\ (4) 
 
 To find how far the body will fall in five seconds, substitute 5 for 
 / in (4); then, s = 16 x 25 = 400. (6) 
 
 Hence, the body falls 400 feet in 5 seconds. 
 
 Also, to find the number of seconds required for the body to fall 
 
250 ELEMENTARY ALGEBRA 
 
 192 yards, or 576 feet, substitute 576 for s in (4) ; then, 
 
 576 = 16 fi. (6) 
 
 Solving (6) for t, t = Q. (7) 
 
 Hence, 6 seconds is the time required for the body to fall 192 
 
 yards. 
 
 EXERCISE 89 
 
 Write each of the statements in the first five examples 
 in the form of an equation. 
 
 1. The area ^ of a circle varies as the square of its 
 radius B. 
 
 2. The volume F of a sphere varies as the cube of its 
 diameter D. 
 
 3. The number iV of articles which can be bought for 
 a fixed sum aS' varies inversely as the cost P of one article. 
 
 4. The weight w of a body varies as its mass m. 
 
 5. The speed v of a falling body starting from rest 
 varies as the time t during which it falls. 
 
 6. If i/ varies as x, and ^=20 when x = 5, find the 
 value of «/ when a: = 10. 
 
 7. If 1/ varies inversely as x, and «/ = 20 when x = 5, 
 find the value of i/ when a; = 100. 
 
 8. If p varies as q, and p = — 2 when ^ = — 3, find the 
 equation connecting p and q. 
 
 9. If m varies inversely as n, and m = 30 when n = 6, 
 find m when n=9. 
 
 10. If 7 bu. of corn are worth 14.90, what are 3 bu. of 
 corn of the same quality worth? 
 
 Suggestion. Employ direct variation. 
 
 11. If 10 men can do a piece of work in 15 days, how 
 long will it take 13 men to do the same work? 
 
 Suggestion. Use inverse variation. 
 
CHAPTER IX 
 
 GRAPHS 
 
 186. Graphical representation of the relation between 
 two variables. It is convenient to represent correspond- 
 ing values of two related variables by points in a plane. 
 Such a representation is said to be graphical. When a 
 sufficient number of points, which represent pairs of cor- 
 responding values of two related variables, have been 
 marked in the plane, a glance at the diagram will reveal 
 to the eye the corresponding changes in the two variable 
 quantities. 
 
 For the purpose of graphical work, paper ruled in small 
 squares, as in the ac- 
 companying diagram, is 
 used. Two of the ruled 
 lines of the paper, one 
 horizontal and the other 
 vertical, are selected as 
 axes of reference. The 
 point 0, at which the 
 axes cross each other is 
 called the origin. The 
 axes of reference are 
 called axes of coordi- 
 nates. The horizontal 
 axis X^X is called the jc-axis, and the vertical axis YY' is 
 called the y-sxis. 
 
 251 • 
 
 
 
 
 — 
 
 — 
 
 
 — 
 
 — 
 
 
 
 Y 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "F 
 
 / 
 
 
 
 
 
 
 
 
 n 
 
 
 
 
 
 
 
 
 
 
 X 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 J 
 
 
 
 _ 
 
 
 
 
 Y 
 
 
 
 
 
 
 
 
 
 
252 
 
 ELEMENTARY ALGEBRA 
 
 Any point in the plane has two coordinates; namely, 
 its abscissa, represented by x^ and its ordinate, repre- 
 sented by y. The abscissa of a point is its horizontal 
 distance from the ^-axis. The ordinate of a point is 
 its vertical distance from the a;-axis. All horizontal dis- 
 tances measured to the right from the y-axis are regarded 
 as positive, and consequently (see section 31), all hori- 
 zontal distances measured in the opposite direction from 
 the y-axis are negative. All vertical distances measured 
 upwards from the x-axis are regarded as positive, and con- 
 sequently, those measured in the opposite direction from the 
 X-axis are negative. 
 
 187. A point fixed by its coordinates. The position of 
 a point in a plane is fixed by its coordinates. 
 
 Thus, the position of the point P whose coordinates are a: = 2 
 and ^ = — 1 is fixed. The sign of its abscissa shows that the point 
 P is to the right of the ?/-axis and the sign of its ordinate shows that 
 
 Y 
 
 P X 
 _,_ 
 
 Y' 
 
 it is below the ar-axis. To actually find the position of P, we count 
 on the ar-axis 2 squares to the right of the origin, since the value 
 of the abscissa of P is 2 ; then we count downward from the a;-axis 
 1 square, since the value of the ordinate of P is — 1. 
 
GRAPHS 
 
 253 
 
 188. Notation. The coordinates of a point are desig- 
 nated by the symbol (x^ y). 
 
 Thus, the point (2, — 3) means the point whose x (abscissa) is 2 
 and whose y (ordinate) is — 3. It should be observed that in such a 
 symbol as (2, — 3) the number written first is always the abscissa. 
 
 189. Plotting a point. The marking of the position of 
 a point on the diagram is called plotting the point. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Plot the point (-2,4). 
 
 Solution. For convenience, we take five divisions on the coor- 
 dinate paper as the unit of measure. The abscissa being — 2, we 
 
 
 
 
 
 
 
 
 
 V 
 
 
 
 n 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^j 
 
 D 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 • 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 n/ 
 
 
 
 
 
 
 
 
 
 
 
 
 . 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 count 10 squares (2 units) to the left of the origin along the a:-axis to 
 the point B ; then, since the ordinate is + 4, we count 20 squares 
 (4 units) upwards from the x-axis, locating the position P of the point 
 (—2, 4). We mark the position of P by a small ring. 
 
254 
 
 ELEMENTARY ALGEBRA 
 
 2. Plot the point (1.25, .5). 
 
 Solution. For convenience, we take 20 divisions on the coordi- 
 nate paper as the unit of measure. Since both coordinates are positive, 
 
 
 
 
 
 
 
 
 Y 
 
 " 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 n 
 
 
 1 
 
 
 
 
 
 
 
 
 - 
 
 "■ 
 
 ~ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 L 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (^a| 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 1/ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 - 
 
 - 
 
 
 
 
 
 
 
 
 we count to the right along the a:-axis and upwards from the ar-axis. 
 Counting 25 squares (1.25 units) to the right along the a:-axis and then 
 10 squares (.5 units) upwards from the a;-axis, we locate the position 
 P of the required point. 
 
 EXERCISE 90 
 
 (If convenient, use graph paper in plotting the graphs in the 
 examples of this exercise. If such paper is not available, draw two 
 axes. Use an appropriate unit of distance, as ^ inch, -J inch, J^ inch, 
 or 1 centimeter). 
 
 1. Plot the points (4, 3); (-4,3); (4,-3); (-4, 
 -3). 
 
 2. Plot the points (-2, 1); (-2, -1); (4, -4); 
 (4,1). 
 
 3. Plot the points (3,0); (0,-3); (0,3); (-3,0). 
 
GRAPHS 255 
 
 4. Plot the points vl(4, 3); ^(-4,4); C(7,0). Con- 
 nect the points A, B^ C with straight lines forming a tri- 
 angle whose vertices are the given points. 
 
 5. Construct the quadrilateral whose vertices taken 
 in order are the points ^(3, 4) ; 5( - 2, 6) ; (7( - 4, — 1) ; 
 i)(4, -2). 
 
 6. On what straight line are all points located which 
 have for their abscissa 0? 
 
 7. On what straight line are all points located which 
 have for their ordinate 0? 
 
 8. What are the coordinates of the origin? Plot the 
 point (0, 0). 
 
 9. Draw the triangle whose vertices are (0,-1); 
 (0, +1); (2,0). 
 
 10. Plot the points (-3, -9); (-2, _6);(-l, -3); 
 (1, 3); (2,6); (3,9). Do these points appear to the eye 
 to be scattered at random over the diagram? How do 
 they appear to lie? 
 
 190. A function. In many of the problems of elemen- 
 tary algebra one or more pairs of related variables occur 
 [see examples 1-5, exercise 89, also section 183]. As 
 another illustration of two related variables, let A repre- 
 sent the age of a boy and W his weight. In general, as 
 A changes it is evident that W also changes ; this fact 
 may be expressed by stating that the weight of the boy 
 varies with, and depends on, his age. 
 
 If one variable varies with another so that when a value of 
 07ie 18 given, a corresponding value of the other is determined, 
 the second variable is called a function of the first. 
 
 Thus, from the indeterminate equation y + 2 x = 5, we derive 
 y = 5 — 2 X. Here y is so related to x that its value is determined 
 
256 
 
 ELEMENTARY ALGEBRA 
 
 for any given value oi x; y is, therefore, a function of x. Also, the 
 area of a circle is a function of its radius. If A represents the area 
 and R the radius of a circle and tt the well-known constant whose 
 value to four places of decimals is 3.1416, then the relation between 
 the area of a circle and the radius is expressed algebraically by the 
 equation A = ttR^ 
 
 191. Graph of a function. When a simple relation be- 
 tween two variables is given, as, for example, the relation 
 expressed by the equation i/ = S x, numerical values may 
 be assigned to x and the corresponding values of y found. 
 On plotting the points which represent the different pairs 
 of corresponding values of the variables, these points are 
 found to lie on a definite straight line or curve ; this 
 straight line or curve is called the graph of the function. 
 It is also convenient to speak of the graph of an equation, 
 an expression which means that the coordinates of the 
 points plotted and through which the line or curve passes, 
 satisfy the equation. 
 
 ILLUSTRATIVE EXAMPLE 
 
 Draw the graph of 3 a; -h 5 for values of x between x= —4 
 and a: = + 3. 
 
 Solution. Denoting the function by y, we have 
 
 y = 3x-\-5 (1) 
 
 We now make a table showing the corresponding values of x and y 
 for integral values of x between a; = — 4 and a: = + 3, as follows : 
 
 X 
 
 - 4 
 
 -3 
 
 -2 
 
 -1 
 
 
 
 1 
 
 2 
 
 3 
 
 3x 
 5 
 
 -12 
 5 
 
 - 9 
 5 
 
 -6 
 5 
 
 -3 
 5 
 
 
 5 
 
 3 
 5 
 
 6 
 5 
 
 9 
 5 
 
 y 
 
 - 7 
 
 -4 
 
 -1 
 
 + 2 
 
 5 
 
 8 
 
 11 
 
 14 
 
 Plotting the points which represent the values of x and y (see 
 diagram), these points appear to lie, and in fact do lie, on the straight 
 
GRAPHS 
 
 257 
 
 line PQ. This straight line is the graph of the function 3 x + 5, and 
 is also the graph of the equation 3/ = 3 a: -f 5. 
 
 Any values of x and y which satisfy equation (1) are the coordi- 
 nates of a point on the graph of equation (1). Conversely, we as- 
 
 
 
 ^ 
 
 
 — ' 
 
 ~" 
 
 
 ■— 
 
 ~" 
 
 
 ^ 
 
 Y 
 
 
 
 ■" 
 
 
 n 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 1 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Y 
 
 
 
 
 
 
 
 
 
 J 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 T 
 
 
 
 
 
 Y 
 
 / 
 
 
 
 
 
 
 
 
 
 
 .J 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 sume that the coordinates of any point on the graph of the equation 
 will satisfy the equation. 
 
 Note. The expressions, construct the graph of, obtain the graph 
 of, draw the graph of graph the function, plot the curve, mean the 
 same thing. 
 
 EXERCISE 91 
 
 Draw the graphs of the following equations for values 
 of X between a: = — 4 and a; = -f 4 : 
 
 1. 1/ = X. 2. y = — X. 3. «/ — 2 a: = 0. 
 
 4. y + 4 a: = 0. 5. y = 3 a;. 6. y = a; + 1. 
 
258 ELEMENTARY ALGEBRA 
 
 7. y = x-\-1. 8. y=-x^-1. 9. y = ^x + 1, 
 10. 2x+Zy=5. 11. 3 2^—2^=1. 12. 5x — 2t/ = 4, 
 
 13. 2:+l/ + l = 0. 14. ^+|- 3 = 0. 15. 3a:-|-4z/ + 7 = 0. 
 
 192. Graph of a linear equation in two variables. Any 
 
 simple equation in two unknowns, as x and 3/, can be re- 
 duced to the form ax-^ by -\- c= 0, The graph of such an 
 equation is always a straight line. It is for this reason 
 that the simple equation ax -\- by -{- c = is called a linear 
 equation. 
 
 Note. It will be assumed here that the graph of any simple equa- 
 tion in two unknowns is a straight line. The proof of this fact is 
 given in analytical geometry. 
 
 193. Graphs of simultaneous linear equations in two un- 
 known numbers. When the straight lines which are the 
 graphs of two linear equations in two unknown numbers 
 are plotted on the same diagram, they will, if the given 
 equations are independent and consistent, intersect in one 
 point. The coordinates of the point of intersection satisfy 
 both equations and, . therefore, together constitute the 
 solution of the given system. 
 
 ILLUSTRATIVE EXAMPLE 
 
 Construct the graphs of the equations 2y — a?-|-4 = 
 and 2 i/ -I- 3 a; — 4 = 0. Estimate from the diagram the 
 coordinates of their point of intersection. Find whether 
 or not these estimated coordinates satisfy the equation. 
 
 Solution. 
 
 2y-a; + 4 = 0. (1) 
 
 2y + 3x-i = 0. (2) 
 
 Solving (1) for y, 2^=1-2- (8) 
 
 Solving (2) for y, y = =^ + 2. (4) 
 
GRAPHS 
 
 259 
 
 Assigning to x in (3) any two values, as a: = 0, and x = 4, we have 
 
 from (3), when:. = 0, 2, = - 2, (5) 
 
 when a; = 4, y = 0. (6) 
 
 Similarly, assigning to x in (4) any two values, as x = 0, and x = 1, 
 
 we have from (4), u n o x^x 
 
 ^ ^' when X = 0, y = 2, (7) 
 
 when X = \, y = \- (8) 
 
 Since we know that the graph of (3) is a straight line [§ 192, 
 note], it is necessary to plot two points only and then draw the 
 straight line through these points. 
 
 
 
 
 Y 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 ■ 
 
 
 
 
 
 ^ 
 
 
 
 
 
 \ 
 
 V 
 
 
 
 ^ 
 
 y^ 
 
 
 
 X' 
 
 
 
 
 
 \ 
 
 ^ 
 
 ^ 
 
 
 
 X 
 
 
 
 
 
 ^ 
 
 ^ 
 
 \ 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 ^ 
 
 
 
 Y' 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 Plotting the points (0, - 2) and (4, 0), the graph of (3) is ob- 
 tained by drawing the straight line through these points. In like 
 manner, plotting the points (0, 2) and (1, |) the graph of (4) is the 
 straight line joining these points. 
 
 By inspection, the graphs of the two equations appear to intersect 
 in the point P {x = 2, y = — 1). Substituting these values of x and y 
 in (1) and (2), we have 
 
 2( - 1) - 2 + 4 = 0. (9) 
 
 2(-l)+6-4 = 0. (10) 
 
 Hence the solution of (1) and (2) is x = 2, y = — 1. 
 
260 ELEMENTARY ALGEBRA 
 
 EXERCISE 
 
 Construct the graphs of the equations in each of the 
 following systems. For each system determine by in- 
 spection approximate values of the unknown numbers 
 which will satisfy the equations. 
 
 "•• \x-7/ = ^. \ y = 2, 
 
 /a;-h2i/=3, r22: + 3y = 3, 
 
 \^x-1y==l, l3a: + 42/ = 4. 
 
 ■3a;-hy = 2, 1^ + 32^ = 6, 
 
 2/=3. • \2: + 2^ = 3. 
 
 3. 
 
 r3a; + 
 I2:r- 
 
 194. Graphs of dependent linear equations in two un- 
 knowns. Two linear equations in two unknowns which 
 have more than one solution in common must have all 
 solutions in common, and, therefore, they are dependent 
 equations. This follows directly from the fact that the 
 graph of any such equation being a straight line, it is com- 
 pletely determined when any two points on it are known. 
 
 Thus, the equations x ■{■ y = 2 and 3 a: + 3 y = 6 have in common 
 the solutions x = 0, y = 2 and x = 2, y = 0, and the graph of each is 
 a straight line through these two points. The coordinates of every 
 point on this line satisfy both equations. 
 
 195. Inconsistent equations. The graphs of two incon- 
 sistent linear equations in two unknowns furnish, a simple 
 geometrical reason for the existence of such systems of 
 equations. 
 
 Thus, the graphs of the inconsistent equations 
 
 x + y = 2 (1) 
 
 x + y = l (2) 
 
 are found to be parallel straight lines. 
 
 In the accompanying diagram two divisions on the coordinate paper 
 are taken as the unit of measure. The graph of (1) is the straight 
 
Ren6 Descartes (1596-1650) was the first of the modern school 
 of mathematics. He was educated at the Jesuit School of LaFleche, 
 France. Descartes is famous as a mathematician and as a phi- 
 losopher. The introduction of the fundamental ideas underlying the 
 graphical representation of related variables was his greatest con- 
 tribution to mathematical science. The introduction of indices as 
 now used was due to Descartes. 
 
GRAPHS 
 
 261 
 
 line which passes through the points (0, 2) and (2, 0), and the graph 
 of (2) is the straight line which passes through (0, 1) and (1, 0). 
 These lines are evidently parallel. 
 
 
 
 
 
 Y 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 ^J 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 \ 
 
 
 
 
 
 X' 
 
 
 
 
 O 
 
 
 \ 
 
 
 \ 
 
 
 
 X 
 
 
 
 
 
 
 
 
 \ 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 \ 
 
 
 
 
 
 
 Y' 
 
 
 
 
 
 \ 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 In order that two linear equations in two unknowns 
 may have a solution in common, it is necessary that their 
 graphs should have a point in common. Two parallel 
 lines have no common point and, therefore, the equations 
 of which the parallel lines are the graphs have no solution 
 in common; they are, therefore, inconsistent equations. 
 
 EXERCISE 93 
 
 Construct the graphs of the following systems of 
 equations. 
 
 a; - 2 = 0, 
 
 z/ - 3 = 0. 
 
 2. 
 
 3. 
 
 x=^2y, 
 
 y - 1 = 0. 
 x-\- y = l, 
 2 2^ + 2 ^/ = 2. 
 
CHAPTER X 
 POWERS, ROOTS, RADICALS, AND EXPONENTS 
 
 196. Rational number. The rational operations in 
 algebra are addition, subtraction, multiplication, and divi- 
 sion [section 79]. Any number which is either a positive 
 integer or can be obtained from the positive integers 
 by the rational operations is called a rational number. 
 Hence, a rational number, when expressed in its simplest 
 form, is either a positive or negative integer or a positive 
 or negative fraction with integral numerator and integral 
 denominator. 
 
 Thus, 2, 3.16, — 7, and — | are rational numbers. 
 
 Also, + y/i is a rational number, since when expressed in its 
 simplest form, it has a rational value, namely, 2. However, finding 
 the square root of a number is not one of the rational operations of 
 algebra. 
 
 197. Irrational number. It is necessary to add to the 
 system consisting of all rational numbers, certain other 
 numbers which are not rational. For instance, in men- 
 suration it is necessary to say that the length of the 
 diagonal of the square is equal to the length of its side 
 multiplied by the V2; yet there is no rational number 
 whose square is equal to 2 ; that is, V2 is not a rational 
 number. We shall assume that there is a definite positive 
 number whose square is equal to 2. We call such a num- 
 ber an irrational number. 
 
 Thus, V3, 1 — V2, — - , V2 + V3 are irrational numbers. 
 
 V2 
 
 262 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 263 
 
 Note. When we say that we assume that there is a definite posi- 
 tive number whose square is equal to 2, we are extending the meaning 
 of the word number. The word number, heretofore, has meant rational 
 number. Now when we use the word number we shall mean irra- 
 tional number as well as rational. 
 
 Remark. We do not call such an expression as V— 2 an irra- 
 tional number. This expression is, for the present, without meaning. 
 The square of either a positive or a negative number is necessarily 
 positive ; hence, there is no positive or negative number whose square 
 is -2. 
 
 Rational numbers and irrational numbers are necessarily positive 
 or negative. 
 
 198. Fundamental identities involving powers. The 
 
 exponents used in the identities of this section are posi- 
 tive integers. 
 
 From section 58 we have, 
 
 (T a" = tf"+". (I) 
 
 By definition, section 14, 
 (a6)"' = ab ■ ab • ab • ■■• to m factors 
 
 = a ■ a a - ••• to m factors x b • b b - .•• tow factors 
 
 That is, {abY = cTlf. (II) 
 
 Remark. Since {abc)'^ =[(a6)c]'" =(a6)'"c'* = a"'6'"c'", it follows 
 that an identity similar to (II) holds for a power of the product of 
 any number of factors. 
 
 By definition,section 14, (cry = a"'- a*" • a*" . ... to w factors 
 
 ffn-\-m-\-rn-\ to fl terms 
 
 = a'"". 
 In like manner, (a")*" = a*"". 
 
 That is, {ary = (a")'" = o^". (Ill) 
 
 From identity (II), (a'"i»*)p = («»") p(5") p. 
 
 From identity (III), (a"*)p(5~)p= a'^pft^p. 
 Therefore, {oTlfy = aTflfP. (IV) 
 
264 ELEMENTARY ALGEBRA 
 
 EXERCISE 94 
 
 (Solve as many as possible at sight.) 
 
 Find the results of the indicated operations by using 
 the identities of section 198. 
 
 1. 23.22. 2. 28.22.2. 3. (2.3)2. 
 
 4. (2.3.5)8. 5. (22)3. 6. (33)2. 
 
 7. (23)3. 8. [(-2)2]3. 9. [(_2)3]2. 
 
 10. [(-2)3]8. 11. (22.33)2. 12. (2.32.52)2. 
 
 13. ai5 . a}^. 14. a^ -a^ • a*. 15. (aby. 
 
 16. (2a)3. 17. (-3a)3. 18. (2aby. 
 
 19. (-3a6)2. 20. (abcdy. 21. (a^^y, 
 
 22. (2253)3. 23. [(-2)8a253]2. 
 
 24. (-3a253^2^4g6)4. 25. [(« + *)2]3. 
 
 26. (a + by(a-{-b). 27. (a-by(a-by. 
 
 28. (a-by(a-{-by. 
 
 Suggestion, (a - hy{a + hy =[(«-ft)(a + &)p. 
 
 29. (a - l)2(a -f 1)2. 30. [2(a H- 6)2((? - (^)8]2. 
 31. [2a5(c+(^)2(e-/)3]3. 32. [(-3)2]3.(33)2. 
 
 33. Show that ( — J = — , and, in general, that (- j =- — 
 
 \bj b^ \bj 6*" 
 
 Suggestion. (-Y = ? . ^ . ? ... to m factors. ^ [§ 139] 
 
 34. Show that (f )8( j)8 = (I X |)8, and, in general, that 
 
 W W \bd) ' 
 Suggestion. {^-\{'-Y^ «« ^ ^ ««^ ^ ^^ 
 
 35. (1)2(1)2(1)2. 36. (|)««(f)^(|)«^. 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 265 
 
 199. Roots. When a number is the product of five 
 equal factors, one of these factors is called a fifth root of 
 the number. In general, when a number is the product of 
 n equal factors, one of those factors is called an nth root 
 of the number. 
 
 Thus, if m^ = a, then m is a fifth root of a, which is expressed by 
 m = Va. 
 
 The two equations, 
 
 m^ = a (1) and m = v'a (2) 
 express, therefore, the same relation between a and m. By substitut- 
 ing the value of m from equation (2) in equation (1), we have the 
 identity (Va)^ = a. In general, when n is any positive integer we 
 have, by definition. 
 
 Note. The expression Va is read the nth root of a. 
 
 200. Radical. An indicated root of any number is 
 called a radical expression, or simply a radical. 
 
 Thus, V3, </27, Vr» v^«T^ are radicals. 
 
 201. Index of a root. The number indicating the root 
 to be taken is called the index of the root. The index of 
 a radical is always an integer. 
 
 202. Radicand. The number or expression under the 
 radical sign is called the radicand. 
 
 203. Like roots and unlike roots. Two roots are said 
 to be like or unlike according as their indices are equal 
 or unequal. 
 
 Thus, y/a and Vb are like roots ; Va and y/a are unlike roots. 
 
 204. Principal root. Any positive number a has one 
 and only one positive nth root. This root is called the 
 principal nth root of a. 
 
 Thus, the principal square root of 4 is 2, the principal cube root 
 of 27 is 3, and the principal square root of 2 is + V2. 
 
266 ELEMENTARY ALGEBRA 
 
 When the index of a radical is an odd number and the 
 radicand is negative, there is one and only one negative 
 root and no positive root. This negative root is called the 
 principal nth root of the negative number in the radicand. 
 
 Thus, V — 8 can have no positive value, since the cube of a positive 
 number is positive. It has, however, one negative value, namely — 2, 
 since an odd power of a negative number is negative. Also, the 
 principal root of \/- 27 is - 3, that of y/ - 82 is - 2, that of \/- 2 
 is — V2 ; and if n be any odd integer that of \/— a^ is — a. 
 
 205. A property of positive numbers. Two positive 
 7iumhers are equal if any like powers of these numbers are 
 equal. 
 
 For, a positive number has one and only one positive 
 root, namely, its principal root. 
 
 Thus, if a is positive and a^ = 3^, then a is equal to 3, since a is the 
 principal cube root of a^, or of 27. 
 
 206. Notation. In what follows in this chapter the 
 letters a, 5, c, and so on, will represent positive numbers 
 or literal expressions which have positive values, except 
 when otherwise stated. By the root of a number we shall 
 mean its principal root ; that is, its one positive root when 
 the radicand is positive and its one negative root when 
 the radicand is negative and the index an odd integer. 
 [See section 204.] 
 
 Thus, 'v^ = + 3, and \/^^27 = - 3. 
 
 207. Surds. A surd is an irrational number which is a 
 root of a rational number. 
 
 Thus, V5 and V3 are surds ; V4 and v^27 are radical expressions, 
 but they are not surds. 
 
 208. Order of a surd. The order of a surd is the index 
 of the root involved in the expression. 
 
 Thus, V2 is a surd of the second order, or a quadratic surd, y/o is a 
 surd of the third order, or a cubic surd. 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 267 
 
 209. Fundamental identities involving roots. Radicals 
 are transformed and combined according to certain rules. 
 These rules may be expressed in the form of algebraic 
 identities. 
 
 (%)" = a (I) 
 
 Identity (I) is simply the definition of a root as ex- 
 pressed in algebraic symbols [section 199]. 
 
 EXERCISE 95 
 
 State at sight the value of each of the following: 
 1. (V2)2. 2. (V3)2. 
 
 3. (V^)2. 4. (^2)3. 
 
 5. (-^31)3. 6. (</5y. 
 
 7. (^310)6. 8. (^/^^)7. 
 
 9. (-s/ly. 10. (</2iy. 
 
 11. (-y/a+by. 12. (Va2_62)3. 
 
 13. (^V^a-^byy, 14. (^/(a + byy. 
 
 15. VT . VT. 16. Vi • Vi. 
 
 17. V^ X ^^ X ^"=7. 18. Vx-^/x- </i. 
 
 ^^='V^P. (II) 
 
 From identity (II) we may infer that : 
 
 The value of a radical is not changed if its index and the 
 exponent of its radicand are both multiplied by the same posi- 
 tive integer. 
 
 Conversely, we may write identity (II) thus: 
 
 From the second form of identity (II) we have : 
 
 The value of a radical is not changed if its index and the 
 
 exponent of its radicand are both divided by the same positive 
 
 integer which is a factor of each. 
 
268 ELEMENTARY ALGEBRA 
 
 Note. An important particular case of § 209, I, is expressed by 
 the identity, 
 
 Thus, Va^P = ^(apy = op. 
 
 Similarly, va* = a^, and Va^ = a*. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Vo^ = Va^ and conversely, V«^ = Va^. 
 
 2. Va^ = a, and conversely, a = Va^. 
 
 3. ^/a3^= a"*, and conversely, a"» = ^/c^. 
 
 The proof of identity (II) is as follows ; 
 
 By identity (I), (Va^)'*^ = 0*"^. 
 Also, by identity (III) of § 198, 
 
 = (a-)p [§209 (I)] 
 
 = a^p. [§198(111)] 
 
 We have now shown that the (w^)th power of the ex- 
 pression in the first member of identity (II) is equal to 
 the same power of the expression in the second member. 
 It follows by the principle of section 205 that the two 
 expressions are equal. 
 
 EXERCISE 96 
 
 (Solve as many as possible at sight.) 
 Simplify the following: • 
 
 1. ^/A 2. -v^. 3. ^'^. 
 
 4. ^(2)8. ' 5. </(a + by. 6. ^/(c-j-dy^, 
 
 7. ^3^ 8. ^/^^r^^. 9. ^^T^. 
 
 10. V-(a + 6)6. 11. .J/^. 12. </(a + by. 
 
 13. \/(a - 6)16. 14. ^^. 15. </a^. 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 269 
 
 Vab=\^a^i. (Ill) 
 
 From identity (III) we infer that : 
 
 An^ root of the product of two or more factors is equal to 
 the product of the like roots of the factors. 
 
 Note. In identity (III) there are two factors and in the statement 
 of the principle that follows there are tioo or more factors. Here, 
 as in all similar cases, what is true of the product of two factors is 
 true in general. This follows directly from the fact that by grouping 
 factors any product may be expressed in the form (a&). 
 
 Thus, in the case of four factors, 
 
 VaM= y/{ah){cd) 
 = VabV^ 
 = (Va Vb) {V~c Vd) 
 — yja Vh Vc yjd. 
 
 Writing the converse of identity (III), we have, 
 
 This second form of identity (III) may be stated in 
 words as follows: 
 
 The product of like roots of two or more numbers is equal 
 to the like root of their product. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. V8 = V4 x2 = V4 V2 = 2V2. 
 
 2. 
 
 ^/54 = V 27 X 2 = a/33 ^2 = 3^/2. 
 
 3. V3a2= VaW3 = aV3: 
 
 4. ^4^2 = ^= ^P = 2. 
 
 5. VaVa^Va^ = -^a x a^ x a^ = \^a^ = a. 
 
 In order to prove identity (III) we show that the nth 
 power of the positive number expression in the first mem- 
 ber is equal to the same power of that in the second. 
 
270 ELEMENTARY ALGEBRA 
 
 Thus, _ 
 
 By identity (I) (y/ahy = ah. 
 
 Also, by identity (II), section 198, 
 
 {VaVby = {y/aY{y/hy, and by section 209, (1), 
 = ah. 
 Since (v^aft)" and {Vay/hY are each equal to ah, it follows that 
 y/ab = V~^Vh. 
 
 EXERCISE 97 
 
 (Solve as many as possible at sight.) 
 Simplify : 
 
 1. VI2. 2. V27. 3. V32. 
 
 4. V128. 5. V2^. 6. V3^4. 
 
 7. V4a3. Suggestion. Vi^s ^ V(4 a2)a. 
 
 8. V4^. 9. V8^3. 10. VT28"^. 
 11. V^. 12. V47m2. 13. Va25M#. 
 14. V72^. 15. V80 ^254. 16. V99^». 
 
 17. V127008. Suggestion. 127008 = 25.34. 72. 
 
 18. V84672. 19. V27 a6V. 
 20. V81 X 5 X «355. 21. V75%2. 
 22. V176 «2j2^4#. . 23. V2(^l^y. 
 
 24. V50(a+5)2(c+(^)4. 25. Vl08a2 (5+^)3. 
 
 26. ^/T6. 27. ^54. 
 
 28. ^2^128. 29. \/625. 
 
 30. ^81. 31. ^S/STS. 
 
 32. ^128l3. 
 
 33. \/— 16. Suggestion. \/^^^T6 = - \/2(2)8. 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 271 
 
 34. V - 54. 35. V - 216 2^, 36. V9 a^y^z^. 
 
 37. V-125a356. 38. ^/U. 39. V64a:5^io. 
 
 40. ^^^. Suggestion, v^^^ = - v/2 [§ 204]. 
 
 41. -s/"^^. 42. x/"^. 43. ^/- 32 2^3^. 
 44. ^^. 45. ^-X^^yz^. 46. ^-2y. 
 
 47. ^-2:10. Suggestion, ^^^^^o = v^(- l)6a;io. 
 
 48. ^32^6. 49. I^^4. 
 
 50. v/?2. 51. ^214 • «!*. 
 
 52. v'- 243^667. 53. ^(> + i)^(<? + dy^. 
 
 54. V28 . 39 . aV^io. 55. </-(^x + yy{x-yy. 
 
 i^ar^Va^. (IV) 
 
 From identity (IV) we- infer that : 
 
 Any power of a radical is obtained hy rkultiplying the 
 exponent of the radieand hy the exponent of the required 
 power. 
 
 Conversely, since Va is one of the m equal factors of 
 (Va)*^, or of its equal -v/o^, it follows that \/a is the Twth 
 root of -v/a^. Hence : 
 
 Rule. To find the root of the radical^ the exponent of 
 whose radieand is exactly divisible by the index of the root,, 
 divide the exponent of the radieand by the index of the re- 
 quired root. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. (</8)3 = (</23)3 = </29 = ^28~x^ = </28 </2 = 22^2 
 = 4</2. 
 
 2. (\^)6 = ^= 42 = 16. 
 
 3. \/^=V^ = a. 
 
272 ELEMENTARY ALGEBRA 
 
 The proof of identity (IV) is as follows : 
 By identity (I), section 209, (v^a«)»= a« 
 Also, by identity (III), section 198, [(v^a)*]" =[(\/a)»]'« = a'". 
 Since (\/a*»)« and [(%)'«]« are each equal to a"*, it follows that 
 (%)"» = v^^[§205]. 
 
 EXERCISE 98 
 
 In simplifying the following, use as many of the pre- 
 ceding identities of this chapter as may be necessary. 
 
 1. (</49)2. 2. (V2)4. 
 
 3. (V3)6. 4. (Va)4. 
 
 5. (V3^3)8. 6. (V2^)4. 
 
 7. (V3^)6. 8. (V3^)5. 
 
 9. (V^ay, 10. (\/^^2^)io. 
 
 11. (-^9)4. 12. [\/2(a+5)]2. 
 
 13. [■^-3((? + C^)2]2. 14. (^/2^)16. 
 
 15. (_^3TS)15. 16. (-^'3T^2)6. 
 
 17. \/Vi. 18. V'V9T2. 
 
 19 
 
 . v^V—S. 20. \/-</^32. 
 
 
 (V) 
 
 From identity (V) we infer that : 
 
 Ani/ root of a fraction is equal to the like root of the nu- 
 merator divided by the like root of the denominator. 
 Conversely, we may write identity (V) thus : 
 
 V6 ^fr' 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 273 
 
 From this second form of identity (V) we infer that: 
 Any root of a number divided by the like root of a second 
 number is equal to the like root of the fraction whose numera- 
 tor is the first number and whose denominator is the second. 
 
 ILLUSTRATIVE EXAMPLES 
 
 3f^ ■^/'^ a2 
 
 2. 
 
 ^6l2 .yp b^ 
 
 2^ = ^/1 = J5=:^=:^ 
 
 • V3 ^3 ^9 V9 3 * 
 
 <^3 ^ /3 ^ 3/12 ^ ^12 ^ </T2 
 
 * ^2 ^2 ^8 ^8 2 ' 
 
 The proof of identity (V) is as follows : 
 By identity (I), § 209, (^^y = |. 
 By identity in example 33, p. 264, 
 
 Since f -^- J and (■;;—) are each equal to -, it follows that. 
 
274 ELEMENTARY ALGEBRA 
 
 EXERCISE 99 
 
 Reduce the following to equivalent fractions having 
 rational denominators [see illustrative examples 5 and 6, 
 p. 273] : 
 
 4. 
 
 7. 
 
 10. 
 
 13. 
 
 16. 
 
 19. 
 
 22. 
 
 25. 
 
 128. 
 
 From identity (VI) we infer that : 
 
 Any root of a root of a number is equal to that root of the 
 number whose index is the product of the given indices. 
 The converse of identity (VI) shows that when the index 
 
 V3 
 
 V2 
 
 2. 
 
 vj. 
 
 3. 
 
 .V|. 
 
 vj. 
 
 5> 
 
 vj. 
 
 6. 
 
 V7 
 V8 
 
 V3 
 
 V5 
 
 8. 
 
 vTi 
 
 V5 
 
 9. 
 
 VI- 
 
 ^a 
 
 11. 
 
 VS- 
 
 12. 
 
 VA- 
 
 yJ^". 
 
 14. 
 
 </2 
 
 15. 
 
 ^s- 
 
 n. 
 
 17. 
 
 
 18. 
 
 
 V?- ■ 
 
 20. 
 
 
 21. 
 
 \j6 
 
 Va6 
 Va 
 
 23. 
 26. 
 29. 
 
 ■V2a? 
 V3a 
 '116 a* 
 
 24. 
 27. 
 30. 
 
 ^86« 
 
 sj-Sa^^ 
 ^ 125^9 
 
 5/32 aW» 
 >'243c6 
 
 /(a + 6)8^ 
 
 ^ie + dy 
 
 
 
 V 
 
 Va='"</a. 
 
 
 (VI) 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 275 
 
 of a given root is a composite number, it is possible to ex- 
 press the given root in terms of simpler roots. 
 
 Thus, \/625=VV625=: \/25=5; also, v^ = Vv^= V^=aV5. 
 ILLUSTRATIVE EXAMPLES 
 
 1. -Wa=</a 
 
 2. -^(a4-^)2 = ^^T^; [§ 209 (II)] 
 or, -^(a+by = \/V(a + 5)2 = -^^:^. [§ 209 (VI)] 
 
 Remark. Since VVa="^a 
 
 and VV^^'^^a, 
 
 it follows that V^ = V^. 
 
 Thus, V^2=v/V5 = ^. 
 
 Also, V V8 a6^»8c9 = V v^8^^6p^ = V2 a%c^ 
 
 = y/a^V2bc = ac^/2bc. 
 
 Remark. The proof of identity VT is left as an exercise for the 
 student. See section 205. 
 
 210. Simplification of radicals. A surd whose radicand 
 is integral is said to be in its simplest form when the index 
 of the radical is as small as possible and when no factor of 
 the radicand has an exponent which is exactly divisible by 
 the index of the root. 
 
 Thus, V2a is in its simplest form : v a^ is not in its simplest form 
 since Va^ = Va^ x a = Va'^Va — « Va ; vTT^ is not in its simplest form 
 since by section 209 (II), Va* = >Ja. 
 
 From identities (HI) and (11), note, section 209, we 
 derive the following rule for simplifying surds : 
 
 Rule. Make the index of the radical as small as possible, 
 then if the exponent of aiiy factor of the radicand is divis- 
 ible by the index, divide the exponent of that factor by the 
 index and remove the factor from under the radical sign. 
 
 Thus, Vbia^-^c^ = y/2x ^^a%^c^a% = 3 a%^c </2~^, 
 
276 ELEMENTARY ALGEBRA 
 
 Conversely, any factor of the coefficient of the radical 
 may be brought under the radical sign and be made a 
 factor of the radicand provided that its exponent be 
 multiplied by the index of the radical. 
 
 Thus, 2 Vd^ = y/WxSTi = Vl2^. 
 
 Any surd whose denominator is irrational is not in the 
 simplest form for the approximate numerical calculation 
 of its value. This may be shown by a particular example, 
 as follows : , 
 
 Find an approximate value of '^-. 
 Solution (1) ^| = V5 = ^ = ?f? = .816^ 
 
 Solution (2) A(| = ^ = l^=•816^ 
 
 Solution (3) ^l = ^l = ± = ^^.MeK 
 
 The actual work required in solutions (2) and (3) is greater than 
 that required in solution (1). The most convenient method of cal- 
 culation, therefore, is that which proceeds by first making the denom- 
 inator of the fraction rational. 
 
 211. Simplest form of a radical expression. A radical 
 expression is said to be in its simplest form when its de- 
 nominator is rational and all integral radicands which it 
 contains are reduced to their simplest forms. 
 
 212. Simplification of a simple fractional surd. A surd 
 
 whose radicand is a fraction in its lowest terms is simplified 
 by multiplying the numerator and the denominator of the 
 radicand by the simplest expression which will make its 
 denominator rational. 
 
 Thus ^^^^ ^ « / "TT ^ i/ 3"^(5^ ^ v^lS^P ^ \/l5^ 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 277 
 
 213. Similar, or like, radicals. Two radicals which, 
 when expressed in their simplest forms, differ only in their 
 coefficients, are said to be similar, or like. 
 
 Thus, V8a^ and — Vl8 ab^ are similar, since their simplest forms, 
 2 aV2ab and — 3 6\/2 ab, differ in their coefficients only. 
 
 EXERCISE 100 
 
 1. Show that V12, V27, v|, VJ, are similar. 
 
 2. Arrange the following radicals in sets so that those 
 in the same set shall be similar : 
 
 ^2ab^; V4a353; yj^-^; ^9a%; W9a%; ^^-^. 
 
 3. Write the coefficient as a factor of the radicand in 
 each of the following: 2V5; 3V2; aV3; 2aV3; 
 
 Suggestion. 2V5 = V22x 5 = V20. [§210.] 
 
 4. Simplify V(a2 — b'^){a + 6), in which expression a is 
 greater than b. 
 
 5. Which of the following radical expressions are surds : 
 
 V2? ^/S? VlH-V2? \/Vl? 
 
 Reduce each of the following radicals to its simplest 
 form : 
 
 6. V72a62c3. 
 9. ^/W¥f. 
 
 12 
 
 ^ y 
 
 15. y/^x^y^-^xh^ 
 
 7. </-82. 
 
 10. V^. 
 
 8. 
 11. 
 
 14. 
 
 -y 
 
 ^ X 
 
 13. \/32*yz«. 
 
 ^X 
 
 Va:^ ^ 2:2^, 
 
278 ELEMENTARY ALGEBRA 
 
 18. VWMK 
 
 17. 
 
 ^ 27 aS 
 
 19. 
 
 Va2«. 
 
 21. 
 
 Va4«+i. 
 
 23. 
 
 Va2»+i. 
 
 25. 
 
 V?- 
 
 37 
 
 Jl?-Z + 1 
 
 
 ^ ix+iy 
 
 29. 
 
 (<^)2. . 
 
 31. 
 
 2 
 
 20. Va2«+i. 
 22. Va;3 + 2^3/. 
 24. Va2"+i2:. 
 
 26. 
 
 ^9^. 
 
 3 
 
 V3* 
 1 
 
 ^8' 
 
 30. 
 
 32. 
 
 Given V2 = 1.414, V3 = 1.732, V5 = 2.236, calculate 
 to two places of decimals the values of the expressions in 
 examples 33, 34, and 35. 
 
 33. ^' 34. Vf 35. V|. 
 
 V3 
 
 36. Simplify V7 x VT x V6 x V2 -s- V3. 
 
 37. Simplify V63 x VT68 -i- V27. 
 
 38. Simplify Vl4 x VT5 -^ VM. 
 
 39. Simplify V35 x V6 -5- V30. 
 
 40. Simplify V12 x V40 x V20 x V48 x V24. 
 
 41. Simplify (^/^)9x(VF)6x(</^)8. 
 
 214. Comparison of surds. For certain purposes it is 
 convenient to express two or more surds in terms of like 
 roots, as, for example, when finding which of two given 
 surds is the greater. 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 279 
 
 Thus, in finding which of the two surds Vo or v^7 is the greater, 
 we make use of identity (II), section 209, to transform both expressions 
 into surds of the same order as follows : 
 
 It is now evident that y/b is greater than V7. 
 
 EXERCISE 101 
 
 1. Express V5 and ^10 as surds of the same order. 
 
 2. Which is greater, V5 or VS? 
 
 3. Which is greater, V3 or V28? 
 
 4. Arrange 2\/27, 3\/4, 2V6 in order of magnitude. 
 Suggestion. Place the coefficients under the radical signs. 
 
 5. When two or more surds are reduced to equivalent 
 surds having a common index, what is their lowest com- 
 mon index? 
 
 215. Addition and subtraction of surds. The algebraic 
 sum of two or more surds is expressed in the simplest 
 form when each surd is in its simplest form and all similar 
 surds are combined by adding their coefficients. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Add 5V8, -4V50, 3V72, -2V98, and VT28. 
 
 Solution. The sum of 5V8, - 4V50, 3V72, - 2V98, and Vl28 
 = 10V2 - 20V2 + 18V2 - 14V2 + 8 VI 
 = (10-20 + 18-14 + 8)V2 
 = 2V2. 
 
 2. Simplify 2Va5^- V9"^2_ yf^^^ gVPi^. 
 
 Solution. 2 y/ab^ - V9 ahy^ - V4 a^xy + 3 x/P^ 
 
 = 2 xVab — 3 yy^ah — 2 ayJxy + 3 h\/xy 
 = (2x-3y)y/ab-(2a-'db)y/xy. 
 
280 
 
 ELEMENTARY ALGEBRA 
 
 Note. The sum of two unlike surds cannot be expressed as a 
 single surd ; such a sum can only be indicated. 
 
 Thus, the expression \/2 + VS is in its simplest form. 
 
 Remark. Particular attention should be called to the fact that 
 V3 -f V2 is not equal to V3 + 2, or VS. 
 
 EXERCISE 102 
 
 Simplify each of the following expressions: 
 
 1. 3V6-2V6 + 5V6. 2. 
 
 3. V50-fV32-VT8. • 4. 
 
 5. 3V60-V240-2V15. 6. 
 
 7. a/250 - 2^16 + \/5¥. 8. 
 
 9. VT8-|V2+V^. 10. 
 
 11. V50 4-VT08+V98 + 3VT2. 
 
 V5i-V24 + Vl50. 
 
 vT5-V27-2Vl2. 
 V9^- VTSP + Va?. 
 
 -i5,_V27+V|. 
 
 V3 ^ 
 
 12. 2V90-V176 + 
 
 V160. 
 
 13. 
 14. 
 
 15. 
 17. 
 
 V4M- V9 ab^-{- 2 V4 ae^ - V9 ab^ + V4 aS-j- V16 ahc\ 
 \ V9^6 - 2 J^- 3x/'? + ^^^^. 
 
 2^4 + 2-\/32-9^I. 16. 2^/4- 5^32 + 3\/108. 
 VT8-A/24. " 18. ^6 + ^1/4-^1. 
 
 19. V6W|-4Vy4-VJf. 20. I^^ + JV^I^. 
 21. H^l'^^^-I^l-I^A- 22. a/T62^ + 2-v/32^. 
 23. x^l5f--V4S¥f-\-xi/V27. 
 V'3^ - 2 V3^ + V3^. 
 
 24 
 
 -• >g-xa+%/i- 
 
 26. 
 27. 
 28. 
 
 ■v/8 + 24 a + VsToTW. 
 
 Vl2^^^^S¥-\- V27 ftV + 18 62 - V48 (^x^-{- 32 c^. 
 3 V3^ - V48^ - V3 a3 4- 6 a2 4- 3 a. , 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 281 
 29. (a -f hy^ab + (a - hyVab - (a2 + 5^) V^. 
 
 30. V2:3_2a:2_V4a:-8-V(a: + 2)(a:2_4). 
 
 216. Multiplication and division. The rules for the 
 multiplication and division of monomial surd expressions 
 are derived from identities (III) and (V), section 209; 
 namely, from the identities, 
 
 VaV5 = ^ah and -;r^ = \t' 
 
 Remark. Observe that these identities contain surds of the same 
 order only. Hence, before two or more surds are combined by mul- 
 tiplication or division, they must, when necessary, be reduced to surds 
 of the same order. See section 214. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. 5 V20 X 3 V45 = 10 V5 x 9 V5 = 90( V5)2 = 450. 
 
 2. 2V^x5v^^ = 2^^3x5-y^454^10^^^ 
 
 = lOah^/ab. 
 
 3. V6^V5=x/^ = 
 6 
 
 
 
 EXERCISE 103 
 
 Simplify each of the following expressions: 
 1. V5xVT0. 2. VlO^VS. 
 
 3. axVa^ x hy^l^. 4. V6 X V2. 
 
 5. V'6 ^ Vl2. - 6. axy/'c^ -j- hyVa. 
 
282 ELEMENTARY ALGEBRA 
 
 7. V5xV20. 8. V90-hVT0. 
 
 9. Va X V6. 10. 2Va X 3V6. 
 
 11. 2^/3x3^/9. 12. ^320-^\/5. 
 
 13. -v^24-^-v/3. 14. 2a/64-4^/8. 
 
 15., ^18x-n/3. 16. </|x^I 
 
 17. V3x\/2. 18. V2xa/3. 
 
 19. a/4xV8. 20. ^x^SPx^^^. 
 
 21. V^lix-s/S. 22. Vi-V9. 
 
 23. Vf-f-Vf. 24. 1-^-5-1^. 
 
 25. 20-5-V|. 26. 8V^-!-4^^. 
 
 27. ^m^2 ^ V8^3. 28. . ^MZ ^ A^ 
 
 29. Va6-i-3V6c. 30. m-^Vn. 
 
 217. Multiplication of polynomials containing surds. 
 
 The product of two polynomials involving surds is found 
 by direct application of the distributive law for multipli- 
 cation ; the operation differs in no respect, from that 
 required in the multiplication of rational integral poly- 
 nomials. Each term of the resulting product must be ex- 
 pressed in its simplest form. 
 
 ILLUSTRATIVE EXAMPLE 
 
 Multiply 2 4- 3 V5 - 2 V2 by 3 V5 - 4 V2. 
 
 Solution. 
 
 2 + 3V5-2V2 
 3V5_4V2 
 6\/5 + 45-6VlO^ 
 
 + 16 - 12V10 - 8\/2 
 6V^ + 61-18VlO-8\/2. 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 283 
 
 Remark. Observe that it is necessary to simplify the terms of 
 each partial product in order that similar surds may be written in 
 the same column. 
 
 EXERCISE 104 
 
 Simplify each of the following expressions : 
 
 1. (2 + V5)(2-V5). 2. (a + 2V3)(2-fV3). 
 
 3. (3-2V2 + V3)(V6+V2). 
 
 4. (aV5 + 6V^)(Va+ V6). 
 
 5. («+ V5-f \/^)(a- V"^+ Vc). 
 
 6. (V^-hV3)2. 
 
 7. (VS_V5)2-(Va + V^)2. 
 
 8. (aVb-b^/ay. 
 
 9. Find the product of 3V2 - V3 and V2 + 3V3. 
 
 10. Find the product of 9 V3 + 2 and 2 V3 H- 9. 
 
 11. Find the product of V2 - V3 + 1 and V2 + V 3 + 1. 
 
 Find the square of : 
 
 12. V3-1. 13. 2 + V2. 
 14. 3V3-V2. 15. \^-2V2. 
 
 Find the cube of : 
 
 16. V3-1. 17. 2 + V2. 
 
 18. 3V3-V2. 19. V5-2V2. 
 
 20. Simplify (1 + 3V3)(9 - V2)(V2 + V3)(9 + V2) 
 (3\/3-l)(V2-v'3). 
 
 218. Conjugate surds. Two binomial quadratic surd 
 expressions which differ in sign of a surd term only are 
 called conjugate surds. 
 
 Thus, 1 + V2 and 1 - V2, also V2 + V3 and V2 - V3 are conju- 
 gate surd expressions. 
 
284 ELEMENTARY ALGEBRA 
 
 219. Product of conjugate surds. The product of two 
 conjugate surds is rationaL 
 
 Thus, (Va + Vb)(Va - y/b) = (V^y - (\/&)2 = a - b. 
 
 Note. When the product of two expressions which contain irra- 
 tional numbers is rational, either expression is called a rationalizing 
 factor of the other. 
 
 ILLUSTRATIVE EXAMPLE 
 
 Rationalize the denominator of the fraction 
 
 1+V2 
 V5 + V2' 
 
 Solution. Since the product of two conjugate surds is rational 
 [section 219], we multiply the numerator and denominator of the frac- 
 tion by a conjugate of the denominator. Then we have, 
 
 l-fV2 ^ (l + V2)(\/5-V2) 
 V5-fV2 (>/5-fV2)(V5-V2) 
 
 ^ V5 -I- Vio _ V2 - 2 
 3 ' 
 
 EXERCISE 105 
 
 Rationalize the denominator of : 
 
 2. 
 
 14.V2 2-V2 2 + V2 
 
 V3-V2 V3-V2 2-Va; 
 
 X . 1-V5 
 
 m 
 
 n + Vw -y/x + V^ 1 + V5 
 
 2 + V3 „ V3+V2 „ yg + v^ 
 
 • 2-V3 V3-V2 Va-V5 
 
 13. _^2_ ^^ 4±V3 ^3 ^-2_ 
 V5-f-V7 4~V3 V12 + V8 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 285 
 Find the sum of : 
 16. _^ + ^ . 17. 9V3 2V2 
 
 V3 + 1 V3-1 * 2V3-f3 3V2 + 2 
 
 Suggestion. Rationalize the denominators. 
 
 18. 4.-^— L. 19. -2=4- ' ^ 
 
 V27 V48 Vl2 V60 Vl5 Vl35 
 
 Rationalize the denominator of : 
 
 4 Suggestion. Multiply both terms by 
 
 20 
 
 2-(-V3-fV5' 2+V3-V5. 
 
 21. 12 ^ 33^ 1 
 
 V2 4-V3+V5 V2 + V5-V8 
 
 220. Division by polynomials containing surds. To 
 divide an expression by a polynomial containing one or 
 more surds, the dividend should be written as the nu- 
 merator and the divisor as the denominator of a fraction, 
 which should be transformed into an equivalent fraction 
 with a rational denominator. 
 
 EXERCISE 106 
 
 Perform the indicated divisions in the following : 
 
 1 « V2 , V3-V2 
 
 3. 
 
 V2-1 V2-I-1 V3-f-V2 
 
 1 , 2-hV3 ^ 3-V5 
 
 V5-V3 1 + V3 V10-V6 
 
 7. Find the quotient of (V2 + V6)-5-(l -}- V3) with- 
 out rationalizing the denominator. 
 
 8. Mention at sight the numerical value of 3V2+y ^ 
 
 1+V2 
 
286 ELEMENTARY ALGEBRA 
 
 V2 + V4-v^ 
 
 H.V2-V3 
 
 9. State at sight the value of 
 
 10. Simplify V3-f '^^~^ » 
 ^ ^ 14-V3 
 
 "• -p>«' (^T-e-^T- 
 
 12. Divide (3 - V2)2 + 1 by 3 - V2. 
 
 13. Divide (V3 + V2)2+l by V3-hV2. 
 
 14. Find the value of ^-^^^-^^-^ ^hen a; = 1 + V2. 
 
 X 
 
 221. Fractional and negative exponents. The following 
 identities from section 198 had a meaning only when the 
 exponents were positive integers. 
 
 «"».«" = «"*+". (I) (a5)'" = aH"^. (II) 
 
 («"»)« = («")»« = a"*". (Ill) (a"^5")p = a*"Pft"p. (IV) 
 
 In section 62 the identity 
 
 — = a»»-", in which m>n (V) 
 
 was shown to result directly from the definition of divi- 
 sion, and in section 63 a meaning was given to the 
 expression a9, where a denotes any number different from 0. 
 That is, it was shown that, 
 
 It is convenient to extend the meaning of the word 
 exponent so that this term shall include, in addition to 
 positive integers, all other rational numbers. 
 
 It is evidently desirable that all exponents should com- 
 bine according to the same laws, and hence to define 
 negative and fractional exponents so that the identities of 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 287 
 
 section 198 may be satisfied for all rational values of w, 
 w, and p. We shall proceed to show that such definitions 
 can be given. In what follows, any base, as a, is supposed 
 to be different from 0. 
 
 222. Definition of a'"". If the identity «»".«»= a^+", 
 in which m and n are positive integers, is also an identity 
 when n is replaced by — w, we shall have 
 
 ^m . ^-m __ ^m+(-m) _ ^0 _ J^ 
 
 We therefore define a~^ by the equation a^ . a~^ — 1 ; 
 hence, a~^ and al^ are reciprocals ; that is, 
 
 cr'" = — ; also, O" = 
 
 It 
 
 223. Definition of af. If the identity («*")*» = a*"", in 
 which m and n are any positive integers, is also an identity 
 
 when m is replaced by ^, where p and q are positive 
 integers, we shall have, by taking n equal to ^, 
 
 p 
 Therefore a^ must represent a number whose qth power 
 
 is equal to a^. Now the principal qth root of a^ is such a 
 
 p 
 number. We therefore define the symbol a^ by the 
 
 equation 
 
 0? = -?/^. (I) 
 
 In this identity, Va^ represents the principal 5th root of 
 aP. In particular, by making j3 equal to 1, 
 
 a^ = Va, (II) 
 
 Since p and q in identity (I) represent any positive 
 integers, it is admissible to change them in this identity to 
 
288 ELEMENTARY ALGEBRA 
 
 pm and qm^ respectively; we then have from identity (I) 
 
 pm 
 
 But, V^=^^ [§209 (II)] 
 
 p 
 
 pm p 
 
 a'""=^a'^. (Ill) 
 
 p 
 From identity (III) it is evident that the value of a^ is 
 
 not changed when £ is replaced by an equivalent fraction. 
 
 9 
 
 p 
 
 Note. In the expression a', the numerator p indicates a power 
 and the denominator q a principal root. In some cases it may be 
 desirable to take the pth. power of the principal ^th root of a and in 
 others to take the principal ^th root of the joth power of a. That, 
 in both cases, the results are the same is evident from the known 
 identity, section 209, IV, 
 
 5 
 
 Thus, in simplifying 8^, either of two solutions may be given, the 
 first being preferable. 
 
 Solution 1. 8' = (8^)6 = (2)5 = 32. 
 
 Solution 2. 8^ = (85)* = K^^yi^ = (2i6)* = 2^ = 32. 
 
 In simplifying 2^ it is evident that the method followed in solution 
 2 is preferable ; that is, 
 
 2^ = v^22 = H. 
 In simplifying 2^ it is preferable to proceed as follows : 
 2^ = 2^"^^ =2 X 2^ = 2v^4. 
 
 _p _p 
 
 224. Definition of a '^ . From section 222, a * must be 
 defined by the identity 
 
 -^ 1 
 
 a </ = 
 
 p 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 289 
 
 225. It will be found that with the foregoing definitions 
 of negative and fractional exponents, identities (I), (II), 
 (III), (IV), and (V), section 221, now hold for aU 
 rational values of w, n^ and p. We shall agree to enlarge 
 the meaning of the word " power " in accordance with these 
 definitions. For example, by the expressions the " one- 
 third power of a" or "a to the one-third power," we 
 
 shall mean the number a^; that is, -^a. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Simplify a^ xa^. 
 
 Solution. a^ X a^ = a^'^^ = a^. 
 2. Simplify ^. 
 
 [§§225,221(1)] 
 
 
 Solution. ^=x^~^=x. 
 
 [§221(V)] 
 
 JC 
 
 3. Simplify (^ah^y. 
 
 
 Solution. {ah^y = (a*)«(6^)6. 
 
 [§ 221 (H)] 
 [§221 (III)] 
 
 4. Write \-^ without using the radical sign. 
 
 Solution. ^}^ ={ ^\ • 
 ^2^y \x»yJ 
 
 [§ 223 (11)] 
 
 5. Simplify mhi-^ -;- m~%*. 
 
 
 Solution. ?n^-^ - m-8n* = !^ ^ ^ = ^. 
 
 n* m* n« 
 
 [§222] 
 
 6. Simplify Q 
 
 
 Solution. (M^Jlii- 1 _ 1 = 
 
 xy 
 
 [Exercise 94, problem 33] 
 
290 ELEMENTARY ALGEBRA 
 
 '• «-^'"' ©' 
 
 Solution. (S?^]^ = («^^^)^ = ^ = MJJ^ 
 
 [Exercise 94, problem 33] 
 
 8. Simplify ^'-^. 
 
 Solution. ^= ( J)*=(,-i)i^(,^)i ^ ,i ^ e,,. 
 
 9. Simplify (^-LW^X^^). 
 Solution i'yP-) (^ ^ ^^) = (4-) ^V 
 
 =^ 
 
 a-iV^ 
 
 =2t 
 
 ia:T^ 
 
 Z X 
 
 10. Multiply a;^ + x^y^ + ?/^ by a;* — y^. 
 Solution. x^ + x*y^ + y^ 
 
 J- 
 
 ,v* 
 
 
 
 J + 
 
 
 xV 
 
 -,v* 
 
 .i 
 
 
 
 -y* 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 291 
 11. Divide x — yhy x* — y^. 
 
 Solution. x^ — y^)x - y \x^ -\- x^y^ + y* 
 
 X — x^y^ 
 
 2 1 
 
 x^y'^ - y 
 
 x^y -y 
 
 X,. Simplify 1 V^.^H.-M)» 
 
 x~^ — y~^ X — y 
 
 Solution. — -, : = T — =• = — — 
 
 x~^ — y~^ 1 _ 1 y — X 
 
 X y 
 
 \X^J X^ 
 
 X^ ^ 
 
 y 
 
 1 y/x^y-^{x 2^2)8 _ a:y y^ x 
 
 p-i_ y-i a- _ 2, ~ y - X x - I 
 
 ^ xy ^ xy ^ xy xy ^ q^ 
 
 y - X X- y y -X y - x 
 
 Check. Let a: = 4 and y = 1] then, 
 
 = -1 + 1 = 0. 
 
292 ELEMENTARY ALGEBRA 
 
 EXERCISE 107 
 
 (Solve as many as possible at sight.) 
 
 Write without negative or fractional exponents, and 
 simplify when possible : 
 
 
 1. 
 
 a-\ 
 
 
 Thus, a-2 = ^. 
 a* 
 
 
 
 2. 
 
 4i 
 
 
 Thus, 4^ = + V4 = 2. 
 
 
 
 3. 
 
 8l 
 
 
 Thus, 8^=(\/8)2 = 22 = 4. 
 
 
 
 4. 
 
 8-^. 
 
 
 T.^8-*=J=i. 
 
 
 5. 2-2. 
 
 
 6. 
 
 9*. 
 
 7. 16i 8. 
 
 4^. 
 
 9. x-^. 
 
 
 10. 
 
 9i 
 
 11. 16^. 12. 
 
 27t. 
 
 13. a;-". 
 
 
 14. 
 
 8-1 
 
 15. 25-i 16. 
 
 64-*. 
 
 17. What verbal statement is expressed in symbols by 
 the formula a*"a" = «"*+" ? 
 
 18. What verbal statement is expressed in symbols by 
 
 the formula — = a*""" ? 
 a" 
 
 19. What verbal statement is expressed in symbols by 
 the formula («"»)"= a*"" ? 
 
 Express without fractional exponents : 
 
 20. ai 21. rpi 22. y^. 
 23. 3a^. 24. 2a;i 25. 4yi 
 26. (3a)i 27. (2 a;)*. 28. (4y)*. 
 29. a^. 30. 2ai 31. (2 a)*. 
 32. a^a;. 33. a^xy^. 34. 2a^a:i 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 293 
 
 1^ 
 35. a"*. 
 
 
 36. 
 
 n+1 
 
 37. 
 
 2y. 
 
 38. 2hxl 
 
 m 
 
 
 39. 
 
 a-b 
 
 40. 
 
 m 
 
 ax^ 
 3 * 
 
 41. ^^". 
 44. a^ + bhK 
 
 
 42. 
 
 Cx-;,-)i. 
 
 43. 
 
 2a¥. 
 
 
 45. 
 
 ©' 
 
 46. 
 
 m— n Bi+n 
 
 Express without the radical signs : 
 
 
 
 47. </i. 
 
 
 48. 
 
 </a. 
 
 49. 
 
 ^?^. 
 
 50. 4^. 
 
 
 51. 
 
 3Vi. 
 
 52. 
 
 5^J. 
 
 53. ^/Sa. 
 
 
 54. 
 
 V7 2:. 
 
 55. 
 
 ^5"c. 
 
 56. V42:3. 
 
 
 57. 
 
 V2a3. 
 
 58. 
 
 2VP. 
 
 59. v21xK 
 
 
 60. 
 63. 
 
 aVa;. 
 Vx, 
 
 61. 
 64. 
 
 as/he. 
 
 62. 7Vi^. 
 
 V2 a'»+». 
 
 65. V3a:3. 
 
 
 66. 
 
 e'</4x^, 
 
 V2 
 
 67. 
 
 ""■v/a-+». 
 
 68. |VF. 
 
 
 69. 
 
 70. 
 
 ^(a+4)» 
 
 71. SVa^K 
 
 
 72. 
 
 V2a:-f Vy2. 
 
 73. 
 
 
 74. -■'^2--'''-^ 
 
 V 
 
 i 
 
 75. ->/^V6^ 
 77. ^V^. 
 
 y?. 
 
 76. aV^ ""'"■v/a;"**-*' 
 
 
 226. Square root of a binomial surd expression. The 
 
 square of a binomial surd expression of the form a + V3 
 is itself a binomial surd expression of the same form. 
 Thus, (2 -\/5)2 = 9 - 4V5 and (a + ^ly = a^ + 6 + 2a\/6. 
 
 When an expression in the form of a -f- V3 is a perfect 
 
294 ELEMENTARY ALGEBRA 
 
 square, its square root may usually be found by inspec- 
 tion. The method may be seen in the solutions of the 
 following : 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Find the square root of 16 -h 6V7. 
 
 Solution. 16 + 6 V7 = 16 + 2v'63 
 
 = 9 -f 2V9 V7 + 7 
 
 = 32 + 2 X 3 X V7+(V7)2 
 
 = (3+V7)2. 
 
 .-. V16 + 6V7 = V(3 + V7)2 = 3 + V7. 
 
 2. Find the square root of 6 — Vll. 
 Solution. 6 - Vll = 6 - 2 V^ 
 
 = ¥ - 2 V-V- vi + i 
 =(V5y-2V-y:V|+(Vjy 
 
 .-. Ve-Vii = V( v^ - Vly 
 
 3. 
 
 Show that V 6 + V35 + V 6 _ V35 = Vl4. 
 
 Solution. Ve + V35 = V 6 + 2 Vy = V| + V| 
 V6-V35=V6-2V^ = V|-V| 
 
 Sum = 2 Vj = vTi. 
 
 227. From the solution of the foregoing illustrative 
 examples we may state the following : 
 
 Rule. A binomial surd expression of the form a-\- 2V6, 
 where a and h are rational numbers^ is a perfect square when b 
 (the number under the radical sign^ is the product of two fac- 
 tors whose sum equals a, the rational term. In this case, the 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 295 
 
 square root of the binomial surd expression is eqyxil to the 
 sum or difference of the square roots of the two factors of 
 the number. 
 
 Remark. Any binomial surd expression of the form a + Vft can 
 
 be written a + 2\-' 
 
 228. (1) A quadratic surd cannot he equal to the algebraic sum of a 
 rational number and a quadratic surd. 
 
 For, if possible, let Va = & + Vc ; 
 
 then by squaring, a = b^ -\- c -\-2 bVc. 
 
 and Vc= «-^'-^ 
 
 2b 
 
 Since Vc is a quadratic surd, it is an irrational number and can- 
 not be equal to the rational number " oa~ ^ ' Therefore, the assump- 
 
 ^ 
 
 tion that Va = 6 + y/c is false. 
 
 (2) A binomial surd expression of the form a + y/b cannot be equal 
 to another expression x + y/y of the same form (where a and x denote 
 rational numbers) unless a = x and Vb = y/y. 
 
 For, if possible, let a +'"\/6 = a: + Vy ; 
 then, Vb =(x — d)-\- y/y. 
 
 By (1) this equation can be true only when a: — a = 0, in which 
 case y/b = y/y. 
 
 EXERCISE 108 
 
 Find, in surd form, the square root of : 
 1. 16-6V7. 2. 6 + VTT. 3. 7-hV48. 
 
 4. 32-V700. 5. 12-6V3. 6. 8-I-2V7. 
 
 1 
 
 7. Simplify 
 
 Vs-i-Vs 
 
 9. Simplify Vn + 12 V2. 
 
296 ELEMENTARY ALGEBRA 
 
 EXERCISE 109. REVIEW 
 
 (Solve as many as possible at sight.) 
 Write without negative exponents : 
 
 1. 
 
 ar\ 
 
 2. 
 
 a%-^. 
 
 3. 
 
 a-^c. 
 
 4. 
 
 x'^yr^. 
 
 5. 
 
 a-^b^. 
 
 6. 
 
 2a-%'\ 
 
 7. 
 
 Sa-H. 
 
 8. 
 
 5-1 a. 
 
 9. 
 
 iaby\ 
 
 10. 
 
 5a-\ 
 
 11. 
 
 3a6-8. 
 
 12. 
 
 x^r^. 
 
 13. 
 
 a^c-K 
 
 14. 
 
 a«6-». 
 
 15. 
 
 3-32^. 
 
 16. 
 
 6-1 
 3 
 
 17. 
 
 aT'^xif^, 
 
 18. 
 
 (5a-i)-i. 
 
 19 
 
 1 
 a-2 
 
 20. 
 
 1 
 
 a;-i 
 
 21. 
 
 1 
 
 X«7. 
 
 (2^)-i 
 
 22. 
 
 a 
 
 23. 
 
 a 
 
 24. 
 
 a 
 
 Sxy-' 
 
 3(xy)-> 
 
 (Zxyy-^ 
 
 25. 
 
 a-i 
 
 26. 
 
 a;«-iy-»-l. 
 (-a)-3. 
 
 27. 
 
 7fl~^, 
 
 (3=^y)-' 
 
 1 
 _1 
 
 a « 
 
 28. 
 31. 
 34. 
 
 ia + x)-^ 
 
 2-8 
 
 16-1 
 (- «)-*• 
 
 29. 
 32. 
 35. 
 
 30. 
 
 33. 
 36. 
 
 37. 
 
 ©"■ 
 
 38. 
 
 <!)"■■ 
 
 39. 
 
 a-^b-^c 
 
 Aft 
 
 
 
 41 
 
 (3 aby 
 
 -8 
 
 n\}. 
 
 • (3a3)-3(2 6*)"^ 
 
 42. 
 
 1 
 
 
 -^ 
 
 -1 
 
 -c-i 
 
 
 a-i + 6-1 
 
 
 AA. 
 
 1 
 
 
 -^ 
 
 
 
 Vm* 
 
 a-i-1 
 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 297 
 
 Perform the indicated operations in examples 46-51 : 
 
 46. a* X a^, 47. 4 a~^ -s- 2 x'K 
 
 48. aT^V'^. 49. (a*)^-5-(a:^)2. 
 
 50. (a^6"^)i 51. (a"^)»x(a^)*x(a~^)-^. 
 
 52. Find the product of a^, a^ a^ a^, a", and a\ 
 
 53. Find the product of rnJ^ m*^ m", m*, m~^ and w"^. 
 
 54. Express 9m^n~^ -i- 15n^p~^ x 20jt?277j-3 with positive 
 exponents, perform the indicated operations, and reduce 
 the result to simplest form. 
 
 55. Express the following with positive exponents, 
 add, and simplify the result : 
 
 x~\yz~^ — y~^z)-\- yr\zx~^ — z~\c)-\- z-\xy-^ — x~^y). 
 
 Simplify the following : 
 
 56. 
 
 9l 
 
 57. 
 
 4i 
 
 58. 
 
 64*. 
 
 59. 
 
 25i 
 
 60. 
 
 -27i 
 
 61. 
 
 (-27)*. 
 
 62. 
 
 (27)-*. 
 
 63. 
 
 (-27)-*. 
 
 64. 
 
 32*. 
 
 65, 
 
 100-^. 
 
 66. 
 
 2 
 4-2 
 
 67. 
 
 256*. 
 
 CO 
 
 (-i)-^. 
 
 69. 
 
 (A%)"^. 
 
 70. 
 
 20 
 
 68. 
 
 (-l)-7 
 
 71. 
 
 516 
 
 512* 
 
 72. 
 
 4-^ X 36i 
 
 73. 
 
 4-2^5-3 
 
 74. 64"^ -64"* +(-64)^ + 64*. 
 
 75. 3-* X Vsi X 5-^ X V5. 
 
 76. (a;-2)3. 77. (^2)-8. 78. (a;-i)-i. 
 
 79. (a3)*. 80. V^. 81. (64*)2. 
 
298 ELEMENTARY ALGEBRA 
 
 1 . 1 
 
 82. —-h — . 83. (64a-6)l 84. (a^)-2. 
 
 85. (-a8)6. 86. (-a3m)4, g^ (^a'^h'^c^^^. 
 
 as. lla-^pp. a. ©". so. (1^)-^- 
 
 91. 8"^ X 16^ X 20. 92. — -i— -. 
 
 2-1 + 3-1 
 
 93. (a + 6)0. 94, 
 
 95. 
 
 
 97. 2«-i x8«+1h-162. 98. ^ — ^. 
 
 99. (a°)". 100. V«~i^^a*. 
 
 101. (af-^yhy^-^y-y. 102. [(e^-e-*)^ + 4]«. 
 
 103. (a;^ + a:^ + 1) (a;* - a;^ + 1). 
 
 104. aizu^ + ^l±L^. 105. [V^Fh^]*. 
 a^ _ ji a^ + h^ 
 
 1 1 
 
 106. [(^0"'] ^-5-[(«"0''] *• 
 
 107. (a^ - ah^ + a^6 - 6^) ^ (a* - 6^). 
 
 108. ix-^y-^ + xy^y, 109. [3V5=V]-*«. 
 
 110. (-V?)io + (2V3)«-a;-Y--?^y. 
 
 111. (a; + 2a;^ + l)i 
 
 112. [a;-2a;^ + 3a^^~2a;^ + l]*. 
 
POWERS, ROOTS, RADICALS, AND EXPONENTS 299 
 Clear of negative exponents and simplify : 
 
 113. .^ ♦ 114. 
 
 m~^ -H ^~^ --- (^m—n)~^ 
 
 Arrange according to descending powers of x : 
 
 115. x^ - 2x^-^4 x^ -{- x^. 
 
 116. Since x^ — i/^ = (a;^)^ — (?/^)2, express x^ — 3/^ as a 
 product of the sum and difference of the same two 
 numbers. 
 
 117. Simplify — i-^+ 1 2 a^ 
 
 a^_l a^ + l a-1 
 
 118. Evaluate a/«2 . ^/^b . Vo^" ^ ^^ when a = 3. 
 
 119. Evaluate 5" • 25"+i . 125-"+i -r- 5^. 
 
 120. Simplify Jl _ [1 _ (1 _ a^yiyi^i, 
 
 121. Evaluate (243)^ + (243)i 
 
 122. Simplify (\2V2V2Vl)^- 
 
 123. Simplify 7/~^[(x + V^)3 -(x- V^)^]. 
 
 124. Which is the greater \/27 or ^16 ? 
 
 125. Simplify (a^"^)' -;- (a:"^)* -5- (2:"^)^ h- (a;"^)i 
 
 a«-6« 
 
 126. Simplify (x^^+f>^' -^ a:(«-*>') ^"* 
 
 127. Simplify ^~^ 
 
CHAPTER XI 
 INVOLUTION AND EVOLUTION 
 
 Involution. The operation of raising an expres- 
 sion to any positive integral power is called involution. 
 
 Remark. The involution of monomials has been explained in 
 section 198. We shall now consider the involution of a binomial. 
 Since by grouping the terms of any polynomial it may be expressed 
 as a binomial, the involution of a binomial is an important topic of 
 algebra. 
 
 230. Binomial expansion. The process of raising a bi- 
 nomial to a power is called expanding the binomial, and the 
 result of the operation is called a binomial expansion. 
 
 Thus, by expanding (a + b)^ we obtain a^ -\- 2 ab -{• b% which is 
 called the expansion of (a + by. 
 
 231. The binomial formula. By actual multiplication 
 we obtain the following expansions : 
 
 (a -f. 5)2 = «2 4. 2 a6 + 62. 
 
 (a + by = a3 + 3 a^ + 3 a62 4 ^,3. 
 
 (a + 6)4 = a* + 4 a% -h 6 a2^>2 + 4 ^J3 + ^4. 
 
 (a + 6)6 = a^ -j- 5 a'^b + 10 a^I>^ + 10 a^^ + 5 aft* + b^ 
 
 If we examine the above expansions we arrive at the 
 following important conclusions which we here assume 
 are true when a binomial is raised to any positive integral 
 power: 
 
 1. 77ie first term in the expansion is a", where a is the first 
 term and n is the exponent of the binomial. The last term is 
 
 300 
 
m^ 
 
 Isaac Newton (1642-1727) was perhaps the greatest mathe- 
 matician of all time. He is best known as the discoverer of the 
 laws of gravitation. In algebra he discovered the binomial theorem 
 and wrote extensively on the theory of equations. His great work 
 " Philosophiae Naturalis Principia Mathematica" appeared in 
 1686-87. 
 
INVOLUTION AND EVOLUTION 301 
 
 6", where h is the second term of the binomial. The number 
 of terms is n + 1. 
 
 2. TTie sum of the exponents of a and b in any term is n. 
 The exponent of b in the first term is zero and that of a in the 
 last term is zero. The exponent of a in the second and fol- 
 lowing terms is one less than in the preceding term. Hence, 
 the exponent of b in any term is one greater than in the pre- 
 ceding term. 
 
 3. The coefficient of any term is obtained from the coeffi- 
 cient of the preceding term by multiplying that coefficient by 
 the exponent of a in that term and dividing the resulting 
 product by ofne unit more than the exponent of b in that term, 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Expand (2 + xy. 
 
 Solution. Since (a + 6)8 = a^ + 3 a^ft + 3 ah^ -I- h\ 
 therefore, (2 + a:)^ = 28 + 3 • 22 . a; + 3 • 2 • ar^ + a:« 
 
 = 8 + 12 a; +6x2+3:8. 
 
 2. Expand (a — 2 ?/)*. 
 
 Solution. Since (a - by = a^ - ^a% + Q a^h"^ - 4 aft8 + h\ 
 .'. (a - 2 i/)4 = rt* - 4 a8(2 2/) + 6 a\2 yY - 4 a(2 yY + (2 y)* 
 = a* - 8 a8y + 24 02^2 _ 32 ay^ + 16 y*. 
 
 3. Expand C^-^Y 
 
 Solution. 
 
 Since (a - hy = a^ - Q a% + 15 a*62 _ 20 a%^ + 15 a%^- 6 aft«+ 6«, 
 
 ^1 6 & J 15 />2 20 68 ^ 15 &^ Q¥ h^ 
 
 /jB n^n n^r^ n^r*^ n^n^ n/^o /.6 
 
302 ELEMENTARY ALGEBRA 
 
 
 
 EXERCISE 110 
 
 
 
 
 (Solve as many as possible at sight.) 
 
 1. 
 
 (x 4- 2/)^ 
 
 2. Cx-yy. 
 
 3. 
 
 (a + hy. 
 
 4. 
 
 (m — ny. 
 
 5. cx-iy. 
 
 6. 
 
 (2 - x-)\ 
 
 7. 
 
 (a + hy. 
 
 8. {x-yy. 
 
 9. 
 
 (X^ - 1)3. 
 
 10. 
 
 (3 a - by. 
 
 11. (3a+2J)3. 
 
 12. 
 
 (5*- 3^)3. 
 
 13. 
 
 (2 m - ny. 
 
 14. (2 a2 - 1)4. 
 
 15. 
 
 (2-3 nfiy. 
 
 16. 
 
 (3 2^ + 2 fy. 
 
 17. (2 a 4- ^>)^ 
 
 18. 
 
 (1 - ay. 
 
 19. 
 
 (_^2_ 2^2)6. 
 
 20. (r2-2)7. 
 
 21. 
 
 (mhfi - 1)». 
 
 22. 
 
 (1^2+1)5. 
 
 23. (1:^2- 1^2)6 
 
 . 24. 
 
 (J^^+3)». 
 
 25. 
 28. 
 
 Qm-hiny. 
 
 »• g-:)* 
 
 27. 
 
 (l-i)' 
 
 
 / 
 29. 1 + 
 
 v 
 
 a- - 
 
 S 
 
 ..)' 
 
 232. Evolution. The operation of extracting an indi- 
 cated root is called evolution. 
 
 233. Square root. In section 88 a square root of a 
 number was defined, and it was shown that any number 
 has two square roots of the same absolute value but with 
 opposite signs. 
 
 From section 92 we have (a -k-hy= a^-\- 2 a5 4- 62; there- 
 fore, 
 
 V«2 + 2ah + l^== ± (a + V). 
 
 From section 209 (V), we have 
 
 therefore, 
 
 J2~vP'" * 
 
 vfiT=V^=-=^-" 
 
INVOLUTION AND EVOLUTION 303 
 
 EXERCISE 111 
 
 Find the following square roots by inspection: 
 1. Vl. 2. Vl6^. 3. V9^2p. 
 
 4. V625 2:y. 5. +Vl6rc^2^. 6. - V324 a*. 
 
 7. +V289^. 8. -VT02l«V. 
 
 9. Va2-2a6+^>2. 10. _Va2 + 4a + 4. 
 
 11. V42:2_4^_^i, 12. Va:2 _ 6 ^ + 9. 
 
 13. Va:V-2a:«/ + l. 14. -\-V9 x^-hl2x^ + 4 f. 
 
 15. -Vl6a2 + 24a^.H-9R le. + V64 x 81 a%^(^. 
 
 17. + V4 X 25 X 256 a^js^ioye. 
 
 18. -V^\ 19. + \/a2"+2^ 20. 4-V«2p^. 
 
 21. 
 
 23. 
 
 25. 
 
 J?^. 24 x/? 
 
 2_2a6 + 52 
 
 729 ^a2^2a^ + ^ 
 
 \a;2_io^ + 25' • ^92:2-18a: + 9 ' 
 
 234. Square root of a trinomiaL The positive square 
 root of any trinomial which is a perfect square may be 
 found by inspection. 
 
 Thus, + Va2 -{-2ab -\-b^ = a-\-b. 
 
 The actual work of finding the square root of a2-f- 2 ab-\-b^ 
 may be arranged as follows: 
 
 a2 + 2 a& + &2 \a-\.b 
 a2 
 
 trial divisor 2a 
 
 complete divisor 2a + b 
 
 2ab + b^ 
 2ab + b^ 
 
 
 The explanation of the method is as follows: 
 
304 ELEMENTARY ALGEBRA 
 
 1. We find the square root of the first term (which is the first term 
 of the result) and subtract its square from the trinomial, obtaining 
 2 a6 + b\ 
 
 2. The second term h of the result may be found by dividing 2 ah 
 by 2 a. We call 2 a the trial divisor. The trial f/irisor /.s double the part, 
 a, of the root already found. After h, the second term of the result, has 
 been found, we add it to the trial divisor, obtaining 2 a -\- b, the 
 complete divisor. 
 
 3. Multiply the complete divisor 2 a -\- bhj b and subtract. 
 
 235. Square root of any polynomial. The process em- 
 ployed in finding the square root of a^ -{-2ab + 1^ is 
 applicable in finding the square root of any polynomial. 
 
 ILLUSTRATION 
 
 Extract the square root oix^ — 2 x^i/ + Sx^i/^ — 2 xi/^ 4- ^. 
 x^-2x^y + S x^y^ -2xy^+ y^ \x^-xy-\-y^ 
 
 1st trial divisor, 2(x^) =2x^ 
 1st complete divisor, 2x^ — xy 
 2d trial divisor, 2(x^ -xy) = 2x^— 2 xy 
 2d complete divisor, 2x'^—2xy-\- y^ 
 
 2 x^y + 3 x^y^ — 2 xy^ + y*, 1st remainder 
 
 2xhi+ a;V 
 
 -f- 2 x'^y^ — 2 xy^ + «/*, 2d remainder 
 
 0, 3d remainder 
 
 Explanation. 1. The square root of x^ is x^ (the first term of 
 the result). The first trial divisor is 2x^ (double the part of the 
 root already found). The first term in the first remainder divided 
 by the trial divisor is — xy (the second term of the result), and the 
 first complete divisor is 2x^ — xy. Here a = x\ 2ab = — 2 x^y, 
 2 a -^ b = 2 x^ — xy. Multiplying 2 x^ — xy hj — xy and subtracting, 
 we obtain the second remainder; this corresponds to multiplying 
 2a + bhj b and subtracting. 
 
 2. The second remainder is the result of subtracting the square of 
 the part of the root already found (x^ — xy)^ from the given poly- 
 nomial. By taking a = x^ — xy, the second trial divisor is found to 
 be 2(x^ — xy)=: 2x^ — 2xy. The first term of the second remainder 
 divided by the first term of the second trial divisor is y^ (the third 
 term of the result), and the second complete divisor is 2 x* — 2 xy + y\ 
 
INVOLUTION AND EVOLUTION 305 
 
 We now multiply 2 x^ — 2 xy + y^hj y^ and subtract. Here a = x^ — ary, 
 2a = 2x^-2xy,b = y\B,nd2a + h = 2x^-2xy-\-yK 
 
 3. The third remainder, 0, is the result of subtracting the square 
 of the part of the root already found (x^ — xy + y^Yi from the given 
 polynomial. Hence, the given polynomial is the square of x^ — xy -\- y^, 
 and x^ — xy -{- y^ is the required square root. 
 
 Remark. The polynomial should be arranged according to the 
 powers of some one letter before the work of extracting the square 
 root is begun. 
 
 EXERCISE 112 
 
 Find the square root of the expressions in examples 1-9. 
 
 1. 9a^-Qx-\-l. 
 
 2. 4:2^-12a^-{-lSx^-6x-hl, 
 
 3. 25a^-20a^ + Ma^-12x-{-9, 
 
 4. 3^-\-22^-Sx^-4:X-\-4:. 
 
 5. 3^-4:a^ + 5x^-2x-\-\. 
 
 6. x^-2x-\-S---h-. 
 
 X x^ 
 
 7. 7^-2x^y-\-b T^y'^ - 14 2:^^3+14 a^»^-20 xf-{-2b /. 
 
 8. x^+2x^-2a^-^x^-2x-{-l, 
 
 9. x^-\-2x^-^l + 2x^ + ^x^. 
 
 10. Show that \ + x^^(2x^- 4)a?» + (a;+ l)(a; + 3) is a 
 perfect square. 
 
 11. For what value of m is the expression 4 a;* — 12 a^y 
 + mx^y'^ — 6 xy^ + y^ ^ perfect square ? 
 
 Extract the square root of the expressions contained in 
 examples 12 to 26 inclusive. 
 
 12. aV+h'^y'^+l + 2ahxy-ir2ax-^2hy. 
 
 13. x^-Qmx^-\- 13 m^x^ - 20 m^a^-{- 16 m'^a^- 6 m^x-\-m^. 
 
 14. (p - qy- 2(jt?2 + fxp - qy+Kp"^ + ^0. 
 
306 ELEMENTARY ALGEBRA 
 
 15. aP-.4x-\--- — 'hlO. 
 
 of X 
 
 -^ m^ . n^ . 2m . ^n . ^ 
 
 16. -5 + — ^ H 1 1- o. 
 
 fir m^ n m 
 
 17. a2j-2 4- J2<?-2 + c2a-2 + 2 a<?-i + 2 <?6-i + 2 har\ 
 
 18. a3 4-2a2 + a + l-2a*(a + l). 
 
 19. (a^ + «2 4- 2 a^ + 2 a + a^ + 1)(«* - l)(a - 1). 
 
 20. if + $2) (r2 + «2) _ (^^ _ ^y.)2. 
 
 21. -^^+^-.^-2+^^ 
 
 22. ^V^ + ^P^-y^ » 
 
 23. 2 0^(2/ 4-2)2 + 2 y2(2 4- :?:)2 4. 2 z\x 4- 3/)2 
 
 4- 4:^5/3(2:4-^4- 2). 
 
 24. (a4-^)(«4-2 5)(a4-3 5)(a4-4^>) + M. 
 
 25. x-\-x^ -\-x^-\-2x^ + 2xi -^2 x^^^, 
 
 26. «-2 4. 6-4 4- c2 4- 2 a-15-2 - 2 a-i(? - 2 6-2c. 
 
 27. Show that a^ ^ 53 _^ ^ _ 3 ^5^ jg ^j^g square root of 
 (^2 _ J^)3 + (52 _ cay 4-(<?2 - a6)3 - 3(a2 - ^>c)(62 _ ^a) 
 
 28. Show that l-\-(x-l)x(x+l)(x + 2} is a perfect 
 square. Hence, write down the square root of 1 4- 2 • 3 • 4 • 5. 
 
 29. If 2:4 H- 4 - 4 ^(^,2 ^ 2) 4- 8 2:2 = (^2 _|_ ^^^ 4. 2)2, what 
 is the value of w? 
 
 30. Find the square root of (-^ - -^V ^^ + 1. 
 
INVOLUTION AND EVOLUTION 307 
 
 31. Prove that x^ -\- (x -h ly + (x -\- 2y -\- {x -\- Sy - 5 is 
 a perfect square. If the sum of the squares of any four 
 consecutive integers is diminished by 5, what may be 
 said of the form of the result? 
 
 236. Square root of arithmetical numbers. We know 
 
 that + VT = 1, + VTOO = 10, + VTOOOO = 100, + VlOOOOOO 
 = 1000. Hence, the positive square root of any number 
 between 1 and 100 is between 1 and 10; that of any 
 number between 100 and 10,000 is between 10 and 100, 
 and so on. That is, the integral part of a square root of 
 a number of two figures contains one figure; that of a 
 number of three or four figures contains two figures, and 
 so on. Hence, to find the number of figures in the integral 
 part of the square root of a given number begin at the 
 units' figure and separate the number into periods, or groups^ 
 of two figures each, the last period containing either one 
 or two figures according as the given number contains an 
 odd or an even number of figures in the integral part. 
 There are as many figures in the integral part of the 
 square root of the given number as there are periods. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Find the positive square root of 9604. 
 
 Solution. Since there are two periods in the given number, there 
 are two figures in the integral part of the root, which we find in the 
 same manner as we did the square root of a^ + 2 ab -\- h\ For a, we 
 take the greatest number of tens whose square is less than 9604 ; that 
 is, 9 tens, or 90. In practice, we take the greatest number whose 
 square is equal to or less than 96 in the first period at the left. The 
 work is completed as follows : 
 
 96^041 90 + 8 
 8100 
 2 a = 180 15 04 
 2a + 6 = 180 + 8 = 188 15 04 
 
308 
 
 ELEMENTARY ALGEBRA 
 
 Remark. 
 
 In practice the work is usually arranged thus : 
 96'04[98 
 
 81 
 
 188 
 
 15 04 
 15 04 
 
 2. Find the positive square root of 56026.89. 
 
 Solution. Observe that if the square root of a number has 
 
 decimal places, the number itself has double that number of decimal 
 
 places. For this reason, in extracting the square root, decimals are 
 
 separated into periods of two figures each beginning at the decimal 
 
 point. 
 
 5^60^26.89 1 236.7 
 
 - 4 
 2a = 2(20)= 40 160 
 2 a + 6 = 43 1 29 
 
 2a = 2(230)= 46 
 
 2 a + & = 46 
 
 2a = 2(2360)= 47 
 
 2 a + & = 47 
 
 3126 
 6 27 96 
 
 20 
 27 
 
 3 30 89 
 3 30 89 
 
 237. Approximate square root. When a number is not 
 a perfect square, we annex as many periods of zeros as are 
 desired and continue the process of extracting the square 
 root. In this way we obtain a rational number which is 
 an approximate value of the square root of the number. 
 
 ILLUSTRATIVE EXAMPLE 
 
 Find to two places of decimals the square root of 10. 
 10.00^00^00 13.162 
 
 9 
 
 61 
 
 [Too 
 
 61 
 
 626 
 
 39 00 
 37 56 
 
 6322 
 
 144 00 
 126 44 
 
INVOLUTION AND EVOLUTION 309 
 
 Therefore, vTO = 3.16, correct to two places of decimals. 
 
 Remark. Had the third decimal figure been either 5 or greater 
 than 5, we should have taken 3.17 as the approximate square root. 
 
 EXERCISE 113 
 
 Find the positive square root of : 
 
 1. 6724. 2. 14884. 3. 5776. 
 
 4. 53361. 5. 110889. 6. 99856. 
 
 7. 591361. .8. 1723969. 9. 146.41. 
 
 10. 91083.24. 11. 100.2001. 12. 493:817284. 
 
 Find, to two places of decimals, the square root of: 
 13. 2. 14. 3. 15. 5. 16. 7. 
 
 17. 11. 18. 13. 19. 12. 20. 15. 
 
 Find, to three places of decimals, the value of: 
 
 21. (l+V2)(l-f-V3). 22. (7V5-2)(V2-f-l). 
 
 23. (2V3-hV5)(l + 3V2). 24. 
 
 V3-V2 
 
 25. — 26. 
 
 V2-1 V3-4 
 
 Find, to two places of decimals, approximate values of 
 the roots of the following simple equations: 
 
 27. (a:-l)V2 = V3 4-2. 
 
 28. a:(lH-V2) = 2a:4-3V3. 
 
 29. V2a:-l = V5 + 2. 30. a.^ V5-l^ 
 
 2 
 Suggestion. 2x-l = 9+4V5. 
 
310 ELEMENTARY ALGEBRA 
 
 238. Equations solved by finding the square roots of a 
 number. Any equation which can be reduced to the 
 form ax^ = 5 in which a and h are positive numbers and 
 
 - is a perfect square, has rational roots which may be 
 a ^ 
 
 found by taking the square root of -. 
 
 ILLUSTRATIVE EXAMPLE 
 
 Solve the equation 4a::2 _ 9 _ q. 
 
 Solution. 4 a:2 - 9 = 0. (1) 
 
 Solving (1) for x^, x^ = |. (2) 
 
 Now a root of (2) is a number whose square is ^ ; therefore, it is 
 
 a square root of f . Hence, taking the square root of each member 
 
 of (2), 
 
 x = ±|. (3) 
 
 Therefore, the roots of (2) are + |- and - |. 
 
 Note. Any root which can be found by the above method can 
 also be found by factoring, as in section 117. 
 
 Thus, 4 x2 - 9 = 0. (1) 
 
 Factoring, (2 a; - 3) (2 x + 3) = 0. (2) 
 
 Equating (2 a; - 3) to 0, 2 a: - 3 = 0. (3) 
 
 Equating (2 x + 3) to 0, 2 x + 3 = 0. (4) 
 
 Solving (3) and (4) for x, ^ = f > ^^d - \. 
 
 EXERCISE 114 
 
 Solve the equations in examples 1-14. 
 
 1. 2^2 = 4. 2. :?^ = 9. 
 
 3. 2 2^2 - 32 = 0. 4. 2^2 = a\ 
 
 5. a2^=^>2. 6. 182^-200 = 0. 
 
 7. 4 2;2= 49. 8. aW^=^\, 
 
 9. (a + 6)22:2=a2_2a6 + 62. 10. {x -f 1)2 = 16. 
 
 11. (aa; H- 6)2 = c2. 12. a:2_f_2a:+l =^2^2 a + 1. 
 
INVOLUTION AND EVOLUTION 311 
 
 13. (2a: -3)2 = (a -2)2. 14. f^±lj = l. 
 
 15. In the illustrative example, page 310, in taking the 
 square root of each member of (2), why is not equation 
 (3) written ± a: = ± | ? 
 
 Calculate, to two places of decimals, the numbers which 
 satisfy the following equations : 
 
 16. a;2 = 2. 17. x^ = S. 18. 0^2 = 5. 
 
 19. 2:2 = 32. 20. 2^=27. 21. 2:2 = 32.16. 
 
 22. Solve the equation ^22:2 _|_ 2 abx -\- b^ = c^. 
 
 23. Solve the equation 4 pV — 4 pqx = p^ — (f, 
 
 4 
 
 24. Solve the equation x= -- 
 
 X 
 
 9 
 
 25. Solve the equation 2: — 2 = 
 
 2:-2 
 
 26. For what values of x will the H. C. F. of 2:^ __ 3 jp2 
 
 4- 2: — 3 and 2:^4.2 2^2 + 2^ + 2 reduce to 5? 
 
 3 2: — 2 7 
 
 27. Solve the equation — - — = • 
 
 ( o X — 2t 
 
 28. If 2: = 2 satisfies the equation 
 
 (a: _ 3)(2: + 1) = 3 2:(2: 4- 2) - (2: + a2), 
 what are the values of « ? 
 
 2' 2 X I 1 
 
 29. Solve the equation -= ^ . 
 
 ^ 2; + 4 2;-2 
 
 30. When 2 2:* 4- 10 2:3 + 13 2:2 - 2: - 6 ig divided by 
 2:2 4- 3 2: + 2, the quotient is 2 2:2 -|- m22: — 3. Find the 
 values of m, 
 
 31. If w2 - 1 = 2:(2: 4- X){x 4- 2)(2: + 3), find the values 
 of m in terms of x. 
 
CHAPTER XII 
 
 QUADRATIC EQUATIONS 
 
 Definition. An equation whose second member 
 has been reduced to zero by transposition of terms is called 
 an equation of the second degree, or simply a quadratic 
 equation in one unknown number, as x^ when its first 
 member is a polynomial of the second degree in x. 
 
 Thus, 5 + 2a;2-7x = 0, and m - nx^ -\- qx ■\- n^ - p^ = are 
 quadratic equations. 
 
 240. Notation. It is customary to arrange the poly- 
 nomial in the first member of a quadratic equation accord- 
 ing to the descending powers of x and to render positive, 
 when necessary, the coefficient of the highest power of x 
 by multiplying both members of the equation by — 1. 
 The first member of a quadratic equation being a tri- 
 nomial of the second degree in rr, has, in general, three 
 terms, a term in x^ with positive coefficient, a second term 
 in X, and a third term which does not contain x and which 
 is the constant term in the equation. In any particular 
 equation, however, not all of these terms may occur, as the 
 coefficient of one or more of them may reduce to zero ; 
 but we shall assume that in a quadratic equation the 
 coefficient of x^ is not zero. 
 
 241. Standard form of a quadratic equation. Any quad- 
 ratic equation can be reduced to the standard for m^ 
 
 ax^ -{-bx-^ c = 0. 
 
 In this standard form a is positive and different from 
 zero, while h and c may have any values, including zero, 
 
 312 
 
QUADRATIC EQUATIONS 313 
 
 Thus, the equation mx^ — 3 mz + 5 = nx^ — 2 n + 3 /)x — 4, when 
 transformed, becomes {m — n)a:2 — 3(7n + i»)2: + (2 n + 9) = 0. In this 
 last equation the a, 6, and c of the standard form have the following 
 values : 
 
 a — m — n^ h = — 3(m + jo), and c = 2 n + 9. 
 
 242. Complete quadratic equation. A quadratic equa- 
 tion in which none of the coefficients a, 6, or e reduces to 
 zero is called a complete quadratic equation. All other 
 quadratic equations are incomplete quadratic equations. 
 
 243. Solution of incomplete quadratic equations. The 
 
 roots of an incomplete quadratic equation can, jn general, 
 be obtained by methods of solution which have been 
 treated in preceding chapters. These methods are 
 applied in the illustrative examples which follow. 
 
 • ILLUSTRATIVE EXAMPLES 
 
 1. Solve the equation 2x^ = 0. 
 
 Solution. 2 x^ = is an example of a quadratic equation in 
 which two of the coefficients, namely h and c, reduce to zero. It is 
 evident that the only value of x which satisfies this equation is 0, 
 since from 2x^ = 0, we derive x^ = 0. We say that the equation has 
 two roots each equal to 0, one root corresponding to each factor of x^- 
 
 2. Solve the equation 2a;2 — 3 a; = 0. 
 
 Solution. 2 a:^ — 3 a; = is an example of a quadratic equation 
 in which the constant term c reduces to zero. We solve such an 
 equation by the method of section 117. Thus, factoring, 
 
 2x^-%x = x(2x- 3)=0. (1) 
 Equating each factor of (1) to zero, we have (2) and (3), 
 
 a: = 0. (2) 
 
 2a:- 3 = 0. (3) 
 
 Solving (3) ar = |. (4) 
 
 Therefore, the roots of 2 a:^ — 3 a: = are and 4, which may be 
 verified by substituting each of these results in (1). 
 
314 ELEMENTARY ALGEBRA 
 
 Remark. Observe that when the constant term c in a quadratic 
 equation is zero, one root of the equation is zero, and conversely, 
 when one root of a quadratic equation is zero, the constant term is 
 zero. 
 
 3. Solve the equation Sa;^ — 2 = 0. 
 
 Solution. 3 a:^ — 2 = is an example of an equation in which i, 
 the coefficient of x, is zero. Such an equation is sometimes called a 
 pure quadratic equation. 
 
 From 3 a;2 — 2 = 0, we derive x^ = |^. 
 
 The values of x which satisfy this equation are evidently the square 
 
 2 /2 "v/fi 
 
 roots of -. Hence, x = i'V- , or ± -7— • 
 o '33 
 
 Therefore, the roots of Sa:^ — 2 = are — and , which 
 
 3 3 
 
 may be verified by substituting these numbers in the given equation. 
 
 4. Solve the equation a^^ _|_ 4 = 0. 
 
 Solution. a;2 + 4 = is an example of a pure quadratic equation, 
 both of whose terms have the same sign. No positive or negative 
 number exists which satisfies this equation; for the sum of two 
 positive numbers, one of which is not zero, cannot be zero. This 
 type of quadratic equation will be considered in section 251. 
 
 EXERCISE 115 
 Solve the following incomplete quadratic equations : 
 
 1. 5a;2 = 0. 2. x^-2x=(i. 
 
 3. 3a;2-.4a: = 0. 4. 4a^2_i,= o. 
 
 5. 5a;2-3a: = 0. 6. 7a:2_15 = 0. 
 
 7. 5a:2~2 = 0. 8. 2a:2-5:r = 0. 
 
 9. 8:c2_9 = o. 10. ax^-hx = 0. 
 
 11. 9a;2~16a: = 0. 12. 3a:2-5 = 0. 
 
 13. Qx^-bx = 0. 14. 20a:2-8a:=0. 
 
 15. 2:^2-11=0. 16. aV-h^^a. 
 
 17. (a+h)x^-(^c-\-d)x=0. 18. (a-h6)a;2-(a~6)2:=0. 
 
QUADRATIC EQUATIONS 315 
 
 19. 5ax^-Sb = 0. 20. ^^^:=0. 
 
 Z b 
 
 21. ax^—a = 0. 22. x^ — 5 = 0, 
 
 23. ^-x = 0. 24. --8a: = 0. 
 
 X X 
 
 25. ^x^ + ^x=0. 26. 19:c2_i2: = 0. 
 
 244. "Completing the square." From section 92 we 
 know that the trinomials x^ -\- 2 ax -\- a^ and x^ — 2 ax-i- a^ 
 are both perfect squares. We therefore infer that both 
 x^-\-2 ax and x^ —2 ax are converted into perfect squares 
 by the addition of a^ to each. Observe in each case that 
 a^ may be obtained by taking the square of one half 
 the coefficient of x. We therefore have the following 
 rule for completing the square of a binomial of the form 
 x^ -\-2 ax, in which the coefficient of a;^ is 1 : 
 
 Rule. Add the square of one half the coefficient of x. 
 
 EXERCISE 116 
 
 Complete the square in each of the following examples, 
 and state the binomial whose square is obtained : 
 
 1. X'^+2X. 2. X^-2X. 3. X^-\-4:X. 
 
 4. x^-^-Qx. 5. x^-6x, 6. x^-\-16x. 
 
 7. x^ - 18 a;. . 8. x^ + 20 a;. 9. x^ + ^x. 
 
 10. x^ — ^x. 11. x^ — ^x. 12. x^-{-^x. 
 
 13. x^-\-2cx. 14. x^ - 2(^a -\- b}x. 15. a;2 + 2(a-6>. 
 
 16. x^ — ax, 17. x^-\-Sx. 18. 2;2 — (a 4- h}x. 
 
 19. x^-^x, 20. x^ + ^x, 21. x^-^^'^x, 
 
 bd c?+ d 
 
316 ELEMENTARY ALGEBRA 
 
 245. Solution of a quadratic equation by ^^ completing 
 the square." When the roots of a quadratic equation are 
 not rational numbers, its solution by factoring [section 
 117] is not, in general, so convenient as that explained 
 in the following illustrative examples : 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Solve the equation Sx^ — 2x — 2 — 0. 
 
 Solution. 3 x2 - 2 a; - 2 = 0. (1) 
 
 Dividing both members of (1) by 3 so that the coefficient of x^ 
 
 shall be 1, x2-|a;-| = 0. (2) 
 
 Transposing, x^ — ^x = ^. (3) 
 
 Completing the square by adding the square of one half the coeffi- 
 cient of X to each member of (3), 
 
 ^^-t^ + i = l + i = f W 
 
 Since the first member of (4) is a perfect square, we have, 
 Extracting the square root, 
 
 Transposing and combining, 
 
 ''- 3 • 
 
 (7) 
 
 Therefore, the roots of 3 a;^ - 2 a: - 2 = are ^ \ and ^ '^ . 
 
 3 o 
 
 2. Solve the equation 4:p^x^ — 4 mpx -{-m^ — m — n = 0. 
 Solution. 4:p^x^ — 4 mpx + m* - m - n = 0. (1) 
 
 Dividing both members of (1) by 4:p^, 
 
 ^2_m^^ m^-m_-n^O (2) 
 
 p 4jo 
 
 m -i- n — m' 
 
 Transposing in (2), x^-'^x = '" -^ ' ~ "" . (8) 
 
 p 4/>^ 
 
QUADRATIC EQUATIONS 317 
 
 Completing the square and combining in (3), 
 
 p^4:p^~ 4j9« • ^*^ 
 
 Since the first member of (4) is a perfect square, we have, 
 
 V 
 
 Extracting the square root, 
 
 2pJ ip' 
 
 
 • 
 
 Transposing, 
 
 2p ' ^^ 
 
 Therefore, the two roots of 
 
 the riven eouation are ^ + ^^ + » 
 
 2p 
 
 2p 
 
 Remark. The solution, example 2, illustrates the solution of a 
 literal quadratic equation by completing the square. Had the object 
 been merely to solve the equation, the solution would be as follows : 
 
 Transposing, i p^x^ — 4: mpx -\- m'^ = m ■}- n. (2) 
 
 Since the first member of (2) is a perfect square, 
 
 (2px - m)2= m + n. (3) 
 
 Whence, 2px — m =± Vm + n 
 
 and X = ^'±^^ + ^, 
 
 2p 
 
 EXERCISE 117 
 
 Solve the following quadratic equations by the method 
 of completing the square : 
 
 1. rr2 -{- 4 re -f 3 = 0. 2. x^ + 4:X-hl = 0. 
 
 3. x^-{-2x-i = 0. 4. a:2 H- 3 2^ + 1 = 0. 
 
 5. x^-\-5x-\-b = 0. 6. 2;2 + 10.r4-15 = 0. 
 
318 ELEMENTARY ALGEBRA 
 
 7. 
 
 2:2 + 112:4-25 = 0. 
 
 8. 
 
 2:2-3a:-l = 0. 
 
 9. 
 
 a;2 - 5 a: + 3 = 0. 
 
 10. 
 
 2:2-7x4-11 = 0. 
 
 11. 
 
 a;2_lla;-l = 0. 
 
 12. 
 
 a:2 _ 13 a: + 30 = 0. 
 
 13. 
 
 x^-15x-5 = 0. 
 
 14. 
 
 2:2-102:4-23 = 0. 
 
 15. 
 
 x^-6x+4: = 0. 
 
 16. 
 
 4a:2_42:-l = 0. 
 
 17. 
 
 Sx^-]-Sx-2 = 0, 
 
 18. 
 
 52:2-52:4-1 = 0. 
 
 19. 
 
 Sx^-lx+S = 0. 
 
 20. 
 
 72;2_7a;-5 = 0. 
 
 21. 
 
 llx^-hlx-S = 0. 
 
 22. 
 
 52:2-32:-5 = 0. 
 
 • 
 23. 
 
 13 2^2 - 13 2; - 3 = 0. 
 
 24. 
 
 5a:2_52:-l = 0. 
 
 25. 
 
 2x^-Sx-i=::0. 
 
 26. 
 
 (a;_5)(2:-3)=l. 
 
 27. 
 
 (x-6)(x-S) = 4. 
 
 28. 
 
 2:2- (a 4- 1)2: 4- a = 0. 
 
 29. 52:2 - «(5 4- 1)2: 4- «2 = 0. 
 
 30. (a 4- 3)2:2 -2(a 4- 4)2; 4- (a 4- 5) =0. 
 
 31. («4-^')2:2 4-(5 4-2)2:-(a-2)=0. 
 
 32. (a - 2)2:2 + ^ __ (« _ 3) = Q. 
 
 33. (2 a- b)x^ -Sax-\-(a-\-b)=0. 
 
 34. 2:2 4-2(w4-l)a:4-m(w + l)=0. 
 
 35. 4 2:2 4- 4^0^ ^ ^^^ 4- m2 4- ri2 _ Q, 
 
 36. 2;2 - 2 aa; 4- ^2 - a = 0. 
 
 37. 42:2- 4^2:4- 52_ 4^ = 0. 
 
 38. (a 4- 5)22:2 - 2(a + 6)22: + (a 4- 5)2 - 2 = 0. 
 
 39. 16 a^V - 8 ab(a + 5)2: 4- (« 4- 5)2 - 16 a^^ = 0. 
 
 40. (« 4- 5)22:2 4- 2(^2 + 52)2: + (a -5)2=0. 
 
 In the following examples, clear the equation of frac- 
 tions and solve the resulting integral equation, checking 
 the roots found : 
 
 12 4 
 
 41. -4. 
 
 X x — 1 3 
 
45. 
 
 53. 
 
 QUADRATIC EQUATIONS 319 
 
 42. -^ §-+J- = 0. 
 
 2:4-1 2: + 212 
 
 43. ^ + 1+ = 0. 
 
 32^-3 ^2x-S 
 
 3 
 
 + 1 - TT^-T^ = 0. 
 
 22: + l 3a; + 2 
 
 12 5 
 
 62: — 5a a a — 6x 
 
 * a;-2 a;+2 * * ix-\-2 3a: + 5 
 
 48 -1^+^^+1 = 1 49 9^+1 , 62^+5^8 
 
 • 6x-5"^4a:H-3 ' 15x + l 3a: + 5 9* 
 
 50. ^±i 2;-2 a;-2 ^^ 
 
 2 2:2+32:-22 2:2 + 2;-1^2;2 + 3a:H-2 
 
 51. ^±^ + _^±8_ ^-4 ^ 
 
 2 2:2 + 52: + 22;2_2._6^.2 2:2-52:-3 
 
 2a: + 2 , 4:x-{-2 ^ 3a: + 3 
 
 rv n "T" 
 
 3a:2 4.a;-2 6x^-Ux+6 2x^-x-^ 
 
 3 ^ = ^ . 
 
 x^ + Sx-\-2 x^-\-7x + 12 x^-^4x-\-S' 
 
 246. Approximations. Many of the problems which 
 occur in physics and geometry give rise to quadratic 
 equations. In general, the roots of such quadratics are 
 irrational numbers which appear in the form of radical 
 expressions. For practical purposes we usually require 
 rational results which give approximately the values of 
 the roots. The method used in obtaining such approxi- 
 mations may be seen from the following : 
 
320 ELEMENTARY ALGEBRA 
 
 ILLUSTRATIVE EXAMPLE 
 
 Approximate to two decimal places the roots of 
 
 25^ + 2 j_47^jf31^^ 
 x+1 2x-S 
 
 Solution. 25X + 2 47.-f31^Q 
 
 x+1 2x-d 
 
 Clearing (1) of fractions and combining, 
 
 (1) 
 
 a;2_30a;_ 8 = 0. (2) 
 
 Solving (2), a: = 15 db V233. (3) 
 
 That is, x = 15± 15.264+ (4) 
 
 Therefore, the roots of the given equation, correct to two places 
 of decimals, are 30.26 and — .26. 
 
 Remark. — In checking the roots of a quadratic equation whose 
 roots are irrational, the expressions in terms of radicals should be 
 substituted for the unknown in the given equation. Thus, in check- 
 ing the result in the foregoing example, substitute 15 ± V233 for x 
 in the given equation. The rational numbers obtained as approxi- 
 mations will not satisfy the equation. 
 
 EXERCISE 118 
 
 Approximate to two places of decimals the roots of the 
 following equations : 
 
 1. a^_4a; + 2 = 0. 2. a^-6x-{-l = 0. 
 
 3. a^-22x-hllS = 0. 4. 2:2_ 20a: + 95 = 0. 
 
 5. 4a;2_i2a:-3 = 0. 6. 2x^-\-Sx-6==0. 
 
 7. 3a:2-5a:-l = 0. 8. 5x^-1 x + 1 = 0. 
 
 9. l2^-15x-\-5 = 0. 10. a:2_32a;-l = 0. 
 
 U. r^-102: + 8 = 0. 12. 3a:2_8a,4.i=:0. 
 
 13. 2x^-lSx-\-l = 0. 14. 3a:2 + 5a:-3 = 0. 
 
 15. ^a^ + lx-2 = 0. 16. 8a^2- 282:4-21 = 0. 
 
QUADRATIC EQUATIONS 321 
 
 247. Irrational equations. The unknown number in 
 an equation sometimes occurs in expressions which are 
 found under radical signs. Such equations are called 
 irrational equations. 
 
 Thus, Qx — y/Wx + 4 = 20 is an irrational equation. 
 
 We shall consider only equations in which the square 
 root of expressions containing the unknown number is 
 indicated. The solutions of the following examples will 
 illustrate the method of solving such equations. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Solve the equation bx — V3 a: + 7 = 11. 
 
 The equation 5 jl- - V3 x + 7 = 11 is an example of an equation in 
 which only a single square root occurs. 
 
 Solution. 5 a; - V3 a: + 7 = 11. (1) 
 
 Transposing terms so that the radical expression stands alone in 
 one member of the equation, 
 
 - v/3a:+ 7 = - 5a: + 11. (2) 
 
 Squaring both members of (2), 3 x + 7 = 25 x^ - 110 x + 121. (3) 
 
 Simplifying (3), 25 x* - 113 x + 114 = 0. (4) 
 
 Factoring, {x - 3) (25 a; - 38) = 0. (5) 
 Therefore, the roots of equation (4) are 3 and |^. 
 Substituting 3 in the given equation, 
 
 15_V9TT:=11. (6) 
 Simplifying (6), 11 = 11, which is an identity. 
 Substituting ^ in the given equation. 
 
 5x ||-V3xff + 7 = 11. (7) 
 
 Simplifying (7), %! = 11, 
 
 which is false. Since ||^ does not satisfy the given equation, it is no^ 
 a root of the equation. It may readily be verified that ||^ is a solu 
 tion of the equation 5 x + V3 x + 7 = 11, which equation differs from 
 the given equation only in the sign of the radical expression. 
 
322 ELEMENTARY ALGEBRA 
 
 Note. In the process of squaring, as in the above solution, we 
 multiply each member of the equation by a factor containing the 
 unknown number. The resulting rational equation may have solu- 
 tions which are not solutions of the given equation. It is, therefore, 
 necessary to test each root of the rational equation by substituting it 
 in the given equation. 
 
 2. Solve the equation Vrr + 5 — V7 a; + 4 -f V2 rc4- 9 = 0. 
 
 (The first step in the solution of an irrational equation of this 
 form is to arrange the terms so that one radical shall stand alone in 
 one member. The process of squaring leads to an equation in which 
 a single radical occurs and the solution proceeds as in that of 
 example 1.) 
 
 Solution. Va; + 5-V7a; + 4 + V2a:-f 9 = 0. (1) 
 
 Transposing, Va; + 5 + V2z + 9 = V7 x -\- 4. (2) 
 
 Squaring (2) and simplifying, 2 a: — 5 = y/(2x-\-9)(x-\-5). (3) 
 Squaring (3) and simplifying, 
 
 2x2 -39x- 20 = 0. (4) 
 
 Factoring, . (2 x + 1) (a: - 20) = 0. (5) 
 
 Therefore, the roots of equation (4) are 20 and — i. 
 Substituting 20 in the given equation, 
 
 V25 - Vl44 + V49 = 0. (6) 
 
 Simplifying (6), 5 — 12 + 7 = 0, which is an identity. 
 
 Substituting - ^ in the given equation, 
 
 V|-VJ-fV8 = 0. (7) 
 
 Simplifying (7), 3V2 = 0, 
 
 which is false ; hence, — 4 is not a solution of the given equation. 
 It may readily be verified that — | is a solution of the equation 
 
 - y/x + 6 - \/7a: + 4 + V2x + 9 = 0. 
 
 EXERCISE 119 
 
 Solve the equations : 
 
 1. V:r-1-1 = 0. 2. V2a:-8-3 = 0. 
 
 1 
 
 ■Vx + l 
 
 = 2. Suggestion. Clear of fractions. 
 
QUADRATIC EQUATIONS 323 
 
 1 
 
 4. V52:H-2 = 7. 5. -5 = 0. 
 
 V2a:-3 
 
 6. x-{-Vx = 12. 7. a;-2Vi-8 = 0. 
 
 8. x + ^Sx-i-1 = 7. 9. a:-V3a:+7=lll. 
 
 10. 5x-\-V5x-^4 = 52. 11. 22:-V2a:-f 3 = 17. 
 
 12. 3a:-|-V3a; + 2=4. 13. 5rr + Va; + 2 = ^. 
 
 14. 3Vi-Va: + 16 = 4. 15. V2a: + 9 + 7 = 3V2^. 
 
 16. Va;-7 + 3=V2^ 
 
 iC. 
 
 17. Va; + 24-V2;-15 = 3. 
 
 18. V92:-f l-V4a;-3 = 3. 
 
 19. V-35-99a: + V27 + 2a: = 13. 
 
 20. V7^T4-f2V3^-VT5^T76 = 0. 
 
 21. V2a:H- ll + 2Va: + 2 = V20a:-19. 
 
 22. V2:-7-V:c-ll=V3a:-29. 
 
 23. V2a;-6-Va:-l+V3a;-15 = 0. 
 
 24. V2a;4-l4-V3a:-ll-V92:-8 = 0. 
 
 25. V7a; + l-V3a: + 10 = l. 
 
 26. a;^ — 12x"^ 4- 1 = 0. Suggestion. Clear of fractions. 
 
 ^2a;H-l ^ 2: 
 
 28. a-\-b — Va^ — a: = Vi^ -|- x. 
 
 29. wax-{-b = — — » 
 
 V22: 
 
 30. 2Vi = Va + V42;— <?. 
 
 31. Vi + Vx -\- a = 
 
 ^x-\-a 
 
324 ELEMENTARY ALGEBRA 
 
 32. V^+V^ '' 
 
 X-4 ^X-\-4: 3 
 
 33.. ■\/x + 2a — Wx-\-2b = 2Vx, 
 34. Vx^-{- 5ax—2a^ = x-{- a. 
 
 35. Va: + a^ = « -h Va;. 
 
 36. ^^ + ^ + ^^^-^=6. 
 ^a-\- x — ^a — x 
 
 248. Solution of a quadratic equation by formulae. Any 
 
 quadratic equation can be reduced to the standard form, 
 
 ax^ -\-bx-\- e=0. 
 
 By this is meant that by assigning particular values to 
 the coefficients a, 6, and o, this equation reduces identically 
 to any given quadratic equation. The solution of this 
 general equation, therefore, contains the solution of any 
 given quadratic equation. The solution of a given 
 quadratic equation may, therefore, be found by substitut- 
 ing in the formulae which give tlie roots of the general 
 quadratic ax^ -\- bx-{- c — 0, the proper values of the coeffi- 
 cients a, 5, and c. 
 
 The derivation of the formulae for the roots of 
 . ax^ -1- 62; -I- (? = is as follows : 
 
 ax^ + fta: + c = 0. (1) 
 
 Since a is not zero, x^ + -x + -= 0. (2) 
 
 (3) 
 
 Transposing 
 
 U 
 
 X 
 
 a 
 
 a 
 
 Completing 
 
 the 
 
 square, 
 
 
 
 
 
 2 . f> 
 a 
 
 ^ia^~ 
 
 b'^-iac 
 4a3 
 
 (*) 
 
QUADRATIC EQUATIONS 325 
 
 Thatis, (x + -) =-^^^. (5) 
 
 Extracting the square root, 
 
 6 ± V62 - 4 ac ,«x 
 
 Transposing and simplifying, 
 
 2a 
 
 (7) 
 
 Representing the two roots of aa^ -\- bx -\- c = hy x^ and 
 x^, we have, therefore, the following formulae, 
 
 _ -fr-V&2_4gg 
 
 Note. When (6^ _ 4 ac) is negative, the given equation, ax^ + hx 
 + c = 0, is satisfied by no positive or negative number. The ex- 
 pression \b^ — 4: ac is, for the present, without meaning when 
 62 _ 4 ac < 0. [See section 251.] 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Solve the equation 12x^ -{- x — 6 = 0. 
 Solution. Here a = 12, 6 = 1, c = — 6 ; hence, by substitution in 
 the foregoing formulae we find the roots to be 
 
 1 + VI + 288 ajj(j - 1 - VT+ 288 
 
 24 24 
 
 which, when simplified, are equal to ^ and — J, respectively. 
 
 2. Solve the equation 2 mx^ — (3 w + l)a7 + (m + 1) = 0. 
 Solution. Here a = 2 m, 6 = - (3 m + 1), c = m + 1; hence, by 
 substitution in the foregoing formulae we find the roots to be 
 
 3 m + 1 + V(3 m 4-1)2-8 7/?(m + 1) 
 4m 
 
 and 3 m + 1 - V(3 m + 1)2- 8 m(m + 1) 
 
 4 m 
 
 which, when simplified, are equal to 1 and ^ "^ , respectively. 
 
 2 w 
 
326 ELEMENTARY ALGEBRA 
 
 EXERCISE 120 
 
 Find the roots of the following equations by substitut- 
 ing, in each case, the proper values of a, 6, and c in the 
 foregoing formulae. 
 
 1. a:2-5a;-14 = 0. 2. x^-10x + 21 = 0, 
 
 3. 6ir2-a^-2 = 0. 4. x^-%x-2^ = 0. 
 
 5. 20a;2-23ar + 6 = 0. 6. 15 a;2- 11 a;- 12 = 0. 
 
 7. 2:2 + 42^ + 2 = 0. 8. a;2-6a; + 6 = 0. 
 
 9. 2^2-10 2: + 18 = 0. 10. a;2-3 2:+l = 0. 
 
 11, 3 2^2 + 4 2^-1 = 0. 12. 25 2^-10 2:- 2 = 0. 
 
 13. 2^2 — (c — d)x — cd = 0. 
 
 14. 2^2 — (2 m - 3 71)2; — 6mw = 0. 
 
 15. abx^-(a^-\-b^)x + ah=0, 
 
 16. 4chy^-{-a^=b^-\-4:acx, 
 
 
 17 1 + 
 
 1 
 
 _-l + l. 
 
 
 2;— a 
 
 2: — 
 
 b a b 
 
 
 18. 6c2:2 + 2 
 
 m2: 
 
 + a^> = 0. 
 
 19. 
 
 a;- 100 = 10 -Vi. 
 
 
 20. 2(2:-l)=V2^. 
 
 21. 
 
 a; + Vi = 0. 
 
 
 22. 2:2-9 = 0. 
 
 23. 
 
 2:2 _ 4 = 0. 
 
 
 24. 2:2-3 = 0. 
 
 25. 
 
 2:2- 3 2:= 0. 
 
 
 26. 2:2_a;V2 = 0. 
 
 27. 
 
 2:2 + a:V3 = 0. 
 
 
 28. 2:2- a = 0. 
 
 29. 
 
 W2:2 — n=0. 
 
 
 30. «2^2_52 3,0. 
 
 249. Relations between roots and coefficients. Taking 
 the standard form of the quadratic equation, we have : 
 
 ax^ -\- hx -\- c = 0. (1) 
 
QUADRATIC EQUATIONS 327 
 
 Dividing both members of (1) by a (since a is not zero), 
 
 (2) 
 
 (3) 
 (4) 
 
 a a 
 
 = 0. 
 
 From the formulae of section 248 we ] 
 
 tiave the roots of 
 
 ^ _ - 6 + V62 - 
 ^ • 2a 
 
 4ac 
 
 2a 
 
 ■4ac 
 
 From (3) and (4) by addition, 
 
 
 - ^■— '^^ 
 
 a 
 
 From (3) and (4) by multiplication, 
 
 
 (5) 
 
 (-b + y/b^-4: ac)(- b - V6'' -4 ac) ' 
 4 a2 
 
 4 a** 
 = _l-[6._*. + 4ae]=£. (6) 
 
 We see from identity (5) that the sum of the roots of 
 equation (2) differs in sign only from the coefficient of x 
 in that equation. Also, we see from identity (6) that 
 the product of the roots of equation (2) is the constant 
 term in that equation. Hence, 
 
 In any quadratic equation of the form ax^ -\- hx -\- c = Oi 
 
 (I) The coefficient of x with its sign changed divided hy the 
 coefficient of x^ is equal to the sum of the roots. 
 
 (II) The constant term divided hy the coefficient ofx'^ is equal 
 to the product of the roots. 
 
 250. Formation of the equation. The principles of sec- 
 tion 249 enable us to form the quadratic equation when 
 its roots are given numbers. For this purpose we suppose 
 
328 ELEMENTARY ALGEBRA 
 
 the equation written in the form x^+px-hq = 0^ in which 
 
 p and q are written instead of - and -, respectively. 
 
 a a 
 
 From identities (5) and (6), section 249, we have 
 ■^i + -^2 = -^=-A . 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Form the equation whose roots are 3 and — 2. 
 
 Solution. Here x^ + x^^^ — 2 = 1= — p; 
 also, XjXg = 3x(— 2) = — 6 = ^. 
 
 Hence, jt> = — 1 and ^ = — 6. 
 
 Substituting these values of p and ^ in x^ -{■ px -{■ q = 0, the re- 
 quired equation is a;^ — x — 6 = 0. 
 
 X -{• 1 _ X 
 
 5 ~6-6a; 
 2 
 is - ; find, without solving the equation, the other root, 
 o 
 
 Solution. Reducing the given equation to the standard form, we 
 >*^^' 6x2+ 5a; -6 = 0. (1) 
 
 From (II), section 249, the product of the roots of equation (1) is 
 equal to the constant term divided by the coefficient of x^ ; namely, 
 
 \ -6 1 
 
 to , or - 1. 
 
 6 
 
 Since | is known to be a root, the other root is equal to — 1 -^ J ; 
 that is, the second root of the given equation is — |. 
 
 We may check thi^ result by making use of (I), section 249, from 
 which we know that the sum of the roots of this equation is the 
 coefficient of x with the sign changed divided by the coefficient of x\ 
 
 which is —5^6, or ^— . Since % is one root, the other root is 
 6 3 
 
 equal to -^-|=-| = -|^, which is in agreement with the pre- 
 ceding result. 
 
 2. Given that one root of the equation 
 
QUADRATIC EQUATIONS 329 
 
 EXERCISE 121 
 
 1. State at sight the sum and the product of the roots 
 of the following six equations. Do not solve the 
 equations. 
 
 a:2_22;-|-l=0. x^-lx-^12 = 0. 
 
 ax^ ~ bx-\- c=0. mx^ -^nx-\-pq = 0. 
 
 2. One root of each of the following six equations is 
 — 2. Find the second root in each case. Do not solve 
 the equations. 
 
 a:2 + 3a:+2 = 0. x^-x-Q = 0. 
 
 3 2;2-h42:-4 = 0. x^-\-4x-\-4 = 0. 
 
 ax^ + (2a-\-b)x + 2b = 0. px^ + (2p - q}x- 2 q= 0. 
 
 3. In a pure quadratic equation what is the sum of the 
 
 roots ? 
 
 Form the quadratic equations which have the following 
 roots : 
 
 4. 2 and — 3. 5. 3 and 2. 6. — 1 and 3. 
 7. 6 and - 3. 8.-2 and - 3. 
 
 9. 1+V2andl-V2. lo. 3 - V2 and 3 -|- V2. 
 
 11. m-\-n and m — n. 12. - + - and 2. 
 
 b a 
 
 13. m -f- Vw and m — Vw. 14. f and — |. 
 15. f V2 and - f V2. 
 
 251.^ Quadratic equations in which bf^ — ^ac is negative. 
 The equation a;2 + 4 = [section 243, example 4], which 
 
 *This section may be omitted, if so desired, until the subject is 
 reviewed. 
 
330 ELEMENTARY ALGEBRA 
 
 is of the form aa? -|- 62: + c = 0, where ^ — 4 ac < [sec- 
 tion 248, note], is an example of a large class of equations 
 which have no rational or irrational solution. There are 
 no rational or irrational numbers which satisfy an equa- 
 tion such as a^ + 4 = 0. 
 
 It is evidently desirable that all equations should have 
 solutions, but this is manifestly impossible so long as the 
 number system of algebra includes only rational and 
 irrational numbers. 
 
 In agreement with the generalizing spirit of algebra, 
 the number system is so extended that all equations shall 
 have solutions. 
 
 252. Pure imaginary number. The square root of a 
 negative number is called a pure imaginary number. 
 
 Thus, V — 4, also V — 2, are pure imaginary numbers. 
 
 253. Real numbers. All rational and irrational numbers 
 are called real numbers. 
 
 Thus, 2, - 3, I, - I, Vi, a/2, are real numbers. 
 
 254. The imaginary unit. The pure imaginary number 
 V— 1 is called the imaginary unit and is denoted by the 
 letter i. 
 
 Thus, I = y/'^\. 
 
 255. A pure imaginary number expressed in terms of i. 
 
 A pure imaginary number is expressed in terms of the 
 imaginary unit i as follows : 
 
 V— a = V(— l)a = V— 1 Va = ^Va. 
 
 Whenever a pure imaginary number occurs in any 
 algebraic work, it is to be expressed in terms of the 
 imaginary unit ^. 
 
.-A 
 
 Karl Friedrich Gauss (1777-1855) is called by general agree- 
 ment the greatest mathematician of modern times. In 1799 he 
 published a proof of the theorem that every algebraic equation has 
 a root of the form a + bi. He introduced the symbol / to denote 
 V — 1 and was the originator of a great part of the modern theory 
 of numbers. 
 
QUADRATIC EQUATIONS 331 
 
 256. Powers of /. We have by definition, (V— 1)2 
 = — 1, or 1*2 = — 1 ; therefore, 
 i = V^T. 
 z2 = - 1. 
 
 ^3 =z iH = (— V)i = — i. 
 1^= 0*2)2= (-1)2= +1. 
 
 iH = 1 ' i = i. 
 
 From these identities it is inferred that any even 
 power of i is a real number, namely, + 1 or — 1, and that 
 any odd power of i is a pure imaginary number, namely, 
 + 2 or — ^. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Find the product of V— 4 and V— 9. 
 Solution. \/31=V(£T)4=V^Vi = 2i. 
 
 v^ = V(- 1)9 = V- 1 V9 = 3 i. 
 
 Hence, v""^ V - 9 = (2 1)(3 i) = 6 i^ = 6(- 1) = - 6. 
 
 Note. An error is often made in finding the product of two 
 imaginary numbers by an incorrect use of the identity y/a Vb = y/ab. 
 In the proof of this identity the numbers a and b ,were limited to 
 positive numbers ; the expressions V — a and V— b were entirely 
 meaningless. This error is avoided by observing that a pure imagi- 
 nary number is to be expressed in the form ai, where a is a real number 
 [§255]. 
 
 2. Simplify — 
 
 Solution. _l__ = l = i = _J_ = _j. 
 
 V3i i i^ -1 
 
 3. Simplify -V^^ x V^Tg -- V^^5. 
 
 Solution. V32 x V^3 ^ V^^ ^iy/2iVS 
 
 iV5 
 
 >/6 6 6 
 
332 
 
 ELEMENTARY ALGEBRA 
 
 257. Simplest form of an expression containing a pure 
 imaginary. An expression which contains pure imaginary- 
 numbers is said to be in its simplest form when its de- 
 nominator is a rational number and its numerator con- 
 tains no real factor under a radical sign which can be 
 removed. 
 
 EXERCISE 122 
 
 Simplify 
 1 
 
 13. 
 
 15. 
 
 17. 
 
 19. 
 
 V- 
 
 -4a2 
 
 v^ 
 
 -5 
 
 V- 
 
 -30 
 
 i- 
 
 «2 
 
 " 3* 
 
 V- 
 
 -2-hV-3 
 
 
 V-2 
 
 V2 
 
 4-V-3 
 
 V2 
 
 "J 
 
 V-3 
 4. v^lV^^. 
 
 6. ^^, 
 V3 
 
 8. v^V^^. 
 1 
 
 
 V-3a 
 
 19 
 
 V-30 
 
 
 V-5 
 
 14. 
 
 v-|. 
 
 16. 
 
 u 
 
 la 
 
 V^a + V^ 
 
 
 V-6 
 
 9n 
 
 V-2 + 3 
 
 22. t^. 
 
QUADRATIC EQUATIONS 333 
 
 
 1 
 ill* 
 
 
 23. 
 
 24. V-125. 
 
 
 25. V- 24x36x53. 
 
 258. Complex numbers. A number which can be ex- 
 pressed in the form a -h 5V— 1, where a and h are real 
 numbers, neither one of which is zero, is called a complex 
 number. 
 
 Thus, 4+V— 1, 2— \/-5, and 5 + 2V— 1 are complex numbers. 
 
 259. Conjugate complex numbers. Two complex num- 
 bers which differ in the sign of the imaginary unit only 
 are called conjugate complex numbers. 
 
 Thus, 1 + iy/2 and 1 - iV2 ; \/2 + V^^a and V2 - V^I^ ; a + 6i 
 and a — hi, are pairs of conjugate complex numbers. 
 
 260.' Sum and product of two conjugate complex numbers. 
 
 Let a + hi and a — hi be any two complex numbers ; then : 
 
 The sum = (a + hi) + (a — hi) = 2 a. 
 
 The product = (a + hi) (a - i{) = a^ - (6i)2 
 
 = a2 _ J2^-2 = a2 _ 52(_ 1)= a2 + 62. 
 
 Hence both the sum and the product of two conjugate 
 complex numbers are real numbers. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Find the product of V2 + V^^ and V3 + V"^5". 
 Solution. V2 + iV3 
 
 VE + 3i 
 
 + ^ VTO + i2Vl5 
 
 V6 + i(3+Vl0)-Vl5 
 Therefore, 
 
334 ELEMENTARY ALGEBRA 
 
 Remark. Observe that the product of the two complex num- 
 bers is a complex number. 
 
 Q 
 
 2 . Rationalize the denominator of the fraction = . 
 
 V5 + V-3 
 
 Solution. Since the product of two conjugate complex numbers 
 is real, we multiply the numerator and denominator of the given 
 fraction by an expression conjugate to the denominator, and have 
 
 V5+V^ VS-V-a 5 + 3 
 
 Note. An imaginary number is sometimes defined as " any even 
 root of a negative number." However, although V— 1, for exam- 
 ple, is evidently not a real number, it is, nevertheless, not a pure 
 imaginary, but a complex number. For, it may be verified that 
 
 (_L + A.,V (±-±-i)\ (-^^±i)\ or i-^-^iY 
 VV2 V2 / W2 \/2 / \ V2 V2 / V V2 V2 / 
 
 is equal to — 1 ; hence, each of the expressions within the paren- 
 theses is a fourth root of — 1. 
 
 EXERCISE 123 
 
 Simplify 
 
 16 
 
 3. 
 4. 
 5. 
 6. 
 7. 
 8. 
 
 + V-36. 
 V3V-2. 
 
 2. 
 
 - a(a > i 
 
 r4)(V: 
 
 -2 + V- 
 
 + 3V- 
 
 • 
 
 -3). 
 
 
 1. V- 
 
 -Tkv 
 
 
 (V-2-V- 
 (l + V-5)2. 
 
 
 9. (2-hV-3)(2-V-3). 
 
 10. q + i^-^^i-iV^). 
 
 11. V— a-j-V— i. 
 
 12. (5 + ^V3)-^(5-^V3). 
 
QUADRATIC EQUATIONS 335 
 
 2 2 
 
 13. = 14. 
 
 -3 + 1 V-2h-V^ 
 
 5 
 
 15. 
 
 V-lO-V-5 
 261. Quadratic equations with complex roots. 
 
 ILLUSTRATIVE EXAMPLE 
 
 Solve the equation 2a;2 — 3a;4-5 =0. 
 Solution by completing the square. 
 
 2x2 -3a: + 5 = 0. (1) 
 
 Transforming (1), a:^ - | a: + | = 0. (2) 
 
 Transposing, x^ — ^ x = — ^. (3) 
 
 Completing the square, a:^ — |-a: + ^ = — f + y^6=— ^q- (4) 
 
 That is, (a:-|)2 = _ll. (5) 
 
 Extracting the square root, ^c — ^ =± 4V— 31. (6) 
 
 Solving (6), ^=i±^El. ^ (7) 
 Solution by formulae. 
 
 _ ^, + V'52 _ 4 ac - 6 - V&2 _ 4 ac 
 
 xi = , x^ = 
 
 2a 2a 
 
 Here a = 2, 6 = -3, c = 5; hence, by substitution in the formulae 
 
 we find the roots to be 
 
 -(-3) + V9-40 ^^^ -(-3)-V9340 
 4 4 
 
 which when simplified are equal, respectively, to 
 
 3+^^=^ and S^l^^Zl. 
 
 EXERCISE 124 
 
 Solve the following equations, which have either pure 
 imaginary or complex roots : 
 
 1. 2:24.1 = 0. 2. ic2 + 4 = 0. 
 
336 ELEMENTARY ALGEBRA 
 
 3. a;2+9 = 0. 4. 3^2^2 = 0. 
 
 5. 2a;2 + 3 = 0. 6. 3a^ + 5 = 0. 
 
 7. 5a:2-h2 = 0. 8. 15a;2 + 17 = 0. 
 
 9. a;2+2a; + 2 = 0. 10. 2^2^43,.^ 5 = 0. 
 
 11. 2^2-62: + 11 = 0. 12. a:2 + 5a; + 7 = 0. 
 
 13. 2a^-32; + 2 = 0. 14. 32^2_3^+2 = 0. 
 
 EXERCISE 125 
 
 1. A man bought a certain number of oranges at 
 75 cts. The number of oranges he bought was three times 
 the number of cents he paid for each orange. How many 
 oranges did he buy ? 
 
 2. The side of one square is three times the side of 
 another and the difference of their areas is 32. What is 
 the side of the smaller square ? 
 
 3. If the edge of a certain cube be doubled, the area 
 of the entire surface of the cube will be increased by 
 72 sq. in. What is the edge of the cube ? 
 
 4. A dealer sold an article at a loss of f 6.25 and 
 thereby lost as many per cent as there were dollars in the 
 cost. What was the cost ? 
 
 5. Find two consecutive integers whose product is 56. 
 
 6. Find two consecutive integers whose product is 
 462. 
 
 7. Find a number whose square exceeds 100 times the 
 number by 2684. 
 
 8. Two odd integers differ by 2 and the difference of 
 their squares is 56. Find the integers. 
 
 9. The quotient of two numbers is 2J and tfieir prod- 
 uct is 756. Find the numbers. 
 
QUADRATIC EQUATIONS 337 
 
 10. The difference of the squares of two consecutive 
 numbers is 197. Find the numbers. 
 
 11. The sum of two numbers is 21, and their product 
 is 110. What are the numbers ? 
 
 12. The difference of two numbers is 4, and their prod- 
 uct is 45. What are the numbers ? 
 
 13. The difference of two numbers is 42, and their 
 quotient is the less number. What are the numbers ? 
 
 14. A dealer sold an article for $ 39 and thereby gained 
 as many per cent as there were dollars in the cost. Find 
 the cost. 
 
 15. The plate of a looking-glass is 18 in. by 12 in. ; it 
 is to be surrounded by a plain frame of uniform width, 
 whose area shall be equal to that of the glass. Required 
 the width of the frame. 
 
 16. Find three consecutive integers the sum of whose 
 products by pairs is 299. 
 
 17. If a body be thrown vertically upward from the 
 ground with an initial velocity of 32 ft. per second, when 
 will it be at a height of 7 ft. ? 
 
 Suggestion. Use the formula s = at — 16 fi, in which a represents 
 the initial velocity and s the height at the end of t seconds. 
 
 18. The denominator of a given fraction is one greater 
 than its numerator ; if ^"^ be added to the fraction, the 
 sum is equal to the reciprocal of the given fraction. Find 
 the given fraction. 
 
 19. The difference between the hypotenuse and base of 
 a right-angled triangle is 6, and the difference between 
 the hypotenuse and altitude is 3. What are the sides ? 
 
 20. A square garden is surrounded by a path. The 
 area of the path is 12,400 sq. ft. The garden is 290 ft. 
 wider than the path. Find the area of the garden. 
 
338 ELEMENTARY ALGEBRA 
 
 21. A field containing one acre is in the form of a rec- 
 tangle I as wide as it is long. The field is enlarged by 
 adding 39,664 sq. ft. in such a way as to increase length 
 and width of the rectangle an equal amount. Find the 
 dimensions of the enlarged field. 
 
 22. From the point of intersection of two straight 
 roads which intersect at right angles, two men, A and B, 
 set out simultaneously, A on the one road riding at the 
 rate of 12 mi. per hour, and B on the other walking at the 
 rate of 5 mi. per hour. After how many hours will they 
 be 65 mi. apart ? 
 
 23. A number of laborers were employed to do a piece 
 of work. If 7 less had been employed, the work would 
 have taken two more days. If 28 men had been em- 
 ployed, the work would have been done in 20 days. 
 How many laborers were employed ? 
 
 24. A gardener planted a certain number of trees at 
 equal distances apart, and in the form of a square. He 
 found on finishing the planting that he had 5 trees to 
 spare. He then added one of them to each row as far as 
 they would go, and found that he needed 10 trees to com- 
 plete the square. How many trees had he ? 
 
 25. Find the price of tea per pound if a rise of 10 
 cents in the price per pound would reduce by 5 lb. the 
 quantity obtainable for $15. 
 
 26. What is the price of eggs per dozen, if a fall of 
 2 cents in the price would increase by one the number 
 of dozen obtainable for i6.84 ? 
 
 27. Two trains travel without stopping between two 
 stations m miles apart. One train goes a miles an hour 
 faster than the other and takes h hours less time for the 
 
QUADRATIC EQUATIONS 339 
 
 journey. Find the speed of each train. What is the 
 speed of each train if m = 40, a = 10, and h = ^? 
 
 28. A man bought a certain number of cows for flSOO. 
 He sold 5 less than the whole number of cows for 1 20 a 
 head more than they cost him and made $100 by the 
 transaction. How many cows did he buy ? 
 
 29. A merchant sold 7 doz. fresh eggs and 12 doz. 
 storage eggs for $5.81, and found that he had sold 1 doz. 
 more fresh eggs for $2.10 than he had of storage eggs 
 for $1.40. Required the price of each kind per dozen. 
 
 30. A merchant draws a certain quantity of vinegar 
 from a full cask containing 63 gallons. Having filled up 
 the cask with water, he draws the same quantity as 
 before. He then finds that the cask contains ^g the 
 original quantity of vinegar. How many gallons did 
 he draw each time ? 
 
 31. A sum of $30,000 is subject to an inheritance tax 
 of a certain per cent, then to a percentage for fees at a 
 rate one half per cent greater than that of the inheritance 
 tax. When the tax and fees are deducted there remains 
 $ 27,504. What are the two rates ? 
 
 262. Utility of the extension of the meaning of the word 
 number. The extension of the meaning of the word num- 
 ber so as to include under the term such expressions as 
 V— 2 and 2 + V— 3 renders possible a greater generality 
 in the statement of algebraic principles and results. As 
 an illustration of this, the statement that except when 
 l^ — 4:ac is negative, the quadratic equation ax^ -\-bx-\- c=0 
 admits of solution [section 248, note], is replaced by the 
 general statement, everi/ quadratic equation has two roots. 
 
340 ELEMENTARY ALGEBRA 
 
 263. Nature of the roots of ax^ + bx+c = 0. In this 
 equation the a, 6, and c represent rational numbers. The 
 roots of ax^ -\-bx-\- c=0 have been found to be, 
 
 a?, = -^—- and x^ = 
 
 An examination of these formulae leads to the follow- 
 ing important principles : 
 
 I. When 6^ _ 4 ac is positive and not equal to zero. 
 
 In this case, 
 
 1. The roots are real and unequal. 
 
 Thus, in the equation Oa:^ — 18a; + 7 = the expression 6^ — 4 ac 
 = 72, and the roots, 1 ± J"^) are real and unequal. 
 
 2. If 52 — 4 ac is a perfect square, the roots are rational ; 
 and, conversely, the roots are rational only when 5^ — 4 ac 
 is a perfect square. 
 
 Thus, in the equation 2ar2 + 5a; — 3 = 0, the expression b^ — iac 
 = 49, and the roots, — 3 and -^ , are rational. 
 
 3. If b^ — 4:ac is not a perfect square, the roots are 
 conjugate quadratic surd expressions. 
 
 Thus, in the equation 9 x^ — 18 a; + 7 = 0, the expression 
 b^ — iac = 72, and the roots 1 + J V2 and 1 — J V2 are conjugate 
 quadratic surd expressions. 
 
 4. If c is 0, one root is zero, and the second root is 
 rational. 
 
 Thus, in the equation a:* — 3 a: = 0, the roots are zero and 3. 
 
 Note. If a and c have opposite signs, 6* — 4 ac is necessarily posi- 
 tive and the roots are always real. 
 
 II. When J^ - 4 acis equal to zero. 
 In this case, 
 
QUADRATIC EQUATIONS 341 
 
 1. The roots are rational and equal. 
 
 Thus, in the equation 9x^ — Qx-^l = 0, the expression b^ ~ i^ac 
 = 0, and the roots are ^ and J. 
 
 2. The polynomial aa^ + 5a; + ^ is a perfect square. 
 Thus, 9a;2-6a: + l = may be written (3 ar - 1)2 = 0. 
 
 3. When c is zero, then b is also zero, and both roots 
 are zero. 
 
 Thus, if 62 _ 4 ac = 0, and c = 0, then b^ = 0, or 5 = ; and the 
 roots of the equation are and 0. 
 
 III. When 6^ — 4 ac is negative. 
 In this case, 
 
 1. The roots are not real. 
 
 Thus, in the equation 3a;2 — 2ar + 4: = 0, the expression b^ — iac 
 
 = ^ 44 and the roots are ~ — and ~ ~ — , which are not 
 
 3 3 ' 
 
 real. 
 
 2. When b is zero, the roots are pure imaginary numbers. 
 
 Thus, in the equation a:2 + 4 = 0, the expression b^ — iac = — i 
 and b is zero. The roots are 2 i and — 2 e, which are pure imaginary 
 numbers. 
 
 3. When b is not zero, the roots are conjugate complex 
 numbers. 
 
 Thus, in the equation 3a;2 — 2a; + 4 = 0, which is the equation 
 given in 1, the roots are seen to be conjugate complex numbers. 
 
 Note. The statement that when b^ — 4:ac = the polynomial 
 ax^ -\- bx -\- c is a. perfect square, may be proved as follows : 
 
 Given ^2 - 4 ac = 0. 
 
 Transposing, 62 — 4 qc. 
 
 Therefore, b =±2y/aVc. 
 
 Substituting, ax^ -{- bx + c = ax^ ± 2VaVcx + c 
 
 = (ar Va ± Vc)2. 
 
a42 ELEMENTARY ALGEBRA 
 
 EXERCISE 126 
 
 Calculate for each of the following equations the value 
 of h^ — 4 ac and determine from principles of section 263 
 the nature of the roots. The principles should not be 
 memorized, but the reason for each statement made should 
 be clearly understood. 
 
 1. a:2H-3a:-4 = 0. 
 
 3. 5a:2 = 0. 
 
 5. 3a;2 + 2a; + 5 = 0. 
 
 7. ic2-a;-l = 0. 
 
 9. 9a:2-12a: + 4 = 0. 
 11. |a;2-f^ + ^Y^ = 0. 
 13. 2a;2_3^_20 = 0. 
 15. a^2_^2«rr + a2+52=o. 
 
 264. Factors of ax^+hx-\- c. By definition, a root of 
 
 an equation is a number which, when substituted for the 
 unknown, reduces the equation to an identity ; that is, 
 satisfies the equation. 
 
 Let Xj represent a root of the equation, 
 
 ax^ -\- hx -\- c = 0. (1) 
 Substituting x-^ for x in (1), 
 
 ax^ + 6^1 + c = 0. (2) 
 
 Solving (2) for c, c = - ax^^ - hx^ (3) 
 .Substituting the value of c in (1), 
 
 ax^ + hx ■\- c = ax^ + bx - ax^ - bx^ (4) 
 
 = a(x^-x,^)+b(x-x^ (5) 
 
 = (x - xi) (ax + axi + b) (6) 
 
 From the foregoing it may be inferred that : 
 if x^ is a root of the equation a^ + 5a; -|- c = 0, the polyno- 
 mial aoi^ 4- ia; + c is exactly divisible by x— x^ 
 
 2. 
 
 2a:2_3_0. 
 
 4. 
 
 a;2 + 1 = 0. ' 
 
 6. 
 
 3a^-2rr-2 = 0. 
 
 8. 
 
 a^-{-x-{-l = 0. 
 
 10. 
 
 fa^-|a; + | = 0. 
 
 12. 
 
 4.9a:2-7.35a;-22.05 = 0. 
 
 14. 
 
 169a;2+442^4.289 = 0. 
 
 16. 
 
 a:2_2aa: + a2 + 52=0. 
 
QUADRATIC EQUATIONS 34$ 
 
 A second root of ax^ + 62: + c = is obtained by equat- 
 ing the second factor of ax^ -\-hx-^ c^ namely, ax + ax^ + 6, 
 to and solving for x [see section 117]. 
 
 Thus, ax + a^i + 6 = 0. (1) 
 
 Solving (1) for x, , x — —x^ • (2) 
 
 Representing the root — x^ — by x,^ we have, 
 
 Transposing, x^-\- x^ = (^) 
 
 Identity (4) is in agreement with (I) of section 249. 
 The expression ax + ax^ + h may be written a[x-\-x^-\--\ 
 
 or substituting — x^ for a;^ + - , we have a{x — x^. There- 
 fore, the polynomial a'39- -\- hx -\- c is identically equal to 
 a{x — Xy){x — x^ where x^ and x^ are two roots of the 
 quadratic ax?' + 6a; + c = 0. 
 
 265. Number of roots of a quadratic. Let x^ be a root 
 of the quadratic equation 
 
 ax^ -\- hx -\- c — ^ \ 
 
 that is, of the equation 
 
 a{x - x^{x - x^=^, 
 
 in which x-^ and x^ are two roots of the quadratic. Ob- 
 serve that the existence of at least two roots was shown 
 in section 248. 
 
 Substituting x^ for x in the given equation, a(x^ — x^^ 
 (a;g — x^) = 0. From this identity it is evident, since a is 
 not zero, that either x^ — x^ = or x^ — x^^O; that is, x^ 
 is equal to either x^ or x^; and, therefore, that: 
 
 Every quadratic equation has two and only two roots. 
 
344 ELEMENTARY ALGEBRA 
 
 266. Quadratic with given roots. The quadratic equa- 
 tion whose roots are x^ and x^ is a(^x — x^^(^x — x^) = [see 
 section 264]. In this equation a may have any constant 
 value. 
 
 Let the two equations ax^ + bx + c = and mx^ + nx + p = have 
 the same roots x^ and x^ We may write [see section 249] : 
 
 b] 
 
 ^1 + ^2 = - 
 a 
 
 and 
 
 •^1 ~f" "^2 
 
 m 
 m 
 
 From these identities, —=- = —. Therefore : 
 m n p 
 
 Two quadratic equations with the same roots have their cor- 
 responding coefficients proportional ; and conversely, if two 
 quadratic equations have their corresponding coefficients pro- 
 portional^ they have the same roots. 
 
 ILLUSTRATIVE EXAMPLE 
 
 P^ind the simplest form of a quadratic equation whose 
 roots are — | and |. 
 
 Solution. The required equation is a{x + J)(a: — ^)= 0. 
 
 That is, a(3^)(5^) = 0. 
 
 Letting a = 15, (3 a: + 2)(5 x - 3) = 0. 
 
 Expanding, 15 a;^ + a; - 6 = 0. 
 
 EXERCISE 127 
 
 1. By inspection, arrange the following equations in 
 groups so that those in any group shall have the same roots : 
 
 5 2:2 _ 15 a; ^. 10 = 0. 10 a^ - 15 a: - 10 = 0. 
 
 a:a~§£-l = 0. m^-Smx-{-2m=0, 
 
 2 
 
QUADRATIC EQUATIONS 
 Find the quadratic whose roots are : 
 
 345 
 
 2. — 5 and 6. 3. | and f 
 
 5. — t and X. 
 
 4. -f and -f 
 
 e. 1±;^ and 3Ll^ 
 
 7. — r— and — -— ■ 
 
 3_ l + 2V-2 ^^^ 1-2V-2 _ 
 
 „ 2 + 3i . 2-3i 
 
 267. Graph of a polynomial of the second degree. 
 
 Graphs of linear functions of one variable and of linear 
 equations of two unknowns were treated in Chapter IX, 
 We shall now consider the graphs of certain quadratic 
 functions. 
 
 268. Graph of x^ — 2 x. Representing the polynomial 
 x^ — 2xhj 1/ ; then, 
 
 i/^x^-^2x. (1) 
 
 We construct a table of the corresponding values of x and y by 
 arbitrarily assigning values to x and calculating the corresponding 
 values of y from equation (1). The table follows : 
 
 X 
 
 -4 
 
 -3 
 
 -2 
 
 -1 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 X2 
 
 16 
 
 9 
 
 4 
 
 1 
 
 
 
 1 
 
 4 
 
 9 
 
 16 
 
 25 
 
 36 
 
 -2x 
 
 8 
 
 6 
 
 4 
 
 2 
 3 
 
 
 
 -2 
 
 -4 
 
 -6 
 
 -8 
 
 -10 
 
 -12 
 
 y 
 
 24 
 
 16 
 
 8 
 
 
 
 -1 
 
 
 
 3 
 
 8 
 
 16 
 
 24 
 
 Plotting the points (x, y) as given in the table and drawing a 
 smooth curve through these points, we have the required graph of 
 the polynomial 3^- 2XfM indicated in the figure, page 346. 
 
346 
 
 ELEMENTARY ALGEBRA 
 
 From the graph of the poly- 
 nomial x^ -2x; that is, from 
 the graph of the equation 
 y = z^ — 2x, the roots of the 
 quadratic equation a:^ - 2 a: = 
 may be found by inspection. 
 Evidently, those values of x 
 which make y equal to zero 
 are the roots of the equation 
 x^ — 2x = 0, for they satisfy the 
 equation. The points on the 
 graph for which y = are those 
 common to the graph and the 
 a:-axis ; for the ordinate y oi a, 
 point is zero only when the 
 point is on the x-axis. The roots 
 of the equation a:^ — 2 a: = are, 
 therefore, the a:(abscissas) of 
 the points in which the graph 
 of the polynomial x^ — 2x in- 
 tersects the X-axis. These 
 points of intersection are x = 
 and X = 2j and the roots of 
 
 the equation ar^ - 2a; = are and 2. 
 
 269. Graphof x2-A:-f 1. 
 
 Let y = x^ —X -h 1. 
 
 A table of corresponding values of x and y is as follows : 
 
 X 
 
 -4 
 
 -3 
 
 -2 
 
 -1 
 
 
 
 \ 
 
 1 
 
 2 
 
 3 
 
 4 
 
 6 
 
 ^ 
 
 16 
 
 9 
 
 4 
 
 1 
 
 
 
 \ 
 
 1 
 
 4 
 
 9 
 
 16 
 
 25 
 -4 
 
 -a;-f-l 
 
 6 
 
 4 
 
 8 
 
 2 
 
 1 
 
 \ 
 
 
 
 -1 
 
 -2 
 
 -3 
 
 V 
 
 21 
 
 13 
 
 7 
 
 3 
 
 1 
 
 \ 
 
 1 
 
 3 
 
 7 
 
 13 
 
 21 
 
 Plotting the points (a:, y) as given in the table and drawing a 
 smooth curve through these points, we have the required graph of 
 the polynomial a;^ — a; -h 1, as in the figure, page 347. 
 
QUADRATIC EQUATIONS 
 
 347 
 
 We observe that the graph of the equation y = x^ — x + 1 does 
 not intersect the axis of x. This indicates that the quadratic 
 equation a;^ — x + 1 = has no real root. The roots of this equation 
 
 are 
 
 1± V- 
 
 that is, conjugate complex numbers. In general, 
 
 when the graph of a polynomial ax^ + bx -^ c does not intersect 
 
 m 
 
 the X-axis, the roots of the equation ax^ + bx -\- c = are either 
 pure imaginary numbers or conjugate complex numbers. 
 
 270. Graph of a system of two simultaneous equations. 
 Given the system of two simultaneous equations, 
 
 
 (1) 
 (2) 
 
 In section 192 we learned that the graph of the linear equation 
 y = 2x is a straight line. It is, therefore, necessary to find the 
 codrdinates of two of its point* only in order to plot the straight 
 
348 
 
 ELEMENTARY ALGEBRA 
 
 line. We observe from equation (2) that the points (0, 0) and 
 (3, 6) may be taken. The corresponding values of x and y in 
 equation (1); that is, of x and a:^, are given in the preceding table. 
 
 The graphs of equations (1) and 
 (2) are seen in the accompanying 
 figure. 
 
 The real solutions of equations 
 (1) and (2) are the coordinates 
 of the points of intersection of 
 their graphs ; for the coordinates 
 of these points satisfy both equa- 
 tions, and the coordinates of any 
 point not on both graphs do 
 not satisfy the equations. The 
 straight line y = 2 a: intersects 
 the graph of y = a;^ in the points 
 (0, 0) and (2, 4). The two solu- 
 tions of the simultaneous equa- 
 tions y = x^ and y = '2x are, 
 therefore, (0, 0) and (2, 4>. 
 
 When the graphs of two equar 
 tions do not intersect, their solu- 
 tions involve pure imaginary num- 
 bers or complex numbers. 
 
 The graphs of y — x"^ and 
 y ■=2x also show by their intersections the roots of the equation 
 a:2 - 2 a; = 0. 
 
 Thus, the ordinate of any point on the straight line whose- 
 equation is y = 2 a: is equal to twice its abscissa, and the ordinate 
 of any point on the graph of y — x"^ is equal to the square of its 
 abscissa. At a common point the ordinate is equal to twice 
 its abscissa and also to the square of its abscissa. The abscissa of 
 a common point, therefore, satisfies the equation 
 
 a;2 = 2 a:, or a:2 - 2 a; = 0. 
 
 In like manner the real roots of aa;* 4- &a: + c = are the abscissas 
 of the points common to the graphs of the equations 
 
 
 L_ Y 
 
 
 , 
 
 L 7 
 
 f I 
 
 4 t j- 
 
 4 ^ ' 
 
 1- ^ 7- 
 
 ' t 
 
 aJ 
 
 X t' 
 
 4 H 
 
 4- l|- 
 
 t t 
 
 X JL 
 
 )^2L 
 
 -^t 
 
 x-j 7"o X 
 
 t 
 
 7 
 
 
 Y'^ 
 
 y = Qx^ and y = - &x - c. 
 
QUADRATIC EQUATIONS 349 
 
 EXEBCISE 128 
 
 1. Construct the graph of the equation y = qc^ — 1. 
 From the resulting graph determine the roots of the equa- 
 tion a;2 — 1 = 0. 
 
 2. Construct with respect to the same axes of reference 
 the graphs of y = 2x^—1 and y = — ^x + l. Estimate 
 from the figure the values of x and y which satisfy both 
 equations. Also obtain approximately the roots of 
 2x^-\-Sx-2 = 0. 
 
 3. Construct with respect to the same axes of reference 
 the graph of y -\-2x^ -Sx -9 = and y-\-x~S = 0. Es- 
 timate from the figure the values of x and y which satisfy 
 both equations. Also obtain approximately the roots of 
 x^-2x-^ = 0. 
 
 4. Construct the graph of x^— 2. Estimate from the 
 figure, correct to one decimal place, the value of V2. 
 
 5. Plot 2y—Sx = 6 and 2x'\-l=4:y — 4:y\ using the 
 same axes, and estimate from the graphs the solutions of 
 the equations. 
 
 6. Graph ^ = 1 + 32^. 
 
 7. Construct with respect to the same axes of reference 
 the graphs oi y = 2x'^ -{-1 and y = x^ -\-S x — 1. Estimate 
 from the figure the values of x and y which satisfy 
 both equations. Also obtain from the figure the roots of 
 2^-Bx-{-2 = 0. 
 
 8. Construct with respect to the same axes of reference 
 the graphs of x^-\-y-5 = and y^-\- Sy — 2x=0. Es- 
 timate from the figure the real values of x and y which 
 satisfy both equations. 
 
CHAPTER XIII 
 SYSTEMS OF QUADRATIC EQUATIONS 
 
 271. Systems of two equations in two unknowns. The 
 
 elimination of one of the unknowns from two equations of 
 the second degree in two unknowns does not, in general, 
 lead to a quadratic equation in one unknown. Certain 
 special systems of two equations, however, neither one of 
 which is of higher degree than the second, are of frequent 
 occurrence and lead to quadratic equations in one un- 
 known. Such systems may be solved by the methods of 
 
 preceding chapters. 
 • 
 
 272. A quadratic and a linear equation. A system con- 
 sisting of a quadratic and a linear equation may always 
 be solved by substitution. The method is indicated in 
 the illustrative examples which follow. 
 
 ILLUSTRATIVE EXAMPLES 
 ,.2 
 
 Solve the system \ „ . ^ „ 
 
 a) 
 
 (2) 
 
 Solution. Solving (2) f or y, y = -Sx + 13. (8) 
 Substituting in (1) the value of y from (3), 
 
 a;a + 2a:(-3a:+13) = 33. (4) 
 
 Simplifying (4) , 5 a;^ - 26 a: + 33 = 0. (6) 
 
 Factoring (5), (x - 3)(5 a; - 11) = 0. (6) 
 
 From (6), ar = 3. (7) 
 
 Also from (6), ^ = ^' (S) 
 
 Substituting 3 for a: in (3), y = 4, (O) 
 One solution of (1) and (2) is, therefore, (3, 4) 
 
 Substituting ^ for a: in (3), y = ^. (10) 
 
 The second solution of (1) and (2) is, therefore, (-y-, ■^). 
 
 S50 
 
SYSTEMS OF QUADRATIC EQUATIONS 351 
 
 Check. Substituting 3 and 4 for x and y, respectively, in (1) . 
 
 9 4- 24 = 33, (11) 
 
 which is an identity. 
 
 Substituting JJ- and -^ respectively, in (1) 
 
 ,.,. ., . W + W = ^^' (12) 
 
 which IS an identity. 
 
 The values of y were obtained from (3), which is another form of 
 
 (2); it is, therefore, unnecessary to substitute in (2), 
 
 2. Solve the system 
 
 2x^-Sxi/-\- f-Bx + l ^ -4:=0, (1) 
 
 4a:+3^ + l=0. (2) 
 
 Solution. Solving (2) for y, y = - ijLtl (3) 
 
 Substituting in (1) the value of y from (3), 
 2x«+3.(i^l) + (i^)'-5.-7(l^)-4 = 0. (4) 
 
 Simplifying (4), 5 a;^ - 8 a: - 4 = 0. (5) 
 
 Factoring (5), (x - 2) (5 a: + 2) = 0. (6) 
 
 From (6), ' a: = 2. (7) 
 
 Also from (6), ^ = - f • (8) 
 
 Substituting 2 for a: in (3), 2^ = - 3. (9) 
 One solution of (1) and (2) is, therefore, (2, — 3). 
 
 Substituting - ^ ior x in (3), !/ = i' (10) 
 The solutions of (1) and (2) are, therefore, (2, - 3) and (- |, •^), 
 
 which solutions should be verified by substituting in (1) the values 
 
 of X and y as found. 
 
 EXERCISE 129 
 
 Solve the following systems, and check each solution: 
 
 *la; = l. ' [^ — x=: 0. 
 
 a; + 2 3/ = 17. ' \x-^ = ^. 
 
 \xy = 2, 
 
 3ir2_^2aj + y-ll = 0, 
 2x-y-\-l = 0, ^ 
 
352 
 
 ELEMENTARY ALGEBRA 
 
 j6a;2^ + 4=0, 
 '' I5a:-5^=21. 
 
 -I 
 
 6 a;y + 4 = 0, ^ f a;^ _ ^ = ^2 _|. ^.^ 
 
 (^ + l)-(2;-l) = 0. 
 1 
 
 10. 
 
 11. 
 
 = 8, 
 xy 
 
 1 + 2^17. 
 a; y 
 
 12. 
 
 13 
 
 a;2_43/2 = 4, 
 2a: + 3«/ + 4 = 0. 
 
 (2;-l)2 + (y_3)2 = 25, 
 
 fa;2_^2^3^^2«/-10 = 0, 
 
 14. 
 
 15. 
 
 16. 
 
 rr2 _ 2 a;?^ + «/2 -f 2 a; - 2 ^ = 0, 
 a;+^-2 = 0. 
 
 2rr2-3a:^+4?/2_2a; + 3^-6 = 0, 
 5a: + 4«/— 1 = 0. 
 
 3a;2+2a;-3^~2 = 0, 
 3a; + ^-4 = 0. 
 
 273. Two quadratic equations, one of which is homogene- 
 ous. When one of the given quadratic equations is homo- 
 geneous, the given system can be replaced by two systems 
 each of which is of the type considered in section 272. 
 The method of solution is as follows : 
 
 ILLUSTRATIVE EXAMPLE 
 
 x^-hx + Qy-lS = 
 
 2x^-xy-15y^ = 0, (1) 
 
 0. (2) 
 
SYSTEMS OF QUADRATIC EQUATIONS 353 
 
 Expressing the homogeneous polynomial 2 x^ — xy — 15 y^ a,s the 
 product of two linear factors, equation (1) may be replaced by 
 
 (x-3y)(2x + 5y)=0. (3) 
 
 Since any solution common to equations (2) and (3) must satisfy 
 either x — 3y = 0, or2a; + 5?/ = 0, we may consider first those solu- 
 tions of the given systems which satisfy 
 
 x - 3 y = 0, (4) 
 
 and a:2 + a; + 6 3/ - 18 = 0. (2) 
 
 Afterwards those that satisfy 2a;+5y = 0, (5) 
 
 and a:2 + x + 6 y - 18 = 0. (2) 
 
 Substituting in (2) the value of x from (4) and simplifying, 
 
 y^ + y-2 = 0. (6) 
 
 Solving (6), y = 1; also, y = — 2. 
 
 From (4), when 3/ = 1, a; = 3 ; when y = — 2, a: = — 6. 
 Therefore, the solutions of (4) and (2) are (3, 1) and (-6, — 2). 
 Substituting in (2) the value of y from (5) and simplifying, 
 
 5x2 -7a; -90 = 0. (7) 
 
 Solving (7), x = 5 ; also a: = — ^-. 
 
 From (5), when a; = 5, ?/ = — 2 ; when x = — ^, y = |-|.. 
 
 Therefore, the solutions of (5) and (2) are (5, — 2), and 
 
 (-¥.|f)- 
 
 Therefore, the solutions of the given system are (3, 1), (—6, —2), 
 (5, - 2), and (-1/-, |f). 
 
 EXERCISE 130 
 
 Solve the following systems : 
 
 a;2_y2^0, (X^-4: 2/^ = 0, 
 
 3a:2 + 7a;?/ = 0, (2x^ + i/^+4:i/-2S = 0, 
 
 x^z-hlx + l y + 49 = 0. *• 13 0^2 + 80:3/ -3 3/2 = 0. 
 a;2 4- 3 rry - 5 ?/2 = 0, f 2 2^2 + 7 a;^/ - 3 «/2 = 23, 
 
 [2/ 
 
 ^-\-4:X2/-14:X-24: = 0. [d X^ - 12 Xt/ -{- 4: t/^ = 0, 
 
354 ELEMENTARY ALGEBRA 
 
 274. Two equations without terms of the first degree. 
 
 A system of two simultaneous quadratic equations, neither 
 one of which contains terms of the first degree, and in 
 which the constant terms do not reduce to zero, may be 
 solved as in the following : 
 
 ILLUSTRATIVE EXAMPLE 
 
 Solve the system (^ ^ "^ ^ ^^ + ^' 7 ^' ^^^ 
 
 •^ 1 3 a^ + 7 a:?/ + 2 3/2 = 2. (2) 
 
 Multiplying both members of (1) by 2 so that the constant term 
 
 in the resulting equation shall be the same as that in (2), 
 
 ix^ + 4: xy + 2 y^ =z 2. (3) 
 
 From (2) and (3) by subtraction, x^ — S xy = 0. (4) 
 
 The given system can now be replaced by the equivalent system, 
 
 x(x-Sy) = 0, (5) 
 
 2x^+2xy + y^ = l. (1) 
 
 The system (5) and (1) can be solved by the method of 
 
 section 273. 
 
 Solving, we obtain the following solutions : 
 
 (0,1),(0, -l),(f,^),(-f. -J). 
 
 Remark. If the constant term in one of the given equations is 
 not a multiple of the other, the equations may be multipUed by such 
 numbers as will make the constant terms equal. 
 
 U2- 
 
 7. 
 
 EXERCISE 131 
 
 Solve the following systems : 
 
 + 42^^/ + 3^2=2, 2 1^2-32:^ + 2^2^3^ 
 
 4a:y+3^/2=12. * U24. 2 2:y - 3 ^^2^ 3, 
 
 1^:2 -h 3 1/2 = 21, g [2a:2_3a:^/ + ^/2 = 3, 
 
 1 2:2 + 2 a:?/- 3^/2= 15. * I a^ + a:!/ + i/2 = 7. 
 
 -3^:2/4-3^/2 = 12, f 8x2 + 11 a:i/+ 8^2 = 12, 
 
 fa:2-3 
 12 2:2- 
 
 2a;a-a:y + 4^2^ 16, Il5a:2+18a;2/ + 12y2=20 
 
SYSTEMS OF QUADRATIC EQUATIONS 355 
 
 275. Particular systems of equations. 
 
 I. When the sum and product of two unknowns are given, 
 either of two special methods of solution may be em- 
 ployed ; thus : 
 
 Solve the system \ ^^ ' )^ 
 
 •^ \x7/=n. (2) 
 
 Solution 1. From section 249 the roots of the quadratic 
 x^ — mx -{- n = are two numbers whose sum and product are equal, 
 respectively, to m and n. The roots of the equation, therefore, satisfy 
 the given system. 
 
 Solving the quadratic x^ — mx + n = 0, we have, 
 
 m ± y/m^ — 4 n 
 
 Therefore the two solutions of (1) and (2) are 
 
 (m + y/m^ — 4 w m — Vm^ — 4 n \ , f m — Vm^ — 4 n m + Vm^ — 4 n \ 
 
 2 ' 2 ) ^"^ \ 2 ' 2 / 
 
 Solution 2. Squaring (1) x^ + 2 xy + y^ = m\ (3) 
 
 Multiplying (2) by 4, 4 xy = 4 n. (4) 
 
 Subtracting (4) from (3), x"^ -2 xy ^ y^ = ni^ - ^ n. (5) 
 
 That is, ix -yy =m^-4: n. (6) 
 
 Therefore, x-y =±^1x1^ - 4 n. (7) 
 
 Adding (1) and (7), 2x = m± Vm^ - 4 n. ' (8) 
 
 Subtracting (7) from (1), 2y = m^ \/m^ - 4 n. (9) 
 
 mi i r . m + Vm^ — 4 n . m — Vm* — 4 n 
 Therefore, when ar is — ■ , y is , 
 
 J , . m — Vm^ — 4 n • w + V/n^ + 4 n 
 and when x is , y is . 
 
 EXERCISE 132 
 
 Solve the following systems : 
 * U?/ = 1. ' U^ = 12. 
 
356 
 
 ELEMENTARY ALGEBRA 
 
 3. i 
 
 la:^ = -28. 
 
 4. 
 
 ^x+y = i, 
 
 
 i^^ = A- 
 
 5. 
 
 
 6. 
 
 a; H- y = 3 a, 
 
 
 a^y = 2 a2. 
 
 7. 
 
 D a 
 , ly = 1. 
 
 8. 
 
 ^xy -\- a^ = 0, 
 
 9. 
 
 ^^ 3-2a' 
 l-5a 
 
 10. 
 
 ' , 2a+ 36 
 ^ + ^ = 2J-3a' 
 
 26- 16a 
 "^=3a-2J- 
 
 II. TFAeTi the difference and product of two numbers are 
 given, the following will illustrate a method of solution: 
 
 Solve the system | _ i o ' 
 
 (1) 
 
 (2) 
 
 Solution. Introducing an auxiliary number 2, defined by the 
 identity y = — 2, the given equations may be replaced by 
 
 a: + z = 1, 
 0:2; = -12. 
 
 (3) 
 (4) 
 
 The system (3) and (4) now belongs to class I and may be solved 
 by the methods employed therein, the solutions being 
 
 ar = 4, 2 
 
 3, and a; = — 3 and 2 = 4. 
 
 Substituting — y for 2, the solutions of the given system are (4, 3), 
 (-3,-4). 
 
 EXERCISE 133 
 
 Solve the following systems : 
 
 \xy + Q = (i, 
 1 8 a;t^ + 1 = 0. 
 
 2. 1^-^^ 
 Uy=3. 
 
SYSTEMS OF QUADRATIC EQUATIONS 357 
 
 I xy = 3. 
 
 a6 ~ I xy 
 
 xy-\-l = 0. 
 
 3a- 26 
 
 X- y = =-, 
 
 a — 
 
 35 
 
 10. 
 
 xy = 
 
 a — h 
 
 a+ 2 
 
 ^ 2g- 2a2 + 2a8 
 "^^ 2 a - 3 
 
 Solve the system { 
 
 III. When the mm of the squares and the product^ mm, 
 or difference of two numbers are given. Typical systems are : 
 
 ic2+3/2=5(l) ^^^2 = 5(1) 2^^^2=5(1) 
 
 xy = 2 (2) 2:+«/=3 (3) a: - y = 1 (4) 
 
 ^2 + 2/^ = 5, (1) 
 
 ^y=2. (2) 
 
 Solution. 
 
 Multiplying (2) by 2, 2xy = 4. (5) 
 
 Adding (5) and (1), x2 + 2 xy + 2/^ = 9. (6) 
 
 Subtracting (5) from (1), x^ -2xy -{• y^ = \. (7) 
 
 From (6), x + y=±^. ' '(8) 
 
 From (7), x-y = ±\, (9) 
 
 Adding (8) and (9), 2 x = ± 3 ± 1. (10) 
 
 Dividing (10) by 2, and combining, a; = 2, 1, — 1, — 2. 
 Subtracting (9) from (8), 2y=±^^l, (11) 
 
 Dividing (11) by 2 and combining, y = 1, 2, — 2,-1. 
 Therefore, the solutions of the system (1) and (2) are 
 
 (2,1), (1,2), (-1,-2), (-2, -1). 
 
 Note. In solving either the system i ^ ~ \J.\ or 
 
 ^ ^ \x + y = ^ (3)/ 
 
 ra:2 + 2/2 = 5 {1)\ 
 
 \x-y = l (4)/^ 
 
 it is evident that if either (3) or ^4) is squared and combined 
 with (1), the value of 2xy will be obtained; it will be found that 
 
358 ELEMENTARY ALGEBRA 
 
 2 a;y = 4, as in equation (5) of the foregoing solution •, so that the 
 solution of the system (1) and (3) reduces to the case of the solution 
 of a system such as is given under I, and the solution of the system 
 (1) and (4) reduces to the case of the solution of a system such as is 
 given under II. Observe, however, that the system of equations (1) 
 and (3), or (1) and (4) may be solved by the methods of section 272. 
 
 1 
 EXERCISE 134 
 
 Solve the following systems : 
 
 ^2+^2 = ¥, 2. |^ + ^' = 4i' 
 
 U + y = A. I 
 
 \x- y = 
 
 9. 
 
 = 290, 
 y = 16. 
 rr2 + ?/2 ^ 13 a2 + 10 a -h 2, 
 X — 1/ = a. 
 
 Xy = l. ' \x^=—4:, 
 
 , o_ 25a2 + l6a4-4 
 "" ^^ - (2a+l)2 ' 
 
 ^ 2a + 1 \xi/=l. 
 
 IV. When the polynomial in x and y {obtained hy omit- 
 ting the constant term) of one equation is a factor of that in 
 the other equation, a solution of the system can often be 
 obtained by quadratic equations, even though one of the 
 equations is of a degree higher than the second. The 
 solution of the following system will illustrate : 
 
 ia;8 I ^8 __ 7. r]\ 
 
 ... , x-^ y = \, (-^) 
 
SYSTEMS OF QUADRATIC EQUATIONS 359 
 
 Solution. 
 
 
 Factoring (1), (x + y) (x^ - xy + y^) = ^. 
 
 (3) 
 
 Substituting ^ior x -\- yin. (3), 
 
 
 J(x^-xy + 3,«) = 2V 
 
 (4) 
 
 Simplifying (4), x'' - xy + y^ = \. 
 
 (5) 
 
 Squaring (2), x^ + 2xy + y^ = \. 
 
 (6) 
 
 Subtracting (5) from (6), ^xy = — J. 
 
 (7) 
 
 Dividing, xy = —^. 
 
 (8) 
 
 Subtracting (8) from (5), x^ -2xy + y'^^l. 
 
 (9) 
 
 Therefore, x - y = ±\. 
 
 (10) 
 
 Addiug (10) and (2) and dividing, x = J or — ^. 
 
 (11) 
 
 Substituting, in (2) the value of x in (11), y = -^ov\. 
 
 a2) 
 
 Therefore, solutions of the given systems are (J, -^), 
 
 , and 
 
 (-if)- 
 
 A solution similar to the foregoing may be given for each 
 one of the following three systems of equations : 
 
 \x-y=ir \x''-^xy + y''=l ]' W-xy + y'' = \ I 
 
 EXERCISE 135 
 
 Solve the systems : 
 
 =-39, 
 13. 
 
 3. 
 
 \oi^ -{- y^ = ^. *U-y = l. 
 
 {x+y = 2, . g {x-y = l, 
 la;3 4-i/3=98. • U3- 2/3 = 127. 
 
 \x^-\-xy + y'^=l. ' Xx^-xy + y^^l, 
 
360 ELEMENTARY ALGEBRA 
 
 EXERCISE 136.— REVIEW 
 
 Solve the following systems of equations 
 
 ^^ Sx^--2f = l, ^ \4.x^ + xy = lb. 
 
 7. 
 
 11. 
 
 16. 
 
 17. 
 
 18. 
 
 19. 
 
 
 rc-f 3«^=34. 
 
 \x^ + xy = Q, 
 
 192^2 + 42/2 = 2, 
 
 — 4 aa; = 0, 
 
 \6xy = l. ^' \y = mx-{-- 
 
 m 
 
 j9x^-^16y^ = l, jy^^Sx, 
 
 1162^ + 9^2 = 1. »• \2;2+^2 = 20. 
 
 ^' ^ " / = 15. ^°- la^-2;y==3«/ + 9. 
 
 jx'- 
 \x^- 
 
 4 2^ + 6 2: 2/ + 2 a; - 6 y + 1 = , 
 2x-{-y-2 = 0. 
 
 12. [^' + 3^+11 = 0, ^^ j2^+^23.74, 
 
 12^2/ = 
 
 \32? + 2y + 14 = 0. 12^2/ = 35 
 
 ^^ r22:2-2;2/ = 2, ^^ f2?+32;+y2-2t/=. 
 
 14. S ^ o -< ^ 15' ' 
 
 12^2_^^=,12. *"• \32;-2t/ = 2. 
 
 l22;-3/ = 
 
 a^_^2_ 3^.^^ + 2 = 
 102. 
 
 22^-30^-292^ + 89 = 0, 
 y-x = 2. 
 
 \Sxy + 2x-Sy-10 = 0, 
 
 \x^-Sxy-^ly^ = 0. 
 
 f22;2-3«/ + 202;-100 = 0, 
 [2x^ + 5xy-7y^ = 0, 
 
20. 
 
 21. 
 
 22. 
 
 23. 
 
 24. 
 
 25. 
 
 28. 
 
 30. 
 
 31. 
 
 SYSTEMS OF QUADRATIC EQUATIONS 361 
 
 '2a;2 + 2a^«/- ^24. 3^_ 22/- 15 = 0, 
 3a;_4y-2 = 0. 
 
 2a;2 4- 5a;^ - 3 ?/ 4- 2a: + 36 = 0, 
 rr+?/=-2. 
 
 3a^ - ic?/ + ^2 _p 22; - 3^ + 5 = 0, 
 3a;2 4.a;^_2i/2 = o. 
 
 fy2 + 2a:y + ^+6 = 0, 
 3a:2 4-lla;y + 6i/2 = 0. 
 
 a;2 _ ^2 _|. 3 a; _ 2 y 4- 237 = 0, 
 52:2 -62;?/ + 3/2 = 0. 
 
 f2iry + 32;-4?/H-19 = 0, 
 
 \l2x-:i/ = l. 
 
 35. 
 
 22;2- 32:^-23/2 = 3, 
 32^2 4. 82:3/ -3^/2 = 8. 
 
 X^-\- y^z= a2, 
 
 a;4-^ = aV2. 
 
 \Q^-xy + y'^ = l, 
 
 X y _b 
 y X b 
 
 27. 
 
 2:8 4. 3/3 = ^37, 
 x + y = -l. 
 
 f2^4-3/'=«', 
 ^^- \2/ = 32;+aVlO. 
 
 32. ^ 
 
 34. 
 
 J + -2=4a2+62 
 
 2^2 ?/2 
 
 xy = 
 
 2 ah 
 
 36. 
 
 2^*4-2:21/24.3/4^91^ 
 2:2 _ 0:3/ 4- y2^ 7^ 
 
 2:4 _ 2/* = 15^ 
 
362 ELEMENTARY ALGEBRA 
 
 37. 
 
 38. 
 
 39. 
 
 f 7a:2 + 34rry + 39 ?/2 + a; + 3^ + 6 = 0, 
 1332:2^140^^ + 1473^2=0. 
 
 f92:2_ii2:^_,. 2^2 = 0, 
 
 1 55a;2 - 542:y - 13 ?/2 + 5a; + «/ + 6 = 0. 
 
 EXERCISE 137 
 
 1. The sum of two numbers is 25 ; the sum of their 
 squares is 457. Find the numbers. 
 
 2. The sum of the squares of two numbers is 225, and 
 the difference of their squares is 63. Find the numbers. 
 
 3. The ratio of two numbers is ^ ; their product is 
 315. Find the numbers. 
 
 4. The product of two numbers is 637 and their 
 quotient is 13. Find the numbers. 
 
 5. The product of the sum and difference of two 
 numbers is 81 and the quotient of their sum divided by 
 their difference is |. What are the numbers ? 
 
 6. Find two numbers whose sum, whose product, and 
 the difference of whose squares are equaL 
 
 7. The area of a right triangle is 6 sq. ft. ; the hypote- 
 nuse is 5 ft. Find the sides. 
 
 8. The diagonal of a rectangle is 25 in. If the rec- 
 tangle were 4 in. shorter and 8 in. wider, the diagonal 
 would still be 25 in. Find the area of the rectangle. 
 
 9. Two integers are in the ratio 2:3. If each is 
 increased by 5, the difference of their squares becomes 
 40. What are the integers? 
 
SYSTEMS OF QUADRATIC EQUATIONS 363 
 
 10. The combined perimeters of two squares are 68 in. 
 One square contains 51 sq. in. more than the other. 
 Find the area of each. 
 
 11. The difference of two numbers is 5 ; the sum of 
 their reciprocals is ^|. Find the numbers. 
 
 12. The difference of the cubes of two numbers is 604 
 and the sum of the numbers is 14. Find the numbers. 
 
 13. The difference of the terms of a certain proper 
 fraction is 8 and the product of this fraction by one whose 
 numerator and denominator exceed the numerator and 
 denominator of the given fraction by 1 and 5, respect- 
 ively, is ^. Find the fraction. 
 
 14. The diagonals of two rectangles are 29 ft. and 5 ft., 
 respectively. The ratio of their bases is 7 to 1 and that 
 of their altitudes 5 to 1. What are the dimensions of 
 the larger rectangle ? 
 
 15. If the length of a rectangle be increased by 4 and 
 the breadth decreased by 2, the area remains unchanged ; 
 if the length be decreased by 4 and the breadth by 2, the 
 area is halved. Find the sides of the rectangle. 
 
 16. The perimeter of a right triangle is 30 ft. ; its area 
 is 30 sq. ft. Find the sides and the hypotenuse. 
 
 Suggestion. Let x — the number of feet in the base. 
 
 Let y = the number of feet in the altitude. 
 
 Then, x + y + Vx^ + y^ = 30, (1) 
 
 and xy = 60. (2) 
 
 Transposing, x -^ y - SO = - Vx"^ + y^. (3) 
 
 Squaring, x^ + y^ + 900 - QO x - 60 y-\- 2 xy z= x^ + y^. (4) 
 
 Simplifying, and substituting value of xy from (2), 
 
 x + y = 17. (5) 
 
 Now solve the system J ^ ^ « * 
 "^ [ xy = 60. 
 
364 ELEMENTARY ALGEBRA 
 
 17. The perimeter of a right triangle is 70 ft. and its 
 area is 210 sq. ft. Find the three sides of the triangle. 
 
 18. The diagonal of a rectangle is 37 ft. If one side 
 were 4 ft. shorter and the other 2 ft. longer, the area of the 
 rectangle would be 14 sq. ft. greater than the area of the 
 original rectangle. Find the sides of the original rectangle. 
 
 19. A page is to have a margin at the sides of ^ in. and 
 one of I in. at the top and at the bottom ; it is to con- 
 tain 48 sq. in. of printing. How large must the page be 
 if the length is to exceed the width by 2 J inches? 
 
 20. The fore wheel of a carriage makes six revolutions 
 more than the rear wheel in going 120 yd. ; if the circum- 
 ference of each wheel be increased one yard, the fore wheel 
 will make four revolutions more than the rear wheel in 
 going the same distance. Find the circumference of each 
 wheel. 
 
 21. The circumference of the rear wheel of a carriage 
 is 2 feet greater than the circumference of the fore wheel. 
 The fore wheel makes 64 more revolutions than the rear 
 wheel in traveling 3496 feet. What is the circumference 
 of each wheel ? 
 
 22. Three men, A^ B, and (7, can do a piece of work to- 
 gether in 1^ days. To do the work alone A would take 
 twice as long as C and 2 days longer than B. How long 
 would it take each to do the work? 
 
 23. A rectangular box is 8 in. long. Its volume is 
 192 cu. in. and the area of its six faces is 208 sq. in. 
 Find the other two dimensions of this box. 
 
 24. The hypotenuse of a certain right triangle is 10 ft., 
 and its area is 24 sq. ft. Find the base and the altitude 
 of the triangle. 
 
CHAPTER XIV 
 PROGRESSIONS 
 
 276. Series. A succession of numbers that proceed 
 according to a fixed law is called a series. The numbers 
 which form the series are called the terms of the series. 
 
 Thus, the sequence of numbers 1, 3, 5, 7, ••• is a series in which 
 the first term is 1, and the second term 3. The law of formation in 
 this series is that any term is obtained from the preceding by the 
 addition of the number 2. 
 
 277. Arithmetical progression. A series in which each 
 term is obtained from the preceding by the addition of a 
 constant number is called an arithmetical progression. 
 
 Thus, — 8, — 4, 0, 4, 8, 12 is an arithmetical progression. 
 
 278. Common difference. The constant number obtained 
 by subtracting any term of an arithmetical progression 
 from the next succeeding term is called the common dif- 
 ference, or simply the difference. 
 
 Thus, in the arithmetical progression 2, 5, 8, 11, the common dif- 
 ference is 3. 
 
 279. General form. The general form of an arithmet- 
 ical progression is a, a -{■ d, a + 2 d, a + S d, "- in which 
 a represents the first term, and d the difference. 
 
 280. The general term. By inspection it is seen that 
 any term of an arithmetical progression is equal to the 
 first term plus a multiple of the difference. 
 
 366 
 
366 ELEMENTARY ALGEBRA 
 
 Thus, in the general form it is obvious that the coefficient of rf in the 
 second term is 1, in the third term 2, in the fourth term 3. In the 
 tenth term the coefficient of c? is 9 and in the nth term it is n — 1. 
 Hence, if the nth term of an arithmetical progression be denoted by 
 a„, we have the formula, 
 
 a„ = a+(n-l)(f. (1) 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Find the fifteenth term of 2, |, 3, ... . 
 
 Solution. Here a = 2, d = ^, n = 15. 
 Substituting in (1), a^g = 2 + (15 - 1)|-, or 9. 
 
 2. Write the first three terms of the series whose twelfth 
 term is 6 and whose thirty-fifth term is ^-. 
 
 Solution. Here ajg = 6 = a + (12 - l)rf, (1) 
 
 and 085 = -%^- = « H- (35 - l)d. (2) 
 
 From (1), a + lie? = 6. (3) 
 
 From (2), a + S4:d = \K (4) 
 
 Subtracting (3) from (4), 23 d = S^. (5) 
 
 Dividing, d - ^' (6) 
 
 From (3), a = 6 - 11 </ = - |. (7) 
 
 Therefore, the required terms are — J, 0, |. 
 
 Remark. Observe that when any two terms of an arithmetical 
 progression are given, the first term and the common difference can 
 be found. 
 
 EXERCISE 138 
 
 1. Find the fourth term of 1, 2, 3, ... . 
 
 2. Find the fifth term of 2, - 2, - 6, ... . 
 
 3. Find the seventh term of 5, 8, 11, ... . 
 
 4. Find the fifth and sixth terms of — 6, — 4, — 2, ... . 
 
 5. Find the fourth and fifth terms of a, a+ b, 
 0+2 5,.... 
 
PROGRESSIONS 367 
 
 6. Find the fourth and fifth terms of rr, 2 rr + 1, 
 
 7. Find the eighth term of 2, — 1, — 4, ... . 
 
 8. Find the tenth term of 2, 7, 12, ... . 
 
 9. Find the ninth term of J, |, 1, ••. . 
 
 10. Find the tenth term of — 2, 0, 2, ... . 
 
 11. Find the ninth term of 2 a;, 4 a:, 6 a:, ... . 
 
 12. Find the ninth term of « + 5, 2 a, 3 a — 6, ••• . 
 
 13. Find the Tith term of 1, 3, 5, ... . 
 
 14. Find the nth term of — 4, — 1, 2, ... . 
 
 15. Find the nth. term of 2, 0, — 2, ... . 
 
 16. Find the nth term oi 2 x, 6 x,S x, »" . 
 
 17. Find the nth term of 3 a — 6, 2 a, a + 6, ••• . 
 
 18. The third term of an arithmetical progression is 6 
 and the eighth term is 16. Find the first term and the 
 common difference. 
 
 19. The ninth term of an arithmetical progression is 
 102 and the twenty-second term is 141. Find the first 
 term and the common difference. 
 
 20. The seventh term of an arithmetical progression 
 is ^ and the twenty -fifth term is ^^-. Find the twelfth 
 term. 
 
 21. The thirty-first term of an arithmetical progression 
 is 46 and the forty-ninth term is 73. Find the common 
 difference. 
 
 22. The sum of the fifth and twenty-fifth terms of an 
 arithmetical progression is 13, and the forty-ninth term is 
 15. Find the series. 
 
 23. The rth term of the series 5, 8, 11, ...is equal to 
 the rth term of the series 61, 57, 53, .... Find r. 
 
368 ELEMENTARY ALGEBRA 
 
 « 
 
 281. Arithmetical mean. When three numbers are in 
 arithmetical progression, the second number is called the 
 arithmetical mean of the other two. 
 
 Thus if a, x, c, are in arithmetical progression, x is the arithmeti- 
 cal mean of a and c. 
 
 If a: is the arithmetical mean of a and c, x may be found in terms 
 of a and c, thus : 
 
 Since x — a and c — x are each expressions for the common differ- 
 ence, 
 
 ' X — a = c — X 
 
 therefore, x = ^ ^ ^ ; that is : 
 
 The arithmetical mean of two numbers is half their sum, 
 
 282. Arithmetical means. In an arithmetical progres- 
 sion, the terms which stand between two given terms are 
 called arithmetical means between the given terms. 
 
 ILLUSTRATIVE EXAMPLE 
 
 Insert four arithmetical means between 2 and 17. 
 
 Solution. a» = a + (n — l)c?. 
 
 There will be six terms in all, of which 2 is the first term and 17 
 is the sixth. 
 
 Hence, On = Oe* ^hich is 17. 
 
 Therefore, 17 = 2 + (6 - l)d. 
 
 Whence, c?= 3. 
 
 The arithmetical progression is 2, 5, 8, 11, 14, 17. 
 
 EXERCISE 139 
 
 Find the arithmetical mean of : 
 1. 12 and 16. 2. — 4 and — 12. 3. a and h. 
 4. x-\- y and x— y. 5. 1 and a. 
 
 6. 2a-\-b and a -{•2 b. 7. | and |. 
 
 8. - and i. 9. ^— and — — 
 
 a b x — y x + y 
 
PROGRESSIONS 369 
 
 10. Insert three arithmetical raeans between 1 and ^. 
 
 11. Insert three arithmetical means between 230 and 
 710. 
 
 12. Insert three arithmetical means between x— y and 
 x+i/. 
 
 13. Insert six arithmetical means between — ^ and J-|. 
 
 283. Sum of an arithmetical series. The sum of n terms 
 of an arithmetical progression may be obtained as follows : 
 
 Representing the sum of the first n terms of the arithmetical 
 progression by S^, we have 
 
 S„ = a + (a + d) + (a + 2d) -h - + K - 2d) + (a„ - d) + a^. 
 or, Sn = an + (an - d) -{■ (an - 2d) + '■' -\- (a + 2d) -\- (a -hd) + a. 
 .-. 25„ = (a + a^) + (a + a^) + (a-^ a„) + + (a + a„) 
 
 + (a + an) + (a + a„) 
 = n(a + an). 
 
 .•• Sn = ^(a + an). 
 
 That is, the formula for the sum of the first n terms of 
 an arithmetical progression is 
 
 S„ = ?(a + <^,). (2) 
 
 Formula (2) may be expressed in words as follows : 
 
 The sum of the first n terms of an arithmetical progression 
 is equal to the arithmetical mean of the first and last terms 
 multiplied hy the number of terms. 
 
 By substituting in formula (2) the value of a from 
 formula (1), section 280, we have a second formula for 
 the sum of the first n terms of an arithmetical progression ; 
 namely, 
 
 S» = ^[2fl+(n-l)<l. (8) 
 
370 ELEMENTARY ALGEBRA 
 
 Note. To find Sn, when n, a, and «« are given, use formula (2); 
 when n, a, and d are given, use formula (3); when a„, a, and d are 
 given, find n by formula (1) and then use formula (2). 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Find the sum of the first n natural numbers. 
 Solution. The required sum = 1 + 2 + 3 + -. •+n 
 
 _ n(n + 1) 
 2 
 
 2. Find the sum ofl+3 + 5 + 7+ + 
 
 Solution. By formula (3) 5„ = | [2 + (n - 1)2] = n^. 
 
 Remark. Observe that the result of example 2 expresses the 
 fact that the sum of the first n consecutive odd numbers is a perfect 
 square, namely, n^. 
 
 EXERCISE 140 
 
 1. Find the sum of 15 terms of — 1, 4, 9, ... 
 
 2. Find the sum of ten terms of f 9 — f ^ — ^, ••• 
 
 3. Find the sum of 12 terms of 3, ^^, J5I-, ... 
 
 4. Find the sum of 9 terms of '^~^ , V3, ^^"^^ ... 
 
 5. Find the sum of n terms of — ^t — ^ — "^ , 1 ... 
 
 n 2n 
 
 6. Find the sum of m terms of (a — 1)^, a^ + 1, 
 (« + l)2, ... 
 
 7. Find the sum of all the even numbers between 1 
 and 101. 
 
 8. Find the sum of all the odd numbers between and 
 100. 
 
 9. Find the sum of all the multiples of 5 between 1 
 and 1001. 
 
PROGRESSIONS 371 
 
 10. Find the sum of all the multiples of 7 between 1 
 and 344. 
 
 11. The 12th term of an arithmetical progression is 35, 
 the sum of the first 12 terms is 222. Find the series. 
 
 12. The first term of an arithmetical progression is a, the 
 last term is ?, and the common difference is 1. Show that 
 n = Z — a + 1. 
 
 13. Find the ratio of the sum of the first n natural 
 numbers to the sum of the first n even numbers beginning 
 with 2. 
 
 14. Show that if each term of an arithmetical progres- 
 sion be multiplied or divided by the same number (zero 
 excepted), the result will be in arithmetical progression. 
 
 15. A body falls 16.08 feet in the first second, three 
 times as far in the next second, five times as far in the 
 third second, and so on. How far does it fall in 10 
 seconds ? 
 
 284. Geometric progression. A series of numbers in 
 which the ratio of each number to the preceding is con- 
 stant is called a geometric progression. The constant ratio 
 is called the common ratio of the progression. 
 
 Thus, the series of numbers, 3, 6, 12, 24, form a geometric progres- 
 sion of four terms, since J = ^^- — ^|. 
 
 285. Increasing progression. The successive terms of a 
 geometric progression increase in absolute value if the 
 ratio is numerically greater than 1, and the progression is 
 said to be an increasing geometric progression. 
 
 Thus 1, 3, 9, 27, 81 is an increasing geometric progression. 
 
372 ELEMENTARY ALGEBRA 
 
 286. Decreasing progression. The successive terms of a 
 geometric progression decrease in absolute value if the 
 ratio is numerically less than 1, and the progression is 
 said to be a decreasing geometric progression. 
 
 Thus, 128, 64, 32, 16, 8 is a decreasing geometric progression. 
 
 287. Form of a geometric progression. From the defini- 
 tion, the general form of a geometric progression is 
 
 a^ ar^ ar^, ar^, ar*, ••• 
 in which a represents the first term and r the common 
 ratio of the progression. 
 
 288. General term. An inspection of the foregoing 
 general form shows that any term of a geometric progres- 
 sion is obtained by multiplying the first term by a power 
 of the ratio. 
 
 Since the exponent of r in the second term is 1, in the 
 third term 2, in the fourth term 3, it is evident that in 
 the tenth term the exponent is 9, and in the general, or 
 nth. term, it is ti — 1. Hence, representing the nth. term by 
 an, we have ^^ — ^^.n-i^ ^1^ 
 
 ILLUSTRATIVE EXAMPLE 
 
 The third term of a geometric progression is ^ and the 
 sixth term is y^^g. Find the series. 
 
 Solution. Here a^ = ar^ = ^, 
 
 and a^ = ar^ = ^j^. 
 
 S=h' • 
 
 or, r8 = ^^, 
 
 whence, when r is real, ^ = §• 
 
 From ar'^ = ^, 
 
 we have i^ = ^j or a = J. 
 
 Therefore, the series is -J, J x §, J x (|)^> ••• *> that is, J, ^, ^, •••. 
 
PROGRESSIONS 373 
 
 Remark. Observe that when any two terms of a geometric pro- 
 gression are given, the first term and the common ratio can be found. 
 
 EXERCISE 141 
 
 1. Find the fifth term of 1, 2, 4, .... 
 
 2. Find the sixth term of 3, |, |, ••.. 
 
 3. Find the sixth term of 1, 5, 25, •••. 
 
 4. Find the fifth term of 2, - 4, 8, .... 
 
 5. Find the sixth term of a, a6, a^, ••-. 
 
 6. Find the fifth term of a, 2a\4:a\ .... 
 
 7. Find the fifth term of m\ m(m — V)^ {m— 1)2, .... 
 
 8. Find the fifth term of 1, V2, 2, .... 
 
 9. Find the fifth term of ^^~^ , V2, 4(1 + V2), .... 
 
 10. Find the nth term of 3, 9, 27, ••.. 
 
 11. Find the nih. term of 1, J, ^, •... 
 
 12. Find the n\h. term of a, Va, 1, •••. 
 
 13. The fourth term of a geometric progression is 24 
 and the sixth term is 96. Find the ratio and the first 
 term. 
 
 14. The third term of a geometric progression is 16 
 and the seventh term is Jg. Find the tenth term. 
 
 15. The sum of the first and fourth terms of a geometric 
 progression is ^Q and the sum of the second and third 
 terms is 24. Find the series. 
 
 16. The sum of three numbers in a geometric progres- 
 sion is 14 and the sum of their squares is 84. Find the 
 numbers. 
 
374 ELEMENTARY ALGEBRA 
 
 17. Each stroke of a certain air pump exhausts one 
 sixteenth of the air in the receiver. How much of the air 
 originally in the receiver is removed in six strokes ? 
 
 Geometric mean. When three numbers are in 
 geometric progression, the second number is called the 
 geometric mean of the other two. 
 
 Thus, if a, x, b are in geometric progression, x is the geometric 
 mean of a and h. 
 
 If X is the geometric mean of a and b, x may be found in terms of 
 a and b thus : 
 
 Since - = r and - = r, 
 
 a X 
 
 we have - = -. 
 
 a X 
 
 Therefore, x^ = a&, 
 
 and X = Va6 ; that is : 
 
 77ie geometric mean of two numbers is the square root of 
 their product. 
 
 Remark. It is evident that the geometric mean of two numbers 
 is the mean proportional between these numbers [§ 175]. 
 
 290. Geometric means. The terms which stand be- 
 tween any two given terms of a geometric progression 
 are called the geometric means between the given terms. 
 
 Thus, in the geometric progression 2, 4, 8, 16, 32, the geometric 
 means between 2 and 32 are 4, 8, and 16. 
 
 ILLUSTRATIVE EXAMPLE 
 
 Insert three real geometric means between 32 and 2. 
 Solution. a„ = ar'^'^. 
 
 There are five terms in all, of which 32 is the first term and 2 the 
 fifth term. 
 
 Here a„ = Og = 2 = ar^. 
 
 .•.82r* = 2. 
 
 Whence^ '^ = :^' ^^^ '* = ± 5 ^^^ ± \' 
 
 lo « « 
 
PROGRESSIONS 375 
 
 That is, the two real values of r are | and — 4. 
 Corresponding to the real values of r, we have the progressions 
 
 32, 16, 8, 4, 2, and 32, - 16, 8, - 4, 2. 
 The required means are, therefore, either 
 
 16, 8, and 4 or - 16, 8, and - 4. 
 
 EXERCISE 142 
 
 Find the positive geometric mean between : 
 1. 3 and 48. 2. Jg and J^. 
 
 3. ^ and -. 4. (x-yy and (x^-^y^y, 
 
 5. 3 a and 27 a^. 6. a and a^. 
 
 Insert three positive geometric means between : 
 7. 4 and 64. 8. 48 and 243. 
 
 9. (x — y') and (^x^ — y^^(x -\- y')^, 
 
 10. - and — . 
 
 a^ 
 
 11. The sum of three numbers in geometric progression 
 is 117 ; the mean is equal to three tenths of the sum of 
 the other numbers. Find the numbers. 
 
 291. Sum of a geometric series. Let S^ represent the 
 sum of n terms of a geometric progression ; then, 
 
 5„ = a + ar + ar2 + ... 4- ar""-^ + ar""-^ + ar''-\ (1) 
 
 Multiplying (1) by r, 
 
 r 5„ = ar -\- ar^ + ••• + ar'^'^ + ar^^ + ar^, (2) 
 
 Subtracting (2) from (1), 
 (1 - r)5„ = a - ar"". 
 Dividing, , 
 
 o„= —1 
 
376 ELEMENTARY ALGEBRA 
 
 That is, the formula for the sum of n terms of a geo- 
 metric progression is 
 
 S„ = ^Z^. (2) 
 
 In an increasing geometric progression the formula 
 obtained by changing the signs of the terms in formula 
 (2) should be employed ; the formula is, 
 
 S„ = ^«. (3) 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Find the sum of eight terms of 2, 6, 18, •••. 
 Solution. Here a = 2, r = S, n = 8. 
 
 We use the formula Sn = — — • 
 
 r — 1 
 
 Substituting in the formula, 
 
 S. = ^'^^-^ = 38 - 1 = 6560. 
 8 2 
 
 2. The sum of the terms of a geometric progression is 
 728, the ratio is 3, and the last term is 486. Find the first 
 term and the number of terms. 
 
 Solution. Since 
 
 «n = ar»~\ 
 
 
 ran = o^", 
 
 and the formula 
 
 s„ = '""°-" 
 
 
 r-1 
 
 may be written, 
 
 -.=?fr- 
 
 Here, 5„ = 728, r = 
 
 = 3, and a„ = 486. 
 
 Substituting, 
 
 ^„o 3 X 486 - a 
 ^^^- 3-1 
 
 whence, 
 
 a = 2. 
 
 From 
 
 a„ = ar«-i. 
 
 we have 
 
 486 = 2 X S»-\ 
 
 or. 
 
 3« = 3«-i. 
 
 Hence, 
 
 5 = n - 1, 
 
 or, 
 
 n = 6. 
 
PROGRESSIONS 377 
 
 EXERCISE 143 
 
 1. Find the sum of 2, 6, 18, •-. to 6 terms. 
 
 2. Find the sum of 4, 2, 1, ... to 8 terms. 
 
 3. Find the sum of — 3, 9, — 27, ••. to 7 terms. 
 
 4. Find the sum of V3, 3 + V3, 6 + 4V3, ... to 5 
 terms. 
 
 5. Find the sum of 1, 2, 4, ••• to 7i terms. 
 
 6. Find the sum of 1, J, ^, ••• to ti terms. 
 
 7. Find the sum of aV5, 6Va, 5V5, ... to n terms. 
 
 8. Find the sum of 3, — 6, 12, ... to 2 m + 1 terms. 
 
 9. If the sum of the series 1 -|- 4 + 16 + ••• is 5461, 
 find the number of terms. 
 
 10. The first term of a geometric progression is 1, the 
 last term is 81, and the sum of the series is 121. Find 
 the ratio. 
 
 11. Find two numbers whose sum is 52, such that their 
 arithmetical mean exceeds their geometric mean by 2. 
 
 12. The first term of a geometric progression is 5, the 
 ratio is 4, and the number of terms is 5. Find the sum 
 of the terms. 
 
 13. The first four terms of a geometric progression are 
 the same as the first four terms of a second geometric 
 progression, but in reverse order ; the sum of the first 
 eight terms of one is equal to 81 times the sum of the 
 first eight terms of the other. Find the common ratio of 
 each. 
 
 292. Infinite geometric series. The terms of a geometric 
 progression in which r is positive and numerically less 
 than 1, become smaller and smaller. By taking n 
 
378 ELEMENTARY ALGEBRA 
 
 sufficiently large, we can make the nth term as small as 
 we desire ; that is, make it more and more nearly equal 
 to zero. 
 
 By this statement we mean that however small a' num- 
 ber we please to mention, we can find a term such that 
 it and each one of the succeeding terms of the series is 
 numerically less than the number mentioned. 
 
 Thus, in the series 100, 10, 1, J^, yoo"' T"^Vo' *" *^® successive 
 terms are becoming smaller and smaller. When n is 10, a„ is y^nToV^nrS"' 
 
 which is a small number; and when n is 103, a„ = , which is an 
 
 exceedingly small number. 
 
 From illustrative example 2, section 291, 
 o _ a — ra^ 
 
 which may be written, aS'^ = aJ ). 
 
 1 — r \1 — rj 
 
 The term is a constant, and the term aJ — - — ) 
 
 1 — r \1 — rJ 
 
 varies with n and becomes smaller and smaller. Hence, 
 
 as more and more terms of the series are added, aS^^ differs 
 
 less and less from . 
 
 1 — r 
 
 The number is called the limit of the sum of n 
 
 1 — r 
 
 terms, as n increases without limit. 
 
 For convenience we shall call this limit the sum to 
 
 infinity of a decreasing geometric progression, and shall 
 
 denote it by the symbol S^ . We therefore have the 
 
 following identity : 
 
 <3 - ^ 
 
PROGRESSIONS 379 
 
 293. Recurring decimal. A decimal in which a figure 
 or set of figures repeats in a certain fixed order is called a 
 recurring decimal, or a repeating, or circulating decimal. 
 
 Recurring decimals are illustrations of infinite geometric series : 
 Thus, each of the following is an infinite geometric series : 
 
 .6666 •••, sometimes written .6, in which a = .6 and r — .1. 
 
 .3434 •.., sometimes written .34, in which a = .34 and r — .01. 
 
 .304304 •.., sometimes written .304, in which a - .304 and r - .001. 
 
 ILLUSTRATIVE EXAMPLES 
 
 1. Sum to infinity the series 2, 1, |, ;^, ••• 
 Solution. Here a = 2 and r = |^. 
 
 2. Sum to infinity the series 6, — 3, 1|, — | •••. 
 Solution. Here a = 6 and r = — -^. 
 
 S - Q - ^ -_L-4 
 
 3. Sum to infinity the series a, —^ — , ^ , .... 
 
 Solution. Here a = a, and r = — 
 
 a;2 + l 
 
 ^-^ 1- 
 
 a:2 + l 
 
 4. Sum to infinity the series .666 • • • . 
 Solution. Here a = .6 and r = .1. 
 
 S = ^ = -^ -'^-2 
 " l-r 1-.1".9~3' 
 
 5. Sum to infinity the series .24545 •••. 
 
 Solution. .24545 ••• = .2 + .04545 •••, in which .2 is not a part of 
 the series. jjere a = .045 and r = .01. 
 
 S ^ q ^ .045 ^.045^ 5 
 * l-r 1 - .01 .99 no' 
 
 ••• •2^5 = 1^ + ^4^, or ^ViJ. 
 
380 ELEMENTARY ALGEBRA 
 
 , EXERCISE 144 
 
 1. Sum to infinity the series 1, ^, J, .••. 
 
 2. Sum to infinity the series 100, 10, 1, •••. 
 
 3. Sum to infinity the series 5, 1, ^, •••. 
 
 4. Sum to infinity the series 1, J, ^, •••. 
 
 5. Sum to infinity the series 6, |, -^^^ •*•• 
 
 6. Sum to infinity the series |, |, f, •••• 
 
 7. Sum to infinity the series 2.5, 1.25, .625, •••. 
 
 8. Sum to infinity the series 18, 12, 8, .•-. 
 
 9. Si;m to infinity the series 3.5, .35, .035, •••. 
 
 Sum to infinity the following : 
 
 10. .5. 11. .54. 12. .61. 
 
 13. .i35. 14. .24. 15. .434. 
 
 16. Show that 4 + 1 + ^/ + — =3 + f + |J + • — 
 
 17. An elastic ball bounces to three fourths the height 
 from which it falls. If it is thrown up from the ground 
 to a height of 20 feet, find the total distance traveled 
 before it comes to rest. 
 
 18. A heavy iron ball at the end of a chain is pulled 
 to the right 1 yard out of the vertical and is then released. 
 It swings to a point 0.9 of a yard to the left of the vertical, 
 then to a point 0.9 of a yard to the right of the vertical. 
 The succeeding swings follow the same law. Including 
 the first movement, find the greatest distance the ball 
 could travel before coming to rest. 
 
 19. Show that before the ball mentioned in example 18 
 passes through the vertical for the seventh time after 
 being withdrawn, it has moved more than half its total 
 movement. 
 
CHAPTER XV 
 GENERAL REVIEW 
 
 1. Evaluate 1 -— when a = 3, 5 = 4, and 
 
 c=l. 
 
 2. Evaluate — — — — ±- i— when x = 2. 
 
 3. Evaluate ^P " ?)' + C + *) % (P + ?)' + C" - O' ^ 
 
 p -{■ q + r -\- 8 p — q-\- r — 8 
 
 when JO = 3, 3' = 2, r = 1, and 8 = ^ . 
 
 4. Write a formula for the area of a parallelogram. 
 Find the area of a parallelogram whose base is 10 inches 
 and whose altitude is 7 inches. 
 
 5. Write a formula for each of two numbers whose sum 
 and difference are known. Find two numbers whose sum 
 is 126 and whose difference is 32. 
 
 6. Write a formula to find the weight of a bag con- 
 taining any given number (n) of bushels of grain, given 
 the weight of the bag and the weight of a single bushel 
 of grain. 
 
 7. Write a formula for the area of the wall of a room 
 of length L and width TF" containing two windows each 
 of length I and width w, 
 
 8. Write a formula to find the number of square feet 
 in the four walls of a room, I ft. long, w ft. wide, and h ft. 
 high. 
 
 381 
 
382 ELEMENTARY ALGEBRA 
 
 Find the sum of the expressions in examples 9-12. 
 
 9. Sa^-\-2xt/-4f + lSz^, 2x^-5x1/-^ 7t/^-^Uz^ 
 and -3a;2+ 2x^-7 y^-5z\ 
 
 10. 8 mn^ — 7 mhi -{- 2n^, 5m^ — 2 mhi — 11 mn\ and 
 m^ — 2 nK 
 
 11. (jE? -h q)a, (^q + r)a, and (r — 5')a. 
 
 12. ix-^^-\-^z,^x + ^y-^z,and2x-ii/-^^z, 
 
 13. Simplify 3(a - b -\- c -\-2d}- 5(a- 2b+ Sc--Sd^ 
 + 4(a-36 + 2(? + 4(^)-2(a-7 6-2c-7c^). 
 
 14. Evaluate (3 a: + 2 a)2 - (2 a; + 5 ay when a; = 7 a. 
 
 15. Evaluate 5(2 m - 37i)(3m - 2 w) - (2/1 + 3 w) 
 (3 w + 2 w) when w = 3 /t. 
 
 16. Simplify 2a; -(- 3^ + 2- jrr- y|)-(3a;H- 22- 
 [-2^ + 32]). 
 
 17. Simplify _ 5 _ [_ (_ « 4. J _ c)]_ cj - { - [- 
 
 18. From the sum of 3 a6 — 2 a;y + 4 and 2 aft — 3 a;^ + 3 
 take the difference between 4 a6 + 3 a;y — 2 and 5 ah — 2 xy. 
 
 19. Add 5jo - (3 ^- 2 r) and - (3 ^ -6 j^)- lOjt? ; from 
 the sum subtract — 4^ — (3 r + 5'). 
 
 20. State what value of x will make the expression 
 3(a; H- 3) — 2(2 a: — 3) equal to twice the value of x, 
 
 21. What number is as much greater than 30 as it is 
 less than 74 ? 
 
 22. Find a number to which if 10 be added the result 
 is equal to 6 times the number. 
 
 23. A man, who rode a motorcycle at the rate of m 
 miles an hour, completed a journey from P to ^ in A 
 hours, during r of which he rested. Find an expression 
 for the distance from P to ©• 
 
GENERAL REVIEW 383 
 
 24. Multiply 5a;3-22^ + 7a;-llby3ar» + 7a;-3. 
 
 25. Simplify (ax + hy)(cx — dy) — (ex — hy)(ax-\- dy), 
 
 26. Multiply b(x + yy - 3(a; + ^) - 2 by 3(a: -f- y)- 
 
 27. Multiply .3 2^2 ^1,2 a; +1 by .52^2 _i. 
 
 28. Multiply a;"* + 2 2^-1 + 3 a^-2 by 2:1; -3. 
 
 29. Multiply a°+i — 2 a« + 3 by 2 a" — 3. Verify your 
 result by putting a = 2. 
 
 30. Simplify ba-\'^a-[2h(p -^ q)-^h(p - q)^\, 
 
 31. Prove by actual multiplication that 
 
 (^a + h+cy-\-a^ + y^+(^=(h+cf+(c + ay+(a^hy. 
 
 32. Show that if 2; = 1 + a, y = 1 + 5, z = l-\- c, then 
 a;2 -(- ^2 ^ 2;2 — ^2 — 237 — 2;^ = a^ _|_ 52 _|_ ^ _ 5^ __ ^^ _ ^5^ 
 
 33. Prove by actual multiplication that 
 (a^ — bey— (h^ — ed)(c^ — aJ)) is equal to 
 
 a(a + h-\- c)(a2 + 5^ + c^ — 6c — ca — ah), 
 
 34. Arrange the expression a^(x — 1) -j- (oc^ — 1)^ 
 — mx (a^ -f 3) in descending powers of x. 
 
 35. Divide4a3 4-a^-i3by a-|. 
 
 36. Divide a^^ + 53^ by a^ + ^. 
 
 37. Divide a;^ + 8 a;^ - 192 a:^ _ 256 x + 1024 by a:^ _ 4 ^^s 
 -16a;+32. 
 
 38. Divide (x-\-6 yY — (y -^ 4: zy hj x •{- 4:(y — 2). 
 
 39. Divide 6x^ -9x^ + 22a^- 4:0^ --^x-lSa^-^lOhy 
 3a:3_2a: + 5. 
 
 40. Divide a^+"6" — 5 ^r+^-^J^n _ 3 ^-|-n-258n 
 + 15 a"»+»-354n lyj fjfiln _ 5 ^n-152n^ 
 
 41. Factor24-822_9. 
 
 42. Factor (m + 2)* - 9(w + 2)2+ 20. 
 
 43. Factor a^ — b^. 
 
384 ELEMENTAKY ALGEBRA 
 
 44. Factor aa^ -\- bx^ -{- a — h. 
 
 45. Factor a^ -\- 2 ab -\- S ac -\- 6 he. 
 
 46. Factor a^-b^- a{a^ - ^2) + 6(a - b)\ 
 
 47. Factor {i^ + 3 2^)2 - (3 x^ + l)^. 
 
 48. Factor ^^-|i. 
 
 «>6 27 
 
 49. Factor^2;2_|.2^a;- 3^ + ^2^^ + 2^x-3^. 
 
 50. Factor 3(a:-3) + a;(rc-3)(3a;+3). 
 
 51. Factor a!^ -2aW ^b^ - a^ -\-2 ab ^ W^. 
 
 52. Prove that {tP' -\- xi/ 4- y^)^- (aP-^XT/- y^y 
 = ^xy\x-^y), 
 
 53. Factor j^^ _ 3 ^2 ^. 4, 
 
 54. Factor 7^ — x^y — xy'^ + y^, 
 
 55. Factor jt?2 _ 6 j9^- 16^252 + 9^. 
 
 56. Factor (x^ -lxy-\- 18(a^ - 7 a;) + 72. 
 
 57. Factor x^^ + m^xy^. 
 
 58. Factor 3(2^+1)84-4(2; + 1)2 + a; + 1. 
 
 59. Express (5 a; - 6) (5 a; + 6) - 4 ^(10 a; - 4 ^) as the 
 difference of two squares. 
 
 60. Resolve a:^ — 13 a;2 + 36 into linear factors. 
 
 61. Show that 
 
 (9 a: ~ 10)2- 2(7 X - 10)2= (x - 10)2 _ 2(3 a; - 10)3. 
 
 62. Find the H.C.F. and L.C.M. of a^s+l and 2 a:2_a;_ 3. 
 
 63. Find theH.C.F. and L.C.M. of a:3 + 22^ + 2a;+l 
 anda:8-.2a;2 + 2a;-l. 
 
 64. Reduce to lowest terms: '^'~^' + f+^^^ 
 
 a^ — c^— 0^ — 2 bo 
 
 TO '' 2 
 
 65. Simplify -j f—J' 
 
. GENERAL EEVIEW 385 
 
 66. Simplify 
 
 x-hB , x + 2 , a; +3 
 
 (2_^)(3-jr) (x-^X^-^} (a;-2)(a:-5) 
 67. Simplify 
 
 + 
 
 (a — 6) (b — e) (h — c)(^c— a) (a — c)(h— a) 
 69. Simplify 
 
 \x—y x^- y aP'-y'^] '\x + y x^y] 
 
 71. Simplify fsx-'5--Ysx-hB--\-^(x--\ 
 
 72. Simplify 
 
 (a— 6)(a--c) (6 — c)(6 — a) (c— a)(<? — 6) 
 
 2:3 + 27 
 
 73. Simplify —^±^—-2^1 ?-V 
 
 „, o- r-P 2 1 2a + 3,l,3«-26 
 
 74. Simplify —---___— n_+ 4- _^ — 
 
 3a 26 Qa^ 2 a o ao 
 
 o- Tx abc—a% ahc—h^c ahc — c^a 
 
 75. Simplify -— — - X —rz — X 
 
 a%e — ^6^^ a62c — abc^ ahc^ — a26c 
 
 76. Simplify [m^^ f V ^2^^2)^/^_^_ + _JL-Y 
 
 V m^—n^J \m+n m—nj 
 
 77. Evaluate aa? + 6v + c when a^ = — — - and 
 
 ma — po 
 _ pc — na ^ 
 
 ph — ma 
 
386 
 
 ELEMENTARY ALGEBRA 
 
 1 r 4 
 
 x-^- 
 
 78, Simplify 
 
 79. Simplify- 
 so. Simplify 
 
 x-^ 
 
 X— 5 
 
 x — b 
 
 a;-3 
 
 X—K> — 
 
 x — b 
 
 a^ — ah b^—bc c^— ca 
 c^ — ac b^—ba c^ _ ^^ 
 
 ^_^_ 2i(a-ft) 
 a-\-b 
 a2 + ^2 
 
 a6 + ^2 
 
 81. Show that ^^-f^ + ^^±^ = (^ + ^)^^±-^. 
 a+6 a-b ^ ^ a^-b^ 
 
 82. 
 
 Simplify f-l^ L + _8^V-^^=^ 
 
 V2a;+^ 2x-y^ y'^-4.x^J (Ix-y) 
 
 (2x-yy 
 
 83. Solve the equation ^^±2 ^ ^±1 ^ 2. 
 
 84. Solve the equation 
 
 85. Solve the equation 
 
 86. Solve the equation 
 
 87. Solve the system 
 
 3 
 
 5 + 4a; 2x 
 
 3a;-4 Qx-1 3a:-4 
 m x X n p 
 
 ^X X 
 
 x-\-2 x+1 
 
 = 2. 
 
 x-^y x-y _ -. 
 3 4 " * 
 
 ~2 3 -^- 
 
 88. Solve the system 
 
 89. Solve graphically the system 
 
 2xy-\-Sx = 6, 
 Sxy + 6x = S. 
 
 '2x-y=:i, 
 
 2x+Sy==12. 
 
GENERAL REVIEW 
 
 387 
 
 90. Divide m into two parts, one of which shall exceed 
 the other by n, 
 
 91. The difference of the squares of two consecutive 
 odd numbers is 96. Find the numbers. 
 
 5 
 
 92. Solve the system 
 
 by 
 
 = -2, x+ly = ^. 
 
 ft3. Solve the system 
 
 94. Solve the system 
 3 . 5 
 
 = 8, 
 
 = 11, 
 
 x-2 y+3 
 2 3 
 
 = 11. 
 
 '2,x + y x — ^ 
 
 2x-\- y X— Sy 
 
 95. Solve the system 
 
 f4a;H-8y-32 = 6, 
 hx-{- y- 2 = 7, 
 [4^ — 5 a; + 4 2 = 8. 
 
 96. Solve the system 
 
 ^_3 y _^ ^_^ 
 
 3+^~7' 4 + 2"9' 5 + x~8' 
 
 97. Solve the system 
 5 .^ 4 
 
 x y z 
 §-2 + 1=12. 
 
 98. Solve the system 
 4 5 
 
 5£:iI + ii^z:^ = i8-5 3,. 
 
 = 1. 
 
388 ELEMENTARY ALGEBRA 
 
 99. Solve for a; and ^: -^ + X. = ^, - + ^=2. 
 
 oa 2 6 a b 
 
 100. A mixture of corn and oats contains 33J % of oats 
 by weight. How many pounds of corn must be added to 
 100 lb. of the mixture so that the resulting mixture shall 
 contain only 20 % of oats? 
 
 101. A dealer bought 2000 lemons, some of them at the 
 rate of 1 J ct. apiece, and the remainder at the rate of 2 ct. 
 apiece. He sold them all at the rate of 27 ct. per dozen 
 and gained $7.50. How many did he buy at each price? 
 
 102. If 7 is added to twice a certain number, the sum is 
 13. Find the number. 
 
 103. If one half of a certain number is added to itself, 
 the sum is 3 less than twice the number. Find the 
 number. 
 
 104. Twice a certain number is 9 less than 5 times the 
 number. What is the number? 
 
 105. One number is 3 times a second number ; the sum 
 of the two numbers is 6 greater than twice the smaller 
 number. What are the numbers ? 
 
 106. The sum of two numbers is 50 ; one of the num- 
 bers is 5 less than 4 times the other. What are the 
 numbers ? 
 
 107. The difference between two numbers is 37. The 
 smaller number plus 3 times the larger equals 163. Find 
 the numbers. 
 
 108. The sum of two numbers is 60 ; one number is 17 
 less than 6 times the other. Find the numbers. 
 
 109. One number exceeds another by 30 ; the smaller 
 is 3 greater than one half of the larger. Find the 
 numbers. 
 
GENERAL REVIEW ^89 
 
 110. One number exceeds another by 101 ; if 3 times 
 the smaller is added to the greater, the result is 201. 
 Find the numbers. 
 
 111. A's share of a business is twice that of his partner 
 B; they sell the business for $12,000. How much should 
 each receive ? 
 
 112. The sum of two numbers is 280, and their difference 
 is equal to one fourth of the greater. Find the numbers. 
 
 113. A house and a garage cost $7000, and twice the 
 cost of the house was equal to five times the cost of the 
 garage. Find the cost of each. 
 
 114. A number is composed of two digits ; the digit in 
 the tens' place is one less than twice that in the units' 
 place. If 27 is subtracted from the number, the remain- 
 der is composed of the same two digits in reversed order. 
 Find the number. 
 
 115. The difference between the squares of two con- 
 secutive numbers is 13. Find the numbers. 
 
 116. If A can perform a piece of work in 3 days, and B 
 in 5 days, in what time should they perform it working 
 together ? 
 
 117. If a man and 2 boys can do a piece of work in 
 5 days and the man working alone can do it in 12 days, 
 in what time can one boy working alone do the work, pro- 
 viding the boys do equal amounts ? 
 
 118. The sum of the two digits of a number is 14 ; if the 
 order of the digits is reversed, the number is diminished 
 by 18. Find the number. 
 
 119. A person has just ten hours at his disposal ; how 
 far may he ride at the rate of ten miles an hour, so as to 
 return home on time, walking back at the rate of 4 miles 
 an hour? 
 
39a ELEMENTARY ALGEBRA 
 
 120. A train travels from Philadelphia to New York in 
 2 hours ; if it had traveled 15 miles an hour slower, it 
 would have taken one hour longer. Find the distance 
 from Philadelphia to New York. 
 
 121. A number of workmen, who receive the same 
 wages, earn together a certain sum. Had there been 6 
 more workmen, and had each received 10 cents more, their 
 joint earnings would have increased by $19.60. Had 
 there been 3 fewer workmen and had each received 10 
 cents less, their joint earnings would have decreased by 
 $9.70. How many workmen are there and how much 
 does each receive? 
 
 122. The total number of boys and girls attending a 
 certain boarding school is 95. If the. number of boys 
 were 30 % less and the number of girls 20 % more, there 
 would be as many girls in attendance as boys. How 
 many of each are there in attendance? 
 
 123. A man has $2.20 in nickels, dimes, and quarters, 
 15 coins in all. If the number of nickels and quarters 
 were interchanged, he would have $1.80. How many of 
 each has he ? 
 
 124. A quantity of wheat sufficient to fill three bins of 
 different sizes will fill the smallest bin four times, the 
 second bin three times, or the largest bin twice with 40 bu. 
 to spare. What is the capacity of each bin ? 
 
 125. A wholesale egg dealer sold on the average 3800 
 dozen eggs a day for cash. He reduced his price 5%, 
 and found that his average daily cash receipts from sales 
 were increased 10%. How many dozen eggs did he sell 
 daily at the reduced prices? 
 
 126. A man has $5000 which he wishes to invest in 
 two enterprises so that his total income will be $ 180 ; if 
 
GENERAL REVIEW 391 
 
 one enterprise pays 4% and the other 3%, how much 
 must he invest in each? 
 
 127. The circumference of the rear wheel of a carriage 
 is 3J ft. greater than the circumference of the front 
 wheel. The front wheel makes 98 more revolutions than 
 the rear wheel in traveling 5600 ft. What is the circum- 
 ference of each wheel? 
 
 128. The sum of two fractions is ^ and their difference 
 is y2_. What are the fractions ? 
 
 129. Find a fourth proportional to a*, ah\ 6 a%, 
 
 130. Find a mean proportional between 32 aV and 2 a^x. 
 
 131. Two numbers are in the ratio oi mi n. If c be 
 added to the first and subtracted from the second, the 
 results will be in the ratio of 4 : 5. Find the numbers. 
 
 132. What number must be subtracted from each of the 
 numbers 6, 9, 15, and 27, so that the resulting differences 
 shall form a proportion when taken in the given order? 
 
 133. If = - , prove that 6 is a mean proportional 
 
 b — c c 
 
 between a and c. 
 
 134. Two numbers have the ratio of 7:8; if 21 be 
 added to each, they have the ratio of 10 : 11. Find the 
 numbers. 
 
 135. Represent by a graph the distance traveled by an 
 automobile at the rate of 25 miles an hour. What is the 
 equation connecting the distance and the time? 
 
 136. Determine the value of x from the proportion 
 
 137. If h is a mean proportional between a and <?, prove 
 that a-26:6-2c = 2a-35:25-3c. 
 
392 ELEMENTARY ALGEBRA 
 
 138. Two numbers, x and y (the first being negative) 
 are in the ratio of 7: — 10. If 15 be subtracted from each 
 one, the resulting numbers are in the ratio of —10:7. 
 Find the numbers. 
 
 139. If x-^-yi a-\-h = x— y:a— h^ show that x varies 
 as «/. . 
 
 140. Expand (2-f a^)*. 
 
 141. Expand ia^^ J by the binomial theorem. 
 
 142. Find the numerical value of '- 
 
 2-2 . 27^ • 50 
 
 143. Express in simplest form with positive exponents: 
 
 24 x~'^yH~^ 
 40 x-'^yH-^ 
 
 144. Find one side of a square whose area is repre- 
 sented by the following expression: 
 
 ^ _ a: + 4 - - + -^. Check the result. 
 4 ic ar 
 
 x^yjx~^y^ 
 
 "^yz^ 
 
 146. Simplify ^ ^"^^ -^ a^iH. 
 
 at(a - ^)o 
 
 147. If a = 64, express each of the following as an in- 
 teger or a fraction: a^; -^a"^; (o^)"^; [(\^a)^]~^. 
 
 ,.« c- ^'f 4tx-^y-^ 6ic2a-i 
 
 148. Simplify -^ X 1-:- 
 
 149. If a + b + c=28 and a =15, h= 14, c=13, find 
 the value of V«(« — a)(« — 6)(« — c). 
 
 145. Simplify 
 
GENERAL REVIEW 393 
 
 150. Simplify ahH^Y ^ C^h-^^-^. 
 
 151. Simplify ( V«*) ^• 
 
 152. Simplify (25-3 -- a-^rc-S)"*. 
 
 153. Evaluate 2^ x 9^ x Srl 
 
 154. Simplify, using positive exponents to express the 
 
 answer, m Vn % 
 
 155. Expand and express in simplest form with positive 
 exponents (m~^a~^ — ma^^^. 
 
 156. Simplify VIOOO; </ST; V{^; -K; (27*)^; 2^ • 4^ 
 
 ■ V2 
 
 m+n m—n 2to— 1 
 
 157. Simplify \_a p x a p >^ «^~"] -p. 
 
 158. Simplify, using positive exponents to express the 
 iswer, f x~'^y^z~^ \^ ( x'^X^ 
 
 V xy-^ ) ^\^y 
 
 159. Simplify VT8 ; 36^ 25"^; ^^x-, -\/W. 
 
 r 
 
 160. Simplify ^ 
 
 161. Simplify 
 
 ^h-- 
 
 3^+1 gor+i 
 
 -l^*+l 
 
 162. Simplify 5 «« - (5 a)0 - l^ + A. 
 
 9^ 
 
 163. Simplify 5V| + V|-V8. 
 
 164. Simplify 3V12+V75-V108. 
 
 165. Simplify ^^-^^. 
 
394 ELEMENTARY ALGEBRA 
 
 166. Simplify '^2-h-V2^2^. 
 
 167. Simplify ^| . ^/IT, 
 
 168. Simplify </2 -\- -^128 ^ <^, 
 
 169. Solve Vrc + 1 - V2 a; - 2 = 0. 
 
 170. Simplify 10Vi~3V||+7Vi-5V2 + 9VS. 
 
 171. Simplify 1 
 
 V18+V27 
 
 172. Simplify ■^_4V| + V^. 
 
 173. Simplify V^-V^ + #^^ + 2-x^!±Zr^. 
 
 y X xy xy 
 
 174. Simplify ^ ~ ^^ - 
 
 175. Solve Va^-8 + a:-8 = 0. 
 
 176. Simplify Vl8 + \/-4- "^ ' 
 
 177. Simplify JV490 + V|-VT60. 
 
 178. Multiply V3 - 1 - V5 by V5 - V3. 
 
 179. Simplify (ViT^- Vr^O^CVrMH- VT^l). 
 
 181. Simplify, using positive exponents to express the 
 answer, r « — , ^ . 
 
 182. Find the numerical value of the following fraction 
 
 to two places of decimals : ^""^^ . 
 
 24-V3 
 
GENERAL KEVTEW W6 
 
 183. Divide 21 a^'' -^ 27 ctf - 26 a:2a ^j. 20 by 5 - 3 ^^ 
 
 184. Simplify . 
 
 r 4 1 1-3 
 
 185. Divide J - h^ by ■y/a--^'h. 
 
 186. Simplify 2 0:^9^1^81 + 27 Vi^TW. 
 
 187. Simplify ^V^)!^5Zi. 
 
 188. Simplify [a + 5(1 + V^^)] [a - h(l 4. V^^)] 
 [a - 6(1 - V^^)] [a + 5(1 - V:r2)] . 
 
 189. Divide Va;^ by Va;^. 
 
 "\/2 -4- 2a/^ 
 
 190. Compute the value of _ _ to two decimal 
 
 places. V2-V12 
 
 191. Simplify i + (i + ^+.v^x^^4-,^^r^-v^^Ti;^ 
 
 192. Find the square root of a^ — 2h^a-{-%h — 2h^a-^ 
 
 193. Simplify ^yC^^-y-^r^-), 
 
 y^ — x^ 
 
 194. Simplify pN^f. 
 
 195. Simplify 7-V2__^ T+V2 ^ 
 
 V34-V5 V5-V3 
 
 196. Simplify '^"^'^ 1 
 
 2-V3 2 + V3 
 
 197. Simplify ^ . 
 
 Vir+1 + V2: — 1 
 
396 ELEMENTARY ALGEBRA 
 
 198. Multiply Jb'^ + 2 + a~h^ by a'h^ - 1 + ah~K 
 
 199. Simplify (V^ + V^'' 
 
 200. Expand by the binomial theorem and express each 
 term in simplest form fa~^ — — ] . 
 
 201. Simplify V ^ + y + ^ _V ^ +y-^ 
 ■y/x -\- y — z ■\/x -\- y -\-z 
 
 202. 
 
 203. 
 
 Expand («^ + -Y- 
 
 Simplify fV3 + V2Y /V3W2y_,3^ 
 VV3-V2/ VV3+V2/ 
 
 204. Write the sum of V28, -V63, and V700 in 
 the simplest form. 
 
 / I i\_£iL _i 
 
 205. Simplify \^a^ -^ a^ j=^+i -f- a v. 
 
 206. Simplify 8^-7V98+i\/i + -i: + ^-i." 
 
 •^- 2^2 V2 3-2 30 
 
 207. Simplify 3*^+1 . 9^^~^ ^ 2T 3 . 
 
 208. Simplify g"*"! . 27"»+i • Sl"^ . 3-~-i. 
 
 209. Evaluate a^Va^ + 3 « Va"^ + 2 aWa^ when a =■ 4. 
 
 Simplify K-r>-yo(^^;-j;j^;)- 
 
 211. Simplify ^-^-2 + 0.^" 
 
 210 
 
 212. Divide 5x^-6x^-4: x~^^ - 4 x~^ - 3 x^ by 
 x^-2x-^. 
 
 213. Show that (29753)2-*. 43 - (29581)2-4- 43 = 237336. 
 
GENERAL REVIEW 397 
 
 214. Solve 5x^-Sx-2 = 0, 
 
 215. Solve in two ways 5 a:^ + 14 a; — 55 = 0. 
 
 216. Solve 8a;- 152:2-1 = 0. 
 
 217. Solve ax^ — hx= c. 
 
 218. Solve a; + 5 + 2 V^+5 = 15. 
 
 219. Solve by factoring, by completing the square, and 
 by f ormulge, 3 a;2 - 26 a: + 35 = 0. 
 
 22b. Solve 4 a;2 -I- 8 mx = 4 mn 4- n?. 
 
 221. Solve m^a^ — mx -\-l = x^. 
 
 222. Solve . ^J ^ =1 ^ 
 
 t^^St + 2 t-2 
 
 223. What values of x will make the expression (a: + 2) 
 (5 — 3 a:) equal to six times the value of a:? 
 
 224. Solve and check -Vx -f 2 — Vx — 6 = Va; — 3. 
 
 225. Solve 5a;2_32;_3V5a^-3a;-13 = ll. 
 
 226. Solve 6(x^-h^-[-6fx-^-\-SS = 0, 
 
 227. Solve 12fx^ + ^-6fx-{--\- 291 = 0. 
 
 228. Solve 4a72 — 7a; + 2 = 0; give both roots correct 
 to two places of decimals. 
 
 229. Solve (l — n^)x^—2mx + m^ = 0. 
 
 230. Solve V^^^ + x-S = 0. 
 
 231. Solve 3VZ--|^ = 8. 
 
 232. Solve V«^H-5 + «^-l = 0. 
 
 233. Complete the square in each of the following ex- 
 pressions : 
 
 a:2-12a;, a;2 + 25, a^ + 9a;, a?^4--^j. 
 
:^I8 ELEMENTARY ALGEBRA 
 
 In problems 234-7, write down the quadratic equations 
 whose roots are given. 
 
 234. 3 and 5. 
 
 235. 1 H- V2 and 1 ~ V2. ^ 
 
 236. — -^ — and — -— . 
 
 2 2 
 
 237. m-i-n-\- Vw — n and m + n — Vm — n(m > n), 
 
 238. Solve the system 
 
 fa; + V^+ «/ = 14, 
 \x^+xy + 1/^ = 84:. 
 
 239. Solve the system 
 
 f a^-y^=^6, 
 
 240. Solve the system 
 
 241. Given p = 9A^^d; find value of d. What is 
 the positive root when t= .21 and j? = 3.2 ? 
 
 242. The area of a mat of uniform width about a picture 
 10 inches long by 8 inches wide is one half the area of the 
 picture. What are the outside dimensions of the mat ? 
 
 243. Solve -l- + _i- ^=0. 
 
 x—a x-\- 2a 2a 
 
 244. Solve --±1-1=^. 
 
 X—6 X-\-l X — 4: 
 
 245. A loop of twine 30 inches long is to be stretched 
 over four pegs so as to form a rectangle whose area shall 
 be 50 square inches. What are the sides of the rectangle ? 
 
INDEX 
 
 [Eeferences are to pages.] 
 
 Abscissa, 252 
 
 Absolute value, 25 
 
 Addition, 24 ; associative law for, 
 37 ; by counting, 30 ; commuta- 
 tive law for, 34 ; graphic repre- 
 sentation of, 30 ; of fractions, 
 164 ; of monomials, 34, 35, 37 ; 
 of polynomials, 38 ; of surds, 279 
 
 Algebra, 1 ; laws of combination 
 in, 9 ; symbols of, 2, 3 
 
 Algebraic expressions, 6 
 
 Algebraic fraction, 153 
 
 Alternation, 242 
 
 Antecedent, 239, 245 ' 
 
 Approximations, 319 
 
 Arithmetic, the notation of, 1 
 
 Associative laws, 37, 55 
 
 Assumptions, 16 
 
 Axes of coordinates, 261 ; of refer- 
 ence, 251 
 
 Base of a power, 11 
 
 Binomial, 13 ; cube of, 115; square 
 
 of, 108 
 Binomial expansion, 300 
 Binomial formula, 300 
 Binomials, product of two, 114 
 Braces, 12 
 Brackets, 12 
 
 Cancellation, 84 
 Checking the result, 7 
 Clearing of fractions, 191 
 Coefficient, 10 
 Common difference, 365 
 Common ratio, 371 
 Commutative laws, 34, 54 
 Completing the square, 316 
 Consequent, 239 
 Constant, 247 
 Coordinates, 252 
 
 Cubes, sum and difference of two, 
 132 
 
 Decimal, recurring, 379 
 
 Degree of an equation, 82, 142, 
 143, 205, 312 ; of an expression, 
 100, 101 ; of a monomial, 100 
 
 Descartes, 261 
 
 Difference, 27 
 
 Distributive law, 58 
 
 Dividend, 33, 72 
 
 Division, 33, 64 ; by zero, 16, 72, 
 154 ; of fractions, 178 ; of mo- 
 nomials, 66 ; of polynomials, 68, 
 69 ; of surds, 281, 285 ; rule of 
 signs in, 33 ; special cases of, 179 
 
 Divisor, 33 
 
 Equations, 15 ; change of signs in, 
 84 ; complete quadratic, 313 ; 
 conditional, 81 ; degree of, 82, 
 142, 143, 205, 312 ; dependent, 
 207 ; equivalent, 82, 208, 219 ; 
 formation of quadratic, 327 ; 
 fractional and literal, 191, 197, 
 222 ; graph of dependent, 260 ; 
 graph of inconsistent, 260 ; 
 graph of linear, 258, 260 ; graphs 
 of, 256 ; homogeneous, 352 ; in- 
 complete quadratic, 313, 314 ; 
 inconsistent, 207, 260 ; independ- 
 ent, 207 ; indeterminate, 206 ; 
 irrational, 321 ; linear, 82, 205 ; 
 number of roots of a quadratic, 
 343 ; number of solutions of, 
 205, 208 ; numerical, 197 ; partic- 
 ular systems of quadratic, 355 ; 
 quadratic, 312 ; quadratic with 
 complex roots, 335 ; rational and 
 integral, 142 ; relations between 
 roots and coefficients of quad- 
 ratic, 326; roots of, 82, 143, 
 342 ; satisfying an, 82 ; simple, 
 82, 205 ; simultaneous, 207, 229 ; 
 solution of, 82, 143, 206, ^316, 
 
 399 
 
400 
 
 INDEX 
 
 824 ; standard form of quadratic, 
 812 ; systems of quadratics, 350 ; 
 systems of linear, 205, 208 
 
 Elimination by addition and sub- 
 traction, 209 ; by comparison, 
 215 ; by substitution, 214, 350 ; 
 by undetermined multiplier, 217 
 
 Euclid, 57 
 
 Euler, 109 
 
 Evaluation of an algebraic expres- 
 sion, 6 
 
 Evolution, 302 
 
 Exponents, 11 ; fractional, 286, 
 
 287, 288, 289 ; negative, 286, 287, 
 
 288, 289 ; zero, 65 
 Expressions, algebraic, 6 ; integral, 
 
 100; mixed, 170; prime, 118; 
 rational, 100 
 Extremes, 241 
 
 Factoring, 118 ; equations solved 
 by, 142 ; remainder theorem in, 
 140 ; special methods of, 134 ; 
 summary of, 136 
 
 Factors, 9 ; found by grouping 
 terms, 121 ; integral algebraic, 
 118 ; monomial, 119 ; of alge- 
 braic expressions, 118 ; of differ- 
 ence of two squares, 126 ; of 
 general quadratic trinomial, 130 ; 
 of integral expressions, 118 ; of 
 monomials, 119 ; of trinomials of 
 the form x^ + ex -\- d, 127 ; of tri- 
 nomial squares, 123 ; sum and, 
 difference of two cubes, 132 
 
 Formula, 17 
 
 Fourth proportional, 245 
 
 Fractional equations, 191, 222 
 
 Fractions, 153 ; addition and sub- 
 traction of, 164 ; change of signs 
 of factors in terms of, 156 ; clear- 
 ing an equation of, 191 ; complex, 
 183 ; continued, 186 ; division of, 
 178 ; laws governing algebraic, 
 153 ; lowest common denominator 
 of, 164 ; multiplication of, 161, 
 178 ; powers of, 176 ; proper and 
 improper, 170 ; quotient of two, 
 178 ; reduction of, 169 ; reduction 
 to lowest terms, 158 ; simple, 
 169 ; signs affecting, 154, 156. 
 
 Function, 255 ; graph of, 256 
 
 Gauss, 330 
 
 Graphic representation, of addition 
 and subtraction, 30 ; of the rela- 
 tion between two variables, 251 
 
 Graphs, 251, 346 
 
 Greatest common divisor, 146 
 
 Highest common factor, 146 ; of 
 monomials, 147 ; of polynomials 
 by factoring, 148 
 
 Identities involving roots, 267 
 
 Identity, 81 
 
 Imaginary imit, 330 
 
 Index, 11 ; law, 55 ; of a radical, 
 
 275 ; of a root, 265 
 Involution, 800 
 
 Least common multiple in arith- 
 metic, 149 
 
 Linear equations, 82, 205 
 
 Literal equations, 197 
 
 Lowest common denominator, 164 
 
 Lowest common multiple, 149 ; of 
 monomials, 150 ; of polynomials 
 by factoring, 151 
 
 Mean, arithmetical, 368 ; geometric, 
 374 ; proportional, 245, 374 
 
 Means, arithmetical, 368 ; geo- 
 metric, 374 ; of a proportion, 
 241 
 
 Members of an equation, 15 
 
 Minuend, 27 
 
 Monomial, 13 
 
 Multiplicand, 31 
 
 Multiplication, 81, 64; associative 
 law for, 55 ; combinations of 
 signs in, 32 ; commutative law 
 for, 54 ; distributive law for, 58 ; 
 of a polynomial, 58, 61 ; of frac- 
 tions, 161, 173 ; of surd expres- 
 sions, 281, 282 ; rules of signs in, 
 32 
 
 Multiplier, 81 
 
 Newton, 801 
 
 Notation, 81, 106 
 
 Number, complex, 333 ; imaginary, 
 880 ; irrational, 262 ; literal, 1 ; 
 prime, 118 ; rational, 262 ; re- 
 ciprocal of a, 178 ; symbols of, 2 
 
INDEX 
 
 401 
 
 Numbers, algebraic, 22 ; conjugate 
 complex, 333 ; negative, 22, 23 ; 
 opposite, 25 ; positive, 22 ; prime 
 to each other, 158; property of 
 positive, 266 ; real, 330 ; scale of 
 positive and negative, 23 ; sum 
 of, 25 ; use of Uteral, 1 
 
 Operations, order of, 7 ; rational, 
 
 100 ; symbols, 2 
 Ordinate, 252 
 Origin, 251 
 
 Parentheses, 12, 48 
 
 Pascal, 204 
 
 Plotting, 253 
 
 Polynomial, 14 
 
 Portraits, Descartes, 261 ; Euclid, 
 57 ; Euler, 109 ; Gauss, 330 ; 
 Newton, 301 ; Pascal, 204 ; Py- 
 thagoras, ii ; Vieta, 33 
 
 Powers, 11 ; ascending and descend- 
 ing, 61 ; fundamental identities 
 involving, 263 ; of a fraction, 176 ; 
 of i, 331 ; quotient of two of the 
 'same base, 64 
 
 Processes, fundamental, 34 
 
 Product, 31 ; of conjugate surds, 
 284 ; of two binomials having a 
 common term, 114 ; of two com- 
 plex numbers, 334 ; of two con- 
 jugate complex numbers, 333 ; 
 of two fractions, 173 ; of two 
 monomials, 56 ; of sum. and dif- 
 ference of two numbers, 112 
 
 Products, type, 107 
 
 Progression, arithmetical, 365 ; de- 
 creasing, 372 ; geometric, 371 ; 
 increasing, 371 ; infinite geo- 
 metric, 377 
 
 Proportion, 240 ; by alternation, 
 242 ; by composition, 243 ; by 
 division, 243 ; by inversion, 241 ; 
 continued, 244 
 
 Proportional, fourth, 245 ; mean, 
 245 ; third, 245 
 
 Pythagoras, ii 
 
 Quadratic equations, 312 ; com- 
 plete, 313 ; incomplete, 313 ; 
 number of roots of, 343 ; particu- 
 lar systems of, 355 ; standard 
 
 form of, 312 ; systems of, 350 ; 
 
 with complex roots, 335 
 Quadratic surd, 266, 295 
 Quotient, 33, 72 ; of two fractions, 
 
 178. • 
 
 Radical, 265 
 
 Radicals, 275 
 
 Radicand, 265 
 
 Ratio, 239 
 
 Ratios, composition of, 245 
 
 Remainder, 72 
 
 Remainder theorem, 139 
 
 Review, 52, 75, 137, 186, 234, 296, 
 381 
 
 Root of an equation, 82 
 
 Roots, 265 ; approximate square, 
 308 ; cube root of a monomial, 
 105 ; found by factoring, 143 ; 
 like and unlike, 265 ; nature of 
 the roots of ax"^ -^ bx -j- c = 0, 
 340 ; principal, 265 ; quadratic 
 equations with complex, 335 ; 
 square root of arithmetical num- 
 bers, 307 ; square root of a bi- 
 nomial surd expression, 293 ; 
 square root of a monomial, 105 ; 
 square root of a polynomial, 304 ; 
 square root of a trinomial, 303 
 
 Rule of signs, in division, 33 ; in 
 multiplication, 32 ; in subtrac- 
 tion, 28 
 
 Series, 365 ; sum of arithmetical, 
 369 ; sum of geometric, 375 
 
 Signs, affecting a fraction, 154 ; 
 change of, in terms of a fraction, 
 156 ; like and unlike, 22 ; of ag- 
 gregation, 12 ; rule of, 28, 32, 33 
 
 Square of a binomial, 108 ; of a 
 monomial, 102 ; of a polynomial, 
 111 ; of a trinomial, 110 
 
 Subtraction, 26 ; of fractions, 164 ; 
 of monomials, 42 ; of polyno- 
 mials, 44 ; of surds, 279 
 
 Subtrahend, 27 
 
 Surds, 266 ; addition and subtrac- 
 tion of, 279 ; comparison of, 278 ; 
 conjugate, 283 ; division by poly- 
 nomial containing, 285 ; multipli- 
 cation and division of, 281 ; order 
 of a, 266 ; product of conjugate, 
 
m2 
 
 INDEX 
 
 '284 ; qnadratic, 266, 295 ; simpli- 
 fication of fractional, 276 ; square 
 root of a binomial expression, 
 
 Symbols, 2 ; of number, 2 ; of 
 operation, 2 ; of relation, 3 
 
 Terms, 13 ; like, 14 ; 6f a fraction, 
 
 163 ; of a proportion, 240 
 Transposition, 83 
 Trinomial, 14 
 
 Type forms, 11© 
 
 Value, arithmetical, 25 ; numerical, 
 25 
 
 Variable, 247 
 Variation, 247 
 
 verse, 248 
 Vieta, 33 
 Vinculum, 12 
 
 Zero, 23, 25 
 
 direct, 247 ; in- 
 
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 AUP 2 n i'i^7 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 LD 21-100/n.-12,'43 (8796s) 
 
 1 
 
rb Jbv^'^ 
 
 541262 
 
 UNIVERSITY OF CAUFORNIA LIBRARY