Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/elementaryalgebrOOnnarsrich ELEMENTARY ALGEBRA BY WALTER R. MARSH HEAD MASTER PINGRY SCHOOL, ELIZABETH, X.J. Of TH€ UNIVERSITY or NEW YOEK CHARLES SCRIBNER'S SONS 1907 M3f MAY 29 191 GIFT COPYRIGHT, 1905, 1907, BY CHARLES SCRIBNER's SONS PREFACE The subject-matter of this text follows the require- ments of the College Entrance Examination Board both as to subjects treated as well as to those omitted, but especial emphasis is placed upon those principles which are the tools of more advanced work in mathematics. The philosophy per se of algebra and all algebraic puz- zles are therefore omitted, to give place to a logical dis- cussion, simply told, of the fundamental principles. The scheme of the whole text is to illustrate the meaning of a principle by carefully selected exercises; every prin- ciple is followed by such a group of examples as will exact a mastery of the principle involved before another topic is taken up. The examples are expressly prepared to illustrate various principles treated in the text. Nearly a thousand of these examples are taken from the most recent college entrance papers. The attention of teachers is especially invited to the use of Graphical Methods throughout the book, the in- troduction of the Negative Number, the treatment of the Graphs of Equations, the introduction of Equations used in Physics, and the insertion of problems from Physics in Ratio and in Variation, and to the treatment of the Progressions and of Permutations and Combinations. It is suggested that paragraphs, exercises, and exam- ples marked by the * be omitted dt first reading. V 219153 VI PREFACE The author begs to acknowledge gratefully the valuable assistance of Professor Charles H. Ashton of the Univer- sity of Kansas, of Miss Mary M. Wardwell of the Central High School, Buffalo, N.Y., and of Mr. Frank C. Rob- ertson of the Pingry School, Elizabeth, N. J., not only for their careful reading of the proofs, but also for their criticisms of the text. CONTENTS CHAPTER PAGE I. Introduction and Definitions 1 II. Addition and Subtraction 19 III. Multiplication and Division 30 IV. Equations and Problems 47 V. Type Forms in Multiplication 65 VI. Factoring 75 VII. Highest Common Factors. Lowest Common Multiples 100 VIII. Fractions 115 IX. Simple Equations ...,.•.. 142 X. Graphs 158 XI. Simultaneous Simple Equations 163 XII. Problems involving Simple Equations . . • . 188 XIII. Inequalities .203 XIV. Involution and Evolution 210 XV. Radicals . 22G XVI. Imaginaries 249 XVII. Theory of Exponents 254 XVIII. Quadratic Equations 268 XIX. Simultaneous Equations solvable by Quadratics . . 299 XX. Problems involving Quadratic Equations . . .318 XXI. Ratio, Proportion, Variation 324 XXII. Progressions 345 XXIII. Permutations and Combinations 362 XXIV. Binomial Theorem 374 XXV. Logarithms 380 vii TEACHERS MAY OBTAIN ANSWER-BOOKS, FOR WHICH NO CHARGE IS MADE, ON APPLICATION TO THE PUBLISHERS. ELEMENTARY ALGEBRA CHAPTER I INTRODUCTION AND DEFINITIONS 1. The science of number includes both Arithmetic and Algebra. Algebra may be defined as generalized Arithmetic. 2. In arithmetic every number represents a definite value. Thus, 4 = 1 + 1 + 1 + 1. In algebra, a set of symbols^ usually letters of the alphabet^ is used to represent numbers. A letter can represent any number wliatever, provided its value does not change during a particular range of operations. \ SYMBOLS OF OPERATION Addition is indicated by the sign +, read "plus.'' ihus, 4 + 1 means the sum of 4 and 1 ; a + d means the \ of a and d, of thd^btraction is indicated by the sign — , read " minus, " "ijius, 3 — 2 means that 2 is to be subtracted from 3 j 6 — c facti^s that c is to be subtracted from b, 1 2 ELEMENTARY ALGEBRA [Ch. I, § 3 Multiplication is indicated by tiie sign X, and by the sign ', each read "times" or, "multiplied by" ; and by the omission of sign. Thus, mxn, m'Ti, and mn all mean the product of m and n, or of n and ?/i. The multiplication sign is never omitted in expressing the product of numbers in the form of digits. Thus, 56 indicates 50 + 6 ; 5*6 indicates 5x6. Division is indicated by the signs 4-, /, :, each read " divided by " ; and by the fractional form. Thus, a-r-b, a/b, a : b, and - all indicate the division of a by b. Equality between two numbers is indicated, by the sign =, read "is equal to." Thus, a = b indicates that a is equal to b. EXERCISE I If tt = 1, 5 = 2, (? =^ 3, c? = 4, find the value of each of the following : 3. 4. a + b c 6. cd 11. abed a + c + d b + d c 7. ad 12. ac + be + a \ a + d c + d a 8. d_b a a' 13. ab + be + cd d + e-b d— a • e 9. ad c b a' 14. ae + ad + cb cd+\ c-—a b • 10. ab + be d • 15. ad+cd— be a+ b + c + d' Ch. I, §§ 4-7] INTRODUCTION AND DEFINITIONS 3 ALGEBRAIC EXPRESSIONS 4. An algebraic expression is a combination of number symbols connected by any of the symbols of operation. Thus, a, 7 — a, 6 +a-h3 + b are algebraic expressions. 5. A term of an algebraic expression is a combination of number symbols not separated by the signs + or — . Thus, in the algebraic expression 6 + a -^ 3 — 6, the terms are 6, a ~ 3, and b. ^-^ 6. When two or more numbers multiplied together produce a certain product, each of these numbers is called a factor of the prpduct. Thus, a, h, and c, are factors of abc. Each of the factprs of a number or the product of any number of factors is called a coefl3:cient of the rest of the term. Thus, in 3 a, 3 is the coefficient of a ; in a6, a is the coefficient of 6 ; in I ahc, f is the coefficient of abc, | a of he, and -| ah of c. The coefficient is generally understood to mean the number placed before the number symbols represented by the letters. If the coefficient he 1, it is always omitted. Thus, a = 1 a. 7. The exponent of a number is the symbol in the form of an integer which represents how many factors equal to the number affected by the exponent are taken. Thus, a^ represents that a has been taken three times as a factor ; or, a^ = a • a • a. 4 ELEMENTARY ALGEBRA [Ch. I, §§ 8-10 The exponent affects only that number symbol which it follows, and at the upper right hand of which it is written. Thus, 3 a%c means that a alone has been taken twice as a factor ; or 3 a?ho = 3 > a ' a ' b > c. If no number symbol be written as the exponent, it is always understood that 1 is that exponent. Thus, in 3 a^bc, 3, b, and c are to be understood as having the exponent 1 affecting each of these numbers ; or 3 a^bc = 3^a-6V. Since the product of a number of equal factors can be called a power of that number, a^ can be read " a with the exponent 3" ; or "a third." Thus, a'^= a ' a ' a - a can be read ^^ a with the exponent 4/' " a fourth/' or, '^ a to the fourth power.'' The distinction between coefficient and exponent should be carefully noticed. Thus, 3a=a + a + a; and a^ = a • a • a. 8. A monomial is an expression containing a single term. Thus, 2 a^, 3 b, and c^ are monomials. 9. Similar terms, or like terms, are those which differ only in their numerical coefficients. Thus, 3 a^b, a?b, and 7 a?b are similar, or like, terms, 10. A polynomial is an expression containing several terms. Thus, 2o?b + 3 ab'^ + 5^ is a polynomial. Ch.I,§§11,12] INTRODUCn^M'N AND DEFINITIONS 5 A polynomial which coiM&|^^'^ ^^^^ terms is called a binomial ; and one which con^i^s three terms is called a trinomial. V^ Thus, o? + 6^ is a binomial ; and a^— a?\+ h'^ is a trinomial. 11. The positive and negative terms of ai3L expression are those which are preceded by the plus and mkius signs respectively. Thus, the positive terms of a^ — 3 o?h + 3 alP' — If are ot^ii^ 3 a6^, and the negative terms are 3 a% and h^, ^B^ « 12. The numerical value of an expression is found by substituting for the letters their values in numbers, and performing the indicated operations. Thus, the numerical value of 2 a, if a = 4, is 8. EXERCISE II If a = 6, 5 = 4, {? = 3, c? = 2, ^ = 1, find the value of each of the following expressions: 1. 2aJ. 11. 2^+3 c2. 21. V,4 + ^2J2 + 54. 2. Zed. 12. 4^2-35^. 22. a^-b^. 3. 4cde. 13. a2-4 6'2. 23. b^+c\ 4. a?d. 14. bad-2h\. 24. b^ - C^ 5 cH. 15. 4:a^-2Pd^. 25. c^ + cd + d\ 6. 4 aHe. 16. ab+hc-}- J2. 26. c'^-cd + d\ 7. 2cH. 17. 2ac-c^+d^. 27. 2a2+52-5(?2. 8. 2hHdH. 18. a^+ab + P. 28. j2_4j + 4. 9. 6 cd^e\ 19. a^-2ab-\-b^. 29. 2a%'^cdh. . 10. 7 ahcdh. 20. a^ + J3. 30. ^3-^2^ + 3 ^^2. g ELEMENTARY AlGEBRA [Ch. I, §§ 13, 14 13. Aggregation, the proces^'^ of taking the result of several operatio7is as a whole^ is ijj^icated by the symbols ( ), { }, [ ], read respectively.^' parenthesis," "brace," "bracket." Thus, aQ)-\-c), a^pj^c], a\h-\-c\ all mean that the sum of h and c is to be ^ijultiplied by a. ORDER OF OPERATIONS 14. In any polynomial in which the various signs of operation occur, the plus and minus signs are used to separate terms. The operations of multiplication and of division are to he performed before those of addition and subtraction. Thus, 28 -T- 4 — 2 X 3 contains two terms, a plus sign being imderstood as preceding 28 ; + 28 -^ 4 — 2 x 3 = first term (28 -- 4) - the second term (2x3); 28--4-2x3 = (28--4)-(2x3) = 7-6 = l. ' Were this problem to be given orall}/ .r. arithmetic, it might be understood: 28'--4 = 7; 7-2 = 5; 5x3 = 15. The difference between the algebraic usage and the arithmetical oral statement is to be carefully noticed. EXERCISE III If a = 1, 5 = 2, (? = 3, cZ = 4, find the value of the follow- ing expressions : 1. a-{-d^b. 6. b(d-ay, 2. 2b^xc-2ab. 7. {Sa-b)(Sa + b}. 3. iaW-exd, 8. (b + a)^ -r-^d - a). 4. Bac'^d-^d'^+Sb^. 9. Sa^d^9bc + b^. 5. (3a + 2^)-lla + J2. 10, 4ax52^2a3^ + 6V. Ch. I, § 15] INTRODUCTION AND DEFINITIONS 7 USE OF LITERAL NOTATION 15. The properties of numbers, whether expressed by integers or by letters, are identical. The advantage, therefore, of representing numbers by letters lies in the fact that the letter, being a general number, often leads to a general conclusion, expressed as a formula. In arithmetic the principle is taught that inter- est = principal x time x rate per cent ; or that i — prt^ whatever may be the numerical values of the letters. Moreover, literal notation is often advantageously used as a sort of shorthand. For example, four times a cer- tain number equals the sum of 60 and three times that number. Expressing the problem in arithmetic, 4 times the number = 60 + 3 times the number. Expressing the same problem in algebraic language, taking x to represent the number, 4a;=60 + 3a;. The advantage of the algebraic form of statement lies in the fact that it is merely a statement in shorthand, where x takes the place of the printed words " the number." EXERCISE IV 1. Express in algebraic form the sum of twice a num- ber, a, and three times that number ; the product of five times a number and four times that number. 2. If 1 barrel of flour costs i5, how much will 2 barrels cost? 3 barrels? a barrels? h barrels? 8 ELEMENTARY ALGEBRA [Ch. I, § 15 3. If 20 barrels of flour cost $ 80, what will be the cost of 1 barrel? If a barrels cost $80, what will be the cost of 1 barrel ? 4. If a man earns 15 a day, how much will he earn in 4 days ? in h days ? in c days ? 5. The sum of two numbers is 20. If one of the num- bers is 8, what is the other number ? If one of the num- bers is a, what is the other number ? 6. If one part of 8 is 6, what is the other part ? 7. If one part of a is 2, what is the other part ? 8. If one part of 2 is a, what is the other part ? 9. If one part of a is x^ what is the other part ? 10. If one part of x is 6, what is the other part ? 11. What is the product of two numbers, if one factor is a and the other b ? 12. What is the divisor, if the dividend is 27 and the quotient 3 ? If the quotient is a ? 13. The divisor of a certain number is a and the quo- tient be What is the dividend ? — 14. How much is 8 increased by 3 ? 8 increased hj a? a decreased by 4 ? m decreased by 2 a; ? 15. By how much does 12 exceed 8 ? 12 exceed a "i a exceed 12? a exceed x? 16. What is the excess of 20 over 11 ? of 20 over x ? of X over 20 ? oi x over i/ ? -^ 17. What is the quotient of 20 divided by the excess of X over 200 ? 18. If X is the smaller part of 5, what is the larger part ? Ch. I, § 15] INTRODUCTION AND DEFINITIONS 9 19. If 10 is the larger part of x, what is the smaller part ? 20. How much does 8 lack of 13 ? of a ? 21. How much does a lack oi x? of 22 ? 22. How much does x lack of 13 ? of m ? 23. If A is 30 years old now, how old will he be in 4 years ? in x years ? 24. If A is now a years old, what would half his age be ? three times his age ? 25. If A is 18 years old now, how old was he 4 years ago ? a years ago ? 26. If A is 25 years old now, what was three times his age a years ago ? 27. What is the average age of two men, the age of the first being 30, and the second being a ? ^ 28. If 3 is the tens' digit of a number of two digits, and a the units' digit, what is the number ? ^ 29. If a is the greater part of a number, and the differ- ence between the parts is 4, what is the other part ? ^ 30. If a is the smaller part of a number, and if the smaller part lacks 4 of the larger part, what is the larger part? 31. If 2 a + 3 represents a certain number, what repre- sents a fourth of that number ? 32. By how much does three times a exceed 22 ? 33. By how much is the third part of a below 9 ? 34. If A has X dollars, B twice as much as A, and C as much as A and B together, how much has B ? how much has C ? 10 * ELEMENTARY ALGEBRA [Ch. I, §§ 16, 17 POSITIVE AND NEGATIVE NUMBERS 16. Up to this time the restriction has always been made that the quantity to be subtracted, the subtrahend, must be less than the quantity, the minuend, from which the subtrahend is to be subtracted. Since 7 is greater than 4, it is possible to subtract 4 from 7. Expressed in arithmetical language, 7 — 4 = 3. Since 4 is less than 7, it is not possible to subtract 7 from 4. But there is a mathematical necessity for making the process of subtrac- tion always possible. 17. It is evident that a new sort of number must be employed if subtractions are always possible. Numbers hitherto employed can be represented as shown in Fig- ure 1. 012345678 ■ : 1 I I L \ 1 1 I I Fig. 1. If a straight line of indefinite length is divided into units of length from zero, the natural numbers can be represented by successive repetition of this unit of length in a direction extending indefinitely towards the right. These numbers will be seen to increase by a unit, count- ing from left to right ; and to decrease by a unit, count- ing from right to left. The addition of 2 and 3 can be illustrated by counting from zero, two units towards the right, and then by counting three more units from 2 towards the right. The subtraction of 2 from 3 can be illustrated by counting three units from zero towards the right, and then by counting two units from 3 in the op- posite direction towards the left. If, however, the prob- Ch. I, §§ 18, 19] INTRODUCTION AND DEFINITIONS 11 lem were to subtract a greater from a lesser number, — for example, to subtract 3 from 2, — the process is: count from zero two units towards the right ; try to count three units from 2 towards the left ; two units can be counted up to zero ; the third unit will seem to be beyond zero to the left. It is evident that the counting cannot continue further unless there- are ne\r units which are different in character towards the left of zero. 18. An abstract number is used without application to things, as 3, 4, 6 ; a concrete number is used with applica- tion to things, as 3 men, 4 inches, 6 cubic feet. Concrete numbers, or quantities, are often opposite in character. The following are examples of opposite concrete quanti- ties: $20 gain and $15 loss; 2 inches to the right and 4 inches to the left ; 10 degrees above zero and 5 degrees below zero ; 25 degrees north latitude and 4 degrees south latitude. If two concrete quantities of opposite kinds be combined, the effect of one is to decrease, destroy, or to reverse the state of the other. For example: $20 gain combined with $15 loss destroys the loss of $15 and leaves a gain of $5. 19. Differences that arise from subtracting quantities from lesser quantities are called negative quantities. Quantities that are not negative are called positive quan- tities. Positive quantities are represented thus : -f- 3, + 5; while negative quantities are represented thus : —3,-5. The former are read : "positive 3," "positive 5 " ; the latter are read: "minus (negative) 3," "minus (negative) 5." The signs -f and — are also used to indicate the processes of addition and of subtraction. Therefore, for the present, positive numbers will be indicated thus: (+3), (+5); and minus (negative) numbers thus: (—3), (—5). 12 ELEMENTARY ALGEBRA [Ch. I, §§20-22 20. The series of positive and negative numbers can be represented as shown in Figure 2 : -9-8-7-6-5-U-3-3-1 0123U56789 I I I I I I I I I, I I I I i I I I I I Fig. 2. Numbers passing from zero in the positive direction increase indefinitely, and numbers passing from zero in the negative direction diminish indefinitely. Positive and negative numbers taken together are called algebraic num- bers. The sign +, indicating a positive number, is some- times omitted ; the sign — , indicating negative numbers, is never omitted. When no sign is written before a number, the plus sign is always understood. 21. The absolute or numerical value of a number de- pends upon the number of units contained in the number, no reference being paid to its sign, or its quality of oppo- sition, that is, its direction towards the right or towards the left. For example : ( + 7) and ( — 7) are equal in absolute or numerical value. 22. A negative number may be considered as indicating a delayed or postponed subtraction. For example : (—1), since it is a difference obtained by subtracting a quantity one unit greater than a second quantity, indicates that (+1) still remains to be subtracted. Since the addition of ( — 1) to a second number means the subtraction of (+1) from the second number, by applying the same principle to any negative number, it is evident that add- ing a negative number to a second number is equivalent to subtracting a positive numl?er {of the same absolute value as the negative number) from the second number. Ch. I, § 23] INTRODUCTION AND DEFINITIONS 13 23. 1. Add (+3) and (+5). -9'8-7-6'5-U-3-2-l 0123U56789 I I I I I I I I i t I I .\ I I I I I I Fig. 2. The sum of (+3) and (+5) is found by counting, from (+3), five units in the positive direction; and is, therefore, ( + 8). 2. Add (-3) and (-5). The sum of (— 3) and (—5) is found by counting, from ( — 3), five units in the negative direction ; and is, there- fore, (—8). 3. Add (+5) and (-3). The sum of (+ 5) and (~ 3) is found by counting, from ( + 5), three units in the negative direction ; and is, there- fere, (+2). 4. Add (-5) and (+3). The sum of ( — 5) and ( + 3) is found by counting, from ( — 5), three units in the positive direction; and is, therefore, ( — 2). If a and b represent any two integers, positive or nega- tive, ( — a) + ( — J)=— a— J, i + a} + (-b}=+a-b, (-^) + ( + J) = -a + 6. Zero may be defined as the sum of that positive and that negative number tvhich are equal in absolute value. 14 ELEMENTARY ALGEBRA [Ch. I, § 24 RULE FOR ADDITION OF TWO NUMBERS If both numbers are positive^ the sum will be positive and^ equal to the sum of the absolute values of the numbers. If both numbers are negative^ the sum will be negative and equal to the sum of the absolute values of the numbers. If one number is positive and the other negative^ the absolute value of the sum tvill be the difference of the absolute values of the numbers^ and will be positive or negative according as the number of greater absolute value is positive or negative. 24. Two operations are said to be inverse to each other when the effect of one is to undo the other. Subtraction is the inverse operation to addition ; and may be defined as the process oi finding from two given numbers a third number so that the sum of the first and the third is equal to the second. The process of subtraction depends upon the principle in § 22. 1. Subtract ( + 3) from ( + 5). I j I I I I J I I I I I I I I I I I I Fio.2.- ^ The result of subtracting ( + 3) from ( + 5) is found by- counting, from ( + 5), three units in the negative direc- tion ; and is, therefore, ( + 2). 2. Subtract ( — 3) from ( — 5). The result of subtracting ( — 3) from ( — 5) is found by counting, from ( — 5), three units in the positive direc- tion ; and is, therefore, ( — 2). Three units are counted from ( — 5) in the positive direc- tion because the subtraction of a negative quantity is equiva- lent to the addition of its absolute value. Ch. I, § 24] INTRODUCTION AND DEFINITIONS 15 3. Subtract ( — 3) from ( + 5). The result of subtracting ( — 3) from ( + 5) is found by- counting, from ( + 5), three units in the positive direc- tion ; and is, there foi^, ( + 8). 4. Subtract ( + 3) from (-5). The result of subtracting ( + 3) from ( — 5) is found by counting, from ( — 5),. three units in the negative direc- tion ; and is, therefore, ( — 8). If a and h represent any two integers, positive or nega- tive, (i + a)-( + h^=+a-h, ( -- ^) — ( — J) = — a + J, l + a)-l-h)= +a + h, Rule for Subtraction of Two Numbers : Change the sign of the subtrahend and add the result to the minuend. EXERCISE V Find the values of the following indicated operations 1. (+ 3) + (+ 5) 2. (- 3)-(- 5) 3. (+ 3) + (- 5) 4- (- 5) + (+ 3) 5- (+ T)-(+ 4) 6. (+ 6)-(+ 7) 7. (+ 9).- (-12) 8. (+12) -(-15) 9. (+ 3)-(- 7) 10. (- !)-(+ 4) 21 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. (+ 8)-(- (+ 4)-(+ 5). (+ 6)- ( + 7). (+ 4) + (+5). (- 6) -(-5). (-f- 4) + (-4). (+ 4) -(+4). (- 7) + (-4). (+ 8) -(+5). (+ 7) -(+8). (4-12)-(-4). 9). 16 ELEMENTARY ALGEBRA [Ch. I, §§ 25, 26 25. The product of two algebraic numbers is a third num- ber whose absolute value is the product of the absolute values of the two numbers ; and is (1) positive if both numbers are each positive or negative^ and negative (2) if one of the numbers is positive and the other negative. The operation of finding the product of two numbers is called multiplication. To find the product of a and b is to multiply a and 5, or to multiply b and a. The product of a and b is indicated thus : (a x S), or (aJ), or ab. Since the arithmetical product of the absolute values of the factors is not determined by the order of the factors, by definition the product of a and b is the same as the product of b and a. If ab indicates the product of a and 6, and ba indicates the product of b and a, ab = ba. (+5)x(+3)=:(+15), (-5)x(-3) = ( + 15), (+5)x(-3) = (-15), (~5)x(+3) = (-15). In general, (+^) x (+J) = (+ ^5), I- a] X (-S) = (+aS), (+a) x(-J) = (-a5), (-a) X (+6) = (-a6). The Law of Signs in Multiplication : Like signs give posi- tive^ and unlike signs give negative products. 26. The absolute value of the quotient of two numbers is the quotient of the absolute values of the numbers; and is (1) positive if both numbers are each positive or negative^ and is (2) negative if one of the numbers is positive and the other negative. The operation of finding the quotient of two numbers is called division. Division is tlie operation inverse to multiplication. Ch. I, § 26] INTRODUCTION AND DEFINITIONS vl^ Since, § 25, (+5)x(+3) = (+15), (+15) (_5)x(-3) = (+15), ( + 15)^(-3) = (-5) ( + 5)x(-3)=(- (_5)x(+3) = (- 15), (-15) (+3)= (+5) (_3) = (+5) 15), (-15)^(+3) = (-5). In general. (+aJ)^(+5) = (+a), C-ab) C-ab) (_6) = (+a), (+a5)^(-6) = (-a). (+5) = (-5).<- The Law of Signs in Division is : Like signs give positive^ and unlike signs give negative quotients. EXERCISE VI Find the values of the following indicated operations: 1. ( + 3)(-2). 2. (-4)(-5). 3. (-8)(-3> 4. (-9)(-4). 5. ( + 6)(-4). 6. (-7)(+3). 7. (-5)(-6). 8. (-8)(+3). 9. (-9)(-5). 10. li. 12. 13. 14. 15. 16. 17. 18. (+6)( + 7). (_9)-H(+3). (_8)^(+4). (+10)^(+5). (_10)^( + 2). (_12)^(-4). (+12) (+15) (-16) (-12). (+3). (-8). 18 ELEMENTARY ALGEBRA [Ch. I, § 27 27. The sign + may be used, § 3, to denote addition^ and, § 19, to indicate positive numbers. In practice, how- ever, the sign + is omitted in indicating positive numbers. Thus, ( + 4), 4, are identical. Henceforth, in this book, positive numbers will be represented by the absence of sign. Thus, 4 means positive 4, and +4 means the addi- tion of positive 4. The sign — may be used, § 3, to denote subtraction, and, § 19, to indicate negative numbers. In conformity with general usage, negative numbers will be henceforth represented by numbers .preceded by the sign — . Thus, ( — 5) and —5 are identical. The sign — , denoting a negative number, is never omitted. EXERCISE VI] [ Simplify the following: 1. (4) + (3). 16. (-5) +.(-7). 2. (4) -(3). 17. (4) . (- 3). 3. (4) + (-3). 18. (-5). (2). 4. (4) -(-3). 19. (- 6) • (- 5). 5. (-4) + 3. 20. (-4). (4). 6. 4 + 3. 21. (-9). 4. 7. 4-3. 22. (_12).(-3). 8. 4 -(-3). 23. 8-5. 9. 4 + (-3). 24. -12-3. 10. 8 + (-2). 25. 24 - (- 3). 11. (-2) +8. 26. _ 36 ^ (- 6). 12. 7 - (- 5). 27. - 54 ^ 18. 13. 7 + (5). 28. _ 39 ^ (_ 13). 14. 7 + 5. 29. - 65 H- 5. 15. 7 + (-5). 30. 50 H- ( - 25). CHAPTER II ADDITION AND SUBTRACTION 28. The addition of two numbers, or quantities, whether positive or negative, has already been illustrated, and the rule' given, in § 23. The sum of three quantities is the sum of the first two quantities and the third quantity ; similarly, the sum of four quantities is the sum of the first three and the fourth quantity. Thu^2 a' + 3bc)\-c' + m = (2 a' + 3 6c) + c^ + m = [(2a2 + 3&c)+c2]+m. 29. Addition is subject to two laws (whose truth is as- sumed), the first of which is the Commutative Law, — the sum of two or more numbers is independent of the order in ■which the addition is performed. Thus, 4 + 5 = 5 + 4; or, in general, a + h^h + a. I| Addition is also subject to the Associative Law, — the sum of three or more numbers is independent of the way in^ which successive terms are grouped in the process of addi- tion. Thus, 4 + 5 + 2 = (4 + 5) +2 = 9 + 2 = 11, and 4 + 5 + 2 = 4 + (5 + 2) = 4 + 7 = 11, in general, a + h+ c^{a + h) + c = a + (6 + c). 39 20 ELEMENTARY ALGEBRA [Ch. II, § 30 The Associative Law gives a short method for combin- ing positive and negative terms. Thus, the sum of the positive terms of 22 — 11 + 12 — 5 + 6 — 17 is 40 ; the sum of the negative terms of 22 — 11 + 12 _ 5 + 6 - 17 is - 33. Hence 22 - 11 + 12 - 5 + 6 - 17 = (22 + 12 + 6) + ( - 11 - 5 - 17) - 40 - 33 = 7. EXERCISE VIII Find the sum of the following numbers : 1. 20-3 + 7-8. 6. 30-14-16 + 5. 2. 16-22 + 12-5. 7. 27-18-17 + 8. 3. 1_ 12 + 13 -7. 8. 6-22 + 33 + 12-6. 4. 8-9-10-11. 9. 24-8-13 + 7 + 5-16. 5. 20-14-13 + 27. 10. 6 + 8-21 + 17-8-5-13.. 11. _ 6 + 5- 19 + 13- 20 + 4+ 7. 12. 9-8 + 15 + 3-19-11-6. 13. 12-8-7-14 + 15-13 + 20 + 5. ADDITION OF LIKE TERMS 30. Like terms can be combined into a single term. Just as in arithmetic, the sum of 4 bushels and 3 bushels is indicated by 4 bu. + 3 bu. = 7 bu., so, in algebra, Sa?b + 5 a% = 8 a%. Hence, to add like terms, add their numerical coefficients^ and prefix this sum as the numerical factor of the literal part. Thus, 3 a^^ + 5 a% + 2 a^^ = 10 a% and — 2a— 3a — 5a= — 10 a, and 262_362 + 1062 = 96l Ch. II, § 30] ADDITION AND SUBTRACTION 21 EXERCISE IX Find the sum of the like terms in the following : 1. 2a + Sa — ia, 2. 6w+5m — 7m. 3. 2c-Sc-4:C. ^ 4. 7a2-4a2 + 2a2. 5. 8ab-2ab -Sab + 12 ab. 6. Q x^ — "ii x^ + b x^ — x^, 7. llxy — 1 xy + 4Lxy — ?> xy. 8. 12bc-?>bc + Qbc-2bc. 9. — x^y — 2 x^y — 3 x^y + 5 ^r^y. 10. 10 b^e-Sb^c-5b'^c + 4.b'^c. 11. 2 62^2 __ 3 J2^ _ 7 J2^2 _ 6 J2^2. - - ^V4^ ^ V * 12. 4iab — 5 ab + 7 ab — 11 ab — 12 ab. 13. 5 m7^ — 4 mn — 6 m?^^ — 7 m/z. — m7^ + 2 m^. 14. 6 a5 — 7 aJ — 2 aJ — a5 + 12 a5 + 22 ab. 15. a:2-lla;2-13ii;2+7^2_5^2_4^2^7^2_9^. 16. mn + 2 mn — 3 mn — 7 m7^ + 13 mn — 14 mn. 17. - a6 + 7 a6 - 13 aJ + 12 (26 - 7 aS - 15 aJ. 18. :i:2_3^2_4^2+7^2_9^2_ll^_4^2+5^^ 19. ?/2-ll2/2_13^2 + 5^2_4^2^3^2_223/2. 20. a2-3a2 + 4a2-6a2-7a2-32a2 + 50a2. 21. — aS + 4 a6 — 7 ah + b ab — 1'^ ab + 11 ab — 56 ab. 22. a -17 a + 33 a -44 a + 109 a- 64 a + 32 a. 23. :i;2y - 3 :r2t/ + 5 rr2^ + 22 :?:22/ - 17 x'^y + 37 x^y. 24. - 17 62 - 33 62 + 105 62 + 62 62 - 109 62 - 56 62. 25. 6 a6 - 17 a6 + 33 ab - 512 ab + 203 ab + 1002 ab. 22 ELEMENTARY ALGEBRA [Ch. II, § 31 ADDITION OF POLYNOMIALS 31. Let A=h+c+d, and let U = m — n—p. The addition of these two polynomials is indicated thus: A + U= (h + c + d) + (jii — n—p}» (1) The parenthesis may be dropped, and the equivalent expression may be written: A + U=b + c + d + m — n—p, (2) Expression (1) indicates the sum of the numerical values of the poly- nomials b+c + d and m — n—p; the numerical value of expression (2) is independent of the order of the terms, and may be considered as the sum of the numerical values of the first three, and the last three terms, which is exactly the result of expression (1). Hence expressions (1) and (2) are equivalent. Whence is the following rule for the addition of polynomials : Write the polynomials in order ^ retaining the sign of each term. If the polynomials contain like terms, these terms should be united. 1. Add ^2 + 2 a5 + 52 and a^-2ah + h^. The work will be simplified by arranging like terms under like terms before combining. 2 a' +252 If the sum^ of more than two polynomials is required, the process is similar. 2. Add a2 - 3 ah, 6 a5 - b^ 11 a^ + 3 ab- 12 b^. a' — 3ab 6ab-^ 52 11 a^ -{-Sab -12 b^ 12a' + 6ab~13b' Ch. II, §§ 32, 33] ADDITION AND SUBTRACTION 23 32.* The process of finding the sum of several poly- nomials containing like terms, may be still further abridged by the method of Detached Coefficients; that is, by omitting the literal parts of several like terms. Thus, in finding the sum oi2m + 3n — 6p, Urn — An -\- 2 p, and —7m + n—p, omit all literal factors except in the .first line and arrange the terms thus : 11 _4 +2 -7 +1 -1 6m —5p The advantage of this method is simply in the labor saved by omitting the literal factors. CHECKS FOR OPERATIONS 33. It is often useful to test, or check, the results ob- tained in the processes of addition, subtraction, multipli- cation, and division with the results obtained in the same operations obtained from the numerical values. 1. (3a^-2x^ + 5x + l) + (3x'-2x + 3)=.3x^ + x^+3x + 4:. In each of the above expressions take aj = 1, then (3-2 + 5 + 1) + (3- 2 +3) = 3 + 1 + 3 + 4, or, 7+4 =11, 11 =11. 2. (5 a^-^ 6'2;2- 3 a-b + m) + (2 6^ + 3a-6 - 4 m) = 5 a^ -^4.l^-3m. In each of the above expressions take a = 6 = m = 1. (5 ...6 -3+1) + (2 + 3 -4) = 5- 4^3, or, * -3+ l = -2, - 2 = - 2. 24 ELEMENTARY ALGEBRA [Ch. L:, § 33 EXERCISE X Add the following polynomials : 1. 2a + 3h + 2c, a+h — c. 2. 8h+ 2c-d, 2b — 5c — d. Z, m-\-n—l p^ —m — n + 1 p, 4. ah + hc + alP'c^ 2 ah — 8 he — 4: ah^o. 5. 2ah + a^ + 3h^ 2ab + Sa^-4: h\ 6. 4:xy — x^ + 4: y\ 6 x^ — 5 x^ — 1 y^. 7. 8a^-bah-\-lG\ 2^2-6^5 + 4^2. 8. x^ — xy ■\- y^^ 8x^ — 6 xy — 4: y^. 9. Im— 2n-{-p + 6q^ 6 m + 5n — 6 p + 2 q. 10. 4:x — 2y + Sz—8, Qx-dy — Sz-^G. 11. a2 + 52, a2-3a6 + J2, 2ah-2h^. 12. m^ + mn +p^ 3 m2 — 2 mn — jt?2, 6 m2 — 3 ^n + 2 jt?2^ 13. 5 m — 10 n + /ip, m — 7 ^ + njt?, 6 m + 12 n — 4 7^p. 14. x^ — xy + 2/2, a;2 + 2 ^?/ + 2/2, — a;2 — 4 rry — 4 ?/2. 15. 12a2-lla6 + 6?)2, -5^2 + 2^6-3 J2, 6a2 + 8a5 + 4J2.: 16. 6 m2 — 3 07171 + 5 7i2, 5 nfi + 8 mn — 4 7^2^ — 10 ^2 + 5 mn + 12 n^, 17. a6 — ac? + acZ, ac — ah + ad^ ad— ac+ ah. 18. rrfi — rfi + jt?2, n^ — rrfi— p^^ p'^ — nfl— r?. 19. a^-ah + h\ h'^ - a^ -^ ah, K- -^ ah - d^. 20. 2a-3c2 + 4(^, J2_3^2_|.2cZ, h'^-a-2a\ 21. 5 a;2 — 11 ^z:?/ + 12 2/^ x^y'^ —?fxy-\- 2/2, a;2 — 2/2. 22. 12 62__i0J^ + 15c;, a2-10 62 + ll5^, d-Uh^-^lla'^ 23. 22 2:2-3% + 42/2, 15 52/-4 2/2-2a;2, 22hy-~y^ + 9x^ 24. 6a26-7a2(?-5e2^+8 52a, 11 c^+S (72^ + 6 ti2^-9 ^2^ 25. 9x32/-a;4-12ii:2^2_i4^^3_,.y^ a^^- 6 ii^^j/ + 10 ^^ ^2yK Ch. II, §§ 34, 35] ADDITION AND SUBTRACTION 25 SUBTRACTION 34. The subtraction of two quantities has already been defined, and the rule given in § 24. SUBTRACTION OF LIKE TERMS 35. Just as in arithmetic the process of subtracting 3 barrels from 4 barrels is indicated by 4 bbls. — 3 bbls. = 1 bbl., so, in algebra, the subtraction of 3 a from 4 a is indi- cated i a — S a = a. But, § 19, 4 a can be subtracted from 3 a, and is indicated 8 a — 4: a= — a; that is, — a must evi- dently be added to 4 a to make 3 a. Similarly, 2a— ( — 5 a) = 7 a i —5a— (6 a)= —11 a. In § 22 it was shown 'that adding a negative number is the same as subtracting that positive number whose abso- lute value is identical. Algebraic subtractions are usually changed into algebraic additions. These operations are equivalent in results, and the change of an algebraic sub- traction of a negative number into an algebraic addition is to be interpreted as illustrated in § 24. EXERCISE XI Subtract the first from the second, and also the second ^ from the first quantity of the following : 1. 2 b, h, 7. 3 m, 4:m. 2. -b,2b. ' 8. 7 c, 4 c. 3. —a, —2a, 9. x-y, —3x^y. 4. — a, 2 a. 10. 7 a%, — 8 a^b, 5. a, 2 a. 11. — ahj, — 3 a^y. 6. a, -2 a. 12. 5 0(?y\ - 13 af/. 26 y '.EMENTARY ALGEBRA [Ch. II, § 36 &. , CTION OF POLYNOMIALS 36. Let A = b + c — d + e, and F= m — n +p — q. The subtraction of the second from the first polynomial is indicated thus: A - F= (b + c - d + e) - (m - n + p - q). (1) Or, A = (b + c-d-^ e) F=(m — n-\-p — q^ A — F= b + c — d + e — m + n —p + y. (2) The quantity A — F must evidently be added to F to produce A ; and the quantity b-\-c— d-\-e — m-\-n —p + q must evidently be added to m^n-\-p — q to make b-\- c — d + e. Expressions (1) and (2) are identical; hence, to subtract a polynomial from a second polynomial : Write the first polynomial after the second^ changing all the signs of the terms of the first polynomial ; combine like terms. 1. Subtract 2a^-5ab-3b^ from a'-2ab + b\ (a^ ^2 ab + b') - (2 a' - 5 ab-Sb") ^d" -^2 ab + b^ --2 a^ + 5 ab + 3b% Or, a2-2a&+ b^ 2a^-5ab-Sb^ The number — a^ must evidently be added to 2 a^ to make a^ ; 3 a6 to —5ab to make — 2 a& ; 4 6^ to —Sb^ to make &-. The work can be still further abridged by the method of Detached CoefiBicients. a2-2a&+ 52 2 -5 -3 Ch. II, § 36] ADDITION AND SUBTRAC ^JION 21 2, Subtract m^ — 2 m + 1 from 3 m^ -^ j^, 37)1^ -7 m + 1 ^v^ ' 1 -2 +1 2 m^ — 5 m (3 m^- 7 m + 1) - (m^- 2 m + 1) = 2 m^- 5 m. In each of the above expressions take m = 1 ; then (3^7 + l)-(l-2 + l)=2-5, or (-3)-(0) =-3, -3 = ~3. The results check, and the subtraction is therefore correct. EXERCISE XII Subtract the first from the second, and also the second from the first expression of the following : 1. x + 5^ x + S. 11. 4:x^ i/ + 5x. 2. rz:— 5, x — 3. 12. a — 5, 5. 3. x + 5, x-S. 13. 7, 2 a + 5. ^. X — 5, x + 8. 14. — a;, — y — 3. ^. 5 + x, 3 + x. 15. a — b, b + a. 6. 5 — .T, 3 — x. 16. 3 — :^, n + 1. 7. b + x, S--X. 17. a — 8, b — 8. 8. 5 — x, 3 + x. IB. 4: — n, n + 4. 9. a, a + 1. 19. _^ 4-8 6, —b — le. 10. a, a — bo 20. a + b — c, 2a + b — c. 21. 5^ + 25 + 6, 7a-b-8. 22. 4^2-752 + 7, 4a2 + 752__i^ 23. Gm — Bn—p, —m — Sn—p — q. 24. ^3k + m — 5n + 4p, Qk--m + 6n + 7p 28 ELEMENTAIlf ALGEBRA [Ch. II, §§ 37-39 AGGREGATIONS 37. An aggregation symbol preceded by the plus sign may he neglected, because the expression within the aggregation symbol is to be added to the preceding number, which number is sometimes 0. a+\h + Si-=a+'b + C] + (a — 6 + c) = a— Z) + c. An aggregation symbol preceded by the minus sign can be removed by changing tJie sign of every term contained within it; because the indicated process of subtraction is the addition of the several terms changed in sign but having the same absolute value by § 24, Thus, 7a-(a-2&--3c)=7a-a + 26 + 3c = 6a + 26+3c. 38. By § 37, the terms of a polynomial can be enclosed by a symbol of aggregation ivhich is preceded by the plus sign without change of sign ; and can be enclosed by a symbol of aggregation preceded by the minus sign if the sign of every term be changed. - 3 i»2 H- 4 o;^ - 2/2 = - (3 a;2 ^ 4 a;?/ + 2/^. 39. An aggregation enveloping several aggregations can be removed by the foregoing principles. Either the inner or the outer symbol may be removed first. Thus, simplify a — [a — J2 a — (3 a — 6) }]. a-[a-{2a~(3a-6)n=^^~[«-12a-3a+&.n, = (X— [a — 2a + 3a— &], = a— a + 2a— 3a + 6| Ch. it, § 30] ADDITION AND SUBTRACTION 29 or, removing first the outer symbol, a- la- [2 a - (3 a -b)l^ = a - a+ \2 a- (S a-b)], = a — a + 2 a — (3 a — &), = a — a + 2a--3a + 6j = - a + 6. EXERCISE XIII Simplify the following expressions : 2. (^a-b) + c-l(d + e)-f-(g-h}]. 3. a-lb-{c-d)]-le + {f-g-)-h}. 4. a-[6-(c + c?) + e]-(/-^) + 7t. 5. [(a - J) + (c - c^)] - [(e +/) + (g- 70]. 6. [(a + 5)-(. + (?)] + [(e-/)-(^ + A)]. 7. [(a_5)-(c-(?)]-[(.-/)-(^-A)]. 8. [(a + J) + (c-cZ)] + [(e-/)-(^-A)]. 9. (3a^ + 52/)-[(7a;-22/)-(8aj-4y)] + (2:-^). 10. (7 m - 4) + 3^- [(8^ + 3j9 - 2) + 5 m - (8^ - js)] 12. a-[2a-(^3a-7a|-3c)]. 13. a-[-(-{-3a-(2a-J)|)]. 14. m — [— 71— [ — 3« — (4m — 6w)|]. 15. a - [ ^ 6 - (c + c * d^. (ahc) • d. The absolute value of the product of three or more alge- hraic quantities is the product of their absolute values^ and is positive when it contains an even number of negative factors^ and negative when it contains an odd number of negative factors. Thus, the product of — a, h, — c, and d is abed ; and the product of — a, — h, — c, and d is — ahcd. Since 0, § 23, =^a-a, a(a- a)= a^- a2= ; a • = 0, 42. By definition, § 7, a^ = aaa,, and a^ = aa. Therefore, a^ x a^= aaa y^aa^ aaaaa = a^ = a"^^. 30 Oh. Ill, §§43,44] MULTIPLICATION AND DIVISION 31 ^4 X a^ = aaaa x aaaaa = aaaaaaaaa = a^ = a^"^^. In the above examples the exponents are positive whole numbers or integers. Restricting, for the present, ex- ponents to positive integers, the product of any two powers of the same letter may be found thus: a"* = aaa taken to m factors, a"" — aa taken to 7i factors, therefore, a^ * a^= (a taken to m factors) x (a taken to n factors), = a taken to (m + n) factors, = a"*+^. In the same way, a"" - a^ - a* =• a'^+^+'' The principle just shown is called the Index Law, — the exponent of the product of ttvo powers of the same letter is the sum of the exponents of the factors. 43. The process of multiplication is subject to three - fundamental laws (whose truth is assumed), of which the first is the Commutative Law, — the product of two or more quantities is independent of the order of the factors. Thus, 2 • 3 = 3 • 2 ; and, in general, a •b :=b » a. 44. Multiplication is also subject to the Associative Law, — the product of three or more quantities is independent of the order in which the factors are grouped in finding the partial products. Thus, by § 41, 5 . 4 . 3 = (5 . 4) . 3 = 20 . 3 = 60, and, by § 41, 5 • 4 • 3 = 5 (4 • 3) = 5 • 12 = GO, and, in general, a * h - c = (a - h) ' c = a (h » c). 32 ELEMENTARY ALGEBRA [Cii. Ill, § 45 MULTIPLICATION OF MONOMIALS 45. 1. Find the product of 2 ahx^y and 5 a%x^. By the associative law, (2abx^y) (5a^bx^) =2 - a - b - x^ - y - 5 - a^ - b - a^, by the commutative law, = 2 * 5 ' a ' a^ - b * b - xr - x^ • y, hj the associative law, = 10 aVx^y. 2. Find the product of —Sx^^ — 5 x^y^^ and 4 xi/z. By the associative law, (_3a)2) (_5ajy) (4.xyz)=:{-3)'x\-~ 5) -a^ -y'^ (A) - x - y - z, by the commutative law, = (—3) (—5) (4-) > x^ - x^ - x » y^ * y - z, by the associative law and law of signs, = 60 a^yh. Hence, the product of several monomials is found 5y annexing to the product of the numerical factors each literal factor^ giving to it an exponent luhich is the sum of the exponents of this factor in the monomials, EXERCISE XIV Perform the multiplications indicated : lo 3 a;^/ • — X7/^. 6. 2a^ ' 4ax ' — 11 a'^x^. 2. -— a • — a^ • — h^, 7. — 772%"^ • — m% • — 7 mn^. 3. ah'Sac.-5hc, 8. ia^ - Gxf- -IBaWrY- 4. 2 mn • — 3 m*^ • — 4 n^, **** 9. a^ • — 5 b^x • — 3 a^x"^, 5. 2c ' 4:xy -7 ab. 10. 11 ac - —lilc^ - —13 a^c^. 11. ah - — ac ' — be - — cd ' — abed, 12. 2 a7 . - 3 a^ . - 3 a^ . - 3 a2 . - 2 ^. Ch. Ill, §§ 46-48] MULTIPLICATION AND DIVISION 33 MULTIPLICATION OF A POLYNOMIAL BY A MONOMIAL 46. An entire expression is an expression no term of which contains a literal quantity in its denominator. Thus, |a^ + 2 ab + b^ is entire. A fractional expression is an expression in which at least one term has a literal quantity in the denominator. 3 Thus, - — ;: + 2 ab + b^ is fractional. 47. The degree of a monomial is found by taking the sum of the exponents of the literal factors. Thus, 3 a^b^G is of the sixth degree ; and 13 x is of the first degree. The degree of a polynomial is found by taking the sum of the exponents in that term in which the sum is greatest. Thus, a^ — oaV + d^e'^ is of the eighth degree because the sum of the exponents of ab^ is eight. A homogeneous expression is one in which the degree of the several terms is identical. Thus, a* + 4 a"& + 6 a^b'^ + 4 aZ>^ + b^ is a homogeneous expres- sion of the fourth degree. 48. The definition of the product of two numbers, § 25, applies to two expressions in the form 3(2 + 4). By definition, § 25, 3(2+4)=3 + 3 + etc. to (2 + 4) terms, by associative law, § 29, = (3 + 3 + etc. to 2 terms) + (3 + 3+ etc. to 4 terms), by definition, § 25, =3.2 + 3.4, similarly, a(b + c) = ab -]- ac. Note : The above law is assumed to hold for positive fractions and negative numbers. 34 ELEMENTARY ALGEBRA [Ch. Ill, § 48 By the commutative law, a (b -\-c) = (b + c) a, by the commutative law, ab-\-ac = ba + ca, therefore, a (b + c) = (b -\- c) a =ab + ac = ba -\- ca. The statement of the foregoing principle is the third law of multiplication, the Distributive Law, — the product of a (^entire') polynomial hy a monomial is found by multiplying each term of the polynomial by the monomial and adding the products thus obtained. 1. Find the product of 2x^ — bxy — 2 y"^ by 3 x. S X (2 x" -5 xy -2 y'') = (2 x'' -5 xy -2 y^) . ^ X, = 6 a;^ —15 x^y — 6 xy^. The work may also be arranged thus : f^ V-- 5xy - -2y' \Sx 1 -15x'y- -6xy^ EXERCISE XV Perform the indicated multiplications : 1. (?(2a + J). 6. xyQx + y). 2. p(3m — 4^). 7. a(^a—2b + Sc}. 3. SxQx — ly^. 8. kmn{ik — 8m — 7n'). 4. 5a(a-6). 9. 6 (2 a + 5b -9 c). 5. Sn^ip-q). 10. (-l)(-5a + 6 6-(?). 11. (a-lb + c^i^n). 12. (lla-8b-5c)(-Sy). 13. C-^ab-Sbc + icdX-^^d)' Ch. Ill, §49] MULTIPLICATION AND DIVISION 35 MULTIPLICATION OF A POLYNOMIAL BY A POLYNOMIAL 49. The product of two polynomials is expressed thus, (a + 6)(6' + cZ). By definition, § 25, (a + 6) (c + d) = (c + d) + (c + d) + etc. to (a + h) terms, by associative law, § 29, = [(c + d) -{- (c + d) + etc. to a terms] + [(c + d) + (c + d) + etc. to b terms], by definition, § 25, z= (c -{- d) a + (c + d)b, by distributive and by commutative laws, = ac + ad + bc + bd. From the foregoing principle is derived the following Rule for the Product of Any Polynomials: Multiply each term of the multiplicaiid hy each term of the multiplier and add the successive products. 1. Find the product of 2x^ — ?»xy + 4: y'^ and x — y. Arrange the work thus : X - y 2 0^ — 3 xhj 4- 4 xy^ — 2 x^y + 3 xy^ — ^y^ 2 X? — ^ x^y + 7 xy^ — 4 ?/^ ^ The product Qii2x^ — Zxy -\-4 y'^ and x is written in the third line, and the product of 2 aj^ — 3 a?^/ + 4 y^ and — 2/ in the fourth line. Like terms are arranged in columns so that they may be united. The product of three or more polynomials is found by taking the product of the first two by the third, and so on. 36 ELEMENTARY ALGEBRA [Ch. lit, §§ 50, 51 50. A polynomial is said to be arranged with reference to a letter when the powers of that letter constantly in- crease or decrease. Any letter can be selected as the letter of order. If the exponents of the letter increase, the polynomial is said to be arranged in ascending order. Thus, x^-\-3 x]f- — 3 a?y — y% arranged with reference to x, in descending order, is x^ — 3 iJi^y -{- 3 xy^ — y^ ; and the same expression, arranged with reference to y, in descending order, is —2/^ + 3 xy^ — 3 xhj + x\ 51.* The application of the method of Detached Coeffi- cients will be facilitated if all of the terms of the expres- sions to be multiplied are arranged with reference to a single letter in the same order, the coefficients of missing i powers of the letter of arrangement being represented hy\ zero. Multiply a3 + a% + aJ2 + l^ by a^ - h\ 1+1+1+1 1 + 0-1 1+1+1+1 1 + 1 + + - 1 - 1 = a^ + a^6 - a&4 - 5^ The result obtained may be checked by substituting a=h=i., (14.1 + 1 + 1)(1_1) = 1 + 1_1_1; 4.0 = 0. Detached coefficients are most advantageously employedi in finding the products of homogenous expressions. The above example also illustrates the following prin- ciple : The product of two homogeneous expressio?is is a homo^ geneous expression whose degree is the sum of the degrees of the multiplicand and multiplier. \ Vn, III, § 51] MULTIPLICATION AND DIVISION 37 EXERCISE XVI Perform the indicated multiplications : 1. a + 2 by a + 3. 8. -x + 2 hj x—7. 2. a — 3 by a — 4. 9. a — 4t hy 2a + 1. 3. m + 4 by m — 3. 10. 2 a + 5 by a — 4. 4. n — 2 by n + 5. 11. 2 a— 7 by 3 a + 6. 5. x-2 hj —x + S. 12. 3a + 4 by —3a + 5. 6. — x—S hj x + 4:. 13. x^ + xi/ + ^^ hj X — y. 7. —^ + 3 by —a; + 6. 14. x^ — xy + y*^ by x + y, 15. 2a^ + ah + lfi by 2a -5. le. ?^x'^-'Qxy + ^ hy -- x + ^y. 17. a2 + a5 + 52 by a2 - a5 + 5^. 18. x^ + x'^y''- + y^ by :r4-:r2/ + /. 19. a^ + 3a26 + 3aJ2 + J3 by a + h + c. 20. a^ - 4 a^i + 6 a2J2 - 4 ^53 _j. 54 ^y ^ __ 5^ 21. 3a2-.4a^ + 5&2 by 2a2-3a5 + 2J2. 22. b x^ — y^ -\-H x^y — 4 2:2/2 by 2 :i;2 + 3 2/2 — :ry. 23. 10a252__i3^4_6^3^_f_g^53+354 by x^-\-f-xy. 24. a*-7a26-8a52-lla36 by a2:?;-. 5 a2:2 4- 7 a^ - 2;3 25. m*^ — 11 aW" — 4 :r2/^ — 3 n^ by ^^ — 7 — 2 alfi + m^. 26. a2 — a6 + a; — ?/ by a2 — a6 — ^ + y. 27. :r3_2:i:2 4-:?;-3 by 2;3_2^2 + ^_3, 28. 2^-3rc2-:r + 2 by 2rz;3-3:i:2 + ^_2. 29. 5a3-4a2J + 2a62--63 by 2 aJ2- J3_ 5 ^3 + 4 ^25. 30. 7? + y'^ '\- z^ by a;2 + 2/2 -f ;22 _ ^^ __ ^^ _ y^^ 38 ELEMENTARY ALGEBRA [Ch. Ill, §§ 52-54 DIVISION 52. In § 26 there was given a definition of division of two algebraic numbers, the rule for finding the quotient, and a statement of the law of signs. If the indicated divisor be zero, since the product of a finite number and is 0, it follows that the quotient can- not be found; that is, cannot be used as a divisor. If the dividend be zero, since a • = 0, - may be defined as 0. 53. Since, by §42, dr . a""^ aj^-^'^. by § 26, a^ ' nnrl "hv S 9.(\ ^m-\-n This principle is called the Index Law, — the exponent of the quotient of two powers of the same letter is the exponent of the dividend minus the exponent of the divisor. Note, m and n are, as in § 42, positive integers only ; and m and n are restricted to such values that m is not less than n, A full dis- cussion will be found in Chapter XVII. DIVISION OF MONOMIALS 54. From §§ 26, 45, and 53, the quotient of two mo- nomials is found hy annexing to the quotient of the numerical factors each literal factor whose exponent is its exponent in the dividend minus its exponent in the divisor. 1. Divide 8a;3 by 2x\ 2ar Ch. Ill, § 55] 2. Divide 12 h^c^m^ by 12 6Vm3 MULTIPLICATION AND DIVISION = — 4 b^~V~'^m^~^ = — 4 b'^cm. 39 — 3 bciri^ 55. Since any quantity divided by itself produces 1, it is evident that a"* -j- a"* = 1 ; and, by the Index Law, it is also evident that —= a'"''' = a^. The quotients just de- rived must be equal, because the dividends and divisors are identical. Hence, an^/ finite quantity/ with the exponent zero may he defined as equal to 1; or, a^ = 1. Divide - 30 a'^b^c by - 6 a^he. - 30 a'b'c ' 6 a'^bc : 5 a^-462-1^1-1^ 5 a%c' = 5 • 1 • & • 1 = 5 6. EXERCISE XVII Perform the indicated divisions : 1, 2^^ "" Z^<^ J 8. 2. - 3^ 10. 4. ■-^^i "• 6. 7. a" 12 a^b -4 6* - 25 a2/)3 5^262 — o o;^?/ "7 20 a%^ ' 39 a3x3 13 a3:r3 - 28 «*x2 - 7 d'x - 64 a562 - 16 a*62 - 30 a*6V. 13. 14. — 6 a^6y — 34 'jfixf'z^ Vlx^yz^ 91 x^y^z"^ 44mV> — 4 TO%2 15. 16. 17. 18. 19. 20. 21. -33ffl769c"i 11 aSJ^cS 60 gS^^cis - 15 a36V2" 84aJ9512g23 - 7 ai253(ji6' 63 'jfiy'^gy^ 7 a^y ai2 ■ - 78 ofiy'z^"' - 13 a;2?/42>*' 42 aWc^^ - 7 a«^*ci5' -b2a^3?y'^z^ - 13 a^xhj^z^ 40 ELEMENTARY ALGEBRA [Ch. Ill, § 56 DIVISION OF A POLYNOMIAL BY A MONOMIAL 56. It has been shown, § 48, that, by the distributive law, ab -{- ac= a(b + c^. By the definition in § 48, if the prod- uct ah + ac^ and the factor a are given, the quotient will be b + c, ^Whence is derived the following Rule for the Division of a Polynomial by a Monomial : Divide each term of the poly- nomial by the monomial and add the quotients thus derived. Divide a^ - 2 a^J + 8 a^J^ by a^, -^ = 1 — 2 a& + 8 arb^. EXERCISE XVIII Perform the indicated divisions : 2. . 5. - . xy — Zx^ ^ 5 m2 + 10 m3 ^6 a^^ 4. g aH^ + 16 ^^ 3. . 6. — — —'. — bm „ . , 2 a;* y - 21 x^y - 91 xf + 56 y^ -ly Q 16 a;8y8 - 48 x'^y^'^ + 112 a;^ 9. 16 xY gilici -13 aWc^ - 21 gS^Sgia ^Q 51a:yi-102a^V^ Cii. Ill, § 57] MULTIPLICATION AND DIVISION 41 DIVISION OF A POLYNOMIAL BY A POLYNOMIAL 57. Since it is always true in exact division that the product of the divisor and quotient gives the dividend, and since (oi? — xy + ^") (x — y^ = x^ —2 x^y + 2 xy'^ — y^, it is possible to take either x^ ~xy + y'^^ ov x — y^ as the divisor, and the other expression as the quotient; while ^3 _ 2 x^y + 2 xy'^ — y^ is the dividend. Take x^— xy + y'^ as the divisor. Then (ofi — 2 x^y + 2 xy*^ — y^^ -r- (aP' — xy + y'^^ = x — y. The quotient x — y\^ derived from the dividend and divisor by the following process : Notice first that the dividend and divisor are both arranged in descending powers of x. The first term of the dividend is evidently the product of the first term of the divisor and the first term of the quotient, the first terms in each case being evidently the term of highest degree because of the order ox arrangement. Therefore, a?^, the first term of the dividend, divided by a^, the first term of the divisor, gives x, the first term of the quotient. Now the first term of the quotient is a multiplier of each term of the divisor, as will be seen by referring to the case, (pi? — xy + y^) (x — y) = x^ — 2 x^y + 2 xy'^ — y^. Therefore the partial products of all the terms of the divisor by the first term of the quotient form a part, at least, of the dividend. That is, (x^ — xy + y^)x = x^ — x^y + xy^ must be subtracted from the dividend since the dividend is the sum of the partial products found by multiplying all the terms of the divisor by all the terms of the quotient. The remainder, so derived, is x^ — 2x^y + 2xy^ — y^^ (x^ — x^y + xy^ = ~-x^y + xy^ — jf 42 ELEMENTARY ALGEBRA [Cpi. Ill, § 57 This remainder may be considered as a new dividend and is the product of the divisor and the remaining term (or terms) of the quotient. As before, the first term of the remainder (new dividend) is the product of the first term of the divisor and the first term of the quotient. Hence — x^y divided by oc^ gives — y, the second term of the quotient. Since the second term of the quotient is a multiplier of each term of the divisor, the product of the whole of the divisor and the second term of the quotient is sought. {x^ — xy + f'){-y) = - x^y + xy^ — y^. Subtracting this product from the remainder, the new remainder will be ; that is, the division is exact. The above explanation may be expressed thus: a^ ^2 Qi?y + 2 xy'^ — y^ = (q(^ — x^y + xy^) + (— x^y + xy'^ — y^), a? — 2 xhj + 2 xy'^ — y'^ _a? — xhj + xy^ — x^y + xy'^ — y^ x^ — xy + y^ x^ — xy + y^ x^ — xy + y^ ' -x-y. It will be noticed that the dividend is separated into such terms that each may be exactly divided by the divisor. The following arrangement is, therefore, more convenient : ividend =^x^ — 2x^y -^2 xy^ — if 0? — xy + y^ — Divisor x^ — x^y + xy^ x — y = Quotient ^ x^y+ xy^ — / If the quotient contains more than two terms, the process of division is the same. Checking the division by substituting x=zy=^ly (qi?— 2 xhj + 2 xy^ — ]f)-^{^ — xy + y^) = 00'—y, (1^2 + 2-1) -(1-1 + 1) = 1-1, 0-1=0. Ch. Ill, § 58] MULTIPLICATION AND DIVISION 43 58. From the foregoing principle is derived the fol- lowing Rule for the Division of a Polynomial by a Poly- nomial : 1. Arrange both polynomials in the descending or ascend- ing order of some common letter, 2. Multiply each term of the divisor by the quotient ob- tamed by dividing the first term of the dividend by the first term. ^/ the divisor, " ' 3. Subtract the partial products so derived from the dividend, 4. With the remainder still arranged in the same order as before^ coritinue the process until there is no remainder^ or until the degree of the first term of the divisor is higher than that of the first term of the remainder. 1. Divide m^ — 3 m^n + 3 mn^ — n^ by m- n. m^ — 3 m^n + 3 mn^ ■ -ii' m — n w? — ni?n w? — 2 mn + n^ — 2 m^n 4- 3 mv? — 2 Kri?n + 2 mii? -n' -n' 2. Divide 2a^-5a% + l a%'^^bab^+2¥ by a^-ab f J2. 2a^-^ a% + 7 a'b' -5ab^ + 2b' 2 g^ - 2 a^6 + 2 a^b^ - 3 a^6 + 5 a'b^ -^5ab^ + 2b' -3a^b + Sa'b'-3ab^ 2d'b'-2ab^-^2b^ 2a'b^-2ab^4-2b^ ab-^ b^ 2a^-3ab + 2l/ 3, Divide 15 x^ -^ 7 x + 7 x^ + 15 x^ + 4: hj 1 + S x^ -\' 2x. Arrange the dividend and divisor in the same order. 44 ELEMENTARY ALGEBRA By the method of detached coefficients : [Ch. Ill, § 58 3 + 2 + 1 5-1 + 4 15+ 7 + 15 + 7 + 4 154-104, 5 - 34,10 + 7+4 •>- 3- 2-1 12 + 8 + 4 12 + 8 + 4 The quotient 5 — 1+4 must have x^ in the first term, and an integer only in the last term ; and is, 5 aj^ — a? + 4. by x^~-?>x^-\-2x + l. l__l_2 + 5-4 + l + l I4-O-3 + 2 + I -1+1+3- _l_0 + 3- -5+1+1 -2-1 1 + 0- 1 + 0- -3+2+1 -3+2+1 1 4.0-3+2 + 1 1 -1+1 x^ — x + 1 The divisor x'^ — 3o(?-\-2x-\-l contains no term in a?\ since^ times X* equals 0, to make the method available, the a^ appears with the coefficient 0. In detaching coefficients, the coefficient of any missing powei of the letter of arrangement is always written as 0. EXERCISE XIX Perform the indicated divisions : 1. a^ + 5a + Q hj a + 2. 2. x^ — 2x — ?>hyx + l. 3. x^ - 16 by x + 4:. 4. 2;2 - 14 2^ + 49 by x— 7. Cii. Ill, § 58] MULTIPLICATION AND DIVK ION 45 5. 4 a2 _ 12 a + 9 by 2 a - 3. 6. 36a^-60a; + 25 by 6a;-5. 7. 4a2 + 12aJ + 952 by 2« + 3J. 8. 9 a;^ — 6 xmn + rw^n^ by 3 a; — raw. 9. 0^-21 by a:- 3. 10. 64 + a^ by 4 + a. 11. 6 ma; — 8 am — 9a;+12a by Sx — 4a. 12. 21 ax — B5 ay + S bx — 5 b^ by 3 a; — 5 y. 13. 20 ac - 15 a(^ - 12 6c + 9 5(^ by 5 a - 3 6. 14. a^ + b^ + c^ + 2ab + 2ao + 2be hy a + b + e. 15. p^ + q^ + r^ + 2pq — 2pr— 2 gr hy p+q — r. 16. ^2_|_ 2 jjg + ^2 _ ^ by ^ + 5' + r. 17. 12a2_452_5^ + 2a5 + 4ac-9Jc by 2 « - 5 - (r. 18. 1 - 18 a2 + 81 a* by 1 - 6 a + 9 a\ 19. - 2 a;li..'La:L4-S2.a;2 + 145 a; + 72 by 9 + 8 a; - a;2, 2a) 216 a^ + 125 by 36 a^ _ 30 « + 25. 21. 1 - S2p^ by 1 + 2jt» + 4 j92 + Sp^ + 16 j9*. 22. 128 a*^-3 - 160 a562 + 2 cjSJ + 15 a^ by 3 a2 _ 8 a5. 23. 5 «c + 75c + 3 a2 _ 7 a5 - 6 52 - 2 c2 by a - 3 6 + 2 c. 24. 44^-30-16^/2 + 3:r + 9a^ by 4?/ + 3x-5. 25. 48 a;2- 192 a;?/ + 192/ -27 22 by 4a;-8j/ + 33. 26. 4 a;* - 197 a;2^2 + 49 ^4 by 2 2^2 + 15 a;y + 7 3/2, 27. 80 6c + 18 a - 64 62 _ 48 6 -t- 9 a2 + 30 c - 25 c2 by 3 a - 8 6 + 5 c. 28. a'^-Qac + 9. 15. 25-6(a;-6) = 20-(2a;-13). 16. 2(9-a;) + 5(2a; + 3) = 81. 17. 6(20+3a;-l)-5(8a;-7) + 19 = 2(a;-72). 18. 3 • 5 (x + 6) + 5 • 7 (1 + 2 a;) - 7 • 9 (a; - 8) = 827. 19. (2a;-l)(3a; + l)=(6a;-12)(a;+3). 20. (5 X + 7)(6 a; - 3) = (10 a; + 2)(3 X + 2) - 9. 21. 7(a;-l)-3(l-a;) = -4(6 + a;). .22. 3(2a; + 7)+4(6 + a;)=-4(a;-3) + 3(2a; + l)-10. 23. 6(2 x- 4) - 3(2 a;- 1) = 7(3 a; + 2) - 8(4 a;- 2). Ch.IV, §69] EQUATIONS AND PROBLEMS 53 69. If the equation contains fractions, it can be simpli- fied by application of Axiom 3. 1. Solve for x^ the equation ^-4 = 10-a;. (1) b Applying Ax. 3 in (1), a; - 24 = 60 - 6 a?, (2) transposing and uniting in (2), 1 x = 84, (3) applying Ax. 4 in (3), x = 12. (4) 12 Verification : -- — 4 = 10 — 12, 6 2 - 4 = 10 - 12, _2 = -2. 2. Solve for x^ the equation 8 4: — X X 9 11 Simplifying in (1), 'lQx + 2 x + 2' 33 9 \(4:-x) ^x 16x + 2 x-{-2 99 3 33 9/ (1) (2) applying Ax. 3 in (2), • S(A-x) = 33x-3(16x + 2)+il(x + 2), ^(3) simplifying in (3), 32-8a; = 33a;-48a;-6 + lla; + 22^ (4) . .f . V transposing and uniting in (4), -4a?=-16, \ (6) applying Ax. 4 in (3), 07 = 4. (ft) ^r 84-44 64 + 2, 6 VEKincATiON : 9 • -JT = 3 - -3f- + 9^ = |-2 + |. 54 ELEMENTARY ALGEBRA [Ch. IV, § 69 EXERCISE XXII Solve for x, the following equations : 6. ^ + ?-6 = l. 8 6 7. - -1. „ 8(2 + 5a;)-5_92; + 2 9 2 7 4. 19-(7+|) = | + 7. 9. |-| + |-£=18. 5. 5x-^=lx-l- 10. 8a;-? = ^ + 153. XT Q a; X + ^ y±. ^ 9~ 3 ■ 1/2. 5 + | = .-5. -'S. ^-5 = a:-23. 4 10 2 7a; 1 17 H/o , in "^- T-4-T8 = 36^^" + '^- 2(7.r-l) _ 3(3rr + 5) . 7a;+13 a; + 8 a^ + ll 13. = • 16 13 8 14. 30(a;-2) + f = ^iJ+30. 15. 1(5 a; + 1) - K4 a^ + 5) = i(3 a: - 1) - 2^(6 a; + 4), Sx + 9 , 5 a; - 33 48 - x , x-\l _'6 + x 16. 72 36 9 4 24 X x-2 a;-22 a;-12 ^ 32-a; ■ 8 5 10 20 40 " ^3_ 3^-5 _ 4(2^+4) _r9-. ^-71 ^_ 15^ 16 9 L 2 ^ 12 J 20. (4;»-l)C5»; + }) = (2» + J)(10a;-J). Ch.IV, §70] EQUATIONS AND PROBLEMS 65 70. The statement of a problem in algebraic language often leads to an equation. The problem is solved by finding the numerical value of the numbers which first appear as unknowns. Certain relations of the unknowns in definite numbers are given ; from these relations the values of the unknowns are determined. Little difficulty need be met in translating the state- ment of a problem into algebraic language if it be remem- bered that every algebraic expression represents some number. EXERCISE XXIII 1. What is the value in cents of 2 two-dollar bills, 3 dollar bills, 4 quarters, and 5 nickels ? oi a two-dollar bills, b dollar bills, c quarters, and x nickels ? 2. If X is the tens' digit and 4 the units' digit of a number of two digits, what is the number ? 3. If 3 is the tens' digit and x the units' digit of a number of two digits, what is the number formed by reversing the order of the digits ? 4. If in a number of three digits the tens' digit is x^ and the hundreds' digit is twice the tens' digit, and the units' digit is four times the tens' digit, what is the number ? 5. What is the cost of 20 articles bought at the rate of 3 for X cents? 6. If x represents a certain digit, what is the next higher digit ? the next lower digit ? 7. If a; is a certain digit, what are the 2 next higher (consecutive) digits? 66 ELEMENTAKY ALGEBRA [Ch. IV, § 70 8. If X is an odd number, what are the next two even numbers ? the next two odd numbers ? 9. If X men contribute equally to a certain fund of % 225, how much does each man contribute ? 10. If a man spends a dollar a day more than on the preceding day, and on the tenth day spends x dollars, how much does he spend on the twenty-third day ? 11. If the price of eggs is lowered 3 cents a dozen from the original price of a cents a dozen, how much does one Q^g now cost ? 12. If the interest on a certain sum of money for a given time is computed at x per cent, what will be one per cent higher rate ? 13. What is the value in cents of the same number, Xy of dollars, cents, quarters, and dimes ? 14. If in a certain number of two digits the units' digit is x^ and the tens' digit is four times the units' digit, what is the sum of the digits ? 15. If a newspaper increased x per cent over the pre- ceding yearly circulation at the end of each year, and if the circulation at the end of the first year was 25,000, what was the circulation at the end of the second year ? 16. If the rate of a stream is 2 miles per hour, what will be the rate down the river of a crew which rows 4 miles an hour in still water ? up the river ? 17. What is the perimeter of a rectangular field whose length is a feet and whose breadth is h feet ? 18. What is the greater of two numbers if the greater is three times the excess of the less number, x^ over 12 ? Ch.IV, §71] EQUATIONS AND PROBLEMS 67 71. After the conditions of a problem have been stated in algebraic language, the next step is to find two equal expressions. In th.e equation formed of these two equal expressions the roots are found by § 68. 1. The sum of a number and its double is 48. Find the number. Let X = the number, then 2 0? = double the number, and x-\-2x = ^x = the sum of the number and its double, but 48 = the sum of the number and its double, by Ax. 5^ 3 0? = 48, by Ax. 4, 0? = 16. Verification: 16 + 2 (16) == 48, 48 = 48. 2. Find that number which lacks as much of 18 as it exceeds 10. Let X = the number, then 18 — ic = the amount the number lacks of 18, and a? — 10 = the amount the number exceeds 10, I but the amount the number lacks of 18 is the same amount that the number exceeds 10 ; by Ax. 5, 18-a; = r»-10, or — 2 0? = — 28, by Ax 4, X = 14. Verification : 18 — 14 = 14 — 10, 4 = 4. 58 ELEMENTARY ALGEBRA [Ch. IV, § 71 3. A's age exceeds B's by 25 years. Five years ago A was six times as old as B. Find the age of each. Let then and and and but by Ax. 6, simplifying, uniting, by Ax. 4, Verification X = B's age, 25 + a? = A's age, a; — 5 = B's age 5 years ago, 25 + oj — 5 = A's age 5 years ago, ^{x — 5) = ^ times B's age 5 years ago, 20 + ^ = A's age 5 years ago, Q(x-S) = 20 + x, 6a;~30 = 20 + a;, 5a; = 50, a; = 10. 6(10-5) =20 + 10, 30 = 30. 4. The units' digit of a number is double the tens' digit, and the sum of the digits is 12. Find the number. Let a? = tens' digit, then 2x = units' digit, and x + 2x = sum of the digits, but 12 = sum of the digits^ by Ax. 5, x + 2 x = 12, uniting, 3 a; = 12, by Ax. 4, a; = 4, by Ax. 3, 2 a? = 8, Therefore the number = 10 (a;) + 2 a; = 48. Verification : 4 + 8 = 12, 12 = 12 Ch.IV, §71] EQUATIONS AND PROBLEMS 59 5. The sum of the third part and twelfth part of a number is 25. Find the number. Let x = = the number, then X _ 3" = the third part of the number, and X 12" = the twelfth part of the number. and X X 3"^ 12" = the sum of the third and twelfth parts. but 25 = = the sum of the third and twelfth parts, by Ax. 5, 3 + 12-^^' by Ax. 3, 4:X + x = 300, uniting, 5 a? = 300, by Ax. 4, x = m. Verification : 60 ,60 ^^ ¥ + 12 = ^^' 20 + 5 = 25, 25 = 25. 6. A man has the same number of half-dollars, quarters, dimes, and nickels. Find the number if he has all together 13.60. Let X = the number of each coin, then 50 a; = the value of the half-dollars in cents, and 25 x = the value of the quarters in cents, and 10 a; = the value of the dimes in cents, and 6 a; = the value of the nickels in cents, and 90 a; = the values of all the coins in cents, but 360 = the values of all the coins in cents, 60 ELEMENTARY ALGEBRA [Cii. IV, § 71 by Ax. 5, 90i» = 360, by Ax. 4, a; = 4. Verification : 90 (4) = 360, 360 = 360. EXERCISE XXIV 1. The sum of a number and three times that number is 48. What is the number? 2. The sum of 10 and twice a number equals four times that number. What is the number? 3. If 13 be subtracted from eight times a number, the remainder equals 86. What is the number ? 4. If five times a certain number is subtracted from 27, the remainder is 7. Find the number. 5. Five times a number exceeds twice that number by 21. Find the number. 6. Find that number th>e sum of whose products by S and 4 respectively equals 119. 7. One number is twice another number and theiu difference is 14. Find the numbers. 8. The sum of 12 and three times a number equals the excess of 39 over six times the number. Find th( number. 9. Twice a number lacks as much of 20 as three timer the number exceeds 20. Find the number. 10. Twelve times a number exceeds 7 as much as tei times the number lacks of 15. Find the number. Ch.IV, §71] EQUATIONS AND PROBLEMS 61 11. The sum of 12 and four times a number exceeds by 2 nine times the number. Find the number. 12. The excess of four times a number over 24 equals the sum of 9 and the number. Find the number. 13. The greater part of 8 equals three times the smaller part. Find the parts. 14. Three times the smaller part of 15 exceeds by 5 twice the larger part. Find the parts. , 15. The sum of two numbers is 47, and their difference is 3. Find the numbers. 16. The sum of two numbers is 26, and their difference is 6. Find the numbers. 17. The sum of two numbers is 120,^ and the greater exceeds the less by 21. Find the numbers. 18. The difference of two numbers is 26 and their sum is 52. Find the numbers. 19. The excess of 7 over the larger part of 5 equals twice the smaller part. Find the smaller part. 20. The sum of three consecutive numbers is 39. Find the numbers. 21. Find the ages of A and B if the sum of their ages is 62 years, A being 16 years older than B. 22. A has four times as much money as B, and both have $ 125. How much has each ? 23. A, B, and C have together I 28. A and B each has three times as much as C. How much has each ? 62 ELEMENTARY ALGEBRA [Ch. IV, § 71 24. A has twice as much money as B, and B has three times as much as C. All have together $150. How much has each^? 25. A has twice as many dollars as B, three times as many as C, and half as many as D. If they all have $92, how much has each? 26. A, B, and together have $ 54. If A has twice as much as B, and G has as much as A and B together, how much has each? 27. A and B together have $ 12 ; B and C, $ 15 ; A and C, $ 19. How much has each ? 28. The same number each of dollars, dimes, and cents amount to $8.88. Find the number of cents. 29. The sum of a certain number of quarters and four times that number of cents is $ 5.80. Find the number of cents. 30. A has ten times as many cents as dimes and eight! times as many dimes as dollars. If he has in all $ 13, findi the number of dimes. M 31. A's age exceeds B's by 20 years. Ten years ago A was twice as old as B. Find the age of each. 32. A is now four times as old as B ; 5 years ago he was seven times as old as B. Find the age of each. 33. A is now five times as old as B ; in 12 years he will be three times as old as B. Find the age of each. 34. Six years ago a father was six times as old as his son, whose age now lacks 30 years of the fathers age. Find the age of each. Ch. IV, § 71] EQUATIONS AND PROBLEMS 63 35. If A is now 52 years old and B is now 12, find the number of years ago that A was five times as old as B. 36. The units' digit of a number of two digits is three times the tens' digit, and the sum of the digits is 12. Find the number. 37. The tens' digit of a number of two digits exceeds by 4 the units' digit, and the sum of the digits is 8. Find the number. 38. The tens' digit of a certain number of two digits is 3 times the units' digit. If 18 be subtracted from the number, the order of the digits will be reversed. Find the number. 39. The hundreds' digit of a number of three digits is twice the tens' digit and four times the units' digit. If 297 be subtracted from the number, the order of digits will be reversed. ' Find the number. 40. A fifth of a certain number exceeds the eighth of that number by 6. Find the number. 41. The excess of a certain number over 8 equals a third of that number. Find the number. 42. The quotient of a certain number divided by 9 exceeds the twelfth part of the number by 1, Find the number. 43. The twelfth part of a certain number is 8 less than the sixth part of that number. Find the number. 44. The eighth part of a certain number is 3 less than the fifth part of that number. Find the number. 45. The ninth part of a certain number exceeds by 1 the tenth part of that number. Find the number. 64 ELEMENTARY ALGEBRA [Ch. IV, § 71 46. The third part of a certain number exceeds 5 by as much as the eighth part is less than 6. Find the number. 47. The fifth part of a certain number exceeds 7 by as much as the ninth part is less than 7. Find the number. 48. Two-thirds of a certain number exceeds one-sixth of that number by 15. Find the number. 49. Three-eighths of a certain number exceeds one- fourth of that number by 4. Find the number. 50. Two-thirds of a certain number exceeds four- sevenths of that number by 2. Find the number. 51. The sum of one-third and one-thirteenth parts of a certain number is 16. Find the number. 52. One and one-half times a certain number exceeds three-eighths of that number by 36. Find the number. 53. The sum of the ages of a father and son is 48 years. How many years ago was the son's age one-seventh of the father's age if the son's age is now 1 2 years ? 54. A has four times as many cents as dimes and twice as many dimes as dollars. If he has in all $5.12, find the number of dollars. 55. Find that number of three digits in which the hundreds' digit is double the tens' digit, and in which the tens' digit is double the units' digit, if the sum of the digits is 14. CHAPTER V TYPE FORMS IN MULTIPLICATION 72. The products of certain expressions are so often required that it is convenient to have a shorthand method of writing the product without performing the multipli- cations as in § 49. These expressions and their products are called type forms. CASE I 73. By multiplication, {a +hy==a^ + 2ab + Ifi, Here a and h represent the sum of any two quantities ; the square of the sum is required. The process may be represented thus : (1st number + 2d number)2= (1st number)^ + 2 (1st num- ber)(2d number) + (2d number)^ Rule : The square of the sum of two quantities is the square of the first quantity^ plus twice the product of the first and second quantity^ plus the square of the second quantity, EXERCISE XXV Write the indicated squares by inspection : 1. (m + ^)^. 5. (c + 2 dy. 9. (a + S2)2. 2. (a + 2 J)2. 6. (2 (? + 3 dy. 10. (2 ^ + ^)2. 3. ic + dy. 7. (6-2 + dy. 11. (4^4- my. 4. (2 c + dy. 8. (a2 + 5)2. 12. (2 a + 12(^2)2^ 65 1 66 ELEMENTARY ALGEBRA [Cii. V, §§ 74, 75 CASE II 74. By multiplication, (a — by = a^ — 2 ah + P. Here a — b represents the difference of any two quantities . the square of the difference is required. Rule : The square of the difference of two quantities is the square of the first quantity^ minus twice the prod- uct of the first and second quantity^ plus the square of the second quantity, EXERCISE XXVI Write the indicated squares by inspection : 1. (m - ny. 5. (7 - 5)2. 9. (5 m- n)2. 2. {n - my. 6. (m-2 dy. 10. (11 m - l)^. 3. (c-dy. 7. {c^sdy. 11. (1-10^)2. 4. (d - cy. 8. (3 tZ - cy. 12. (2 m - 3 d^^y. CASE III 75. By multiplication, (a + &)(a — J) = a^ — P, Here the product of the sum and difference of the same two quantities is required. Rule : The product of the sum and difference of the' same two quantities is the difference of the squares of the' first and second quantities. exercise XXVII Write the indicated products by inspection : -v 1. (a 4- c}(a - c). d ^ C 4. (2 c + d)(2 c - dy^ 2. (m-n)(m + n). yv^-^ 5. (a^ -\- b) (a'^ - b) , \ 3. (^d + e)(id ^ e}. 6. (2c-d?)(2c+d:^^. Cii. V, § 76] TYPE FORMS IN MULTIPLICATION 67 76. It is sometimes possible to arrange the terms in both multiplicand and multiplier to take the form of Case III. 1. (a + b + c)(a + b-c)=l(a + b) + c]l(a + b)-cl = (a + by-c^ = a' + 2ab-\-b"-c'', 2. (a-b + c)(a + b-c) = la-(b-c)\ja + (b-c)l = a?-{b.-cf, ' + 2bc- ■c^ The rule of Case III applies to the product of terms so arranged. EXERCISE XXVIII u Write the indicated products by inspection : 2. (^ni — n + p} {ni + n— p^ 3. (m — n—p^(m — n+p) 4. (^m —n —p^(7n + n +p^ 5. {2a + b + c^)(2a + b-e^} (2 a2 + 3 a5 + 52)(_ 2 a^ + Sab + 52). (^2 _ 2 ^i + d^) (6-2 + 2 ^^ - 6^2), (c2 -ab + 5d)(c^ + ab + 5 d). (^s — sa — sb')(s -{- sa + sF), (6-2 _ s^a + s25)(s2 + s^a - s%). (xY -Sxf + i 2/0(^y + Sxf-4: 2/*). L'lS. (So? - 2x9j + f - iX^x^ - 2xy -^ 2/^ + 4). 14. (a^ - 3 a J + 52 - 2 6) ( - a3 - 52 _ 2 5 - 3 ab). 6. 7. 8. 9. 10. 11. 12. 68 ELEMENTARY ALGEBRA [Cii. V, § 77 CASE IV 77. By Case I, p, = (a-^by-\-2(a + h)c + c% = a' + 2ab + b^ + 2aG-{-2bG + c^, = a^ + 6- + c^ + 2 ab + 2 ac + 2 be. By Case 11, (a-6-c)2=f(a-5)-cp, = (a-6)2_2(a-5)c + c2, = a^-2ab + b^-2aG + 2bc + (^, = a^ + b'-{-c'-'2ab-2ac + 2bc. Rule : The square of any 'polynomial is the sum of the squares of the several terms^ and twice the product of every i term by every term that folloivs it^ giving to every product the proper sign, EXERCISE XXIX Write the indicated squares by inspection : 1. ^a + h-cy, 8. (1 + 2a + 3^2)2. 2. l-a + h + cf. 9. (2a + 5-3c)2. 3. Q^a-h + cy. 10. (2a^-{-^hx + x^y, 4. (^a^h-.cy. 11. (1+2 2: + 3:^:2)2. 5. (_^_J_^)2. 12. (2 a2 - 3 a?> - 5 />2)2. 6. (2a + J + ^)2. 13. [(a4-fi)4-^ + 2 6?]2. 7. (^_(. 26 + 7^)2. 14. (2 77^2 - 3 7^2 + 4 mn)2, 15. [(2a-5)-(?+36?]2. 16. \Jom — lbmn + n(n — m-yy. 17. (2 ^3 - a25 + 3 a62 _ J3-)2. Ch. V, § 78] TYPE FORMS IN MULTIPLICATION 69 CASE V 78. By multiplication, and {a - J)3 = a^ - 3 a% + 3 aS^ _ J3. Rule : The cube of the sum of two quantities is the sum of the cubes of the quantities plus three times the product of the square of the first quantity and the secondJ plus three times the product of the first quantity and the square of the second. Rule : The cube of the difference of ttvo quantities is the difference of the cubes of the quantities minus three times the product of the square of the first quantity and the second^ plus three times the product of the first quajitity and the square of the second. The result of the two rules can be shown thus : (a ± 6)3 = aS ± 3 a% + 3 aJ^ ± J3^ where the sign ±, read "plus or minus,"means that in the cube of a 4- J the signs are all plus ; and that in the cube of a — J the signs are alternately plus and minus. Write by inspection (a — 2 6)^. (a-2 6)3= (a)3-3(a)2(2 &) +3(a) (2 6)^- (2 h)\ EXERCISE XXX Write the indicated cubes by inspection : 1. (x + yy. 4. (2 a + 6)3. 7. (w-5n)3. 2. (x-yy. 5. (2^:2 + 3 2/2)8. 8. {la^-l ab'^)K 3. (x^J^y'^y. 6. (2x^-''&xyz)\ 9. (1-52^2)3. 70 ELEMENTARY ALGEBRA LCh. V, § 79 CASE VI 79. By multiplication the product of two binomials of the form x + a and x + b can be determined. (:r + 2) (a; + 3) = a;2 + 5 a: + 6. (^x + 2)(x-S) = x^- x-6. (ix-'2)(x + n} = x'^+ x-6. (a;-2)(a;-3) = 2;2-5a: + 6. Rule: The product of any two hinomials whose first terms are identical is the product of the first terms of the hinomials^ the algebraic sum of the second terms as the coefficient of the common term^ and the product of the second terms of the hinomials, EXERCISE XXXI Write the indicated products by inspection : 1. (x + V)(x + 2^. 8. (:ry-3)(rr2/ + 4). 2. (a; + l)(^-2). 9. (a:2_3)(^2 + 4). 3. (m + 5)(m-4)- 10. i^ — l xy^(^- xy^. 4. (^_3)(^__4). 11. (^2_4)(^^2+6). 5. (m-7)(m + 3). 12. (ax + lV)(iax + V), 6. (x-bn)(x + ?^n^. 13. (a2-21)(a2 + 3). 7. (x^^b^(x^-S). 14. (xy-l)(xy + ^y 15. (16-5a;^)(16-2:r^^). 16. (5 m^n — 3 ny'^^ (5 rrfin — ny"^^. 17. [(a + 6) + 5] [(a + 6) -3]. 18. [l-(:r + 2/)][l-4(a; + 2/)]. 19. [(a;-2/) + 2][(a;-2/)f 7]. :h. V § 80] TYPE FORMS IN MULTIPLICATION 71 CASE VII 80. By multiplication the product of two binomials irhich contain the same letters can be determined. (x-2y)(2x+?>y) = 2x^- xy + Qy"^. (x + 2y)(2x-Zy)=-2x^+ xy-Qy^. (x-2y)(2x^?>y^=^2x'^-lxy + Qy'^. Rule : The product of tivo binomials which contain the ame letters is the product of the first terms of the binomials^ he algebraic sum of the cross products^ and the product of he second terms of the binomials. Write by inspection (3 a; + 7 ?/)(2 a: — 4 y). {3 x + 1 y){2 x-^ 4:y) = ^ x" + 14.xy -12 xy --2S y\ The cross products, as 14 xy and — 12 xy, are usually com- ined, without writing in full, into the middle term of the •roduct. EXERCISE XXXII Write the indicated products by inspection : 1. (2x-a)(?>x + a^. 7. (x-b y)(2x -3 y^. 2. (2m + a)(m-2a), 8. (2 ^— m2)(3a;- m^). 3. (2x + aX?^x-a). 9. (:r + 1)(3 a:-4). 4. (2m~a)(m-2a). 10. (5 a- 2 5)(2 5 + 5 a). 5. (2x—a^(Zx-a^. 11. (a — 11 (?)(2 a — + 1 = (13 a6 - 1)(13 db - 1). EXERCISE XXXVI Factor : 1. 2? + 14 a; 4- 49, 9. ^Ix^-IUxy + imy^ 2. a:2_l4a; + 49. 10. 484 a:* - 44 a;2y2 + ,^, 3. a;6-4a;3 + 4. 11. 256 a;2«/2-96 y). Since the first term of the trinomial is the product of the first terms of the binomials, the first terms of the binomials must be 2 a; and x ; since the last term of the trinomial is the product of the last terms of the binomials, the last terms of the binomials must be 3 2/ and y. The sign of the last term of the trinomial is minus ; hence the last terms of the binomials must have opposite signs. By trial the factors are now found as given above. If the trinomial contains no common monomial factor, the binomial contains no common monomial factor. The middle term is found by multiplying the first term of the first binomial by the second term of the second binomial, ar ^ by multiplying the second term of the first binomial by the ^ term of the second binomial, and taking the algebraic sum . these products for the middle term. The process is represented: 2x^ + ^xy-?>f={2x-y)(x^2>y). Writing the possible factors of 2 oi?^ 2x^ + 5xy-3y'^ (2 x )(x ), and in the parentheses writing also the possible factors of — 32/^, the factors of 2xi^ + 5xy — 3y^ are 86 ELEMENTARY ALGEBRA [Ch. VI, § 92 either (2^ + 3?/) (x - y), (1) or (2x-3y){x + y), (2) or (2x + y)(x-3y\ (3) or (2x-y)(x + 3y). (4) Each, of the possibilities (1), (2), (3), (4) must be tried by actual multiplication until the proper factors are discovered. 2. Factor 2^:2 + 5 2:2/ -12/. The possible factors of 2 x^ are 2 x and x ; the possible factors of 12 y^ are 12 y and y, y and 12 ?/, 6y and 2 y, 2 y and 6y, 4:y and Sy,3y and 4 ?/. That is, 2 a; and a? must be tried with each of the six possible factors of 12 y^. (2x 12 2/) (X y)> (1) {2x y)(x 12 2/), (2) {2x 6y)(x 2 2/), (3) (2x 2y){x 6 2/), (4) (2x 4.y)(x Sy), (5) (2x 3y)(x iy)- (6) Possibilities (1), (3), (4), and (5) are immediately eliminated because the binomials contain a factor which is not a factor of the trinomial. By trial, 2 x' + 5 xy ^12 y' = (2 X- 3 y)(x + 4:y). / 3. Factor 2 a;2 - ax + 4tlx^-Qal. 2 a;3 _ 3 ^^ + 4 5^ _ 6 a& = (2 ar^ - 3 aoj) + (4 &a;2 - 6 ab\ = a;(2 a;2 - 3 a) + 2 &(2 i»2 - 3 a), = (x + 2 2>)(2aj2-3a). 3. Factor 2 0^-4:1:2^ -3 a: + 6 2/. 2^-4.x'y-^x + Q>y=(2o?-A.x'y)-{^x-Qy), = 2a^(x-2y)-S(x-^2y), = (2x''-3)(x-'2y). 4. Factor 12 a%^ - 42 h^c + 16 a^ - 56 ahc. 12 a^b^ - 42 b'c + 16 a^ - 56 abc = 2 [3 b\2 a^ - 7 &c) + 4 a(2 a^ - 7 5c)], = 2(3 62 + 4a)(2a2~7 5c). Ch. VI, § 94] FACTORING 91 EXERCISE XLV Factor : 1. mx + am + nx + an, 2. mx — am ■—nx + an. 3. 2 hx^ — ahx + 4:Cx—2 ac. 4. m^n — 3 abn — 2 m?^ + 6 ahp. 5. x^ + ax + bx + ah. 6. x^ — ax — hx + ah, 7. 6 hx — 15 ah — 4: dx + 10 ad, 8. dmnx — ac^dx + mnrs — ac^rs, 9. —2an + Sap + 2hn — S hp. 10. 6 eg — 9 c?^ + 4 a^? — 6 acZ. 11. a4i2 - 2 a264 + ^^5 - 2 aS^. 12. 2 5(?m — 4 ah^c + 7 am^i — 14 a%n. 13. 2 T/i^/i — 2a'^h — 8 cm^n + 3 a^J^?. 14. — a%x^ — 4 Je — 2 ^2^2^ — S cy. 15. 8 aa2 + 10 art + 12 6s2 + 15 hrt. 16. — mnx — 2 mn + p^x + 2 jt?^. 17. 14 a^e^f + 35 ^2^62^ _,. q ^2^^^ + 15 52^^^^ 18. 10 a(? + Jc - 110 ad - 11 5c?. r 19. rs + a^w — 3 c^rs — 8 a^t^n. 20. 8 acajy — 14 o^xz + 21 acc^s — 12 c^cZy. 21. 6 a3 - 8 a262 _ 15 ^5^ + 20 Jfic. 22. 6 a^ - 33 aca; - 8 ca:^ + 44 ^^2. 92 ELEMENTARl ALGEBRA [Ch. VI, §§ 95, 96 95. A theorem is a statement of a general truth which requires demonstration. THE FACTOR THEOREM I 96. If any expression containing x reduces to when a is substituted for x, then x— a is a factor of that expression. Let E represent the expression. Divide E by x — a until the remainder does not contain any power of x. Let R be the remainder, and Q be the quotient. Then E= QQc-^a^ + R. (1) Equation (1) is always true whatever may be the value of x. Take x = a^ and substitute in (1) ; 0=^(a-a) + i2, (2)| 0=^(0) + i2, (3) = 72. (4) In (2), E becomes 0, because the expression is taken as one containing x^ which becomes when a is substituted fora; ; also in (2), a — a = 0, and ^0 = 0. Li (4), R be- comes ; or, in other words, there is no remainder. Con- sequently a; — « is an exact divisor or factor of E. Since a; + a = 2;— (— a), the theorem holds true if (a) be replaced by (— ^) in the statement, thus: if any ex- pression containing x reduces to when (— <^) is substi- tuted for Xy then a;— (—a), or x + a^is a factor of that expression. The Factor Theorem has a wide application, and maj be applied as a check to most of the preceding cases, and to many forms which are not included in those alreadj iiyivea Cii. VI, § 9b'] FACTORING 93 1. Factor a^ + 1. Take x = l, and substitute in o;^ + 1 : 1^ + 1 = 1 + 1 = 2. .' I ere the expression oc^ + 1 does not become 0, or vanish, and .s( ) cc — 1 is not a factor. Take x= —1, and substitute in ar^ + 1 : ( - 1)^ + 1 = — 1 + 1 = 0. The expression vanishes, and x + 1 is, therefore, a factor of a?^ + l. By division the other factoi or factors may be established. 2. Factor afi + y^. Take x = ?/, and substitute : (yY + y^ = 2y^; ^ — y is not a factor. Substitute x— —y: (— ?/)^ + 2/^ = ; a; + 2/ is a factor. ^ j^ 'if z= {x + y) (xl^ — Q(?y +x^y^ — xy^ + y^)o The second factor, x'^^ii(?y + Q(?y'^ — xy^ + y^^ is found by division. 3. Factor :i:5 + 32 /. By substitution, x + y and x — y are shown not to be factors. Try x = 2> y '. (2 yy + ^2 if = Q>4.y^', x — 2y is not a factor. Try i7; =: — 2 7/ : (— 2 2/)^ + 32 ?/^ = ; cc + 2 1/ is a factor. By division x^ — 2 x^y + 4 xPy^ — 8 a?^/^ + 16y^ is the other factor. a;^ + 32 2/^=(a: + 2 2/)(a;*-2a^i/ + 4a^/-8aji/^ + 16^^). 4o Factor :2;3 + ^_7^ + 2. The substitution of x = l, x= —1 both give remainderfe t now try a;=:2: (2)^ + (2)'- 7(2) + 2 = 8 + 4 - 14 + 2= ; a; — 2 is a factor. By division, x^ + 3 a; — 1 is the other factor. ar* + ^- 7 aj + 2 = (a;- 2)(a;2 _j_ 3 ^ _ j^)^ The factor obtained by division must be carefully inspected to determine if it is prime. y4 ELEMENTARY ALGEBRA [Ch. VI, § 96 EXERCISE XIiVI By use of the Factor Theorem, separate in factors: 1. ^ + f. 9. a2-16a + 64. 17. a}-W. 2. afi+y^. 10. a^— 25. 18. 1 — Sa:^^ 3. a*-l. 11. a;2-57a; + 56. 19. 2l3^-f. 4. a^ + 8. 12. 343-a;3. 20. 32a;5+j^. 5. 1 + a^. 13. a--*— 81. 21. a^ — y^. 6. a^-27. 14. a;5 + 243. 22. a^^ + ^e, 7. 64-a;3. 15. 82a^+b^ 23. a^^ + ^/S, 8. 32 -a^. 16. a' + b''. 24. x^ + yo^ 25. a^-a;2-a; + l. 32. a;3_^2_ 2 a;_ 12. 26. a:3-a;2-3a;-l. 33. a^+ 2 a;2- 3 a;+ 20. 27. a^-2a^-6a;-2. 34. 3^ + :^-inx-21. 28. a^-3a;-2. 35. a^-S x^+12x+ 9. 29. a:3-a;2-a;-2. 36. a^+lB3^+ iBx+ 6. 30. a^ + a^ - a; + 2. 37. ai^+Za^-x^-2x+l. 31. a;3_5a; + 2. 38. a*-a^+5 a^+14:a-lQ. 39. 2a^-5a^ + lSx^-9x-l. 40. 3a:*+8a^ + 8a^ + Ta;-2. 41. 2a4-7a3+8a2_6a + 4. 42. a* + Qa^ + lla^-a-21. 43. a*-Sa^ + n a^-Ua + 8. 44. 2a^-13a^+16a;2-6a;+5. 45. 3^-4:x^ + 10a^-5x^-4:X+2. 46. 3a^ + 3a;* — 5ar^-6a;-4, Ch. VI, § 97J FACTORING 95 HINTS ON FACTORING 97. It is impossible to give any definite method of attack in factoring. A monomial factor should at once be removed. Every factor should be carefully inspected for further factors. It may happen that an expression can be factored by different methods. If the expres- sion can be factored as the difference of two squares, it is generally preferable to do so. 1. Factor a;^ — 2/^. By Case III, by Case VIII, ={x + y) (x^ ~xy + y^) {x - y) {a? -{-xy + y'^). By Factor Theorem, x^ — y^= (x — y)(x^ + x'^y +x^y^ + x'^y^ + xy^ + y^), = (x-y) lx\x + y) + x^y\x +y) +y\x-]r y) ], by Case IX, ={x — y){x + y) (x^^ + o?y^ + y^), by § 88, =^{x-y){x + y)(x^ + xy + y'^(x'--xy + fy 2. Factor afi + y^. By Case VIII, x' + y'= (xy + (y'f = (x^ + r) (x' - xhf + y'). 3. Factor 2:12 + 64. By Case VIII, x"' + 64 = {xy + (4)^ = (x' + 4) {x^ -^x' +16). 4. Factor x^ — y^. By Case VIII, x'-y'=: (xy - (f)^ = (o^- f) (x' + x^f+ y% = {x-y)(x' + xy + f)(x' + :^f + yy 5. VsLCtoT x'^^ + y^^. By Factor Theorem, 96 ELEMENTARY ALGEBRA [Ch. VI, § 97 6. Factor 3 (a - 6)3 - a + 5. S(a-by - a + b^3(a- by - (a-b), by Case IX, =(a-&)[3(a-6)^-l], = (a - 6) (3 a=-6 a6 + 3 6^- 1). 7. Factor a^ + b^ + c^ - 8 abc. a3 + 53 ^ c3 _ 3 abc = (a^ + 6») + (c' - 3 abc). (1) Now a' + b^^ia + bf-Sa'b-Sab'-^ia + by-Sabia + b). (2) Substitute a' + 6^ = (a + bf -3ab(a + b) in (1), a3 + 63 + c'-3«6c = (a + 6)'-3a& (a + 6) 4-<^-3 a6c, (3) = [(a + &)* +c'] - 3 a6 (a + &) - 3 abc, (4) by Case VIII, =(a + b + c)l(a + bf -c(a + b)+ c^J -3a6[a + 6 4-c], (5) by Case IX, =(a+b+c)l(a+by-c(a+b)+c''-3ab2, (6) = (a + 6 + c) (a^ + 2 ab + b^-ac-bc + c^ -Sab), (7) = (a 4. 6 + c) (a^ + 6^ + c= - ab -ac- be). (8) REVIEW EXERCISE XL VII Factor : 1. a;2-22« + 121. 4. 343:r3-l. 2. 4a:2_49^-(a_J)2. 105. ^4 + J4 ^ ^4 _ 2 ^2J2 _ 2 a2c?2 + 2 62^2. 106. 4J4c4-J4_2 52^2_^4, 107. x^+a^+x^y^ — y^ + y^. 108. 3a;3_|.^2(2a-9) + :?^(3-6a)+2a. 109. a;* — a;3 — :i:2 ^ 3 ^ _ 2. 110. (a+hy-c\a+by-c(a+by+(^. 111. rz;4-2a;3_2a;2-22:~3. 112. l + b^+c^-3bc. 113. a4+2a2 + l-5(a2 + l)+6. 114. a;* - 2 a;2 - 5 ^ + 2. 115. (x-yy-x + y. CHAPTER VII HIGHEST COMMON FACTORS. LOWEST COMMON MULTIPLES THE HIGHEST COMMON FACTOR 98. A common factor of two or more algebraic expres- sions is an exact divisor of each of the expressions. Two expressions are said to be prime to each other when they have no common factor other than 1. The highest com- mon factor of two or more algebraic expressions is the product of all the common prime factors. Thus, a^ and 2 are common factors of '^c^x and 6 a^J, and the highest common factor is 2 c?. 99. The highest common factor — abbreviated H. C. F. — of several monomials is readily found by inspection. Thus, find the H. C. F. of 6 T?y\ 12 x^y\ 40 x^y\ 6a^z/2 = 2 . 3 . a;3 -2/2, 12 0^^ = 2 .2 .3 .aj2.2^3^ 40a;y = 2 .2. 2 .5 .0^.?/', H.C.F. =2 .0^ .2/'=2,t2|/2. Note. The H. C.F. of two algebraic expressions has reference to the degree of the factor ; the greatest common divisor of two arith- metic quantities has reference to value. The H. C. F. of a and a^ is a r if a is any common fraction, and equal, say, to |, the greatest common divisor of a = J and a^ = ^j is ^j. The terms H. C. F. and G. C. D. are not, therefore, interchangeable. 100 Ch. VII, §100] HIGHEST COMMON FACTORS 101 THE H.C.F. BY FACTORING 100. lo Find the H. C. F. of a^+ah, a?+hK d^-\-a'b = a{a-\-h), H.C.F. = a + 6. 2. Find the H. C. F. of d^ - h\ a? - h\ a* _ a^-b'= (d + W) (a + b)(a- b\ H.C.F. =a-6. 3. Find the H. C. F. of nfi - 27, m^ - 6 m -}-9, nfi + m -12. m^-2T= (m - 3) (m^ + 3 m + 9), m^ — 6 m + 9 = (m — 3) (m — 3), m^ + m — 12 = (m + 4) (m — 3), H.C.F. = (m-3). 4. Find the H. C. F. of 4 ^^ - 7 x^ + 3 ^\x^ - x^t/ -xf ^- 2/^ x^ — 2 x?y^ + i/, 4 x^ - 7 a^2/' + 3 ^' = (4 a;2 _ 3 y") ix"- - f) , x?-o(?y^ xf- + f = o?{x-y)- 2/2 (x — y), = (^^ - 2/') (^ - 2/) = (^ + y) (^ - 2/) C^-2/). oc^ — 2 x^y^ + .V* = (0^2 _ ^2)2 ^ (^ ^ y^ Q^ _ 2/) (aj + 2/) (^ - 2/) j Il.C.Y, = (x + y)(x-y), The H. C. F. of several algebraic expressions is found hi/ taking the product of the common prime factors the least number of times they occur in any of the given expressions. 102 ELEMENTARY ALGEBRA [Ch. VII, § 100 EXERCISE XL VIII Find, by factoring, the H. C. F. of the following ex- pressions : 1. ab + a^ b^+b. 4. am—an + bm — bn^am—an, 2. 15 2; - 9, 6 - 10 X. 5. (x + 1)2, ^2 ^ ^^ 3. ax^ — 2 axy^ 2 ax^ — axy. 6. a2 — 4 a + 4, 3 a6 — 6 6. 7. a2 - 6 a + 9, a6 - 3 a - 3 6 + 9. 8. a;2 — 1, ^2 — 07. 9. x/^-y\ x^ + y^. 10. a^ + l, ^2 — 1. 11. 25^2-9(5-1)2, 6J-10a-6. 12. a;2 + 3a;+2, a;2 + a:— 2. 13. a;2__9^ + 20, 2)2-16. 14. 4a2-5a&-6J2, 8a2 + 2a5-3J2. 15. x^ + x^y'^ + y^, afi + y^. 16. ri;2_8a: + 12, a:2_7^ + 6^ ^_216. 17. 3(x - 1)3, a:* - 1, a;3 _^ ^2 __ 2. 18. a^ — 2/^, a;^ + a;2^2 + ^4^ 2)^ + 4 r?;2^ + 4 a:j/2 -f. 3 ^/S. 19. 2:r2 + 172: + 21, 82)3 + 27, 2ri:2+5a;+3. 20. m2 - 4 m + 3, m2 - 6m + 9, m^ - 9m2 + 27m - 27. 21. a^-16, a3+2a2+4a+8, 3 a^- 2 a2+ 12 a~ 8. 22. nfi— 9^^, m^— 3 m2^2_j, 3 mn^—n^^ m^— mn^+m^n—ri^. 23. a;3_27, 2)4+92:2+81, x3+22;2+6a:-9. 24. Safi+2x^-2x-l, 2)4-1, ^2;+a-J2)— J. 25. (a- 6)3+1, (a-_j)2_l^ (^__j)2_2((x-J>-3. ]i. VII, §101] HIGHEST COMMON FACTORS 103 THE H. C. F. PARTLY BY FACTORING 101. If difficulty be met in factoring one of the expres- luus, the factors of this expression may often be found y trial of the factors of the other expressions. Find the H. C. F. of 1- - h^\ 2 «6 - 3 a56 _ a*62 + 3 ^353 + ^254 -Za¥' + 6«. By division, a- + b^ is not a factor of 2 a" — 3 a% — aV + ; (I.V + a^5* — 3a¥ + b^; and a* — d^W + ¥ is a factor, pro- lucing the quotient 2a^ — 3a6 + 6l !> a« - 3 w'b - a'b^ + 3 a%^ + a}¥ - 3 at' + &" = (a* - a'b'- + b*) (2a'-3ab + b^, = (a* - d'W + b*) (2a-b)(a- b), ' H.C.F. = (a*-a262 + 6«)(a-6). EXERCISE XLIX Find the H. C. F. of the following expressions : 1. a^ + 5a; + 4, 3?-x^-^x-l. 2. 2:3 - 2 a;2 4- 1, x* - 1. 3. 2a;2 + 24a;4-70, x* + 7a^-2;2_6a; + 7. 4. a*-2a3 + 4a2 + 6a-21, 3a*-lla2 + 6. 5. l + 2x^ + 23^ + 2x^ + A 1+a^. 6. 3^—2 x^t/ + 2 xy^ — y^-, ax + ix — ay — hy. 7. a!^j^y'i-2xy + z{-2-x')-y(2-z)+2x,a?- 4+y^ -2xy. 8. a^+a^^+b\ a\2m-Sn^+m(2b^-2ab} + UnCa-b^. 104 ELEMENTARY ALGEBRA [Ch. VII, §§ 102, 103 THE H. C.F. BY DIVISION 102. * If the expressions are such that they are not readily factorable, the H. C. F. can be found by a process which depends upon the following principles : 1. A factor of an expression is also a factor of any mul- tiple of that expression. Let a be contained h times in R. Then B=ah. Let mR be any multiple of R. Then mR = mah ; that is, a is a factor of mR. 2. A factor of two expressions is a factor of the sum^ or of the difference^ of any two multiples of these expressions. Let a be contained b times in R and c times in S. Then Rz=ah and S=ac; or, applying the preceding principle, mR = mah and nS= nac. Adding or subtracting the two last equations, mR ± nS = mab ± nac = a(^mb ± ne^ ; that is, a is a factor of mR ± nS. 103.* Let A and B be any two expressions, arranged in descending order of the same letter. Let A be contained m times in B^ with a remainder of ; let C be contained n times in A^ with a remainder of D; let J) be contained exactly p times in 0, A)B(m mA ~0)A(n TlC D)Oip pD eH.VlI,§103] HIGHEST COMMON FACTORS 105 Since D is contained p times in (7, pD = C ; since the dividend equals the product of the quotient and divisor, plus the remainder, and since is contained n times in A^ with a remainder B^ A = nG + D\ since A is contained m times in B^ with a remainder C^ B ^ mA + 0. That is, C^pB, (1) A = nO+B, (2) B = mA + 0. (3) D may be shown to be a factor of each of the equations (1), (2), and (3). B has already been shown to be a factor of (7, since it is contained p times in 0. Substitute the value of from (1) in (2), A = npB + B (4) = BCnp + 1) (5) Substitute the value of A from (5) in (3); and the value of from (1) in (3) also, B = mB(np + l)+pB (6) = B(mnp + m+p}. (7) Hence, B is a common factor of A, B^ and C, Moreover, B is the Jiighest common factor of A and B. From (3), B-mA = (7, (8) from (2), A-nO=B. (9) By § 102, 2, a factor of A and 5 is a factor of B — mA^ or of (7; and a factor of A and C' is a factor oi A — nC^ or of B. That is, a factor of A and B is also a factor of B, Since there can be no factor of B of higher degree than B itself, B is the highest common factor of A and B. 106 ELEMENTAllY ALGEBRA [Cn. VIl, §§ 104, 105 104.* From § 103 is derived the statement of the Rule for finding the H. C. F. by division : arrange the expressions 171 the descending powers of the common letter ; remove a monomial factor^ if any^ from either expression^ and if the monomial factors so removed have a common factor write such a factor as a factor of the H, (7. F, subsequently found ; divide the expression of higher degree by the remaining expression until the remainder is of less degree than the divisor ; con- tinue the division with the remainder as a divisor^ ajid the former divisor as a dividend^ as before ; the last divisor ivill be the H, O, F, if there is no common monomial factor ; but if there is a common monomial factor^ the H, 0, F, is found by multiplying the last divisor by that factor. 105.* The H. C. F. of two expressions remains un- changed if either of the expressions be multiplied or divided by a quantity which is not common to both ex- pressions, since, by definition, the H. C. F. is the product of all the common prime factors. Thus, at any stage in finding the H. C. F. by division, a factor not common to both expressions may be removed by division ; or, if at any stage the expressions are such that the first terms are not exactly divisible, they can be made so by multiplying either of the expressions by a quantity which will make them divisible — thus avoiding the use of fractions — without altering the value of the H. C. F. 1, Find the U. C,F, oi x^ + 2x^-2x^ + 4x^ + Sx and x^+2x^-2x'+4.x^+3x=x{x^-i-2 0:^-2 x^+4.x+S), x^+2x^-x''+8x^+5:x?-3x=x{x'+2x''-x'+Sx-+5x '[]). Cn.Vn,§106] HIGHEST COMMON FACTORS 107 The factor x is common to both expressions ; therefore cc is a part of the H. C. F. x^ + 2x^-2x- + 4.x + S a^ + 2x*- x^ + Sx' + Sx-S x^ + 2x^-2a^ + 4:X^ + Sx af + 4:X^ + 2x-S The remainder is now of lower degree than the divisor. a^ + 4:i^ + 2x-3 x^ + 2a^-2x^+ 4.X + S x'^ + 4:a? + 2x'- 3x ^2a?-^x'-\- lx + 3 4a;2-fllaj-3 The remainder is now of lower degree than the divisor. x-2 4.x^ + llx-3 x^ + 4.x' + 2x-3 4 4ar^ + 16i»2+ 8a?-12 4a;^ + lla^^- 3a; 5a;^ + lla;-12 _4 20aj2 + 44a;-48 20x'-ir55x-16 -11 lla?-33 x+ 3 To avoid fractions, multiply the expression aj'+4a^+2a;— 3 by 4. This will not alter the H. C. F., because 4 is not a fac- tor of 4 a;^ + 11 a; — 3. Multiply the remainder, 5 a.*^ + 11 i» — 12, by 4 to make the expressions exactly divisible. Divide the remainder —11a; — 33 by —11. This will not alter the H. C F., since — 11 is not a factor of 4 a;^ + 11 a; — 3. x + 3)4.x' + llx-3(^4.x-l 4:X^ + 12X — a5 — 3 -a;-3 Therefore, the H. C. F. = a;(a; + 3). m PAAmmrAnt ALGt^.^nA [Cn.VIT, §105 2. Find the H. C. F. of \x^ -It^ -Vlx^ -\^x-\^ and 3 :r^ 4- 3 r?^^ + 9 a;2 + 9 a: + 12. 3 x^ + 3 aj^ + 9 a;2 + 9 aj + 12 = 3(i^ + ^^4 _^ 3 a;^ + 3 .^ + 4). The monomials removed contain no common factor. 2 a;4 + 3a;2 + 3a; + 4 2a^+2a!^ + 6a^ + 6a; + 8 2a^- a;^- 6ar'- Sa;^- 5a! 3a,4+ 6ar'' + 14a;2 + lla; 2 + 8 6 a;^ + 12 ar* + 28 a;2 + 22 a; 6a;*- 3«3_i8x2-24a! + 16 -15 15ar' + 46a;^ + 46a! + 31 15aj3+4Ga;2+46 0^+31 ^4-a? + l 2a.^4_^3_0^,2_g^_5 30i»^- 15i»^^- 90a;2-120a;-75 30aj^4- 92 ar^+ 92 a^^.^, 52 a; -107ar^-182a^-^-182aj-75 -15 1605 x" + 2730 aj2 +2730 aj+ 1125 1605 ar" + 4922 0? +4922 aj+3317 -21921-2192 a?2-2192aj-2192 15a^ + 46aj2 + 46a; + 31 15a^ + 15a;- + 15a; 31a!2 + 31a? + 31 31ar^ + 31a: + 31 x^-\- x+\ 15aj + 31 107 Therefore, the Yi,Q,Y, ^x" + x-\-l. Ch. VII, §105] HIGHEST COMMON FACTORS 109 3. Find the U.CF. oiia^ - 2a^-2x^ -^Sa^-^^x^-ex and i x^ — 4 x^ + x^ — x'^ — 6 X — 9. Ax^-2x^-2x^+8x^-2x'-6x=2x(2a^-x^-a:P+4.x'-x-3). 23(^--x'-x'+4.x^-'X-3 4:X^—4:xP-\- x^— x^—6x —9 4.x^-2af-2x'' + Sa?-2a^-6x -2a^+3x^-Sx^-{- x'-d — 2a^+ a;^+ i^—4:X^+x+S 2x'-9x' + 5a^-x-12 2x'-9x'+5x^-x-12 2x'- x'- a^+ 4.x^- x-3 2a.^_9a)H- ^^x^- x'-12x 2a;-l X + 4: 8 a;^-36 0^4-200^- 4a;~48 15 1 30 ar^- 15 ^^+15 a; 4-45 2af'_a;2+a;+3 2aj^- 2a;^ 2ar^- a;2_^ -9ar^+5a;2- a;-12 x+ 3 a;— 4 -8a^+4aj2_4a;_i2 -8aj3 4-4i»'-4i»-12 Therefore, the H. C. F. = 2 oj^ - i^^ + a; + 3. A more compact arrangement of the above example is the following : 2x5- X*- a:^^. 4^2_ aj_3 2x5-9a:t+ 5x'*^ -12x 8x4- 6x3+ 5x2+llx-3 8x*-80x3 + 20x2- 4x-48 15 1 80x3-15x2 + 15X+45 2x3— x2+ x+ 3 4x6-4x5+ x*- x2-6x -0 4x6-2x5-2x4 + 8x3-2x2-0x -2x5+3x4-8x3+ x2-9 -2x5+ x4+ x3-4x2+ x+3 2x4-9x3 + 5x2- X- 2x4- x3+ x2+3x 12 -8x3+4x2-4x-12 -8x3+4x2-4x-12 2x-] x+4 X- 4 The H. C. F. of three or more expressions is found by division by first finding the H. C. F. of the first two expressions; and then finding the H. C. F. of that result and the next expression. 110 ELEMENTARY ALGEBRA [Ch. VII, §105 EXERCISE L* Find the H. C. F. of the following expressions : 1. 4a? i-Sx-lO, 4x^+1 x^-3x-15. 2. 3^ + 2a? + 2x + l,3^-2x^-2x-B. 3. 4a^-6a?-4:x + 6,123^-2x'^-20x-6. 4. 6a? + 1 x^- 5 X, 15 a^ + Bl3^ + 10 a?. 5. 2x* + a?-9a? + 8x-2,2a^-7a? + lla?-8x + 2. 6. 4a^ + na?+4x-S,2a^-Bx? + 2a?-2x-S. 7. 8a? + 2a^ + a?, 8x^ + 2a?- Sx^ + 2x-l. 8. 2a^-5a;2-22a:-15, 6x*-21a?-41 a?-14x-80. 9. 4a?+14ai^ + 20a?+7Qa?, 8x'' + 2Safi- 8a? -12x^ + 563?. 10. 2a?-nx^-9, 4 a^ + 11 a;* + 81. 11. x* + 23? + 9, x*-4a? + 10x?-12x+9. 12. 6x*-5a?-10x'^ + Bx-10, 4a? -4x^-9x + 5. 13. 6a^-lSa? + Sx^ + 2x, 6a:*- 9x3+ 15^^- 27 a;- 9. 14. 3a;*-a;3_2a-2 + 2a;-8, 6ar5 + 13a^ + 3 a; + 20. 15. 9a?-7a? + 8x^ + 2x-4, 6x^-7 a? -10x^ + 5x + 2. 16. 6a?-2x^-lla? + 5x^-10x, 9a? + Si*-lla? + 9a?-10x. 17. x'^ + Ba? + Sx^ + 9a?-4x^-12x,a? + Sa?-x?-3x'^. 18. 2ai^ + a?-8x'^-x + 6, 4x'^ + 12a? -a?-27 x-^ 19. 6a;6-9a;* + lla:3^.6 2:2_i0a,, 4a:5 + 10 a^ + 10a;3 + 4a;2 + 60a;. 20. 4a? + a?-nx^+9x-9, 2a?+3x* + la?-Qx^-9x-27. 21. 43^-6a? + 9a?-5x + 8, 8a? + 8a? + 9. Ch.vii,§§io6,io7] lowest, common multiples 111 the lowest common multiple i06. A multiple of an algebraic expression is an ex- pression which contains all the prime factors of the first expression and is therefore exactly divisible by it. A common multiple of two or more algebraic expressions is an expression which contains all the prime factors of each expression. The lowest common multiple of two or more expressions is that expression which contains, only, all the prime factors of each of the given expressions. Thus, 2 a^b is a multiple of 2 a6 ; 6 a^x^ is a common multiple of 2 a and 3x^i 6ax^ is the lowest common multiple of 2 a and 3 x^. 107. The lowest common multiple — abbreviated L. C. M. — of several monomials is readily found by inspection. 1. Find the L. C. M. of 4 a^, 6 a% 12 b^ 4:a^b = 2'2'a''b, 6 a% = 2 . 3 . a^ . &, 12b' = 2.2.3'b% L. C. M. = 2 . 2 . 3 . a' • b^ = 12a''b^ 2. Find the L. C. M. of 8 x^i/, 10 xY, 15 a^f. 8 x'y = 2 . 2 . 2 . a;2 . 2/, 10:x^7/ = 2 '5'X*> y\ 15 x^y^ = 3 ' 5 - x^ - y^y L.C.M. = 2 . 2 . 2 . 3 . 5 . a;* . 2/' = 120 xy. Note. As in the case of the H. C. F. there are two forms of the L. C. M., one being the negative of the other. 112 ELEMENTARY ALGEBRA [Ch. VII, § 108 THE L. CcM. BY FACTORING 108. 1. FindihGL.C.M. oi8(^a^-P),4:a^ + 8ab + 4:b% a^^2ab + b\ 3(a2-62)=3(a + 6)(a-6), 4 a- 4-8 a& + 4 6» = 2 . 2(a + &)(a + 6), a^_ 2 ab + W=(a-h)(a- h), L.C.M. = 2 .2.3. (a + 6)(a + 6)(a-&)(a-5), = 12 (a' -by. 2. Find the L. C. M. of x^- 8a; + 15, a;^- 3 a;- 10, x^ — X — 6, a^ — 6 x^ — X + SO. ix^-Sx + 15=(x-3)(x-5), a;2-_3a;_10=(a; + 2)(a;-5), x''-x-6 = (x + 2)(x-3), a?-.ex^-x + 30 = (x + 2)(x-3){x-5), 'L.C.M.= (x + 2){x-3)(x-5). 3. Find the L. C. M. of x^ + xY + 2/^ ^ + y\ x^-f. a;4 + r^y'^ + / = (a;- + xy + if) {x^~xy + y'^), ^-\-y^={^ + y) C^*' - ^y + y% ^ — f=^(x — y) (aj2 + xy + t/^) , L. C. M. = (a; + 2/) (^ — y) ('^" + i»2/ + 2/^ (^ - «^2/ + 2/^- Rule : Separate each expression into its prime factors^ and write the product of all the different prime factors, giv- ing to each priyne factor the highest exponent which it has in any of the given expressions. Ch. VII, § 108] LOWEST COMMON MULTIPLES 113 EXERCISE LI Find, by factoring, the L. C. M. of the following ex- pressions : 1. 2 :^y^, 3 o?y\ 5 x'^y, 7 xy"^. 2. 4 Q(?y^ 5 2;2z/3, 6 xy^^ 15 y^, 3. 4 a%, 6 a^J^ 18 a^J^, 36 a%^, 4. 5 a^JV, 7 ^2^7^^ 91 ^453^^ 65 d}¥(f^. 6. (a + S), (a2 - 52), ^2 + 2 a6 + 52. 7. a;2 — 2/2, o;^ + 2/^ ^^ — y^* 8. a* - h\ a^ + 2 aW + 5*, a^ + 5^. 9. (m — n)^, m^ — 9^i-^, n& — 7i*. 10. o;^ — 2/^9 ^^ + 2/^ ^^ "" 2/^* 11. m2 — 2 m — 3, m^ — 27. 12. ofi — Ixy -\- 2/2 — 1, (2; — 2/)^ — !• 13. :i:3 _j_ 64^ ^2 + ^ _ 12. 14. ^12 — 2/12, ^6 _|_ yc 12a36% 20 a'^hfi 352;/ 2a-2h a^-^ah + ¥ (a — b)c — (a — h)d (a + b)c — {a + h')d ae — ad + bc — bd ac + ad + be + bd T^ + x^ — x—1 3(2^-1) a2-2a6-362 a^^4,ab + 3P 72 2^3^ V' 8a + 85 9a + 9j^ mx — my mp + mq c? -f- ah IS. 19. 20. 21. 22. 10. 11. 12. ax -{-a h + lx 52 + 5 1 + 6* x^--Y 3(2^-1)' 4(2/ + 1)" 4a3.,_5«5_662 8a2 + 2a6-362 a4 + a2 + i a3_l a2_J2_^+26e a2 + 52 _ ^2 + 2 a6 a6(2;2 + ?/2) + xyifj^ + J2) abix^-y^y + xyicfl -62) afi + x^--[Sx-4: ^ + 2x^-16x-5 By means of § 104, reduce the following fractions to >vvest terms : or 23.^ 24.^ 25.^ x^-6x^-^16x-15 0:^-6:1:^ + 12 a;2_9^-10 2:^4^ 5^_5:^2__82:-4* m* ^ — 5 ^^ + 5 m^ + 4 rw^ — 5 m + 6 a , — a --V b~^ b ~ a ^(—1) b 5(-l) — a -b a —a — a a b -b b -b 118 ELEMENTARY ALGEBRA [Ch. VIII, §§ 118 11 THE LAWS OF SIGNS IN FRACTIONS 113. Since a fraction is an indicated quotient, the laws o signs are derived from the laws of signs in division, § 26 Therefore, By §106, Hence, From the foregoing laws is derived the Rule for Change* of Signs in Fractions : The value of the fraction is not changea if (1) the signs of the numerator and denominator an changed simultaneously^ or if (2) the sign before tht fraction and the sign of either the numerator or the de- nominator are changed simultaneously. 114. If the numerator and denominator are expressed in factors, since by the laws of signs in multiplication, § 25, the product of an even number of positive or nega- tive factors is positive, and the product of an odd num- ber of negative factors is negative, the value of the fraction is not changed if (1) the signs of an even number of factors in the numerator^ or in the denominator^ or in both of them., are changed ; and if (2) the sign before the fraction and' the signs of an odd number of factors in the numerator., or in the denominator., or in both of them., are changed, 1 (^ — a)(6 — g) a»^ _ (a — b)(a — b) __ 1 (a — b){a— b)(a — h) (a — b)(a — b)(a — b) a — b 2 & — a)(c ~ d)(m — n) _ _ ( a — b)(c — d)(m — n) __ _ ^ (a — b)(d — c) (n — m) (a — b)(c — d) (m — n) Cm. VIII, §114] FRACTIONS 119 [The numerator and denominator, or either of them, %j consist of several terms. A change of sign of the jmerator or denominator means a change of the sign of lery term of the numerator or denominator. Thus, ■x^-\-2 xf/— 7r _ a;- — 2 xy + y^ — X- + y' x' — 7/ __ __ x^ — 2 xy -\-y'^ __ _ —x^+2xy — y^ -x'-^t x^ — y^ EXERCISE LIV Reduce the following fractions to lowest terms 1. 5. x^- 6. ax a52- aC" (c-b)(c-d) m^ TV" — 7W (h — a)2 — x^ a? — ac — ah + he he — ah + ac — a^ 9. 10. l2-x-x^ 6a;2+22:-60* a?-4:x^ + x + Q ofi — ^aS^ + Wx—Q {a-h^(h-e^(ie-d^ (a + h^(h + e^{d — c) lio + ^x-^x^-^' 11. 12. 13. 14. {x— a^(x— h^(x— g)(g— ^) (a;+ a^(h — x)(^e— x^(c — x^ (^a^b)(h-e)(e-dXd-a^^ (a + h)(e - h)ld - e){d + a') — m'^+2 m^n — 2 mn^ + n^ (n — m) (n — rn) (n — m) (m — n) (a^-h) (h^ - e) (g2 - d) (d^ - a) lb - a2)((? - h^Xd - e^Xd? + a^ 120 J:L^MENTARY AtG^^BliA [Cii.Vm. §§115,1 H 115. An integral expression is one that does not contaii any literal quantity in the denominator of any term. Thus, 2 a^ + 3 ab'^-\-- is an integral expression. A fractional expression is an expression which contains a literal quantity in the denominator of one or more oi its terms. Thus, x^ + xy-{-^ is di> fractional expression. A mixed expression consists of an integral expression and a fraction. Thus, a + - and oc^ + xy + y^ ~ are mixed expressions^ b ^ x^-^2 If the numerator of a fraction is of higher degree than the denominator, the fraction is called an improper fractions if the numerator is of lower degree than the denominaton the fraction is called a proper fraction. Thus, — -^ is an improper fraction ; and "^ is a proper fraction. "*" "^ REDUCTION OF IMPROPER FRACTIONS TO INTEGRAL ORt MIXED EXPRESSIONS 116. If the denominator is a factor of the numerator? the quotient is an integral expression ; if the denominatoi? is not a factor of the numerator, the quotient is a mixeci expression. ^3 7.3 Thus, = a^ + a6 4- &^ is an integral expression, a — 6 ^3 I 7.3 O 7.3 and "^ = a^ + a6 + 6^ H is a mixed expression. a — 6 a — 6 Ch, VIII,§I16J FRACTIONb 121 Rule for Reduction of an Improper Fraction to Integral or Mixed Expression : Divide the numerator by the denumi- nator. Thus, ^^±jl±l = :,^ + ^ + l. 1. Reduce ,^ ^ :^ ■ — to an integral or mixed expression. 2x' + x-l 2x^ + x-l 4rx'-{-2x^-2x^ — 6a^H-5a;^ + i^ + l ^ex'^Sx'+Sx Sx'-2x + l Sx' + 4.x-4: 2x'^-Sx + 4: -6x + 5 4.x'-Ax^ + 3x'' + x + l ^^^ 3a; I 1 I -^^ + ^ 2x^^x-l "^ "^2x^ + x-l Or, by § 113, =2x'-3x + 4.- 6 x — 5 2 a;2 ^ a; - 1 EXERCISE LV Reduce the following improper fractions to integral or mixed expressions : 2. X^ — X1/ + ^2 X 2x^ + Ax + \ 2x aP' + ixy- .«/2 x+y 3 to2 + 4 TO + b s. 6. TO + 1 ^ — y^ x+y 2:2+2 a^-23? + 2x^ + x-l _ x^ + x-1 a^-Sa^ + 2a-l 122 ELEMENTARY ALGEBRA [Ch. VIII, § 117 REDUCTION OF FRACTIONS TO EQUIVALENT FRACTIONS HAVING THE LEAST COMMON DENOMINATOR 117. As in arithmetic, the least common denominator — abbreviated L. C. D. — of a number of fractions is the L. C. M. of the denominators. w T3 -, 2 m 6 m^ 12 mn , • i i. j ^ • 1. Keduce •— — , -:: — , _,^^ ^ to equivalent fractions 6a^ b a 10 a^ having the least common denominator. The L. C. M. of the respective denominators is, § 107, 30 a?. Take 30 a^ as the L. C. D. ; and divide 30 o? by the respective denominators, 3 a?, 5 a, and 10 a^, thus obtaining the respective quotients, 10 a, 6 a^, 3. 2 m(10 g) ^ 20 am . ^ mH% a^) ^ ^Q a^m^ ^ 12 mn(3) ^ 36 mn 3 a^lO a) ~ 30 a^ ' 5 a{Q a') 30 a' ' 10 a\3) 30 a^ * 1 1 2. Reduce — and to a" — an — ac + be oc + ac — ao — e^ equivalent fractions having the L. C. D. Factor each denominator, and simplify by § 114, if possible : 1 ^ 1 a^— ab — ac+ be (a — 6)(a — c)' be + ac — ab — c^ {b — c)(c — a) (c — b)(a — c) The L. C. M. of the denoininators is (a — b)(a — c)(G — b). Divide the L. C. D. by the factors of the respective denominators, thus obtaining the respective quotients, c — b and a — b. (a - b)(a - c)(c -b) (a - b){a - c)(c - b)' 1 (a— b) a — b (c — b)(a — c)(a — b) (a — b)(a — g)(g — b)' Cii. VIII, §117] FRACTIONS 123 Rule for the Reduction of Fractions to Equivalent Frac- tions having the L. C D. : Simplify each fraction^ and express the denominator as the product of prime factoids. Take the L, C. M, of the denominators as the L, O, D. Multiply both terms of each fraction by the quotient found by dividing the L, C, D, by each denominator, EXERCISE LVI Reduce the following fractions to equivalent fractions having the L. C. D. : 1. J_, A., i. 6. -i-. ^±^. 2. 10 a 4 a^ a^ x + y X — y 20^ 5x 6 „ a — c a + c , . • c— d c + d 8y 2/ 12 y^ 3x- y, ?/-^^. la^ 4: am a—h a + b 5xz \2x^ a a a + b a — b 11. — 1 9 8. 5. -•» -• 10. x^ -{- y x-\- y^ ^ x^ — y'^ X -\- y x^ y'^ a? — y^ x^ — y^ Sx — 4:y 5y — 8x x^ — y^ y^ — a? 12. 13. 14. (a — b^(m — n) (J — a}(m + ?i) a^ V^ (^a-b^(b-ci (b-a){c-b^^ cP" — ac b^— be (a - J)(J + c}Qa -c) (a - 6)(J - c}(c - a) 2m-3 3m4-7 2m^-5m-{-o' 3 m^ - 2 m^ - 18 m -f- 7 * 124 ELEMENTAllY ALGEBRA [Ck, VIII, §§ 118, 119 ADDITION AND SUBTRACTION OF FRACTIONS 118. By§56, ^+^ + ^ = %l^ + g. *^ a a a a Therefore, tlie sum of a number of fractions having a common denominator is a fraction whose numerator is the algebraic sum of the numerators and whose denominator is the common denominator of the fractions. By the law of signs in fractions, § 113, a fraction in the form — -- may be changed to the equivalent form, -] — — -. b h Find the alerebraic sum of - — h — "" 3 a 3 a 3 a ^ By§113, -A== + =:^. I 3 a 3a The algebraic sum of the numerators = 2 x + ^x^ — h. The common denominator = 3 a. 2x bx'' -b^ 2x + 5x^-b 3a 3a 3a 3a 119. If the denominators are not common, the L. C. D.| may be found by § 117 and the fractions added as before. 1. Find the algebraic sum of ^^ \-- tt' 3 a 2x a^x The L. C. D. = 6 a^x. 2x _2ss(2ax) iaa^ Sa 3 a (2 ax) 6a'x 3a _3a(3a') _ 9a^ 2x 2a;(3o^ 6a'x' 1 _-l(6) . * -6 a^x a'x{&) 6a^x 2x , 3a , -1 4.ax^-^9a^-6 J 1 = — ^* 3 a 2 X a^x 6 a^x Ch. VIII, § 120] FRACTIONS 125 Rule for Addition (or Subtraction) of Fractions : Reduce the fractions, in their lowest terms^ to equivalent fractions having the least common denominator ; the sum of the frac- tions is a fraction tvhose numerator is the algebraic sum of the Jiumerators and whose denominator is the least common denominator of the fractions. 120. It should be carefully noticed that the sign of division in fractions is a sign of aggregation. o? 4- ob 4- h^ Thus, ' — means that the whole of the numerator, a^-2 ah + Ir a^ -\-ah + b^, is to be divided by the whole of the denominator, d^ — 2 ah + Ir. If the fraction ^ ^ be preceded by the a- — 2 ab -{-¥ minus sign, the whole process is indicated : - (a^ -\-ab + b") ^(a''-2ab + b% That is, the minus sign before the fraction is to be inter- preted as affecting the whole of the numerator. Thus, a^b a—b _ (a+-b) — (a—b) ^ 2b a'+ab + b'^ a:'+ab+b^~ a'+ab-[-U' '^ a^+ab+b^' or, by § 114, a-\-b a—b a+b . —a-\-b __ 2b a^+ab+b^ a^+ab + b^ a'+ab + b^ a^+ab + b^ a'+ab + b'' 1. Find the alsrebraic sum of — : The L. CD. = 12 a. a-\-b ^ 4:(a + b) ^ a—b ^ S(a — b) 3 a ~~ 12 a ' 4 a "" 12 a * a-{-b a-b _ 4:(a + b)~3(a-b) __ 4a-\-'ib — 3a+3b _ a-\-7b 3a 4a ~ 12a "" 12a *" 12a ' 126 ELEMENTARY ALGEBRA [Ch. VIII, § 120 EXERCISE LVII Find the algebraic sum of the following fractions i 4. ^ + A. 7. 7 + 1. OX 4:X a 1. i_£. m m 2. 3 a a 7 7 3. m n ^ a^ X 1 . 1 a — a; 5. _^ + -i^. 8. ^^^ — ^ + 1. 6a 96 X 6. 4-^- 9. 1- ^ ^ xy a-\-h ba-Yll lh-2a a-4b 16 "^ 12 8 ■ .. 2a;-5.y-3 3a;-8.y + 45 , ^ "• 15 25 "^"' a + 4 6 2(a-36) ll(a + 6) ■ 10 15 20 ' 13 3(x-y) 5(2x-3.y) 7(x-2,y) _ 4 6 8 5^ + 16 6-14 , a-66 + 7 3a + 14J-15 14 33(a + l) 36(a + l) 44(a + l) 4a4-7 6' 3 45-3a 3a-2c 6 be c 2 ah ac 5a; + 3y y + 2g 3a: + 4y x + ?,z 1 ?>x^ iyz ^xy 4xz 4 a; ^^ a2 + 4 5c 62_3ac (16 + 4^2 ^ , ^ . Sac 6a6 46e 3a 2b' 2 2(6 a- b) 1 a + h Sa^-b^ ■ 5a 15 a6 3 J 10a2 10a26 * a-2b Sa-4b 1 Ga-5b a^ + 2b'^ ' 4ab 6a2 2b 20b^ Za% Ch, VIII, § 120] FRACTIONS 127 20. 4:(ah + xy^ ay—^lfi a^ — 5bx h y 15 hx ^ab 6 ax 2a 3 6 10a; 21 (1 + ^)^ (l + y) " I 2a;-2.y + l I 2x+^/ -1 -^ -- ^ ^y7 b 22. - H -• 27. x^ y' 2 3 + 1 a — \ a a + m h 1 + m X _^_ X — ■J.. y X x + y xy xy'^ 23. ---^ 28. 24. - — ^ — "T ^. 29. 3 2 a; — 11 a; -7 3(x + 2) cc + 5 b(x 2x -2). + 1 x-13 X -18 25. . 30. 1-x 1 + x 10(a;-3) 14(a;-2) 26. -1^--U. 31. '^ 3 a-6 a + b Zx-S 2x-2, 2.t;-13 3a;-16 O^. 10a; + 10 15a; + 45 33. 3 5 2 2a;-4 6a;-12 3a; + 6 34. 1 1 ^-y 2 . y x^ + xy x + y 35. X- y X — a y — b 7 ' .7 36. 37. xy ax -\-bx ay + by 5 . 3 13a + 7 6 4(a + 6) a-b 4{a^-b^) 5 7 x-4 x — 2 x — S a^ — 5x + 6 ,„ 21a; + 13 5x , 16a;-3 OO. — —— • j- 12a; + 24 Sx-d ix^ + ix-^ 1 128 ELEMENTARY ALGEBRA [Ch. VIII, § 121 121. If some of the factors of the denominators are alike except that their terms are not arranged in the same order, they may be made to take the same order by § 113. 1. Find the algebraic sum of 1 1.1 1 1.1 (a—b)(b—c) (6— c)(c— a) (a—c){a—h) 1 1 {a — b)(b — G) (b — c)(G—a) {c — a)(a — b) 9 _ c — a — (a — b) — (b — c) 2 (c — a) (a — b)(b — c)(c — a) (a — b){b — c){c — a) 2 (a-b)(b-c) 2. Find the algebraic sum of ' +=j — A^.+ * x+l 1 — x — 1— a^ 1 + a^ 11 2,4 a; + 1 ic - 1 — 1 - a;2 1 + a;* a;-l-a!-l 2 4 ^ -2 2 4_ -1) ar' + l a;^ + l a^-1 a^+1 oc* -i- I (a; + l)(a!-l) ar' + l a;' + _2a^-2 + 2a^-2 4 ^ -4 4 x*-l "^x^ + l a;^-l a;* + l' -4a;<-4+4a;^-4 _ -8 _ 8 a^ — 1 a^ — 1 1 — a;* Ch. VIII, § 121] FRACTIONS ' 129 EXERCISE LVIII Find the algebraic sum of the following fractions : ^ _L_ 1 L- x^ — 1 x + 1 1—x 7? 1 , x^ 2. 3. x^ — ]f y — ^ y^ — ^ a __ 2 a 3 db a — h a + b IP" — c? x — Zy Sy + x 9 y^ — 3^ 1 1 (a;-l)(a;-2) (2-x)(S-x) . (l-a;)(a;-3) 6. I I + — J . (a — h^(h — c) (b — a'){a — c} {c — J) ({? — a) c + a b + c a + b (a — b^Q) — c) {a — b}(^a— c) (^c — b)(^c —a) 8. ^! + ^! + ^ . («— 6)(a — c) (h — a)(h — c) (^c — a)(c — h) (a—m')(b — m') {h — m)(c—in} (c—m')(a — m') (a — c)(b — e) (b — a)(c — a) (c— 6)(a— 5) 10. a . h {a-b^ia-cXa-d) (h-a)(b-c')(b-d) (c-a)(c-fi)(c-(;) (c;-a)(cZ-J)((Z-c) a b-\-a b—a a— "lb —lb — a -1^2 1 _^ 1 . \x 32 :?^ a + 2^ 2;?; — a a2 + 4;i;2 a^ + 16^* 130 ELEMENTARY ALGEBRA [Ch. VIII, § 122 REDUCTION OF MIXED EXPRESSIONS TO FRACTIONS 122. Since a-h 1 = a^ any integral expression may be written in the form of a fraction having the integral expression for a numerator and 1 for a denominator. Thus, a + 6 = | + | = ^. Hence, the Rule for Reduction of Mixed Expressions to Fractions : Write the integral part of the expression as a fraction having the integral expression for a numerator and I for a denominator^ and add the fractions, 1. Reduce a — h ■^-—- to a fraction. a + a-\-b 1 a + b a + b EXERCISE LIX Reduce the following mixed expressions to fractions : 1. a + b —--- 6. — \-2 — m- a + b 3 2. m + -—' 7. -— — a + b* m — 3 a — b 3. x^ + xy + y'^ • 8. x + y ^• X — y x + y 4. x^ — xy + y ^ 9. 2a — 6b — ^f — -• ^ ^ x + y Sa-2b 5. a^ + ab + b^-- — -• 10. m^-n^ iT~"2' a^--ab + b^ m^ + n^ Cii. VIII, § 123] FRACTIONS 131 I MULTIPLICATION OF FRACTIONS a c 123. Let 7 and -- be any two fractions ; and let the b d product of these fractions be P. Then, a c p (1) multiplying (1) by 6, ah c rjy (2) multiplying (2) by d^ a (3) simplifying (3), dividing (4) by bd^ ac = bdP, (4) ac _ -p (5) applying Ax. 5 to (1) and (5), 1 a c ac b d bd (6) r The product of three fractions can be found by multi- plying the product of the first two by the third, and so on. Rule for Product of Several Fractions : The product of any number of fractions is a fraction whose numerator is the product of the numerators^ and whose denominator is the product of the denominators of the given fractions. 1. Find the product of — ^ x —^ x ^• o y^ 8 10 2o(? 5 xy ?/^ __ 10 x^]f _ x^y 32/2' "s" * 10""240^~24* 132 ELEMENTARY ALGEBRA lCh. VIII, § 122 In multiplying several fractions the process may be simplified by cancelling the common factors before finding the product. 2. Find the product of ^^ • ^+/-^t -^ • ^P^- ^ — y^ XT — y^ x^ — y^ ■y? + ]f 3? + xy 4- if {x — yf (x-yy (x^)(x^J^j:^-r-f){:x + y){x,^^{o^A^){x + y){x^^ _ {x — yY _ fx — y"^^ ~ (x + yy~[x + y^ EXERCISE LX Find the product of : 3 3^ a^b^ ' ofi + y"^ x-^ y ^* 9/* 20a^3" ■ a3 + 63* 12a* g 4 m^n^/) ^ 2S a^y^ _ ^^ 2x + U ^ a^-9 7 it^y'^z m^^p ' a — 3 3 a; + 21 7 g^m^c* 26 a^y%^ (x - ,y)* _ a^- j/* 13^^7 ■ 21 a%*c3' ■ a^+'i/2 ■(^_^)S . 44 a25a^ 26 m% ' 12 '^^ + ^''' '^ + "^ ■ Qbanfi 33 aJa^ a2_i6 ^2_3^^ q ®" 32 a*5*~ ■ 81 d^-2x-x^ Zx^-8x + ^' 2x^ + bx-12' x^-'^x + ^' W^ )\?>a-b A ?>a + h) 2^:2 + 2^2:- 8:1^- 8 a 2^:2-2 t«.r-3 a; + 3 a x + 2 x^ — ax + 2x—2a x^ + ax — 4:x — 4:a 2x — S a:3_2a;-l x^-x^ + 2x^-x-\-l x + 1 x^ + 2x^ + 2x + \ ofi-\ aj2 + l ^ + ^2^1 *^2 + l' ^_1 7yi2 — a2_|_2 ^—1 (??2 + a + l)2 m — 1 7Y?-\-2am-\-<£^ — \ m^ — am + a — 1 m — a + 1 7722 + 3 m — 4 73i2 + am + 5m + 4: a + 4: 134 ELEMENTARY ALGEBRA [Ch. VIII, §§ 124, 125 DIVISION OF FRACTIONS 124. The reciprocal of a fraction is a fraction formed by interchanging the terms of the given fraction. Thus, the reciprocal of - is -: the reciprocal of a is — ha a The product of a fraction and its reciprocal is 1. Thus, = — = 1. n m nm 125. Let y and - be any two fractions; and let 0' a ' M=<>. (1) Then, by §26, f=^-5' (2) multiplying (2) by ^, f • ^ = ^ • J • ^ = 35(a2 - 62) ^ 28(a - h-)\a + J) 12(a!-iy . 15(1 -a;) 4 4 __ n(J'-^0--x) 28 (a-^)(»-^(^H^^ 6 5 _ 16(l-a;)(a-6) 25 Cn. VIII, § 125] FRACTIONS 135 1. EXERCISE LXI Find the quotient of : S ax 5 a 8 55 a 24^' 12 xy 15 aV 36^ _^ 24^3^ 27 a2J3 ' 81 a% * 40 a^h^c^ _^ 35 a%^c\ 22nfixh^ ' 88m^xz^' 'a^ + h^ ^ 9yCa-4:b) _^ 6y\4h-a) 22 a\v{a + J) 55 ax'^Qa — 6) 6. — ^(a— 6). 8. (la + i5)-(|a+|6> a2 + aJ + Sn . «* - ^* 10. 11. 12. 13. 14. 15. 16. 17. 63 (a2 -68 62)2 raM-63 [a* + 6* U2 + 62 ■]■ a + 6 a* + 6* a* — 68 V ja + 02 _ c2 _^ a2-_(6-c)2"] (a + 6 + g^ . [(J + c)2_a2 ■ c2-(6-a)2j(a-6-c)2' 2)3 _ 2 ^2 - 2 a; + 1 a;2 + a; + l "| y?~\ -1 J(^ + l/ a;2 - 2 X + 1 2?- .y)''-g^ (ir + g)2- o2j- t/2-(a;-g)2_ L8x2-2a;-l ' 3a;-l-2a^J" 5a;-l 2/ + 2)2 (a;-^ + 2)2j • a;2_(^+2)2 + 2a^ r- m^ — m — 6 2 m^ - m- m^ — 5 m + 6 6 m^ — 11 m + J^ 2 m^ — m — 3 3 m^ — 10 w + 3 136 ELEMENTAKY ALGEBKA [Ch, VIII, § 126 COMPLEX FRACTIONS 126. A complex fraction is a fraction having one or but! of its terms in the form of a fractional expression. Thus, -; — -T-, and -' are complex fractions. X X The process indicated is merely one of division, — aftei the numerator and denominator have been simplified. Hence, the Rule to Simplify Complex Fractions : Dividi the numerator hy the denominator. 1. Simplify a — b b 1- a b a- -b a\ f a — h \ —br{-r} 1 b—a a—b -^. = -1. If the L. C. D. of the denominators of the several frac tions is easily found by inspection, it is sometimes prefer able to simplify the complex fraction by multiplying botj numerator and denominator by that L. C. D. -? bii-'^ hj h — a___A '{^) " Cn. VIII, § 127] FRACTIONS 137 i27. A continued fraction, that is, a complex fraction in the form , is simplified by beginning at the last fractional expression and working up^ 1. Simplify j . 2x ^ 1_ 2a?-l 2x • 2x 2x 2{x-l) 2a;-l 2x-'l 1 2(2 g; - 1) "" - 2 ' x — 1 x — 1 -2 Note. In § 52 it was stated that can never be taken as a divisor; I hence;; if h = 0, the form y may be considered impossible. The defini- tions of fractions hitherto given must be understood to exchide as a denominator. 138 ELKMKNTAUY ALGEBRA [Ch. VIII, § 127 EXERCISE LXII Simplify the follow kviiig fractions : 1+1 X 1 » h a 3 r y ^* h c m X n y 2. ^. X m n y 5. h ' a — h h a — h a a -\- b y 1 + (" ^-yy 4x1/ 1- X — 3y 7 x + y Bx + y -3 x + S X- X — -y . 2 8. X — 3 x + i 1 1 7 9, x-n [ 11 4 12. 1-1 X 13. 1- 1 —2 2 3 lOrr x-y 14. X a%^ 23 62 - 9 ~r 1- 10, 1+1 1 5 6 a62 « a X X ,2 J2 15. ,^ a — ha + b q,1 11, _ . t) -J — +—— 1-1 (a - hf (a + iy- X Ch. VIII, § 127] FRACTIONS 139 REVIEW EXERCISE LXIII 1. Simplify the following fractions: 5 a:^ + 4 a: - 1 a^+Sa + 2 5. 1 12a;2 + 24a:-15 a^+1 a + 12 y x^ — y^ 3. ■ 1 __ 1 X? ■\' y^ ofi—^ y^ x-\-y y x x^ — y'^ x^ + ^xy + y"^ 3 o;^ + 6 a;^ — 3 a; — 6^ xy*^ + y^ ^ x^y'^ — x^^ x^ + S x^ + 2 X x^ + xy + y^ y^ ^ ^^ 9. 10. 11 — ! a _b a2_52 £ — 1 cfi + b^ b a^ — x^ , a^x + x^ a^—2ax + x^ a? — x? x^-^x + 20 :r2-13a: + 42 Q X x^ — 5 X lo /"^ I ^ — ^A^ fi b'-a\ 1 ^ _^ f a + x __ g — a^ Y a — x) \a — X a + xJ f x + y o^_+jf\ _^ f x + y ___ ^Mv^Y x^ + y\x — y x^ — y^J \x — y x^-^y^J 14. (?-- 1-4 a;?/ 140 ELEMENTARY ALGEBRA [Ch. VIIL § 127 16 3 2:^;-! ^^ Sx'^+2x-l "^+2 + 2 2m- 3+1 18. —^ :j 19. 20. 21. ^4 _ ^4 ^2 _ a;2 x3 + a;2- X- -1 a2 + 62 b a a? -J2 1_1 b a + 52- 2:i;2^5r?: + 2 , 2 2;2 4.9a; + 4 ^2 — 4 * a; + 4 1 X 22^ ^ - y a:^^ - y^ ^ ofi-2xy + y'^ ^ X y xy xy + y^ x^ + xy /^^_jf4 ^ a; + y \ _^ fo^_+^ _^ x + y \ \x^ — y'^ x^ — xy) \x — y xy — ^v /-J ah V-| a h \ ^ a^ — < • V a^^ah^hV\ a? + 2ah + hV ' a^-{-. a^ — h e V^ + ca c^ + ah 25. 7Z~z r — 1 — 71 . , , T" ~r ' ?6. (a — 5)(^ — ^) (5 + c)(ft--a) ((? — ^)((?H-i) a;2 — ^^^ y"^ — zx ^ — xy {x + ^/)(a: + ;2) (?/ + 5:)(2/ + :r) (aJ + x^\z + j/) 1 a 27 5 15 28. ^2 ^ + 1 1 a_ X + -" 1 i- a; — 1 a — 1 Ch. VIII, § 127] a — h FRACTIONS 5-. 29. \-\-ab 1 + hc . (^a — b)(b — e^ ~ {l + ab){l + be} 30. ^24. M ^2+52 (^2-52) a+b '2\ a+ b 31. (a2 + 62). J4 J2- a + b a — b -1 32. 33. 34. 35. 36. l-\-'l/\l-^X 1- x^+2/'^~x + y \ 1 - y 1-2/^ a+b a—h a — b a + b. 1 1 . (a + h a-l\ ' \a-b a + b) X y 141 a — x a — y (a — x)^ {a — y)"^ 1 1 {a — y)(a — x)^ (a — x)(^a — y)^ ■X , x-y , C.y-g)(g-a:)(a;-y) a^ — he . Ifi— ca , e^ — ah + + X y , z 37. If — — =a, 3/4-2 a; + 2 h. x + y = (?, find the value of 1+a 1+b 1+0 CHAPTER IX SIMPLE EQUATIONS 128. Some forms of equations have already been defined and discussed in Chapter IV. As before, § 64, the last letters of the alphabet are used to represent unknown quantities, and the first letters are used to represent known quantities. 129 An integral equation is one which does not contain the unknown quantity in any denominator. A numerical equation is one which contains the unknown quantities and numerical quantities only. A literal equation is one which contains other literal quantities than the unknown quantity. Thus, 2 a; + 3 = 11 is both integral and numerical ; a + x = b 2 4 is both integral and literal ; - + 3 = - is both fractional and X X numerical ; - + 6 = c is both fractional and literal. X 130. The degree of an equation in one unknown quan- tity depends upon the highest degree which that unknown quantity may have in any term. If the equation, in its simplest integral form, contains the first degree of the unknown number as the highest degree, the equation is said to be of the first degree, or a simple, or linear equation. Thus, ax + b = c is a, siini)le equation. 142 (H. IX, §131] SIMPLE EQUATIONS 143 NUMERICAL FRACTIONAL EQUATIONS 131. Two equations are said to be equivalent when the roots of the equations are idefitical. The general method for the solution of simple equations consists, as in § 68, in the transformation of the original equation into a series of equivalent equations, until such a simple form s obtained that it contains as a left member only the un- known quantity, and as a right member only the known quantity. n . 1 2 a;- 3 1. Solve 7^.-ij + jL,= f:^. (1) Simplifying i,i (1), 7 a, -I + 1 = ^^. (2) Multiplying (2) by 36, the L. C. M. of the denominators, 252a;-T(4)+3 = 6(2a;-3), (3) liniplifying in (3), 2r)2 x - 28 + 3 = 12 a; - 18, (4) ransposing in (4), 252 a; - 12 a.' - 28 - 3 - 18, (5) uiiting in (5), 240 x = 7, (6) lividing (6) by 240, '^ = 2i0* ^^^ 2. Solve 5 + ^:^-1- = 15. (1) Multiplying (1) by 28 x, 28(8) +4(5 + 0;) -14(3) =7a;(15), (2) implifying in (2), 224 + 20 + 4,0; -42 = 105 a?, (3) ransposing in (3), 4 o; - 105 x= - 224 - 20 + 42, (4) niting in (4), - 101 a; = - 202, (5) ividing (5) by - 101, a; = 2. (6) 144 ELEMENTARY ALGEBRA [Ch. IX, § 131 To solve a simple equation in the fractional form and con- taining one unknown quantity • Multiply every term of each member of the equation by the L. (7. M of the denominators ; transpose the unknown terms to the left member^ and the knoivn terms to the right member ; unite similar terms. Divide every term of each member of the equation by the coefficient of the unknown quantity. Multiplying by the L. C. M. of the denominators is called clearing the equation of fractions. EXERCISE LXIV Solve the following equations : X X 3 42 1 1 _40 ' X X 3x 3 . u?+?=l. XXX 4. ^ ^ -1. 5x lOz 10 9 X X S X 9 X 6. J- + J- + -1 + J— l^=.o. Qx 12x Sx 24:x 72 ,. 4 + ?-^-UI = 0. X 11 :z: 6 X 5 97 5(11 -3a;) ^ 7-9 X X 1 6x 2x Ix ^ Sx + 1 5:?:-l ^ 9a; + 5 ^21 5 4:x 8x bx 5 sc 10. _L_Jl + i3_ = 4_l 3. Sx 12a; l(Ja; 3 6 8 Ch. IX, § 132] SIMPLE EQUATIONS 145 132. The method of procedure in case the denominators contain several terms is the same as in § 131. 1. Solve ^_---ll- = 13. (1) Factoring the denominators in (1), r 11 :13, (2) 2(4a; + l) 6(4a; + l) multiplying (2) by 10 (4 a; + 1), 5 (7) -2 (11) = 13 (10) (4 a; + 1), (3) simplifying in (3), 35 - 22 = 520 x + 130, (4) transposing in (4), -520 a; = -35 + 22 + 130, (5) uniting in (5), - 620 a; = 117, (6) dividing (6) by - 520, a; = - ^^^ = - ^%. (7) 2. Solve ^^_^ = 4. (1) x — 1 x + 1 Multiplying (1) by {x -l){x-\- 1), {5x-\-l){x + l)-{x-^){x-l) = 4t{x--l)(x + l), (2) simplifying in (2), 5a;2 + 6a; + l-a;2^10a;-9 = 4a;2-4, (3) transposing in (3), 5x^-x'-4.x^ + ^x + 10x:=-l + Q-4.y (4) uniting in (4), 16 a? = 4, (6) dividing (5) by 16, a; = J. (6) Although (3) contains x^, yet the equation can be solved as a simple equation because the simplified form, (5), contains only the first power of the unknown quantity. Note. Each term of a fractional equation which is in the frac- tional form should be reduced to its lowest terms. 146 ELEMENTARY ALGEBRA [Oh. IX, § 132 EXERCISE LXV Solve the following equations : x + 2 3(a; + 2) 3 3(^-7) 6 2a;-14 2 5 3 ^5 ^^ 120 ^ x-\0 x + \0 x + 1 2x + 2 2' ■ 144-x2 12 + x Vl-x 9 7 _13 ^^ 5x , 7 + 4a;_-, 6-5a: 2a; + 2 3a; + 3 12 a;-3 4a;-7 a;-3 1 1 ^1 2-3a; 4-6a; 6' 12. 5. JL^— 11- = .. 13. 3a^ + 42 ^ + 5 1 6a:2+7a ^ + 8 2 x-^ ^ 10 + x-A = 2. a; + l 2a; + 2 2 x 4 , 31 _1 ^^ 5a;2+7,y+4_3a.2+63;+7 2a; + 2 3a; + 3 6 Ibx^+x-ij 9a;2+6a; + 3 7 13 7 ^ ^ ^5 7 3^-2 2r?: + r> ^ lly+3 a;- 2 5a^-10 ' ' 5x+3 3a: + 9 15a; + 9' 8 15 ^ ^ 11 16 5a: + 2 3x + 1 ^ 3a: + 2 "3 — 2a; 6 — 4x * '4a; + 3 Ga; + 2 4x-t)' 1 11 a; 4 17. 18. 2a; + l 12 2a: + l 3(2a; + l) 5a: + 4 3(a;-7) ^^3 9 a;— 1 o(a;— 1) 5(a;— 1) 19 5_+J ,2x-l 7a;4-5_a;2-25 20. 4 3a;-12 8a;-32 4a:-16 13a; + lQ 2ri0a;+l) 7-lla; ^ 28a;-32 49a;-56 35a;-40 1 + a; , r , 2x2 _,, \ + x 21. 2::li:i + ,5 + -j^2=3-^, . 3^ -^ 9 "^ 3 ' *^ Cii. IX,§133] SIMPLE EQUATIONS 147 133. If the equation contains several terms in one (k nominator and several simple denominators, the pro- cess of solution is much simplified by first multiplying every term of each member of the equation by the L. CM. (){ the simple denominators and then simplifying the resulting equivalent equation. 1. Solve- -^ — = — -^ ^ — (1) x + 4: 15 5 3 ^ ^ Multiplying every term of each member of (1) by 15, I^-(16x + 59) = 3(3x + 2)-5(5x + l), (2) X-\-4: simplifying in (2), X-{-4: transposing integral terms in (3), 75 a; x + 4: = 16a; + 59 + 9a; + 6-26a;-5, (4) 75 a; uniting integral terms in (4), = 60 (5) X ~\~ t: dividing each member of (5) by 15, ^ = 4, (6) multiplying each member of (6) by a; + 4, 5 a; = 4 a; + 16; (7) transposing and uniting in (7), a; = 16. (8) Some equations which appear to be higher than first degree equations may be solved, by various devices, ns first degree equations. 148 ELEMENTARY ALGEBRA [Ch. IX, § 133 2. Solve -- = 7 -• (1) x-2 x-8 x-4: x-~5 ^ ^ Uniting the members in (1) and simplifying, -1 = ~'~ (2) (x-2){x—3) (x-4:)(x-5y ^ ^ multiplying each member of (2) by the L. C. M., (x _ 4)(a; - 5) = (x - 2)(x - 3), simplifying in (3), 0(y^ — 9x-{-20 = x'^-~5x + 6 transposing and uniting in (4), — 4 a; = — 14, dividing each member of (5) by — 4, x = |-. 3. Solve 2^1+^ + ^^^ = ^^+1^. x—1 x—2 :r + 3 Eeduce each fraction in (1) to a mixed expression, x — \ X — 2 a; + 3 uniting integral terms in (2), 5 . 10 ^ 15 a;-l"^a;-2 .^ + 3' dividing every term of each member of (3) by 5, 12 3 a;-l a?-2 aj + 3' ^ ^ multiplying every term of each member of (4) by the L. C. M., (^_2)(a^ + 3) + 2(a:-l)(a; + 3)=:3(a;-l)(a;-2), (5) simplifying in (5), a;2_|-a;_6 + 2a^ + 4T-6 = 3a;2_9aj + & ('6) transposing and uniting in (6), 14x = 18, (7) dividing each men^ber of (7) by 14, a; = -f-. (8) :ii. IX, § 133] SIMPLE EQUATIONS 149 EXERCISE LXVI Solve the following equations : ^ 2x+7 Sx + 8__4:x+J^ 2. 3. 5. 6. 10. 11. 12. 4 5x + 2 8 Qx + l lla;-l_2x + ll 15 4a;+3 5 5x + l 3x + 2 15a;-39 12 5a;-8 36 2a;+3 2a;-l x + S 10 42^+2 5 a:j— 3 x — 4: x—5 x — 6 x — 4: x—5 X— Q x—1 X+1 _ X+2 ^ X+8 __ X + 4: x + 2 x + S'^ x + 4: x + 5 21a; + 13 8a; + 13 ^g 3^+1 4:i; + l 9x+2 5x+2 ^Q Sx — 1 x + 1 2ir + 3 , 4:x + 5 = 6. x — 4: X — Q x + 8 __x + 4 ^ x + 5 __ x+ Q x+4 x+5 x+ 6 x+ 7 4a; + 5 _ 14a;+3 ^ 16a: + 3 9 35a^+l 36 X -10 13a;-2 47 2:-! 6rr + 7 ^. 7 10^ + 7 35 5 13 3rr+8 5a: + 8 10a; + 27 ^ ^g 'a; + l :i: + 2 a; + 3 150 ELEMENTARY ALGEBRA [Ch. IX, § 134 LITERAL FRACTIONAL EQUATIONS 134. Literal fractional equations are solved by the Rule given in § 131. 1. Solve a + ^- = ^-h. m XX Multiplying each term of (1) hj x, ax-{-b = a — hx, (2) transposing in (2), ax + hx = a— h, (3) factoring in (3), x{a-\-'b) = a—'bf (4) dividing each member of (4) by a + 6, x = ^^^ — (5) 2. Solve ^^^^^ + d = x + a. (U c ^ ^ Multiplying (1) by c, ax—b + cd = cx + ac, (2) transposing in (2), ax— cx = b — cd + ac, (3) factoring in (3), x(a — c) = b — cd + ac, (4) b —Cd + aC yr^. x = • (5) a — c ^ 1. (1) Multiplying (1) by a^—b'^, bx= (x+3 b)(a-b) - (a^-b'), (2) simplifying in (2), bx=ax—bx^3ab—3b^—a^-{-b'^, (3) transposing iu (3), bx—ax-\-bx=3 ab—3 b^—a^-{-b^, (4) uniting in (4), 2bx — ax = — a^ + 3ab — 2b^y (5) factoring in (5), x(2 b — a)= — (a — b)(a — 2b)j (6) dividing (6) by 2 6 - a, x = - (a - 5)(-- 1), (7) simplifying in (7), x = a — b. (8) dividing (4) hj a — c, 3. Solve Z"^, = x + Sb a + b Ch. IX,§134] SIMPLE EQUATIONS 151 EXERCISE LXVII Solve the following equations : ax M ^ m , n 1. —=1. 5. —+— = (?. ax ox ^^ i^ V ^^ X , X , X 2. — = -• 6. — H f- = a. m n p ^ w -, ^ ^ ax bx ex . 3. --1 = 7. — + — + — = c?. " - m n p a b e ^ 8. — + — 4._ = ^. ma: nx xp ax T'' = 1. a bx'' _b^ a a 1 = a — 1 x X X ,^_ - a. m n I 9. ^4.A + _^=a2 + 52 + ^. 6(?a; acx aox ,^ ab , ae , be 1,1,1 10. _++_= + +. ex ox ax a- o^ c^ ^^ rr+^ , 7 __ 2(x - b) ^ ^ b 3 Sa ^2 SCx-4b) 5Cx-Sa) ^^ a 46 __ ax — l . bx — la^ + 13. [■ 14. 15. b a €^b^ ex + ab j_ax+^ cfi^ a be b b'^ a 1 a^ ax d^ 2bx 2 2 62 b-\-e a + e a + b _2 box aex abx c __ ax 1 17. r = l— ;c. 18 X b + x a + b a 19. a + x ^__ 1 b + e a 20. ab — c^x a __ ex a-b Ze~3b 21. X a b + ex e(l + a} 22. ax bx 9 = a^ - a—b a—b a+b 152 ELEMENTARY ALGEBRA [Ch. IX, § 134 X — 2 a + Sb _ 5 b ax + be _bx — ac _^c ' x + 2a — Sb'~ ia — b ' b + c d^-^V^~ a X — a . x-\-3b Q „^ ctx^ + bx + 1 a a — 5 a + b bx^ + ax + 1 b X 9 X— 2bc 28. 31. 33. 34. 36. fx a[ i 27. a --b a -{-b a — b 2bx , ^^ — ^ __ ^(^ — 2 ab^ ___ 7 ab + cd a ab — cd ^ X — ab , X— ac , x — bc o be + a^ , c 29. 1 H d = ■ + -. ae be ab ac (b — e)x , ex + b^ 2b — c 30. v^ ^ -I = . bx — e^ e(x + J) b ax + be bx + a{e — 1) _ t^ — 5 , (a — 5)(a; — g) + 5 bx + ae ax— be a bx + ae a , X — ae 1 bx— a 4 / ' 1 -• r ! / / ' /^^ J 7 i 7 / / kj / r M EXERCISE LXXIII Construct the graphs of the following equations : 1. y=^X, 3. X+9/=:0, 5, SX+8 9J=0. 2. 5y = lx. 4. 2a;— 5y=0. e. x—5y = 0. 162 ELEMENTARY ALGEBRA [Ch. X, § 14( 145. Any simple equation in x and y^ wliicli contains: a known quantity, can be reduced to the form y — ax + h, If the graph of the equation y — ax is plotted, and f romi every point on this line lines parallel to ^ Z'and equal in length to h are drawn, the extremities of these lines will evidently be the points whose coordinates satisfy the equa- tion y = ax-\-h. These points are also on a straight line. It will be noticed that the graph of every equation of the first degree in x and y is a straight line. To find the graph, it is only necessary to determine two points and draw a line through them. These two points are usually taken on the axes. For example, the equa- tion 2a;— 32/ + 6 = is satisfied by (—3, 0) and (0, 2) ; its graph has the position of MN in Fig. 6. The pupil should assure himself by trial that this line contains every point which satisfies the given equation ; for example, the points (1, 2|), (2, 3i), J K X \ f/ X h y ^ \y^ 1 y .^;,<>^ ! > y ^!^ < y ■^ "r A /" > X X \y ^ \ y y 1 ^ ^ y y ^ ^ y ^M y y K Y\ Fig. G. (4, 4|), (-1,11), etc. EXERCISE L.XXIV Construct the graphs of the following equations z 1. x + y = 8. 5. 8x-\-4:y = 21. 2. x + Dy= 16. 6. 4:x-^5y= 25. 16. 3. 4 2: + ^ = 10. 4. Sx+2y=:13. 7. x-\-()y = 20. 8. Sx + 2y = 24:. CHAPTER XI SIMULTANEOUS SIMPLE EQUATIONS 146. Two or more simultaneous equations are those Whioh can be satisfied by the same values of the unknowns. Thus, 2 X + 3 y = S, and 3x + 2 y = 7, are simultaneous sim- ple equations, since each equation is satisfied if x=l, and2/=2. Similarly, x + y-\- z=6, 2x — y + z=3, and 3x + 2y—4: z = —5, are simultaneous simple equations, since each equation is satis- fied ii x = l, y = 2, and z = 3. 147. Two or more equations are inconsistent when they cannot be satisfied by the same values of the unknowns. Thus, x+y=5, and x+y=4:f are inconsistent equations, since the unknowns cannot have the same values in both equations. 148. Two or more equations are dependent when each equation can be derived from the others. Thus, x + y==4:, and 2 a; + 2 2/ = 8, are dependent equations, since when the second equation is divided by 2 it gives a;+?/=4, identical with the first equation. Dependent equations, though simultaneous, are redu- cible to a single indeterminate equation. 149. Two or more equations are independent when none of them can be derived from the others. Thus, 2 X + y = 5) and x + 3 y = 10, are independent since neither can be derived from the other. 163 164 ELEMENTARY ALGEBRA [Ch. XI, §§ 150,, 151 150. A system of equations is a group of two or more equations. A solution of a system of equations is a set of numbers which satisfy each of the equations in that system. The process of finding the solution of a system of equations is called solving the equations. GRAPHS OF SIMULTANEOUS SIMPLE EQUATIONS 151. Graphs of simultaneous simple equations in two unknowns can be constructed by the method of §§ 144 and 145. Consider the simultaneous simple equations : r^+2y = 4, (1) U+ 2/ = 5. (2) In (1), B = (0, 2), A = (4, 0) ; in (2), D = (0, 5), C= (5, 0). In Fig. 7, the location of the points A and B gives the line AB] and the loca- tion of the points (7 and i> gives the line CD. The lines AB and OD cross, or intersect, at F ; and since P is on both lines, its coordinates must satisfy both equations. Hence its coordinates are the values of x and y which would be determined by solving the two equations simultaneously. These are found by measurement to be x = 6 and y = — l. Two lines which intersect represent simultaneous equations which have a single solution. 's -n - Y N \ •v X \ ^ ^ \ J) ^ *s \ •?i V V, \ V. ^ ^ b\ \ *>a h^ ^J \ JC •^ SI k c X ^ <: ^ P s \ "v V, \ \ Y Fig. 7. Ch. XI, §§ 152, 153] SIMULTANEOUS SIMPLE EQUATIONS 165 GRAPHS OF TWO INCONSISTENT SIMPLE EQUATIONS 152. Inconsistent equations, § 147, may be shown to have no common solution by constructing their graphs. Thus, find a solution, if possible, of '2x + y = 4., (1) \2x + y = S. (2) In (1), a B = (0, 4), ^=(2,0);in(2),D=(0,8), (7=(4,0). In Fig. 8 the graphs of (1) and (2) are such that they never meet ; that is, AB and CD are parallel lines. Hence there is evi- dently no common solu- tion of (1) and (2). V ^ ^ - \ ^^> ^ s^ 5 5 ^ ^ ^^-^ v^V 5^fe ^1 V^ J \jyJ \C X " i v=^ 3 3 V V 3 3 r r 3 3 ^ ^ \ \ i ^ ^ Y \ \ Fig. 8. GRAPHS OF TWO DEPENDENT EQUATIONS 153. Two dependent equations, § 148, may be shown to be reducible to a single indeterminate simple equation by constructing their graphs. Thus, find a solution, if possible, of 2x + 3y = S, (1) (2) 3 2 3 In (1), B = (0, I), A = (4, 0) ; in (2), D = (0, |), C= (4, 0). Since B and D and A and C have respectively the same coordinates, the graph is a single line ; and the given equations are therefore reducible to an indeterminate simple equation whose graph has been shown, § 145, to be a line.crossing the axes. 166 ELEMENTAKY ALGEBRA [Cii. XI, § 153 EXERCISE Determine the nature of the tions by the graphical method : LXXV following systems of equa- 1. 2. 3. 5. 6. 8. 9. 10. 11. 12. r ^ + ^/ = 8, - 2: 4- 3/ = 1, x + y = 14, 2x + y =6. 2x+Sy = 12, ^x+5y = 20. x + y=^5, 2x + y=6. 8x + 2y=7, [2x+Sy = 8. 2x-Sy = 2, X — 5 y = — 5. 2x + 3y = 5, Sx + 2y = 5. x-2y = 4:, [2x-'4:y = 8. ^5x-Sy=--2, 4:x+2y=-6. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. Sx+2y=6, 9a;+6^ = 18. Sx-\-4:y=% 8x+4y = 12. {4:x+2y = 8, 2x-Sy = 2i, 2x-8y=::6. 2x+5y=:10, ix-Sy = 12. ( 5 X — 6 y = S^ [l0x-12y = 6. 2x-8y = 0, 8 a; - 4 ^ = 0. 4:x — 5 y = 1^ ^5 x — 4: y=9. 5x + S y = 5, 9x + 4:y = 9. r6x-5y=-l, .4 a:+ 3^ = 6. Sx-5y=12, x-10y = 24:. It; Ch. XT, §§ 164, 155] SIMULTANEOUS SIMPLE EQUATIONS 167 154. Elimination of one of two or more unknowns in a l^ystem of simultaneous equations is the process of com- ining the equations in such a way as to obtain fewer quations containing less unknown quantities. The quan- ity which has been caused to disappear is said to be liminated. TWO UNKNOWN QUANTITIES I. Elimination by Addition or Subtraction 1 155. 1. Solve 1 ^- + 2, = 12, (1) (2) Multiplying (1) by 2, 6 a; + 4 ?/ = 24, (3) multiplying (2) by 3, _ 6 a; + 9 ?/ = 15, (4) ,dding (3) and (4), 13 ?/ = 39, (5) Lividing (5) by 13, 2/= 3. ■ (6) Substituting y from (6) in (1), 3 a; + 6 = 12, (7) ransposing in (7), 3 a; = 6, (8) ividing(8) by 3, x= 2. (9) Verification: 6 + 6 = 12; —4 + 9= 5. The above equations can be solved by this method by ultiplying the first equation by 3 and the second equa- ion by 2, and subtracting the equivalent equations thus .erived. It is to be noticed that the equations are checked by ubstituting the values of the unknowns in the original quations. 168 ELEMENTARY ALGEBRA [Ch. XI, § Kd 2. Solve! 11 ^ + ^^^ = 2^' (1) l9^ + y=8. (2) Multiplying (2) by 2, l^x + 2y = 16, (3) rewriting (1), llx-\-2y = 23 , (4) subtracting (4) from (3), 1 x = -l, (5) dividing (5) by 7, x = -l. (6ji Substituting x from (6) in (1), - 11 + 2 2/ = 23, (7) transposing in (7), 2 2/ = 34, (8) dividing (8) by 2, ^ = 17. (9>| Verification: -11 +34 = 23; -9 + 17= 8. The above equations can be solved by this method by multiplying the first equation by 9 and the second equa^i tion by 11, and subtracting the equivalent equations thusi derived. That unknown is preferably chosen for eliminatior whose coefficients are such that they can be made equa! by the smaller multipliers. Rule for Elimination by Addition or Subtraction : Mah equal the coefficients of one of the unknowns in each equatio7 hy multiplying one or both of the equations by the necessan numbers. Add or subtract the resulting equations accordim as the equal coefficients have unlike or like signs. Find th other unknown number by substituting the value of the un known already found in that one of the given equations whic) has the least coefficients, Ve7nfy the solution by substitution in each of the given equations. Cn.XI,§155] SIMULTANEOUS SIMPLE EQUATIONS 169 6. 7. 10. 11 12. 3. 4. EXERCISE Solve the following systems ^ (2x + y = l, I -2a; + 3 3/ = 13. 7 X - 3 1/ = 15, 5a; + 6y = 27. 8 a; + 17 «/ = 42, 2a:+193/ = 40. 4 a; + 63/ = 40, 6 a; — 7 «/ = 2. 17a;-18i/ = 15, 5a; + 12y = 39. 28 a; + y = 33, - 21 a; + 11 ^ = 34. |33:r-(y+9) = 23, l44a;+3(y + l) = 50. 3a;-7«/ = l, 5a; + 3«/=2. 9a;-6«/ = 2, 45a; + 8 = 72^. f 19a; -16?/ = 91, ■ I 27 a; -20 3/ =130. |8a;-9?/ = 34, i9a:-8^ = 17. LXXVI of equations : f6a; + 5« = 68, 13. \ L4a;-13«/=78. fl8a; + 5w = 38, 1 12 a; — y= — 5. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. r8x + 9y = 26, . 32 a; - 3 y = 26. 33 a; + 54 2/ = -24, .44a;-80«/ = 44. 21a;-232/ = 2, .7a;-19y = 12. 15 a; + 28?/ = 157, 20 a; + 21 ^ = 144. 65a;+68t/ = -3, 39a;-119t/ = 158. 63 a; - 46 «/ = 29, 42 a; - 69 «/ = 96. 27 a; - 5 ?/ = - 37, 81a;- 7?/ = -151. 13 a; -15?/ = 11, ll2a;-7y=17. 11 a: + 13,?/ = -9, 15 a;— 14?/ = — 44. 19a;-23t,- = -ll, 22 a; +25?/ = -10. 170 ELEMEXrAUY ALGEBRA [Cii. XI, § 1G(; II. Elimination by Substitution 156. Solve|-^ + ^^^ = ^^' ('-) I -4 a: + 21 ^ = 55. (2i Transposing in (1), 2x=13-3y, (3) dividing (3) by 2, x = ^^^~^^, (4) substituting x from (4) in (2), _4(^l^^/^ + 2l2/ = 55, (5) simplifying in (5), ^^^^^ + 21, = 55, (6) multiplying (6) by 2, -52 + 12^ + 422/ = 110, (7) transposing and uniting in (7), 54 2/ = 162, (8) dividing (8) by 54, ?/ = 3. (9) Substituting y from (9) in (1), 2 a: +9 = 13, (10) transposing and uniting in (10), 2 a; = 4, (11) dividing (11) by 2, a; = 2. (12) Verification: 4 + 9 = 13; -8 + 63 = 55. It is to be noticed that the above equations may also be solved by the Addition and Subtraction method. Rule for Elimination by Substitution: In one of the equations find the value of one ujiknown quantity in terms of the othe7\ Substitute the value thus obtained in the other equation. Reduce this equation. Verify the solution iv each of the given equations. Cii.XI,§160] SIMULTANEOUS SIMPLE EQUATIONS 171 EXERCISE LXXVII Solve the following systems of equations by substitution ; 2. 3. 4. {^ 8. 9. 10. 11. 12. 2x — t/ = 0. -2x+t/ = -B, -3a7 + 4«/ = 8. [Sx + z/=lS. r2a; + 3«/ = 46, 4x + ?/ = 23, 3a;-2y = 9. .2x — y =■ 15. 42; + 3«/ = 81, -2;+2«/ = 21. 4a;+2«/=:38, 3 a; -3 2/ = 6. f2a; + t/=20, l4a;+32/ = 70. p-9y = 0, .4 a; -2/ =70. f4a;-5«/=3, l8a; + 2«/ = 66. f 2 a; - 2/ = 10, l3«/ + 17a;=177. 13. 14. 15. 16. 17. 18. 19. \ 20. 21. 22. 23. 24. l7a; + 32/ = 82. |3rc-6y = 2, l4a; + 72/ = -93. J4a;+3y = 4, l-7a; + 5 2/ = 75. f 8 a; — 5 «/ = 6, i7a; + 10«/ = 149. r3a; + 12«/ = 57, l2a; + ?/ = 10. r 7 a; + 4 y = 95, .a;— 2y= — 7. r27a; + 14^ = 41, 1 36 a; + 51 3^=87. f 100 a; -143 2/ = 757, .llx-91«/ = 8. 55^ + 31z/ = 171, 27 a; -11 2/ = 18.4. r 109 a; + 110 2/ = 86, ,107a; + 146«^ = 98. 83a; + 25«/ = 4, 21a; + 85y = 6. [39a;- 98^^ = 3, .51 a; + 182^ = 63. 172 ELEMENTARY ALGEBRA [Ch. XI, § 157 III. Elimination by Comparison 157. Solve i ^^ ^ ^ [ x+ 2/ = 18. (2) Transposing in (1), 2 a; = 16 — 3 ?/, (o) dividing (3) by 2, x = ?^^^, (4) transposing in (2), x = l^ — y, (5) comparing x in (5) and (4), 18 — t/ = — i""^? (^) multiplying (6) by 2, 36 - 2 2/ = 16 - 3 2/, (7) transposing and uniting in (7), ?/ = — 20, (8) substituting ?/ in (5), x = 38. (9) Verification: 76-60 = 16; 38-20 = 18. Rule for Elimination by Comparison: In each equation find the value of one unknown in terms of the other. Place these values equals and solve the resulting equation. Verify the solution in each of the given equations. exercise lxxviii Solve the following systems of equations by comparison, and check the results on the graph : ^ r5^+^=7, ^ |:^,+ 4^ = 7, 2 '^•f2/ = 0, 2x+?^y = l. '^+3?/ = 3, ^x+^y = -l. 2x-\-^y = 10, ^3^+2y = 0, 6. rAx [?>x + 22/ = 10. [2x-y=--l Ch.X1,§158] simultaneous SIMPLE EQUATIONS 173 158. If either, or both, of the equations in a system of equations contain aggregations or fractions, it is, in gen- eral, best to simplify the equations before elimination. Solve 4(:^;-3^/) = 8, (1) ^^ = 3. (2) Sim plif yiiig in (1), 4 a; - 12 ?/ = 8, (3) multiplying (2) by i» — 2 y, a; + ?/ = 3 a; - 6 2/, (4) transposing and uniting in (4), — 2 a; + 7 ?/ = 0, (5) multiplying (5) by 2, — 4 ic + 14 ?/ = 0, (6) rewriting (3), 4 a; - 12 y/ = 8, (7) adding (6) and (7), ^ 2/ = 8, (8) dividing (8) by 2, ^ = 4. (9) Substituting y from (9) in (3), 4 a; - 48 = 8, (10) transposing and uniting in (10), 4 a; = 66, (11) dividing (11) by 4, a; = 14. (12) 14 + 4 o Verification : 4(14 — 12) = 8 ; 14-8 EXERCISE LXXIX Solve the following systems of equations, selecting the best method : i2(5rr-y)-32/ = 5. 174 ELEMENTARY ALGEBRA [Ch. XI, § 158 f 7 _ 1 ^ 3. l2x — t/ x — y'' g |5:r-(3./-i)=f, '^±1=6, x-y x + 1 ^1 x + ly 3* 6a; — y _ 1 8. ^ 6. \i) + x 10 + ?/ 17 3 = 0, 9. Ix—by x+S 5(a; + 2 2/) 29 5a; + y ^2 f£+j_+4^g^ X — Z/ + 6 2a;-.y + 7 _ ^ a;-2«/ + 7 10. J 11. ' 8a;-3,y _^ 5x-2«/ + 3 4a; + 2y + ll ^ 4 6x— 7i/ + 6 3 3.y-Tx io_ a;-.y .y. 4 8 3 12. . 13. J a^ + 1 1 y + 2^ 2C.y-a;) ^ 4 10 5 a;-l ,y-2 ^ 3.y-8a; . 4 12 18 * f 4a; + .y-4 6a; + 2y-7 _» 3 9 2a: — y + l 10 a; — 4y _j^ 8 3 Ch.XI,§]68] simultaneous SIMPLE EQUATIONS 175 14. -! ^x + y + o 2x-y + 5 ^ x + 2i/ + l 9 6 2 ' 2x — t/-\-7 _ 4x-'dt/-l _ 5x + 8y + S 8 4 ~ 16 * 15 |(3^+8)(4^-3) = (2:r + 9)(6y-5), l(2x-l)(12y-l) = (3a: + 8)(8.y-7). 16. f7.v 5.y + 22 _a; 55-8.y 10 7 5 6' — ! — ^ = —rX + zy— 19. 7 11 11 "^ f 2x + 7.y + 5 Q^_4x+ll.y + 5^ 17. ^ 18. o 5a;+3.y _ llrc-14,y+241 ^"^ ir"~ 154 o , 4,y+5a: , 9g+8.y-12 _ 1 , lla:+6.y4-l SX+ ^—+ - -+ - 7 12 8 19. X — x + 2y — X 22 x+1 20- 49-2a; 9 6 r 20. J 3 "^ 4 6 ' 2y+7 3a;-y _ 3.y— 2a;4-4 8 7 ~ 8 * 21. 3.y-2 _ x-5 ^ r _ 2a;+3.y-l ^ 4 2 8 ' 5a;+6«/-3 2^+9w-2_ „ 176 ELEMENTARY ALGEBRA [Ch. XI, § 159 159. It is often convenient in simultaneous equations containing fractions to eliminate one of the fractions. 1 . Solve ■^ _ 2/ _ 1 2 i~ ' U 12" ''• 4 12 Multiply (1) by i, 1-1 = 1' subtracting (3) from (2), 12^8 2' multiplying (4) by 24, -102/ + 37/ = -72--12, uniting in (5), — 7 ^ = — 84, dividing (6) by —7, * 2/ = 12. Substituting 2/ from (7) in (2), |-5 = -3, transposing and uniting in (8), t = 2, multiplying (9) by 4, x = 8. .J , '8 12 -, 8 60 o Verification: - — —==1'^ -——= — 3. ^4: 4: liiJ This method is especially valuable in solving, by the foregoing methods, equations which conta^'a the unknowns in the denominators. Ch.XI,§159] simultaneous SIMPLE EQUATIONS 177 2. Solve ^- + 1=5, . (1) 1^-^ = 2. •• (2^ X y 16_28^2 X y Multiplying (1) by 7, ?i + — = 36, (3) X y adding (2) and (3), — = 37, (4) multiplying (4) by x, 37 = 37 a;, (6) dividing (5) by 37, x = l. (6) Substituting x from (6) in (1), 3 + ^ = 5, (7) y transposing and uniting in (7), i-2, (8) multiplying (8) by ?/, 4 = 2 y, (9) dividing (9) by 2, 2/ = 2. (10) Verification: ? + ^ = 5; ^-?5 = 2 Although equations (1) and (2) can be solved by first multiplying each equation by xy^ and the.n multiplying the resulting equations by 2 and 5 respectively and next 3ubtracting these last equivalent equations, this method is not recommended. If equations are solved by the latter method, it may happen that roots are introduced which do not verify. 178 ELEMENTARY ALGEBRA [Ch. XI, § 159 EXERCISE LXXX Solve the following systems of equations by eliminating the fractions : 2. 3. 4. \ 5. 6. ^1+1=7, f?+5 = 6, 6 5 7. ' X y ^-4=7. M_21^3. 3 16 I a; y | + ?=11, ff + '' = 18, 3 4 8. 4iX by 1 + 1 = 5. r+o' = 12. l7 8 l3a; 9^/ '2-f = 6, r 3 , 5_11 6 4 ' 9. 2a; y 4 f-| = 4. .'+o'=l- ,7 2 13a; 9y ff + '^/ = 17, [f /=7, 5 8 10. 7x by ? ^2^= 7. 13 4 11 13 19 16a; by 3* x , w 63 fo ,8 44 3a; + -^= — , 9 7 10 11. by 3- X 52,y 392 a; 1 4 I 3 ' 56 10 l4 ^y 9 \ll '^/ = 16, ^i+i=9, 10 7 12. X «/ 5? ^y=\Q X 8 I 8 35 l^^ 7' Ch.XI,§160] 8IMULTANE07JS SIMPLE EQUATIONS 179 160. Literal simultaneous equations are solved in the same way as are numerical equations. Especial care should be taken to express the values of the unknowns in terms of the knowns; and to that end the known terms should always be transposed to the right member of the (equation. ' 1. Solve P + r'? ^'^ Transposing in (2), 4:X + y = 5 a, (3) rewriting (1), x + y = 2 a, (4) subtracting (4) from (3)> 3 a; = 3 a, (5) dividing (5) by 3, x = a. (6) Substituting x from (6) in (1), a + y = 2 a, (7) transposing and uniting in (7), y = a. (8) Verification : a + a = 2 a; a = 5a— 4 a. 2. Solve ,«- + ^^ = *^' (1) _bx + ay = h^. (2) Multiplying (1) by h, abx + % = a% (3) multiplying (2) by a, ahx + ary = ab^, (4) subtracting (4) from (3), b^y — ary = a'b — aW, (5) factoring in (5), y (b^ — a?) = ab {a? — W), (6) dividing (6) by W-o?, y = - ab. (7) Substituting y in (1), ax — ab^ = a% (8) dividing (8) by a, x—b^ = a\ (9) ti-ansposing in (9), x = a^ + b^ (10) Verification : a(a' + b') + b(-ab) = a^', b(a' +b') +a(-ab) = b\ 180 3. Solve < ^A + -l = 62 + c2, ax by . hx cy Multiplying (1) by | multiplying (2) by -, subtracting (4) from (3), r ALGEBRA [Ch 9 XI, §160 (1) (2) X Q-y (3) 1 W _^-i^ W X acy a ' (4) ac W _ac^ b^ b^y acy b a' (5) multiplying (5) by ab^cy, (j^(? + 6* = Q?-b(?y + b^cy^ (6) factoring the right member in (6), aV + ?>'* = bey (aV + 6^), (7) dividing (7) by (aV + b% 1 = bey, (8) dividing (8) by 5c, ^ be (9) Substituting y from (9) in (1), ax b ' be (10) simplifying in (10), A + c^==.^ + c^, (11) transposing and uniting in (11), ax ' (12) dividing (12) by 6, ax ' (13) multiplying (13) by ax^ 1 = abx, (14) dividing (14) by a&, 1 X=-T' n.n (15) Verification : - + ^ = ?y^ + c^ ; - — - = a^ - Cii,XI,§lCO] SIMULTANEOUS SIMPLE EQUATIONS 181 EXERCISE LXXXI Solve the following systems of equations and verify the results : 2. 3. 5. 7. i 8. 9. 10. ' x+ ai/ = a% X— hy = h\ a; + hy = 1)^, ax+hy = (?, X m ' a(x+y) + h(x-y^ = c, X _m ^y" n ax+hy = e^ a^x + ^1 = c^y. ax+ly = 2, ab(x+ y^= a + h. {ax = h{y-2), a2+ J2 y-x-- ah 11. 12. 13. ' a(x+c)+h(]j—c)=a?—b\ y—x=2c. J2_^2 14. x — {a+ V)y= (h — a)x + ahy = V^, a h cfi—V^ . (a+V)x+(a—K)y=:^a+K ax . hy ,7 6 a X y __a^ +h^ a b c?}?" 15. a+b bx+ a2 a5, = 1 + a^/ = ^(a; + 1) — a, (a+6+^X^-a) a^n; — b'^y = a + J, 5:z; — a^/ = — 1 . a:r — (a — b^y= (a— 6)2, bx—y = b(a — b—V). 16. 2/— 6 a— c X — b __a + c ^y — a b — c y 17. \ b -\- c a + € a — C'^ .-,-^ = 5. a-\- b b + c 182 ELEMENTAUY ALGEBRA [Cii. XI, § IGl THREE OR MORE UNKNOWN QUANTITIES 161. Three simultaneous equations containing three unknowns are solved by the elimination of one of the unknowns between a pair of the given equations, and by the further elimination of the same unknown between a different pair of the given equations ; the resulting equa- tions are then solved as in §§ 155-7. Elimination is performed by the addition and subtrac- tion method. That quantity is generally chosen for elimi- nation whose coefficients are smallest. It is evident that of three given equations the first may be combined with the second, the first with the third, and the second with the thirde 'x + 7/ + Z=U, (1) 1. Solve i4:x + 27/ + z=:4:S, (2) [Qx + 5y + z=88. (3) Subtracting (2) from (1), -.3x-y = -29, (4) subtracting (3) from (2), — 5 cc — 2/ = — 45, (5) subtracting (5) from (4), 2x = 16, (6) dividing (6) by 2, a; = 8. (7) ^ Substituting x from (7) in (5), _. 40-2/ = -45, (8) transposing and uniting in (8), 2/ = ^- (^) Substituting x from (7) and y from (9) in (1), 8 + 5 + ^ = 14, (10) transposing and uniting in (10), z = l. (11) Verification : 84-5 + 1 = 14; 32 + 10 + 1 = 43; 72 + 154-1=88. Ch.XT, §102] SIMULTANEOUS SIMPLE EQUATIONS 183 162. Four or more simultaneous equations containing four or more unknowns are solved by the elimination of one of the unknowns between three or more pairs of the given equations, in the resulting equations another un- known is eliminated between two or more pairs of the resulting equations, and the process is continued until three resulting equations are obtained. These latter equa- tions are solved by the method shown in § 161. Care must be taken to keep the same number of equa- tions as unknowns ; otherwise, dependent equations will be obtained. x + y + z + w = -ig^ (1) 2x + y + ^z + 2tv = l, (2) ^x + 2y + 2z + ^'W=:^^, (3) 4a:+3J/ + 4^ + 6^^ = \^. (4) 1. Solve Eliminate y Subtracting (2) from (1), —x — 2z — io=:^ — ^-, (5) subtracting (3) from (1) x 2, — a; — 6 ic = — |, (6) subtracting (4) from (1) x 3, —x — z — ^w^^ — ^-^. (7) Eliminate z Subtracting (7) x 2 from (5), a? + 5 lo = -i/, (8) rewriting (6), — oj — 6 2^ = — J. (6) ' Eliminate x ' Adding (8) and (6), ~ ^w = -\, (9) dividing (9) by - 1, w = \. (10) Whence, by substitution, a;=l, 2/ = i z = \,w^. 184 ELEMENTARY ALGEBRA [Cii. XI, § 16^ EXERCISE LXXXII Solve the following systems of equations : 1. '4:X + 5?/ + 92 = 13, 5x + y+2z = -5, Ix -5^-82 = -31 2. 3. f x+ 5y — 2z = 5^ I Sx-\- 8?/ + 4^ = 31, [1 x+2oy-4:z = 4:5. 2x-9y+10z = 55, llx—'dy— 5z=7^ ISx — iy— 62^ = 1. 4. z — -, t) 5. '5x+Si/—2z = %, 4 x (Sx-^-l/- ^-12 ^i2;+Jy+|3 = 14, s^J + i?/- ^2 = 5, i^+ y-j2 = 12. la;+32/-fi 16, 6. i 2x-y + lz = 25, ' lx-iy+ s = 17|. I 3 '1-1.2, X y 1-1 = 8, y 2 1 + 1 = 9. _ Z X 10. fl - + X 1_ = 7, - 1 - + y 2_ 2"" = 14, X 8_ 2 21. fl - + X 3 - + y 4_ 2 8, 4 x-^ 5 2_ 2 16, 7 X y 4_ 21. f3 - + X 4 8_ 2~ 15, 6 a; 1 2y 2 + - 2 = h 9 [4a; 8 + - y +1. 2 = 13. x + y-- = 2« ', ay + z-. = a2. hx — Z : = 52. 11. {ax + y=^l, 12. -! hx-\-z = l, I cz + X = bo. bx+ay=2 ab, 13. ^ cy+bz = 2 bc^ ^ ex+ az = 2 ac. Ch.XI, §162] SIMULTANEOUS SIMPLE EQUATIONS 185 15. 16. 17. i x+ ay =^a(a-\-h^^ 14. \ a^z — bx = a^, y = z-a. ' ax + y -\- z = ahc + a(b-\- 1?), - x-{-hy -\- z = ahc + b(a + c?), x+ y -{- ez = ahc + c{a + 6). x-\- y + z + u = 65^ x+2y — z — u = l^ 2x+3y-j-2z-u=m, ^Sx—2y-{-2z + ii = 54:. ' x+2y-{- z— 21 = 10, X— y + 2z + u = 23, x+Sy + 4:z-2u = 89, X— 5y — 4:Z— 8u = 41. ^x+ y+ z-\- u = 10, x+ 8y+ 5z+ lu = SO, x+ 6^+15^ + 28^^ = 80, ^ + 10 2/ + 35 ^ + 84 ^ = 188 ' {b+ c)x+by = c^ (a -\- c)y + cz = a, (^a + b)z + ax = bo 18. 19. 20. 21. 'a;+5^=23, y + 4z = -l, z+Su= 20, u-^2v = 3. V + x = 6. \y + lz- ^21 = 16. 2x+3y = 57, 5x-iz = 20, 22. -! 32+2^ = 48, 4y+3v = 68, 1 u — 6v=^ 15. 186 ELEMENTARY ALGEBRA [Ch.XT,§162 REVIEW EXERCISE LXXXIII Solve the following simultaneous equations: 2. 4. ^ ' ax + by = 1, hx— ay = 1. r ax = Jy, hx + ay = e. X 15 4' ,x 2 '2 3 , - + - = 4, X y 1+1 = 6. [X y 6. %x 2y-5_ 4-9a; ^l a;-2 3 lx-3y=10. x + y = 2a, [ (a - h)x = (a + h')y. — +^ = e, 1 = (i. 2x-Z y-2 , 7 2x—y '2y—x _ _49 3 4 " 12* 9. 10. hx ay '3^ + 2^-42=15, 5^-3^+255=28, .3?/ + 42 — a;=24. + -^ = 2a, 11. 12. 13. 5(rr-2i/)-(a;-2/)=-24, [ll(2:z;+32/)+(2:^-2/)=200. qx — rb = p(a — ?/), ql+r = p(i + ^ (^D-K^D=*. a+ 6 a — 6 Oh. XT, §1(52] SIMlTLTANi:0lJ8 SIMPLE EQUATIONS 187 14. 15. 16. (a - b)x - (^ + h)7/ = 2 a2 - 2 b% (a + b)x — {a — b)y = 4 ab. ax—by = €? —5^—2 a6, bx + ay = 2ab + a?'— b\ y^=b^a. a + b b+ c y , ^ c+ a z c— a X = b- 17. \ b — c a— b r {a + />) (a; + 2/) - (^ - ^) (^ - ^) = «^ (a - />)(:?; + ^) + (« + ^)(^ - 2/) = ^*^- + ~-^ = 2a, 18. ^ ^ — y _ ^ + y , 19. ^ 20. I 2a^> a2+ J2 a; + y - ^ = 7, y + 2 - w = 9, z + u—.x=19^ .u+ X — y = 13. ^ + 2/ + ^ = 0, (^ + h)x + (a+ c)y + (b + a)z = 0, Qc — b)x +(a — c)y + (J - a^z = 2(a J + «(? + bd) -2(a2+62+^2), CHAPTER XII PROBLEMS INVOLVING SIMPLE EQUATIONS EXAMPLES 163. 1. The sum of two numbers is 27, and if the greater be divided by the less, the quotient is 1 and the remainder is 5. Find the numbers. Let X = the greater number, and y = the less number. By the first condition, by the second condition, Solving (1) and (2), Verification: .^ , _, , It should be noticed that in this, as in many of the fol- lowing problems, one, two, or more unknowns may be employed to find the solution. Let X = the greater number, and 27 — a? = the less number. By the second condition, —^ — = 1. (1) Solving (1), x = 16 2ind 27 -x = 11. In general, if an equation can be solved with a single un- known, this method is preferable. 188 x + y = 27, (1) a; — 5_-. y (2) a; = 16 and y = ll. 5 + 11 = 27:^^7^ = = 1. Ch. XII, § 163] SIMPLE EQUATIONS 189 2. The width of a rectanguhir room is f of its length. If the width were 5 feet more, the room wotdd be square. Find the dimensions. 5 X Let X = number of feet in the length, and -— = number of feet in the width. By the conditions, — ■ + 5 = x, (1) Solving (1), a; = 30 ; hence ^ = 25. Verification : 25 + o = 30. 3. A's age is ^ of B's age, but 5 years ago A was ^ as old as B. Find their present ages. Let X = the number of years in A's age, and 5 x = the number of years in B's age. By the conditions, 9(x — 5) = 5x — 5. (1) Solving (1), a; = 10 ; hence 5x = 50, 4. A can row 4 miles an hour down a stream, and 2 miles an hour against the stream. Find A's rate in still water, and the rate of the stream. Let X = A's rate in still water, in miles per hour ; and y = rate of stream, in miles per hour. By the first condition, x + y = 4:] (1) Dy the second condition, x — y = 2. (2) Solving (1) and (2), x=3, y = l. 5. At what time between 2 and 3 will the hands of a clock be («) together ? (J) exactly opposite ? In the same period of time the minute hand moves twelve times as fast as the hour hand. Thus, the minute and hour hand cover in an hour respectively 60 and 5 minute-spaces; and in 12 minutes respectively 12 and 1 minute-spaces. 190 ELEMENTARY ALGEBRA LCh. XII, § 163 Let X = number of minute-spaces passed over by the minute hand in given time, and — = number of minute-spaces passed over by the hour hand in given time. (a) Since the minute hand starts at XII and moves to A, where it meets the hour hand which starts from II, which is 10 minute-spaces from XII, and in the same time moves to A, by the conditions, aj = 10 + 12 Fig. 9. Solving (1), X = 10|^. (1) (b) Since the minute hand starts at XII and moves to B, where it is exactly opposite the hour hand, which starts from II, 10 minute-spaces from XII, and in the same time moves to Ay by the conditions, ^=10 + j|+30. Solving (1), X = 43^p (1) Fig. 10. 6. The sum of the two digits of a number is 6, and if 36 be added to the number the order of the digits is reversed. Find the number. Since in arithmetic, position indicates the value of the digits in a number, (56 =1 10 • 5 + 6), let Cii. XII, § 163] SIMPLE EQUATIONS 191 X = the digit in the tens' place, and y = digit in the units' place, and 10 x-{-y= the number. . By the first conditions, aj + ?/ = 6, (1) Dy the second condition, 10 x + y + 36 = 10 y -{-x. (2) Solving (1) and (2), x = 1, y = o] hence the number is 15. 7. A can do a piece of work in 5 days, and with the help of B can do it in 3 days. How long would it take B alone to do the woi*k ? Let X = the number of days it takes B alone to do the work, then - = part that B can do in 1 day, X and - = part that A can do in 1 day, and - = part that A and B can do in 1 day. By the conditions, _ _|. _ = _ . (1) O X o Solving (1), X = 71 8. A train runs 84 miles in the same time that a second train runs 96 miles. If the rate of the first train is 3 miles per hour less than that of the second train, find the rate of each. Let a; = rate of the first train, and x + 3 = rate of the second train. By the conditions, — = -^^' (1) X X -f- o Solving (1), 07 •■= 21 ; hence x-{-3 = 24. 192 ELEMENTARY ALGEBRA [Ch. XII, § 163 9. A number of 4 ^o bonds were sold at 90, and the pro- ceeds invested in 3J ^ bonds at 75, the par value of each bond being $100. If the gain in income is 14, find the number of 4 ^ bonds. Let X — the number of 4 % bonds, then 4 a; = the income in dollars of the 4 % bonds, and 90 cc = the value in dollars of the 4 % bonds, 90 X then — — = the number of 3|^ % bonds, /90 x\ and Z\ [ ] = the income in dollars from the ?>\ % bonds. By the conditions, 3| (^^^-^ - 4 a; = 4. (1) Solving (1), a; = 20. EXERCISE LXXXIV 1. The sum of half a number and its third part is 135. Find the number. 2. The difference between the third and seventh parts of a number is 40. Find the number. 3. The excess of the sum of the fourth and twelfth parts over the ninth part of a number is 8. Find the number. ' 4. The excess of the sum of the fifth and seventh parts over the difference of the half and the third parts of a number is 259. Find the number. 5. Find that number which is 1^ times the excess of the number over 2. 6. The sum of two numbers is 32, and their difference is 8. Find the numbers. |(Jh. XII, § 163] SIMPLE EQUATIONS 193 7. Tlie difference of two numbers is 13, and if 144 be subtracted from 8 times the first, the remainder is 56. Find the numbers. 8. The fourth part of the larger of two consecutive numbers exceeds the fifth part of the smaller by 1. Find the numbers. 9. The sum of two numbers is 18, and if the greater number be divided by the less, the quotient is 2. Find the numbers. 10. Find the two numbers such that their difference is 20, and the quotient of the greater divided by the less is 3. 11. The sum of two numbers is 26, and if the greater number be divided by the less, the quotient is 1 and the remainder is 4. Find the numbers. 12. The difference of two numbers is 9, and if the greater be divided by the less, the quotient is 2 and the remainder is 2. Find the numbers. 13. The difference of two numbers is 18, and if the less be divided by the greater, the quotient is ^. Find the numbers. 14. The sum of two numbers is 22, and if the less be divided by the greater diminished by 7, the quotient is -|-. Find the numbers. 15. The sum of two numbers is 200, and their difference is I of the less number. Find the numbers. 16. The sum of two numbers is 59, and if the greater be divided by the less, the quotient and the remainder is 4. Find the numbers. 194 ELEMENTARY ALGEBRA [Ch. XII, § 163 17. The difference of two numbers is 16, and if the g'reater be divided by the less, the quotient is 2 and the remainder is 4. Find the numbers. 18. If 59 be added to half of a certain number, the sum obtained is 1^^ times a seventh of the number. Find the number. 19. A number is 10 times a second number. The quo- tient of the first number divided by 22 exceeds by -^^ the quotient of the second number divided* by 3. Find the numbers. 20. If a certain number be added to the terms of |, it becomes |. Find the number. 21. Find the fraction such that if 1 be added to the numerator it becomes ^ ; but if 1 be subtracted from the denominator it becomes \, 22. Find the fraction such that if 3 be added to the numerator it becomes | ; but if 1 be subtracted from the denominator it becomes ^. 23. Find the fraction sjach that if 4 be subtracted from its terms it becomes J ; but if 5 be added to its terms it becomes |. 24. The sum of two fractions whose numerators are respectively 7 and 9 is '^-^-^-; but if the numerators be interchanged, the sum of the fractions is ^f . Find the fractions. 25. A certain fraction becomes y^g ^^ ^ ^® subtracted from the numerator, and becomes ^ if 4 be added to tlie denominator. Find the fraction. Ch. XII, § 163J SIMPLE p:quations 195 26. If 3 be added to the numerator and 1 be added to the denominator of a certain fraction, it becomes | ; but if 1 be subtracted from the numerator and 3 be subtracted from the denominator, it becomes ^-. Find the fraction. 27. The sum of two fractions whose numerators are each 1 is ^|-. The first fraction exceeds the second by J-^, Find the fractions. 28. The width of a rectangular room is | of its length. If the wddth were 3 feet more, the room would be square Find the dimensions of the room. 29. The dimensions of a rectangle are respectively 12 feet more and 8 feet less than the side of an equivalent square. Find the dimensions of the rectangle. 30. The length of a rectangular floor exceeds the width by 6 feet. If the width be increased by 3 feet and the length b}^ 2 feet, the area is increased by 134 square feet. Find the area. 31. A square contains the same area as a rectangle whose dimensions are respectively the half and the double of the side of the square. If the width of the rectangle be increased by 3 feet and its length be diminished by 5 feet, the area is increased 34 square feet. Find the side of the square. 32. Seven men and 5 boys earn $11.25 per day, and at the same wages 12 boys and 4 men earn $11 per day. What are the wages per day of a man ? 33. A sum of money is divided equally among a certain number of men. If there were 4 more men, each would receive $1 less; if 5 less men, each would receive $2 more. Find the number of men. « 196 ELEMENTARY ALGEBRA [Ch. XII, § 163 34. A could have bought 5 more oranges, each at half a cent less, for the same amount of money that he could have bought 3 less oranges, each at half a cent more. Find the cost of the oranges. 35. A's age is ^ of B's. Five years ago A was ^ as old as B. Find their present ages. 36. A's age is five times B's. In 12 years B's age will be ^ of A's. Find their present ages. 37. A is 50 years old, and B is 25. In how many years will B be ^2 ^s old as A ? 38. A's age is twice that of his son, but 10 years ago it was three times as great. Find the present age of each. 39. If A was four times as old as B 7 years ago, and if A will be twice as old as B in 7 years, what is the present age of each ? 40. If A is ^ as old as B, and if he was eight times as old as B 20 years ago, find the present age of each. 41. A's age exceeds B's by 21 years. In 8 years A will be 1| times as old as B. Find the present age of each. 42. A's age exceeds B's by 12 years. Twelve years ago A's age was ^ of B's age. Find the present age of each. 43. Find three numbers such that the sums of the num- bers in pairs of two are 6, 8, and 12. 44. A has $15 more than B ; B has $5 less than C; A and B and C together have $65. How much has each ? 45. A and B and C have $54. A has six times as much as B ; B and C together have as much as A. How much has each ? / Ch. XII, § 168] SIMPLE EQtiATlOKS Oil 46. A and B have only | as much money as C ; B and [C together have six times as much as A ; B has $>680 less than A and C together. How much has each ? 47. A can row 6 miles an hour down a stream, and 2 miles an hour against the stream. Find A's rate in still water, and the rate of the current. 48. A crew can row 20 miles in 2 hours down a stream, and 12 miles in 3 hours against the stream. Find the rate of the current, and the rate per hour of the crew in still water. 49. A man can row Si miles down a river in 56 minutes. If the river has a current of 2 miles per hour, find the rate of the man in still water. 50. At what time between 3 and 4 will the hands of a clock be together? between 7 and 8? between 9 and 10? 51. At what time between 5 and 6 will the hands of a clock first be at right angles ? between 6 and 7 ? between 10 and 11 ? 52. At what time between 12 and 1 will the hands of a clock be exactly opposite ? between 4 and 5 ? between 11 and 12? 53. At what time between 8 and 9 is the hour hand of a clock 20 minute-spaces ahead of the minute hand ? 54. At what time between 4 and 5 is the minute hand of a clock exactly 5 minutes ahead of the hour hand ? 55. The sum of the two digits of a number is 9, and if 9 be subtracted from the number the digits will be reversed. Find the number. 198 ELEMENTARY ALGEBRA [Cir. XII, § 1(53 56. The tens' digit exceeds the units' digit of a number of two digits by 1, and if 9 be subtracted from the num- ber, the digits will be reversed. Find the number. 57. The sum of the digits of a number of three digits is 17 ; the hundreds' digit is twice the units' digit ; if 39() be subtracted from the number, the order of the digits will be reversed. Find the number. 58. The sum of the digits of a number of three digits is 5 ; the hundreds' digit is | of the units' digit ; if the number be divided by the sum of the digits, the quotient so derived is 8S^ less than the number. Find the number. 59. A number is expressed by three digits whose sum is 18. If the digits in the hundreds' and units' places be interchanged, the number will be diminished by 792. The digit in the tens' place is |- of the sum of the other two digits. Find the number. 60. A can do a piece of work in 3 days, and B can do it in 5 days. In how many days can A and B, working together, do the work ? 61. A can do a piece of work in 3 days, B in 7 days, and C in 5 daj^s. How many days will it take all together to do the work ? 62. A can dig a ditch in Ij- days, B in 5^ days, and C in Q^ days. How many days will it take all together to do the work ? 63. A and B together can plough a field in 15 days, while A and C together can plough it in 18 days, and C in 30 days. In how many days can B and C together plough the field ? :n. XII, § 1G8] SIMPLE EQUATIONS 199 64. A and B can build a walk in 6 days, B and C in ^i days, and A and C in 10 days. How many days will t take A, B, and C together to build the walk ? ; 65. A and B can do ^ of a piece of work in 2 days ; B '.an do I of it in 6 days. How long will it take A alone o do J of the work ? 66. Two pipes, A and B, can fill a cistern in 70 minutes, ^ and C in 84 minutes, and B and C in 140 minutes, low long will it take for each alone to fill it ? 67. One tap will empty a vessel in 80 minutes, a second n 200 minutes, and a third in 5 hours. How long would b take to empty the vessel if all the taps were open ? 68. A and B can do a piece of work in m days, B and C a n days, and C and A in p days. How many days will : take A, B, and C, all working together, to do the work ? 69. A cistern can be filled by two pipes in 5 and 7 hours espectively, and can be emptied by a third pipe in a hours. n what time can the cistern be filled if the first two are unning into, and the third is emptying the cistern ? 70. A train runs 100 miles in the same time that a 3Cond train, whose rate is 3| miles an hour less, runs 0. miles. Find the rate of each train. 71. Two trains leave A at the same time, and run in pposite directions. The first train runs at a rate, in liles per hour, j faster than the second. How man)^ ours will each train have run when they are 425 miles part, if the distance covered by the first train in 10 hours xceeds that covered by the second train in 8 hours by 120 liles ? 200 ELEMENTAUY ALGEBRA [Ch. Xll, § 72. A and B are 240 miles apart. If at the same time a train leaves A and B, and runs for the other place, how far from A will they meet if the train from A runs at the rate of 45 miles an hour, and the other ^ as fast ? 73. A leaves the place X at 8 a.m., and 2 hours latei B leaves Y, 100 miles from X, and meets A at noon. Ij A had left at 8.30 a.m., and B at 9 a.m., they would alsc have met at noon. Find the rate of A, and of B. 74. A is 100 units east from B. If A and B mov( toward each other, they will meet in 4 minutes ; but i:. both move west, A overtakes B in 20 minutes. Find theii: rates of speed. 75. A left a certain town and travels at the rate o a miles an hour, and in n hours was followed by B at th« rate of h miles an hour. In how many hours did B over take A ? 76. A leaves New York and travels at the rate of 1. miles in 5 hours ; 8 hours after, B leaves New York, anc travels after A at the rate of 1*3 miles in 3 hours. Ho^^ far must B travel to overtake A ? 77. A and B run a mile. First, A gives B a start o 44 yards and beats him 51 seconds ; in the second heal A gives B an allowance of 1 minute 15 seconds, and is beatei by 88 yards. Find the time it takes B to run a mile. 78. A fox is pursued by a hound. The fox takes leaps while the hound is taking 3|^. Four of the hound'; leaps are equivalent to 7 of the fox. The fox has 45 c her own leaps the start. How many leaps will each mak before the fox is caught ? Cii. XII, § 163] SIMPLE EQUATIONS 201 79. Find the principal upon which the simple interest for 3 years and 3 months at 3|^ is $93.60. 80. Find the time required for $2275 to amount to ^2378.74 at 3|/o. 81. Find the rate per cent at which $20,000 doubles itself in 27 years, 9 months, and 10 days. 82. A sum of money at simple interest in 5 years amounted to $2400, and in 7 years to $2560. Find the principal. 83. A has twice as many 4:Jo bonds as 5^ bonds, whose [)ar values are each $1000. The bonds produce an annual income of $1950. Find the number of 4^ and of 5^ l>onds. 84. A has $20,000 invested between real estate and stocks, the par value of each share being $100. On the r(jal estate he nets, at 5| /o, $440 ; on the stocks, at 3|^, he nets $8 less than on the real estate. Find the amount ill stocks. 85. The sum of A's income for 3 years at simple in- terest on $12,500, and on $15,000 for 4| years at simple interest, is $4020. If the rates of interest were inter- changed he would receive, in the same time, $3975. Find the different rates. 86. The sum of the capitals of A, B, and C is $120,000. A's capital is invested at 3|^, H's at 4/o, and C's at 3|/o, and the sum of their incomes is $4530. If the rates at which A's and B's capitals are invested are interchanged, the income of all is $30 less. Find their capitals. 202 KLEMEN TAIIY ALGEBRA [Cii. XIT, § IG:] 87. A mass of gold and silver which weighs 10 pounds loses, when weighed in water, -^^ of itself. If gold loses ^ig, and silver ^^ of its weight, when weighed in water, how many pounds of gold and silver are there in tlie ^ mass ? 88. A mass of tin and copper, which weighs in air 687 pounds, weighs in water 608| pounds. If one pound of tin loses ^|§ of a pound, and one pound of copper loses 2^2^2 ^f '^ pound, when weighed in water, how many pounds of tin and copper are there in the mass ? 89. If a number of soldiers be formed in a solid square, 24 men fail to get places ; but if another solid square be formed, with one more man on a side, there are 29 places unfilled. Find the number of soldiers. 90. How many ounces of 14 carat gold must be mixed with 40 ounces of 15 carat gold to make a mixture of 14 i carat gold ? 91. Five pounds of gold 840 points pure are melted with 7 pounds of another sort, and produce a mass 700 points pure. How many points pure is the second sort ? 92. How many quarts of water must be mixed with 250 quarts of alcohol 80 ^/o pure to make a mixture 75^ pure? 93. A piece of work can be done by 20 workmen in 11 days, and by 30 master workmen in 7 days. In how many days can the work be done by 22 workmen and 21 master workmen ? 94. At a gathering of 14 men and 23 women the ratio of unmarried men to unmarried women is 2 to 5. Find the number of married couples present. CHAPTER XIII INEQUALITIES 164. The signs > and < express inequality : a>h is read "a is greater than S" ; a b^ (3) ah^ a—h is positive; when a J, then a — 5 > ; and, since all negative quantities are less than zero, if a < 5, a — J < 0. 165. An inequality is a statement that one of two ex- pressions is not equal to (that is, is greater, or less than) the other. The first member of an inequality is the ex- pression to the left of the sign of inequality ; and the second member is the expression to the right of that sign. Thus, o? -f- W is the first, and 2 ah the second, member of the inequality, o? +'b'^ ^2 ab, A term of an inequality is any term of either the first or second member. Two inequalities subsist in the same sense when they have the same sign of inequality. Thus, a > 6 and c> d are inequalities subsisting in the same sense. 203 204 ELEMENTARY ALGEBRA [Ch. XIII, § 106 Inequalities subsist in the opposite sense when they have opposite signs of inequality. Thus, a> b, c < d, are inequalities which subsist in the opposite sense. 166. The general principles upon which inequalities rest are : I. If equals be added to unequals^ the sums are unequals subsisting in the same sense. If a>b, (1) then a-b>0. (2) Now, (a+c^-(b + e} = a-b, (3) or, substituting (3) in (2), (a+c)-Cb + c^>0, (4) or, rewriting (4), a+ c ^b + c. (5) II. If equals be subtracted from unequals^ the remainders are unequals subsisting in the same sense. If a>b, (1) then • a-J>0. (2) Now, (a— c) — (b — c) = a — by (3) or, substituting (3) in (2), (^a-e^-(b-c)>Q, (4) or, rewriting (4), a— c^h— c. (5) Application of I and II : Any quantity in an inequality may be transposed from member to member if the sign of that quantity be changed. If a—c^b^ (1) by I, a>b^c. C2) Cii. XIII, § 166] INEQUALITIES 205 If a+h>c, (1) by II, a>c-b. (2) If the sic/7is of all the terms of an inequality be changed^ the sign of iiie quality must be reversed. If a — J>c — c?, (1) transposing all the terms in (1), d—e>b — a^ (2) or, rewriting (2), b—ac, (1) rewriting (1), 6 — ^ > 0, (2) changing all signs in (2), c — J < 0. (3) Now, (a-b) + (-a+c)== -b + c, (4) substituting (4) in (3), (^a-b) + (-a + c}<0, (5) rewriting (5), a — b <, a — c. (6) IV. If unequals be multiplied by positive equals^ the products subsist in the same sense. I If a>b, (1) then a-b>0. (2) Let m be any positive quantity. Then m(^a — J) must \)o a positive quantity, since the product of two positive ([iiantities must be positive. Tlierefore, m(a — J) > 0, (3) or, rewriting (3), ma — mb:> 0, (4) or, ma > mb. (5) 206 ELEMENTARY ALGEBRA [Ch. XIII, § 166 Since the process of division is multiplication by the reciprocal of the divisor, it follows from IV that if unequals be divided by positive equals the quotients subsist in the same sense. Application of IV : To clear an inequality of fractions multiply each term by the L, C, D, taken as a positive quantity. Thus, if _^4_^>^ (1) multiplying (1) by 24, —6x + Sx>x. (2) V. If unequals be multiplied by negative equals^ the products subsist in the opposite sense. If a>b, (1) then a-b>0. (2) Let —71 be any negative number. Then —n(a—b) must be a negative quantity, since the product of a nega- tive and a positive quantity is a negative quantity. Therefore, - n (a - J) < 0, (3) or, rewriting (3), — /^^ + n5 < 0, (4) or, —naK— nb. (5) Since the process of division is multiplication by the reciprocal of the divisor, it follows from V that if un- equals be divided by negative equals the quotients sub- sist in the opposite sense. Henceforth, in this chapter^ literal quantities are used to represent only positive and unequal quantities. Tliis fact must be kept in mind, for otherwise the proofs will not hold. jCh. XlII, § 167] INEQUALITIES 207 167. A conditional inequality is true only for some value or values of the letters involved. An absolute inequality is true for all values of the letters involved. Thus, 2a; — 3>a; + 2 is a conditional, and a^ + b^>2 ab is an absolute, inequality. A. Prove that a^+b'^>2ab. Either (1), a-b>0, or (2), a-b<0. 1. If a-^>0, (1) multiplying (1) by itself, a^ - 2 ab + b^>0, (2) transposing in (2), a^ +b^>2 ab, (3) 2. If a-b0, (2) (1) is negative : multiplying a negative number by itself is, by V, an inequality subsisting in the opposite sense. Transposing in (2), cr + ^- > 2 ab. B. Prove that a^ + 6^ > ab (a + 6). Now, a^-2a^4-^'>0, {A) transposing —ab in (J[), a^ — ab + 6^ > ab, (1) multiplying (1) hj a-\-b, (a + b) (a? -ab-\- Ir) >ab(a + 5), (2) a^ + W->ab{a + b). (3) C. Prove that a^ -\- b'^ + c^>ab + be + ca. Now, by A, a' + b'>2 ab, (1) and, by A, 52 _,_ ^2 ^ 2 be, (2) and, by ^, c^ + a'>2ca, (3) adding (1), (2), and (3), 2 (a' -{-b'+c'):>2 (ab + bc + ca), (4) dividing (4) by 2, a^ + b"+ c" >ab + bc + ca, (5) 208 ELEMENTARY ALGEBRA [Ch. XIII, § 168 B. Prove that a^ + b^+ c^>Z abc. ISTow, by (7, o? + h^-\-c^> ah + hG + ca, (T multiplying (1) by a, a^ + ab^ + ar > crb + abc + a% (2 multiplying (1) by b, a^b -\-b^-\- bc^ > ab' + b^c + abc, (3 multiplying (1) by c, ah + Wc + c^ > abc + 6c- + c^a, (4 adding (2), (3), and (4), and uniting, a^ + W-\-&>^abc. (6 The type forms, A^B^ (7, and 2>, should be remembered* 168. The solutions of various problems in conditiona inequalities are illustrated in the following problems. 1. In the conditional inequality, 3a; + |>a; + 8, fine one limit of x. Let 3 a; + 1 > a? + 8. (1) Multiplying (1) by 3, 9 a; + 4 > 3 a; + 24, (2)' transposing and uniting in (2), ^x'> 20, (3) dividing (3) by 6, x> 3i (4) 2 X 2. In the conditional inequalities, (1) :?: + 7 > — - + 9, o (2) — - < - + 2, find the integral values of x. Multiplying (1) by 3, 3 a? + 21 > 2 a? + 27, (3) transposing and uniting in (3), a: > 6, (4) multiplying (2) by 20, 8 a; < 5 a^ + 40, (5) transposing and uniting in (5), 3 a; < 40, (6) dividing (6) by 3, x< 13\, (7) From (4) and (7), x lies between the amits 6 and 13^ ; and may therefore take the integral values, 7, 8, 9, 10, 11, 12, 18. Ch. XIII, § 168] INEQUALITIES 209 EXERCISE LXXXV 1. Between what limits must x lie, to satisfy the in- equalities 2a;-3>20 and 32;-7<22: + 6? 2. Given 2x—^9, find the limits of x. 4. Given 3a;— 5>2a;+l, and 3a;+15>4:r+5, find the limits of x. Prove the following inequalities, the letters being ])ositive and the sign ^^ being read, "not equal to '^ 5. f + ^>2,ifa=^5. a 6. a^>2ah-V^, ii a--^h. 7. 77^2 _|_ ffi-\. p^::>mn +mp + np^ ii m^n, n^p^ m^p, 8. a%^ + Pc^ + (T^a^ > 3 aWc^, if a^-h, a=^c,h^c. 9. an + hm < 1, if a^ + J^ = 1 and if nfi+rfi^l and if a^n^ and h4^m, 10. aa;+ %<15, if ^2+ 52= 25 and if a;2+ /= 5. 11. 2a3 + 63^a(a2 + a5 + ?^2)^ if ^-^5. 12. a3-J3>3^j(^_5), if ^>6. 13. (a + 5)3+ ((?+d:y>(a + 5 + c?+(^)(a+J)(c? + c?), if (a + 6):?t(c + d). 14. ^2 + 4 62 + ^2 ^ 2 aJ + 2 5^ + ^c,if a^2h,a^c,2h-^c. m n p n ^ m , p .J, ^ ^ -, ^ 15. _ J |_^->_j f--^ if m>n^ n>p^ and m>p^ p m n p n m CHAPTER XIV INVOLUTION AND EVOLUTION INVOLUTION 169. The operation of raising an expression to any given power is called involution. An expression is said to be expanded when the indicated multiplications have been performed. Thus, (a)^ and (a + &)- have been expanded when the re- spective products have been found to be a^ and a^ + 2ab + W, MONOMIAL!^ 170. Involution of monomials is subject to the follow- ing Index Laws, in the proofs of which a =?^ 0, and m and n are restricted to positive integers. I. {a'^'Y = a''''\ By definition, (a"")" = [(a to m factors) to n factors], by associative law, = a to mn factors, by definition, = cf''. The exponent of the powei* of any given monomial is found hy multiplying the exp>onent of the given monomial by the index of the required power, II. (ahy^=aH'^. By commutative and associative laws, (ab)"" = (a to m factors) (b to m factors), by definition, = a"*6"*. 210 (II. XIV, §171] INVOLUTION AND EVOLUTION 211 Similarly, (abc)"^ = a'^b"'c'^. The mill power of the product of two quantities is equal to the product of their mth powers. \bj 6^* ' By commutative and associative laws, j - J = (a to m factors) -f- (b to m factors), by definition, = a"* h- Z>"* = — • The mth power of the quotient of two quantities is the quotient of their mth powers, 171. Involution is also subject to the Law of Signs. (— a)( — «) = ( — ^)2_^2^ (a) (a) = (a)2 = a^, ( — a)( — ^)( — a) = ( — a)^= — a^, etc. All even powers of a negative moyiomial are positive^ while all odd powers of a negative monomial are negative ; all powers of a positive monomial are positive. EXERCISE LXXXVI Expand the following expressions : 1. (a^^. 6. -(-4^(^)^ ^^ (-11^53)4 2. ia^y. ■7- (2a;V0^- ' i^a%y 3. (- a^y. «• (- 2 "^"y^'y- 12. - r^-^Y- 9. _(_ 4:^5^)6. \ 2ac j 3 212 ELEMENTARY ALGEBRA [Ch. XIV, §§ 172, 173 BINOMIALS 172. The expansion of binomials may be shortened by employment of the Binomial Theorem, a proof of which is given in Chapter XXIV. The use of this theorem is evi- dent from the following type forms, which are derived by multiplication : (a + hy = d' + 2ah + h^, (1) (a + 6)3 = a^ + 3 arh + 3 aZ^^ + W, (2) (a + hy = a^ + 4^ a% + 6 a^^ + 4 aW + h\ (3) (a + 6)^ = a^ + 5 a% + 10 a%' + 10 a%^ + c>ah^ + h\ (4) (a + 6)« = a« + 6 a% + 15 a'ly" + 20 a^l/ + 15 a^¥ + 6 a6^ + h\ (5) Similarly, it may be shown that the expansion of the binomial (a — J) gives, if the exponents are those of the left members respectively, the results in (1) to (5), except that the signs of the terms are alternately plus and minus, the first term being plus. 173. Examination of the expanded forms shows, if n be the exponent indicating the pow6r, and a and b are respec- tively the first and second terms of the binomial, that 1. The number of terms in the expansion is ti + 1. 2. Every term, except the last, in the expansion con- tains a.; and every term, except the first, contains h. 3. The exponent of a in the first term is n^ and decreases by 1 in each succeeding term ; the exponent of h in the second term is 1, and increases by 1 in each succeeding term. 4. The first coefficient is 1, the second n ; the third, and any subsequent coefficient, is derived from the preceding Cft. XIV, § 178] INVOLUTION AND EVOLUTION 213 term by multiplying the coefficient by the exponent of a and dividing this product by the exponent of h increased byl. Any binomial may be expanded by this method if in the right member a equals the first term and h equals the second term. 1. Expand {a^-1iy. By type form (3), § 172, (a2 -2 &y = (a2)4 _ 4 {a^f (2 6) + 6 {(j?)\2 Vf - 4 (a') (2 by + (2 b)* = a« - 8 a% + 24 a'b' - 32 a'b^ + 16,b\ In a similar way, a polynomial, in the form of a binomial, may be expanded. 2. Expand Cx-2y+S z^. By type form (2), § 172, l(x-2y)+3zJ=(x-2yy+3(x-2yy(3z)+3(x-2y)(3zy+(Szy = a^-6 x^y +12 xy^-Sf +9 afz-36xyz+36y''z +21xz^-54.yz'+21z\ EXERCISE LXXXVII Expand the following expressions : 1. Qp + qY^ 8. (2a+iy, 2. (^x+yy. 9. (x+2yy. 3. (1 + ay, 10. (x^-y'^y. 4. (p+qy. 11. (1-qy. 5- (x-yy, 12. (2x-3yy. 6. (h+iy, 13. (3:^-22/2)4. 7. (x + yy, 14. (3 mn — ipy 15. (2a;2_5y)5. 16. (2a^-8b^y. 17. (a — S + 6?)3. 18. (^a-b-2ey. 19. (22:-y+3^)3. 20. (a — b— cy. 21. (2/2^2— 3:i:^+?/2/. 214 ELEMENTARY ALGEBRA [Cn. XIV, §§ 174-170 EVOLUTION 174. The operation of extracting a root of an expressio] is called evolution, and is indicated by the radical sign, V The quantities whose roots are to be extracted, calle< radicands, are written after the radical sign. The par ticular root to be extracted is indicated by a small number called the index of the root, written above the radical sign The index 2 is generally omitted. If the index of the root is an even number, the root is called an even root ; if an odd number, the root is called an odd root. Thus, V4, -\/81, V?", are even roots ; "v/g, -^^^243, '""^^?^, are odd roots. 175. If a quantity can be expressed as the product of two equal factors, one of these factors is called the square root of the quantity ; one of the three equal factors of a quantity is called the cube root ; and, in general, one of the n equal factors is called the /ith root. Since involution and evolution are inverse processes, 176. The one positive root of a positive number is called its principal root ; the one negative root of a negative number is called its principal odd root. The radical sign will be used to indicate the principal roots only. Thus, V4 means the positive square root of 4;. that is, V4=+2; similarly, V25= +5; -^^:::27=-3; ^^=^243=^3; \j a'' = a. Note. Only expressions whose exponents are multiples of the indices of the roots will be discussed in this chapter. Oh. XIV, §177] INVOLUTION AND EVOLUTION 215 MONOMIALS 177. The Index Laws for the evolution of monomials are the inverse forms of tlie Index Laws for involution. I. -v^'o^ = d"". By I, § 170, (d'y = d^\ )y definition, -v^a^*^ = d^. 11. ^al^h'^e^abc. By II, § 170, (obey = a^5V% 3y definition, ^ d'lfc'^' = ahc. From I and II is derived the Rule for the Root of a Monomial in the form of a Product: Divide the exponent jf each factor hy the index of the required root. IIL 4f=t By III, §170, (|)" = |, 3y definition, j^lw^ __ a From III is derived the Rule for the Root of a Monomial n the form of a Quotient: Divide the exponent of each factor in the terms of the fraction hy the index of the 'e quired root. 1. Simplify x/^^^i^ 343 ai2j9 ^f6 4xY ^ A T'^y^ J2?xy^ ___ ^:xf ^'343a^%'^ ^Va^'^W^ ' " 7 a'b' 7 a%^ 216 ELEMENTAKY ALGEBRA [Cu. XIV. § 178 EXERCISE LXXXVIII Simplify the following expressions : 1. V^. 9. Vy a%*c^ 17. Va\x - yf 2. V4^. 10. V64 a%^(fi. 18. V-^3^f\ 3. VmV. 11. -^21 a%K ,y-^ 4. v^^SP^. 12. ^16^*P. ^®- \ 16^' 12 5. -^ 8-27. 13. Via+by. ^1 6^a%^ 6. -^^85^2. 14. Va^-2ab + b^, ^^' ^~M3^ 7. represents the part of the root already found and if u represents the next term of the root. CH.XtV,§178i INVOLUTION AND EVOLUtloN ^it 1. Extract the square root of 4:x^+ 4x1/ + y^. Let t^ + 2tu + u^ = 4: x^ + 4.xy + y^, (1) by(^), t = 2x, (2) squaring (2), f = 4:X^, (3) subtracting (3) from (1), u(2 t + u) = Axy + y^f (4) by (5), u = y. (5) Substituting t = 2xj and u = y, in (4), u{2 t + u) =?/(4 x + y)r=4.xy + y\ (6) Since V4 aj2 -f. 4 a;?/ + ?/2 = V^^ -}- 2 ^^^ + ^^^ ^ ^ + u, (7) and since ^ = 2 ic, and u = y, V4 x^ -]- 4:xy -\- y' = 2x + y: (8) The work may be more compactly written : t = 2x 4.x' + 4.xy-hy'\2x + y 4.x' 2t = 4.x u = y 2t-\-it = 4:X-{-y u(2t + 2i) = y(4.x-{-y) 4.xy + y^ 4 xy + y^ The terms of the polynomial should be arranged either in ascending or in descending order of some one of its letters ; otherwise the formula method is not available. If the polynomial contains more than three terms, it should be carefully noticed that the part of the root already found in every case is represented by t. Since (a + h -cy=\_a^(h - c)]^ = [(a + ?>) - cf, and since {t + iif = (a + 6 — cf, t is represented successively by a and a-^b. 218 ELKMENTAUY ALGEBUA [Cii. XIV, § 178 2. Extract the square root of a^ + i c^ + P—2ab + 4: be — 4: ae. t = a \a — b — 2c a^-2ab-4:ac + b'- + 4:bc-\-4 c' a' 2t = 2a u = ^b 2t-Yu = 2a-b u(2t + u)==-b(2a-b) ~2ab-4 ac + b^ + 4:bc + A r ~2ah +b^ 2t = 2a-2b u = -2c 2t + u = 2a-2b-2c u(2t+u)=-2c(2a-2b-2c) — 4:ac + 4 &c + 4 c- — 4 ac + 4 6c + 4 c- In the above example, after the second term of the root has been found, the first two terms are together equal to t.. Since t=(a—b), andi has been squared and subtracted, the remainde': again corresponds to the expression u(2 1 + w). EXBBOISB liXXXIX Extract the square roots of the following expressions : 1. 25a2-70ac + 49c2. 2. a^ + 2ab + b^+2ac + 2bc + (^. 3. b^ + 2hc + e^-2ab-2ac + a^. 4. 4: a^ + 12 ab + 9 P + 16 ac + 2i be +16 c^. 5. 49a^ + ib^-28ab + 42ac + 91 and <100 has one digit, the square root of a number >100 and < 10,000 has two digits, the square root of a number > 10,000 and < 1,000,000 has three digits ; and so on. If, therefore, tlie number be separated into periods of two digits each, running from right to left, the number of periods will equal the number of digits in the root. Thus ViT64 has two digits, V811,801 has three digits. 182. Every integral number may be considered as made up of tens and units. Hence (t+ u}^^ where t represents the part of the root already found and u represents the next term of the root, will correspond to any integral number in the form of a perfect square. 42 = 40 + 2 = ^ + ^^, (1) squaring (1), (42)^ = (40 -j- 2)^ ={t + u)', (2) simplifying (2), 1764 = 1600 + 160 + 4: = f + 2tu + u^ (3) indicating square roots in (3), V1764 = V1600 + 160 + 4 = -y/t' + 2fAi + u\ (4) 1. Extract the square of 1764 = 1600 + 160 + 4. ^ = 40 f = 1600 2^ = 80 u= 2 2t + u==:S0-\-2 i((2t + u) = 2(S0 + 2) 1600 + 160 + 4 I 40 + 2 = 42 1600 160 + 4 160 + 4 Ch.X1V,§182J involution AND EVOLUTION 223 The work necessary in writing a number in the form f -\-2tu+ (r is tedious, and may be abridged; the preceding written in the abridged form is: < = 40 17 64 1 40 + 2 = 42 f- = 1600 16 00 2t = 80 164 u= 2 2t + u = S2 u(2t + tt)=2(S2) 164 In the above example, if ^ = value of the digit in the tens' place, and u = value of the digit in the units' place, t is the greatest multiple of 10 whose square is < 1764 ; that is, ^ = 40. Subtracting ^- = 1600, the remainder is 1G4. Dividing X64 by 2 if = 80, the quotient is 2, which is u. Hence u(2t-{- u) = 2(80 4- 2) = 164 is to be subtracted from the remainder, 164. The remainder being 0, the square root is 40 + 2 = 42. In the above example the work may be further abridged by omitting the two zeros in the square of 40. 2. Extract the square root of 4,414,201. t = 2 4 41 42 01 <2 = 4 4 2< = 40 41 u= 1 2 < + t< = 41 u{2t+u) = \{Al) 41 2 < = 420 42 u= 2 « + w = 420 m(2« + It) = 0(420) 2 1 = 4200 42 01 u= 1 2t + u = 4201 ?«(2< + m) = 1C-^201) 42 01 2101 224 ELEMENTARY ALGEBRA [Ch. XIV, §§ 183, 184 183. Since VO.Ol = 0.1, VO.OOUi = 0.01, VO. 000001 = 0.001, etc., the square root of a decimal in the form of a perfect square has half as many decimal places as the number itself. A decimal is therefore separated into periods of two digits each, running from left to right. After pointing off the decimal, the square root is extracted as if the decimal were an integer. 1. Extract the square root of 0.01301881. <=1 0.01 30 18 81 1 0.1141 f=l 1 2t = 20 30 u= 1 2t + u = 21 u(2t + u)=l(21) 21 2 < = 220 918 M= 4 2 < + M = 224 m(2< + m) = 4(224) 8 96 2 t = 2280 22 81 u= 1 2t + u== 2281 M (2 < + «) = ! (2281) 22 81 Since there are eight decimal places in the number there are four decimal places in the root. 184. The approximate square roots of numbers, whether integral or decimal, or both, not in the form of perfect squares, may be found by annexing zeros to fill out the periods of two digits each until the number of periods equals the number of root digits required. Ch. XIV, §184] INVOLUTION AND EVOLUTION 1. Extract the square root of 7.1 to three decimals, 225 t = 2 7.10 00 00 t' = i 4 2< = 40 310 ti= 6 2 t +11=4:6 u(2t + u) = 6(i6) 2 76 2i = 480 34 00 u= 6 2 < f M = 486 m(2< + m) = 6(486) 2916 2 < = 4920 4 84 00 M= 9 2t + u = 4929 m(2< + k) = 9(4929) 4 43 61 EXERCISE XOI Extract the square root of the following numbers : 1. 361. 6. 136,161. 11. 0.1369. 2. 1681. 7. 3,404,025. 12. 0.134689. 3. T396. 8. 1,225,449. 13. 0.094864. 4. 71,824. 9. 3,466,383,376. 14. 8476.0436. 5. 15,129. 10. 0.0081. 15. 2499.700009. Extract the approximate square root to four decimals )f tlie following numbers : 1.6. 2. 19. 6. 22. 0.831. .7. 3. 20. 7. 23. 10.4. .8. 5. 21. 10. 24. 32.701. CHAPTER XV RADICALS 185. The quantity Va has already been defined, § 17t as the quantity whose nth power is a, or (V^)'* = ^. I a is an exact n\h power, the existence of such a quantity i at once evident, as V8 = 2. But if a is not an exact ni power, it becomes necessary to prove the existence of -yja Such a proof is beyond the province of this book ; an^ a simple numerical example must suffice. It is not pos sible to obtain exactly the value of V2, since there is n number, integral or fractional, whose square is exactly 2 ^^^' (1.4)2 <2<(1.5)2, (^ (1.41)2< 2 < (1.42)2, (^ (1.414)2< 2 < (1.415)2. {G In (J.), since 2 lies between (1.4)2 and (1.6)2, V differs from 1.4 and 1.5 by less than they differ from eac' other : that is, since 1.4 and 1.5 differ from each othe by 0.1, V2 differs from either by less than 0.1 ; similar! in (^), V2 differs from 1.41 and 1.42 by less than 0.01 and in ((7), V2 differs from 1.414 and 1.415 by less thai 0.001. Continuing the process shown in (^), (^), an( (6^), a number may be found which will represent a close an approximation of V2 as is required. p b l5 c D S X Fig. 11. 226 (ii.XV, §18G] RADICALS 227 The value of V2 may be represented graphically. On the lino OX, Fig. 11, let equal distances be laid off from toward the right, and OA represent the number 1, OB the number 2, etc. Then 1.4 will be represented by OC, 1.5 by OD. The numbers 1.4, 1.41, 1.414 will be seen to be represented by lines whose terminal points move toward the right, while the numbers 1.5, 1.42, 1.415 will be represented by lines whose terminal points move toward the left. The terminal points representing these two sets of numbers will approach each other, but no terminal point in either set can cross into the region of the other. Yet the numbers show that the terminal points may be made as near to each other as may be required. There will be some point P which will be the limiting posi- tion of both sets of terminal points; and the line OP will represent V2. 186. An indicated root of a quantity is called a radical. Thus, Va, V27, are radicals. An expression which is composed of radicals is called a radical expression. Thus, -Vx + V27, Va — V^, are radical expressions. All integers and fractions are called rational quantities. All other numbers are called irrational quantities. The simplest class of irrational quantities consists of indicated roots which cannot be extracted. Thus, 2, and |, are rational ; V2, Vl + V2, are irrational. An expression which contains rational quantities only is called a rational expression. Thus, a + f is a rational expression. An expression which contains an irrational quantity is called an irrational expression. Thus, a + V2 is an irrational expression. 228 ELEMENTARY ALGEBRA [Ch. XV, §§ 187-189 187. A radical whose radicand is rational and whose root is irrational is called a surd. Thus, -Va and -v^4 are surds ; while v 1 -f V3, being the in- dicated root of a quantity not rational, is not a surd. The order of a surd depends upon the index of the root. A quadratic surd, or a surd of the second order, has 2 for the index of the root ; a cubic surd, or a surd of the third order, has 3 for the index of the root ; a biquadratic surd, or a surd of the fourth order, has 4 for the index of the root, etc. Thus, Va, V^, Vc, are respectively quadratic, cubic, and biquadratic surds. 188. A rational factor of a surd is called the coefficient of the surd. Thus, f is the coefficient of | Va5. Surds which have 1 as a coefficient, expressed or im- plied, are called entire surds. Thus, -Vay and Vi are entire surds. Surds which have other coefficients than 1 are called mixed surds. Thus, 2VS and SVa — b are mixed surds. A surd is called a monomial surd if it consists of a sin- gle surd. Thus, a/^ and 5V3 are monomial surds. The sum of a rational, and a surd quantity, or the sum of two monomial surds, is called a binomial surd. 189. The difference between algebraic and arithmetical irrational quantities should be noticed. Such a quantity . XV, § 100] RADICALS 229 1,8 V2 is an arithmetical irrational quantity ; similarly, quantities such as Va are considered algebraic irrational [uantities, although if a = 4, Va is an arithmetical rational [uantity. In this, as in the preceding chapter, the principal roots mly are discussed, and the quantity under the radical lign is restricted to positive values. Thus, VJ =^±2, but Vi = 2. This fact must be kept in nind, for otherwise some of the proofs of the principles will lot hold. PRINCIPLES OF RADICALS 190. I. The product of the nth roots of any number of quantities is equal to the nth root of their products. By II, § 170, C^a Vb V~cy = abc, oj definition, "Va VJ ^c = -Vabc. If the radicand contains a factor whose exponent is a multiple of the index of the root, the surd may be simpli- fied by I. Since -^'a"" = a^ by I, V^ = V^-^ = a^. 1. Simplify Vl6. ^/16 = -v/2^ = -v/2^ -v/^ = 2-v/2. 2. Simplify 'l25c2 3. vi. 16. €• 4o 5. 17. ^ll- 6. ^. 18. \ 2. (a/3)2. 5. (-v/8, ^/5>V2. EXERCISE XCVIII Reduce the following surds to equivalent surds of the same order : 6. -v^T^, ■\/Jab\ 7. 4 V2 T^y^ ■\/2 xy\ 8. W^, -sjWf. 1. V^, -V7\ 2. Va, Va. 3. V2:r, ^\x\ 4. ^a%, ) + V4 ah. Before finding the square root it is necessary that the term corresponding to 2^ ah shall be writter with the coefficient 2. 1. Extract the square root of 8 + V 60. Vs+Veo = Vs + 2 Vl5 = V3 + V5. 2. Extract the square root of V24 + V25. V V24 + -yj'lh = V2vfT5 = V3 + V2. EXERCISE CIV Extract the square root of the following binomial quad- ratic surds : 1. 3+2V2. 6. 9-2VI4. 11. V121-V120. 2. 4 + 2 V3. 7. 11 - 2 V2i. 12. V64-V28. 3. 7 + 2 VlO. 8. 11-2V28. 13. V256-V156. 4. 8 - 2 Vl5. 9. V121-2V10. 14. J^— JVI4. 5. 6 + 2 V5. 10. V8T-V80. 15. 2 -Vs. H. XV, §203] RADICALS 245 RADICAL EQUATIONS 203. An equation which involves the indicated root of le unknown is called a radical or irrational equation. Thus, V3 -\-x = 2 is a radical equatiouo A radical equation which involves square roots only can ften be solved as a simple equation by isolating one or lore of the radicals and rationalizing the resulting equa- on by squaring. But since two equations with different gns may give the same result when squared, the solution btained by solving the squared equation does not neces- irily satisfy the given equation. It is necessary to test the olution in every case hy substituting in the given equation. If the equation contains a single radical, it is simpler to jolate the radical and then square the resulting equation; t the equation contains two or more radicals, the more ivolved radical is isolated. The squared equation should [len be simplified, especial care being taken to reduce ae resulting equivalent equation to the simplest integral Drm. 1. Solve the equation : Va;+ 6 + ■\/x— 2 = 4. (1) Transposing in (1), Vaj + 6 = 4 — -yjx — 2, (2) ^uaring (2), x + 6 = 16- 8Vaj-2 + x-2, (3) L'ansposing and uniting in (3), 8V^^=^ = 8, (4) ividing (4) by 8, V^^=^ = 1, (5) quaring (5), a; — 2 = 1, (^) ransposing and uniting in (6), x = S, (7) Verification : V9 + Vl = 4 ; therefore 3 is a root of (1). 246 ELEMENTARY ALGEBRA [Ch. XV, § 203 ; 2. Solve the equation : ■Vx+ 6 — -Vx —2 = 4. (1) Transposing in (1), Vif + 6 = 4 + Vx — 2, (2) squaring (2), x + 6 = 16 + 8 VoT^ + x-2, (3) transposing and uniting in (3), 8V^^=^ = -8, (4) dividing (4) by 8, Vx^^ = - 1, (5) squaring (5), a? — 2 = 1, (6) transposing and uniting in (6)jX==3, (7) Substituting in (1), V9 — Vl ^ 4 ; therefore 3 is oiot a root of (1). 3. Solve the equation : ('a — b}\——- + b = a. (1) a — b Transposing in (1), (a — h) \/ = a—b, (2) dividing (2) by a - 6, yj^ = 1, (3) squaring (3), — ^ = 1, (4) multiplying (4) by a — 5, x = a—b, (5) Verification : (a—b) \h^-^ — |-6 = a; a — b + b = a. ^a — b 4. Solve the equation : V2 + a;+ Va; — 3= V4a;— 3. (1) Squaring (1), 2 + a^+2Va.•--.^'-6+.^-3 = 4a;-3, (2) transposing and uniting in (2), 2-\/x^—x—6=2 x—2, (3) dividing (3) by 2, V? - ic - 6 = a; - 1, (4) squaring (4), x' — x — Q> = x- — 2x-^ 1, (5) transposing and vmiting in (5), x = 7, (6) Verification : VO + V4 = V25. Cii. XV, § 203] KADICALS 247 EXERCISE CV Solve the following radical equations : 1. -Vx+5 = S. - 4. V7:r+2 = 4. 2. 6Va: + 4 = ll. 5. ■V5-{-x=3-^/x. 3. 7 = 3V^-4. 6. Vl5 + a;=3V5-V^, 7. V22:+ll + V2:^-5 = 8. 8. V27:r+1 = 2-3V3^. 9. V4 + a; V24 + 2;2 = :?; + 2. 10. V^_ + V^^5. 11. Vx + Va + a; : 12. V^ + V3 + a; : . -y/S + x 13. Va; + 4 a6 = 2 a + V^. 'Va + X 24 14. V:r+ ^ = a — -\/x — a. 15. h — a-\/x = ^/€fix. 16. — = — ' ^x+'i ■\/x+Q 17. x= a — ^ a?" — x-y/x'^ + 8 a?. 18. V5 + 22:= V2(8 + 92:)-Vl + 8a;. 19. 3Vl + 2:i:- V8:t:-15= V2(a; + 6). 20. V9a:-14 + 3V^+2 = 2V92:-2. 248 ELEMENTARY ALGEBRA [Ch. XV, § 203 REVIEW EXERCISE CVI Simplify the following expressions : 1. ^f 16. ^|. 29. (^/ax'y. '■4- 17. €■ 30. -^/=^=V25^ = 5^, VZTie + V"^^^^ + V^=^ = (4 + 5 -j- 6)/ = 15 i. (11. XVI, § 208] IMAGINARIES 251 3. Multiply V^^ + V2 by 2 V^4 + 3 V2. 2i+V2 4 i +3V2 + GV2i + 6 8 »2 + 10V2 i + 6 :-8 + 10V2 i + 6 = 10V2^• - 2. 4. Divide 1 by V^^ + V- 3. 1 V2 1 - V3 1 ^ V2 ^• - V3 ^• ^ V2 I - V3 1 V2i+V3*'V2j-V3i~ 2i^-3t^ -2 + 3 ' =(V2-V3)i. 5. Expand ( V^^ + V^^y. ( V3 i + V5 if = [?■( V3 + V5)]2 = *2(a/3 + V5)2 = _1(3 + 2Vl5 -f- 5) = - 8 - 2Vi§. 6. Extract the square root of 1 + 2V— 6. Vl + 2V6i=V2«- + V'3. EXERCISE OVII Reduce the following pure imaginary quantities to the ;ypicai forms : 1. V^^25. 4. yP:^. 7. V-225/. 2. V^;^. 5. V- x!^y\ 8. V-484a:8. 3. V-100. 6. -\^--i:3^y\ 9. V-625a^«/2. Simplify the following expressions : 10 V^;n^ + V-25 + V-a4-5v'-ioo. 11. V^;^ly + V-121-V-169-V-196. 252 ELEMENTARY ALGEBRA [Ch. XVI, § 20$ 12. V^a;2-V-4a;2- V-9 a:^ + V- 25x2. 13. 3+V^^ + 5V^n^ + 16 + 7V^"225. 14. a + by/-x^+2a- bV- x^ + Sa-i 6V- x^ 15. ^^^ • V- 9. 18. V-i-V-i. 16. V^:^25 • V^^36. 17. V^) • V^^16. 21. (V^a, + - 22. ( V^^ + 19. — V- v/- rr* 20. V— a^^^ • V— ab*. ^)(2V:^+3V^). 23. V— a^J • V— ^252 . y_ ^53, 24. V^2 • V^^ • V^^. 25. (i+v^r4)2. 26. ( V^^ + V^)2. 27. (2 V^^ + 3 V^=^)2. 32. (1+V^^)5. 30. (V=^+l)8. 31. (1 + V^=T6)*. 28. (3V-9-4V"^)2. 29. (1-V^^)3. 33. (V-| + V-i)a ^7 V^^Ti V-9 1 38. V-16 Q 39. V-9 4.0 9 34. (V-J^ 35. (V^:4+V^^+V-16)2. 36. ( V - a^ - V^p _ ^Zr^yi, Va i)^ 41. 42. -\/ — 7? 43. V-49 V^:^ 44. -V~25 1 45. 46. 47. 48. 2 + V- -9 a + V- ^h a — V- ~-b a-V- rft a+ V- ^6 1 V2+V-2 Ch. XVI, § 208] 54.VZ3 3+2V3-5' 8 - 5V^^ 6 + SV^s' IM AGIN A HIES 253 49. 50. 51. 52. —= V2+V3 + V-5 53. 54. ViTi V3 + V7+V-10 l + V2 + V^:^* 55. (2V3-V^)(4V3-2V^^). 56. (a;-5 + 2V^)(a;-5-2V^^). 57. (V^--2 + V^3)(V^ + 2-V^^). 58. (a;-2V5 + 3V^=^)(a;-2V5-3V^^). 59. (2 - V^ - 3 V^^)(4 V^^ + 6 V'^2). 60. {x-\^ 1 V^) {x-\- 1 V^3). 61. (a;-2-V3)(a;-2 + V3)(a;-3 + V^^) (a;-3-V^=n[). 62. (a;-l-A/^2)(x-l+V^^)(a;-2 + V"^) (a;-2-V^=^). 63. Vi + v^^ . Vi_v^T • V3_v^2 • V3+V^2; 5.3.4 64. : + : + 4-V-4 ' 1+V-l ' 1-V^l 65. (V^T)2+(v^^)3+(v^^)4+(V^=i:)5 +(V-i)«+(V-iy+(V-i)8. CHAPTER XVII THEORY OF EXPONENTS THE EXPONENT IN THE FORM OF A POSITIVE FRACTIO^^ 209. In § 177 it was shown, if m and n are integers and n. 71 is a multiple of m, that Va" =^ a"^. If, however, n is not an exact multiple of m, there can be no meaning attached n to a"* according to the previous definition, § 7, of an ex- ponent. Thus, it is impossible to speak of a^ as meaning a taken three-fourths of a time as a factor. The definition of an exponent is therefore extended to include the expo- nent — , it being understood that a'^ (where n and m are m positive integers and a is a positive real quantity) is siinply an alternative way of writing V^% or the principal value of the wth root of the nth power of a. This extension of the definition of an exponent is valid only in case exponents in the form of a positive fraction conform to the laws of exponents which have been shown to hold for positive integers. That is, exponents in the form of a positive fraction must be shown to obey the laws, a'^ay = a^+^, I a'^ -^ a'^ = a'^-y, II (a-^)?/ = a^y, III (ahy^^^a'^h''. - IV 254 Cii. XVII, § 210] THEORY OF EXPONENTS 210. I. n r n r By definition, « r a'" a' = Va" Va'", by V, § 196, by I, § 190, = V««'v'a'»'-, = Va«»+""', ns+mr by definition, = a ^* , n r or, = a^'* *. II. n r w r By definition, n r by V, § 196, by II, § 193, = - n, there is no difficulty : but if m-3. 7. 86-i 11. (a; + «/)*. 4. 9i 8. GV)"^. 12. 3(a + 25)-^ Change each of the following radicals into expressions containing exponents in the form of fractions : 13. /9^+VF^. 24. ;-:r-l). 10, (a-3^-56^)~(a-^-62«). 13. (^f _2/^)_j.(^i_^i). 11. (ii;*-|-:r^- 6)^(2:*-2). 14. (p^ -q)-^(p^-q^). 15. (a^-jt)^(a*+a*5i+J*). 16. (a;3__^2>^^(^J + ^i), 17. (2 a5J-3 - 5 a^}-^ + 7 a^J-i - 5 ^2 + 2 aJ) 18. ( -v^^* - 4 rry + 4 y -v/^ + 4 2/2) -^ ( -^^ + 2 x^y^ + 2 «/) . Extract the square root of the following expressions : 19. x-^ — Gx-^+llx-^—Gx+x^. 20. 4a;" + 9a;-" + 28- 24a; 2_16a;^ 21. 1 +4a:-^- 2a;~*- 4a;-i+ 25 a;"^ - 24 a;"* + 16 a;-2. Cii. Xvn, § 217] THKORY O^ EXPONENTS 263 217. By the principles of tlie preceding articles many expressions may be simplified. {a-b)-' "■ \a bj ^ \ ab ) = -(a-b)\ 2. (a-'-^)"+^ + -^^^ = a"(--^) + - = a*^ -" + « = ^ + a. ^ ^ a a a" 2"+^ • 2"-^ "" 2'^' "" 2^' EXERCISE ex Simplify each of the following expressions, giving each result in a form free from radicals and from negative exponents : 1. 2. ( ■4a-'xh^ „ ^'"^^^"-^ 8. ■^a^6*c-i 2* • 9^ • 4^' 8a2 N-i ,« 2 a;2;y ^ 6« \27 a-^W Sa^-^ 7^^i Va:a-i a: = 4, c = 0; 4a;^ + 4a; + 3 = 0isa complete, or affected, quadratic equation in which a = 4, 6 = 4, c = 3. PURE QUADRATIC EQUATIONS ^2. (1) Extracting the square roots in (1), ± x = ± a, (2) (+x=+a, (3) The complete form of (2) is — x= — a, (4) — x=+a, (5) x = — a. (6) A value of —x is not required; therefore, multiplying (4) by — 1, x= a, (7) multiplying (5) by — 1, x = — a. (8) It is evident that (3) and (7) are indentical ; and that (6) and (8) are identical. Hence, if the double sign be used only in the right membei^, the roots are not altered in value. Thus, Extracting the square roots in (1), x = ± a. Verification : a^ = a^, 2. Solve: ^-20 = ^. (1) 4 5 -^ (Clearing of fractions in (1), 5x^ — 400 = 4 x^, (2) transposing and uniting in (2), a? = 400, (3) extracting square roots in (3), a; = ± 20. Vkhikication: 4p0_20 = 400_ 4 5 Note. , If x^ is negative, the signs of all terms must be changed, since the square root of a negative number cannot be obtained. 270 ELEMENTARY ALGEBKA [Ch. XVIII, § 22j EXERCISE CXII Solve the following equations : 1. ^2=169. ^^ x+5 _ 2x + 7 2. x^-a^=0. ^ + 1-3 3a; + 18 3 3. :?;2_81 = 0. 12. f+^-^=-19i+: 4. 3^:2=48. 13. 1 'T-l ^+ 9" . ^ 5. 25x'^-b^ = 0. x^y x + ^ x^-l 6. a^x^=b^x^, -. r?: + a,a;— a 7 14. ^ -| =6. 7. lla;2=36 + 22:2. ^""^ ^ + '* , 8. x^ = a^+2ah + b^ 15. 2^ = ?^. (?:r + a ax — 9. ax2 — ah — 2 ax^. ^ t\ i a{x — 0) _a_^bx 10. (7 2;)2=296-(5:i;)2. • 6a; ""6 a' 17. ^—2 + 2 = 3a* + l. 12/ . . 2 \ 5 - f (^->+4i) a; + l SOLUTION OF QUADRATIC EQUATIONS BY FACTORING 221. If the product of two quantities be zero, either of the two quantities may be taken as equal to zero. When the left member of a quadratic equation, reduced to the general form, can be factored, either factor may therefore be taken equal to zero, or equated to zero. The roots of the factors are therefore the roots of tlie equation. The Factor Method holds for all forms of quadratic ; equations, botli complete and incomplete. Til. XVIIT, § 221] QUADRATIC EQUATIONS 271 1. Solve by factoring : 9 x^= 36. (1) Dividing (1) by 9, x'=:4:, (2) transposing in (2), a^ — 4 = 0, (3) factoring in (3), (x + 2) (x-2) = 0, (4) equating each factor in (4) to zero, \ ' (5) ^. X ~^-~ ij — Uj transposing in (5), x = — 2, or x = 2. (6) Verification: 9(-2/ = 36: 9(2)2=36. 2. Solve by factoring : ax^+ hx = 0. (1) Factoring in (1), x{ax + ?>) = 0, (2) ^ (3) ax+'b = 0, transposing in (3), a; = 0, ax = — b, (4) dividing ax = — bhy a, x = • (6) Verificatiox : a(0)+5(0)=0; af -^^ + bf --\=^^' -- =0. \ aj \ (^J <^t ^ 3. Solve by factoring : x^-4:x-21 = 0. (1) Factoring in (1), (x - 7) (a; + 3) = 0, (2) (-/^ 7 = equating each factor in (2) to zero, / ^ (3) \^ X -j~ o -^ ", transposing in (3), x = 7, or x = — 3. (4) Vekification : I (7y-4(7)-21 =49-28-21=0. I (_3)2_4(_3)_21= 9 + 12-21 = 0. 272 ELEMENTARY ALGEBRA [Ch. XVIII, §§ 222, 223 EXERCISE CXIII Solve the following equations by factoring : 1. 2^2+7^+12 = 0. 8. 3:z:2_25^ + 28 = 0. 2. x^+x-m = 0. ^ 15:^2+23a:-28 = 0. 3. ^2_^_i2 = 0. ^^ -63a.2+16:?;-l = 0. 4. :z;2 _(. 9 ^ + 20 = 0. 5. :^2+2a;-224 = 0. ^^- ^'"2!^ 20^^* 6. :^2_7^_260=0. 1 12. 4:0x^-x- — =0. 7. 2:i;2+9a;-5 = 0. 20 13. x^ — (a+h)x+ab=0, 14. :z;2-a;(2j9 + 5^) + 10jt?^ = 0. NUMERICAL COMPLETE QUADRATIC EQUATIONS 222. If the coefficients of the equation ax^-{-bx+c = are numerical, the equation is called a numerical complete quadratic equation. Thus, 5x^ + 7 X — 3 = is a numerical complete quadratic equation. 223. Solution by completing the Square. By §§ 73 and 74, (ir ±7^)2 = x^±2nx+ n^. The third term is evidently the square of half the coefficient of x. If the left member of a complete quadratic equation contains the unknowns only, and the right member the absolute term, the equa- tion may be made to assume the form x^ ±2 nx by dividing the equation by the coefficient of x"^. The left member may be put into the form of the square of a binomial by adding tlie square of half the coefficient of x^ a process t Ch. XVIII, § 223] QUADRATIC EQUATIONS 273 which is called completing the square. This process is best understood hy examples. 1. Solve the equation: x^—6x= 16. (1) The left member is already in the form 3if—2nx] that is, the coefficient of x^ is unity. Half the coefficient of x is — 3 ; ( — 3)^ = 9. Therefore, adding 9 to both members of (1), so as not to destroy the equality, or, completing the square in (1), x^ — 6x + 9 = 2o, (2) extracting the square roots in (2), a; — 3 = ± 5, (3) transposing and uniting in (3), it? = 3 + 5, or, ic = 3— 5, - combining in (4), x=8, or x = — 2. (5) Verification : (8)2_6(8) = 16; (-2)2_6(-2) = 16. • 2. Solve the equation : a;^ _ 14 ^ __ 1][ _. 0. (1) Transposing in (1), cc^ — 14 a; = 11, (2) completing the square in (2), o?^ — 14 a; + 49 = 60, (3) extracting the square roots in (3), x — 7 = ±2 VT5, (4) transposing and uniting in (4), x = 7 + 2 Vl5, or, aj = 7— 2V15. '} (4) (5) IFICATION : + 2V15)2 - 14(7 + 2Vl5) - 11 = 49 + 28 Vl5 + 60 - 98 - 28 Vl5 - 11 = 109 - 109 = 0. (7 - 2 Vl5)2 - 14(7 - 2Vl5) - 11 = 49 - 28Vi5 + 60 - 98 + 28v'15 - 11 = 109 - 109 = 0. 274 ELEMENTARY ALGEBRA [Ch. XVIII, § 223 3. Solve the equation : x^—Sx=4:. (1) Completing the square in (1), a;^ — 3 a; -f f = 4 + | = ^^\ (2) extracting the square roots in (2), x — ^=± |, (3) transposing and uniting in (3), x = 4, or it^ = — 1. (4) Verification: (4)2-3(4) = 4; (- 1)^- 3(- 1)= 4. 4. Solve the equation: 1 :!_==_. ^ 22: + ! 3-2: 6 Clearing of fractions in (1), 5(3 - x) + 5(2x + l)=6(2x + 1)(3 - x), simplif3dng in (2), 15 - 5 x + 10 X -{- 5 =- 12 x^ + 30 x + 18y transposing and uniting in (3), 12x'-^25x = -^2, dividing (4) by 12, x^-^j^ = -^, completing the square in (5), ^_25_^' /25Y^/25Y__ 2 529 12 V24y V24y 12 576' ^ ' extracting square roots in (6), ^ 2.5. L. 2 3 /7> transposing and uniting in (7), x=2, or x — -f^, (8) Verification : 1 j^6. _1_,_1_^6 12^42^6 5"^ 5'J. + l"^3-iV ^^'^ ^^^ ^'' Rule for solving Numerical Complete Quadratic Equations : After clearing the equation of fractions (if any exist) ^ trans- Ch. XVIII, § 223] QUADRATIC EQUATIONS 275 pose the unknow7i8 to the left member and the absolute term to the right member ; divide the equation by the coefficient of x^ ; complete the square by adding to each member the square of half the coefficient of x ; extract the square root of each member; solve the simple equations thus derived, EXERCISE CXIV Solve the following equations by completing the square : ' 17. 2 a; + 14 + - = 0. 2. 3. a:2 + 12x=13. x'^ + x-2--^ 0. 18. X 4 4. 5. x^ + x-^i^ 0. :0. 19. X"^ X 9, 2a; 8 A on 113.r— :^i'r2— _ 411 7. x2 + 2x + 40 = 0. 8. 2:2-32:+l = 0. 9. 2^2 + 5^-7 = 0. 10. ?>x'^+bx=2, 11. 3 2;2-7a;=16. 12. 2a;2-52:+3 = 0. 13. x(x+ 1)^12. 14. a: + 3 = 21. 2 13 a;-l x + 3 8 22. 5 2_ 14 x+ 1 X a; + 4 23. 5 3 2 4«2_i 2a;+l 3 *>A a; + 3a;-3 2a;- 3 x-\-l x-2~ x-\ 25. x^+Q 3 _, 7 « a;2_4 2-ir x + l 15- ^ + " = 2- 26. 9^a;2-90^a:=-19.5. 2 , - 5 a; „„ 17 32-II.t h 4 = 27. — : 3a; 62^ + 7 X "6 x^ 276 ELEMENTARY ALGEBRA [Ch. XVIII, § 224 LITERAL COMPLETE QUADRATIC EQUATIONS 224. If the coefficients of the equation ax^ -\-hx + c = () are literal, the equation is called a literal complete quadratic equation. Thus, 2 ax^ + mx + 6 n = is a literal complete quadratic equation. The solution is found in the same manner as in the pre- ceding paragraph. 1. Solve the equation : x^ — hx — cx=(a + l))(a — c). (1) Factoring in (1) to show coefficient of x, :^^x{h + c) = {a + h) (a- c), (2) completing the square in (2), ;^-x(b + c) + (^^ = (^±^+(a + h){a-c), (3) ^bplifying the right member in (3), ^._,(,+e)+(^j^J-*«^+^«*-^;''+'^- -2hG + G^ (4) extracting square roots in (4), "" 2 ~^ 2 ' (5) transposing and uniting in (5), x = a-\-h, ov x = c — a. (6) Verification : ( (a+by-(a+b)(b-^c) = (a+b)(a-c), (a-\-b){a — c) = {a-{-b){a — c), (c-a)"-(c-a)(b-\-c) = (a-\-b){a-c), (c-a)(-a-5) = (a + 6)(a-c). Ch. XVIIT, § 224] QUADRATIC EQUATIONS 277 2. Solve the equation : ax^ -{- be — hx = acz. (1) Transposing in (1), aaj^ —bx — acx = — be, (2) factoring in (2), ax^ — x(b + ac)=^ — be, (3) dividing (3) by a, a^ - x f^+^^\ = - -, (4) \ a J a completing the square in (4), simplifying the right member in (5), extracting square roots in (6), ^_b + ac^ b-m ,^. 2a 2a ' ^ ^ transposing and uniting in (7), x = -, ov x = c. (8) Qj Verification^ : . \ay \aj \aj a a ac^ + bG — bc — a(?. The left member should always be factored to show the coefficients of :jfi and of x, EXERCISE CXV Solve the following equations by completing the square i 1. a;2 + 4 J:r = — 4 J2. 4. ^ :i^—%'pq=z'i'pq~Zqx. 2. 2;2 — 5 aa: + 6 ^2 = 0. 5. hx^ + ac = (a + bc}x. 3. x^ + ax-'2a^ = 0. 6. x'^ + Qa + h)x + ab = 0. 278 ELEMENTARY ALGEBRA [Cii. XVIII, § 224 7. x^ + ax = €?, 1 , 1 8. T + - = ^ + — X a 9. Ix^ + mx + ^i = 0. 10. x^ — lax + h^^. 20 x—h X-— a 21. :r (1 — x^ = ax^ + b. 22. 1 — 7 x/^=2ax — 5:r^. 11. ^LZl^_^+^_5 ^^ (^_a)2 ^ :r + a 12. ax'^ — 2bx-i'C = 0. x — a ^ 24. - + = a: + -. 14. ^ ^ >T + ^ ^ 25. a22'-2/>2=^J.^-±_^. 15. aa;2+a = (^2_|_i^^^ a; + l 16. x^-lax-Va^-^W-^^. 2g^ ^ 1 & ^ 2a * J + a; a + 2^ 2a-/> 17. -1 — ^=^±4- a-x a + x a^-x^ ^_a x-2h b ^ 27, = — — • 18. mqx^ — mnx + pqx = np. ^ x— o a + o 4^2^j2 -^- ^+2 2a;-l a-\'X a — x__ ^bQa-\-b^ 30 a — J + 2.T a + S — a^ 31. cx'^-{a + b-\-c)x-\-{a-\-b^==0. 32. mTz:?;^ — (m + ?i)(m7Z + l)x^ +(m + 7i)2= 0. 33. 2x^a^-b'^)--{?>a^ + b'^)(x-l)=^{W^^a^^(x-\-r) Cn. XVIII, § 224] QUADRATIC EQUATIONS 2Yi) 34. + h + X X— a 2(a + 5) 35. x\a 4- 5)2 - x(a2 - J2) = ah. 36. 37. 2(a + ^>) . 2 5 ^ 3(^-5) X — h X — a ir — 35 a:— ^ X _ 2(x + g) x-^^h x — h x+ ^a — 4:b 33^ ajl-^x^) ^ (2a-b)x ^ ia hx a a + b 39. \(x+l) + -(x-\^ 2a^-l 40. {n — x)(\- Za + Zx c-h y^A X l-2a ■ 2c (c-3)- l+g 1-ic 41. x+h , 2a X— 2a -b \ X — b J 42. (a2 + 62)(4 a;2 + 1) + 2 a5 (4 x^-l) = 'ix(a^~ b^). 4 43. ax ax + b lb\l+x)x-a\l-x')'\ = b. ^ ^a;-l a + lV a; a(a;-l)y 2 rt(a + 5) - b^x 2 45 46. Jx— 2 a lf^-2^ " b\2a J b_n _ _y a^\x 2 a, 2x] r 4a il-i 1_1 .a; a X b_ = 0. 280 ELEMENTARY ALGEBRA [Ch. XVIII, § 225 SOLUTION OF QUADRATIC EQUATIONS Bf A FORMULA 225. Every quadratic equation may be reduced to the general form, ax^ + hx + c=^Q. Solve the equation : ax'^ + hx + c = Q. (1) Transposing in (1), ax^ -f 6a; = — c, (2) dividing (2) by a, ^ + ^Q = _ £, (3) completing the square in (3), \aj 4 a"^ ^a^ a simplifying the right member in (4), \al Aa^ 4a2 extracting square roots in (5), '+r„=*^^^' ('') transposing and unitmg m (6), jr = ^ y x = — a) 2a The values of x in (7) are general values. The values of the roots in any particular equation are found by sub- stituting in the formulas in (7) the values of a, 6, and c in any particular equation. 1. Solve by the formula : 2 a;^ -- 5 ir = 3. (1) Putting (1) in the general form, 2 a?^ — 5 a? — 3 = 0. (2) In (2), a = 2, b = -5, c=-3. (3) ( 11. XVIII, § 225] QUADnATlC EQUATIONS 281 Substitute for a, h, and c their values from (3) in the formu las in (7), 5+V(-5r-4(2)(-3 )^3 2(2) or, _^_ 5-V(-5y-4(2)(-3) _ , 2(2) ■''■ (4) Verification : 2(3)2 - 5(3) - 3 = 18 - 15 - 3 = 0, .2(-i)^-5(-i)-3 = i + |-3 = 0. The formulas are written in the more compact form, x- 2a EXERCISE CXVI Solve the following equations by the formula : a^ - 10 a; + 25 = 0. 5a;2+ii^ + 83 = 0. . (6) f V4T2i - V2T7 = 2 ; 2 is a root of (1). VERIFICATION : \ ^ ^ ^ V- 12+21- V-0+7=2; --G is a root of (1) Ch. XVIII, § 220] QUADUATIC EQUATIONS 283 EXERCISE CXVII Solve the following equations and verify the roots : „ Vx +1 — 2 o /- ^ 3. y'x + 5 + x = 7. 7. ;. =2V^+1. 2Va:;— 1 4. 4 a; — Va; + 3 = a; — 5. 5. 9x--\/'^x + l = 2x-l. ' a; — 1 X 9. Va; + 1 + V5(a; + 2) = 3. 10. V2a;-7 + V7¥+8 = ll. 11. VB a; + 4 +V5(x + 1) = 9. 12. Va + X + V6 — x = Va + 6. 13. 2V3¥+l-3V^H^ + 2 = 0. 14. 3V3x-4 + 4x=10(a;-l). 15. V5^-2x+l^v^^3-3. 2Va; 16. gVx+S- Vi^=2V2 3;+2. 17. 3V3a;+l-2Vx+3=V2(a;+l). 18. y/Tx+l + 3V9a;-2 = 5Vox-l. a; , a; 8 a 19 Vx+Va— a; Vx— Va— a; 3Va; 284 ELEMENTARY ALGEBRA [Ch. XVIII, § 227 SOLUTIONS OF EQUATIONS IN THE QUADRATIC FORM 227. An equation which contains only two different powers of the unknown quantity one of which is double the other is said to be in the quadratic form. The general type of equations in the quadratic form is ax^" + bx'' + c = 0. Equations in the quadratic form may be solved like quadratics. 1. Solve the equation : x^ -2x^ + 1==Q. (1) Writing (1) in the quadratic form, (aj2)2-20xj2)+l=O, (2) factoring (2), {x^ - 1) (x" - 1) = 0, (3) equating the factors in (3) to zero, W_i=o, ^^ transposing in (4), a? = 1, x^ = l, (5) extracting square roots in (5), a: = ± 1, or a; = ± 1. (6) Verification : 1 — 2 + 1 = 0. 2. Solve the equation: a;^ — 9 a?^ + 8 = 0. (1) Writing (1) in the quadratic form, (a;5)2-9(aj^) + 8=0, (2) factoring in (2), {x^ - 8) {x^ - 1) = 0, (3) equating the factors in (3) to zero, transposing in (4), x^ = 8, x"^ — 1, Cii. XVIII, § 227] QUADRATIC EQUATIONS 285 raising each equation in (5) to | power, {x')i = sK (xi)^^(l)i (6) simplifying in (6), x = 16, or x = 1. (7) Verification : (16)"2 - 9(16)^ + 8 = 64 - 72 + 8 = 0, 1-9 + 8 = 0. 3. Solve the equation : x^-lx- Vx'-lx + lS = 12. (1) Adding 18 to each member in (1), (a;2 _ 7 a: + 18) - Va;2 - 7 X + 18 = 30, (2) writing (2) in the quadratic form, (Va;2-7:^ + 18)2-Vi»'-7a; + 18 = 30, (3) transposing in (3), (^/af -7 X + lSy --Vx' -T X + 1S - 30 = 0, (4) factoring in (4), ( Va;' - 7 a; + 18 - 6) ( Va^' - 7 a; + 1 8 + 5) = 0, (5) equating the factors in (5) to zero. transposing in (6), -^ay'-7x + lS-6 = 0, Vi^ - 7 a; + 18 + 5 = 0, Solving Vaj2 _ 7 a; + 18 = 6, x = 9, or -2. (6) ■y/x^-7x-}-lH = 6, Vaj2-7a; + 18=-5. (7) Var^ — 7 a; + 18 = — 5 is impossible since the radical cannot equal a negative quantity. Vkkificatiox: on substitution in (1), both 9 and —2 are roots. 286 ELEMENTARY ALGEBRA [Ch. XVIII, § 227 EXERCISE CXVIII Solve the following equations : 2. a;4_5^__i26 = 0. 5. x^-6xi = lG. 3. x*-30it^ + 125 = 0. 6. 2x-^-x-i-6 = 0. 7. :.-3_ -3^_ 7 ^ — X — g^ 8. {x + -] —x—l2 = -' \ xj X 9. 55 ix + iy (^ + 7) 11 10. ^x^-d^ = '2 » 11. V^T12+a/^"T12 = 6. 12. a; + Va;2 — ao; + J'^ = \-h. a 13 3? + %X+U X+4: 14. A/^^n[ + 2 v^^=n: - 1 = 0. 15. a;2-2Va^ + 4a;-5 = 13-4a;. 16. 49a:2 + 42a;4.9 = l_(7a;-|-3). 17. 3 a^ + 15 ^ - 2 Va^ + 5 x + 1 = 2. 18. 2a^ + 3a;-5V2a? + 3a; + 9 = -3. 3 2 19 - + 1 = 0. (a^ _ 5 iP + 7)2 a^ _ 5 a; + 7 3 20. 4a;2 + 22.'r-3V2a;2 + lla; + 13 = 78. 21. -\/x-^ + ox+2S + 7x^ + 5(x+80) = 0, 14. a^-Sx^-15a? + 85x^ + 54:x-12 = 0. Ch. XVIII, § 229] QUADRATIC EQUATIONS 289 CHARACTER OF THE ROOTS 229. The roots of the general form ax^ + bx + c=0 have been found, S 225, to be: x = ; — ^^^-^ Upon the nature of '\/}fi—^ac will depend the character of the roots. The quantity, 6- — 4 ac, is called the discriminant. (1) If 6^ — 4ac is positive, that is, if J^— 4a(?>0, the roots are real and unequal, and either (jx) rational or (6) irrational. If the discriminant is {a) a perfect square, the roots are real and rational ; if (&) not a perfect square, the roots are real and irrational. Thus, in the equation 6 a?- + 5 ic — 21 = 0, since a=^%^ 6=5, c= —21, the discciminant is 529 = 23^. Therefore, the roots are real, rational, and unequal. In the equation 2a;^4-5a: — 4 = 0, since a = 2, 6 = 5, c = — 4, the discriminant is 57. Therefore, the roots are real, irrational, and un jqual. (2) If 6^ — 4 ac is zero, that is, if J^ = 4 ac^ the roots are real, rational, and equal. Thus, in the equation 4 a;^ — 12 a:; + 9 = 0, since a = 4, 6 = — 12, c = 9, the discriminant is 0. Therefore, the roots are real, rational, and equal. (3) If 6^— 4a(? is negative, that is, if IP'—^acK^^ the roots are imaginary and unequal. Thus, in the equation a^ — 2aj + 4 = 0, since a = 1, ?>= — 2, and c = 4, the discriminant is — 12. Therefore, the roots are imaginary and unequal. The character of the roots of any given equation may therefore be found by evaluating the discriminant. 290 ELEMENTARY ALGEBKxV [Ch. XVIII, § 229 The following summary will be found useful : (1) If b- — iac> 0, the roots are real and unequal. (2) If b^ = i ac, the roots are real and equal. (3) If 6- — 4 ac < 0, the roots are imaginary and unequal. 1. Determine, without solving, the character of the roots of 2x^-lx + 5 = 0. a = 2,b = — 7, c = 5. b'-4.ac = A9 - 4(2)(5) = 9 = 31 Eoots are real, rational, and unequal. 2. Determine, without solving, the character of the roots of 9:i;2_i2a: + 4 = 0. a = 9, 6 = - 12, c = 4. b'-4.ac = 144 - 4(9) (4) =0. Eoots are real and equal. 3. Determine, without solving, the character of the roots of 4 x^- 4 a; + 5 = 0. a = 4, ?> = — 4, c = 5. 52 _ 4 ^^c = 16-4(4) (5) = -^ 64. Koots are imaginary and unequal. 4. For what value of m are the roots equal in the equa- tion 8x^+4x+ m = ? a = 3, 6 = 4, c = m. If the roots are equal, ¥ — 4 ac = 0, 16-4(3)m = 0, 16 - 12 m = 0, 12 m = 16, m = |. Ch. XVIII, § 230] QUADRATIC EQUATIONS 291 EXERCISE CXX Determine by the use of the discriminant the character of the roots in the following equations : 1. x^-4:X+4: = 0. 8. a;2-7:i;+12 = 0. 2. x^-5x+Q = 0. 9. 3x^-ix+l = 0. 3. ^2_2:^;_1 = 0. 10. 2:^:2-13:r+5 = 0. 4. 2a:2-3a;+5 = 0. 11. 3 2^2 - 4 2:+ 12 = 0. 5. 5:i:2_2a;+l = 0. 12. 2it2_5^_5 = 0. 6. ^2_3^_,_1^0. 13. 8x^~5x=2, 7. 2^2-4 2;+ T = 0. 14. a;2__2ax=(J+a)(S-a). Determine the value of w for which the roots are equal in the following equations : 15. 2x^ + 4:x+m = 0. 18. 16x^+8mx+l = 0. 16. mx^ + Qx+S = 0. 19. 4:x^—12x+m = 0. 17. 8x^+4:x— m = 0. 20. ma;2— (8 + m):?:^ + 9 = 0. RELATION BETWEEN ROOTS AND COEFFICIENTS 230. It is convenient to derive the formula for the general equation ax^ + bx+ c = 0^ where the coefficient of x^ is unity. Dividing the general equation by a, x^-\ f- - == ; in the last equation, putting /> = -, and a a a q = -^ the equation is x^ + px + q = 0. a The roots of x^-{ px+ q = are found to be 292 ELEMENTARY ALGEBRA [Ch. XVIII, § 231 Let ^■^ -p+Vj>^-4g ^ (1) and ^^ -V--/¥^^^ ^ (2) adding (1) and (2), « + ^ = -j), (3) multiplying (1) and (2), a^ = q, (4) Hence, in the equation x^+ px + q=0 : (1) The sum of the roots equals the coefficient of x with its sign changed, (2) The product of the roots equals the absolute term, 231. Since the equation x^ + px + q=Q is the general form of complete quadratics, the sum and the product of the roots of any complete quadratic may be found by inspection. 1. Find by inspection the sum and product of the roots of '2:^2 + 3^_,_l = 0. (1) Dividing (1) by 2/ a? + ^x + \ = (), (2) if a and /3 are the roots, by the rule, a -f- y^ = — f , (3) a^ = \, (4) The equation x^ + px + q = 0^ wherein a and y8 (read respectively, '-'- alpha " and " beta ") are the roots, may be written, jr^ - (a + p)jr + ap = 0. 2. Form an equation whose roots are — 2 and 3. Take oe = - 2, and ^ = 3. Then, « + ^ = -2 + 3 = l; ,x^ = (- 2) (3) = -6. Substituting for « + j8 and a/? these values in x^—{a-\-P)x+a^^^^ a;--(l)a;-6 = 0, or, x'-x-^-^O, CiT. XVIIT, § 231] QUADRATIC EQUATIONS 293 3. Form an equation whose roots shall be the squares of the roots of the equation x^ + px-\- q = 0. (1) Let a and ^ be the roots of (1). By the conditions the required equation is, x'--.(a' + /3')x + a'^' = 0. (2) Now a/3 = q] hence, a^^^ = (f. Again, a -f y8 = — /) ; hence, oC- + 2a^-\- ^^ =p^, a2 + ^2^p2_2g. Substituting a^ + ^''=p^-2q and a^p'' = q', in (2), x^-{lf-2q)x + q'=:z0. 4. Form an equation whose roots are reciprocals of the roots of the equation x^ + px + q=0, (1) Let a and y8 be the roots of (1). By the conditions, the required equation is, \a (3 J a/3 \ a/S J a^ Now, a + p = —p) and aji = q. Substituting a + /3 = —p, and a^^q, in (2), q q ' qa^+px + 1 = 0. The results obtained in the preceding examples may be verified by solving the equation. 294 ELEMENTAHY ALGEBRA [Ch. XYIII, § 231 EXERCISE CXXI Form that equation whose roots are respectively; 1. 2 and 3. 6. 4 and — |. 2. 3. 4. 5. 5 and 2. 6 and — 2. — 3 and — 6. I' and J. 7. - 5 and |. 8.-4 and -|. 9. 2 + V3 and 2 - VS. XO. 1+^^andl-A 2 2 XI. -1 -2V5 3 , 1 + 2V5 and 3 . X2. 1 + 2V^ -3andl-2V-3. 13. 2+V-2and 2-V-2. 14. a + V& and a — VJ. 15. a + V — 6 and a — V—b. 16. — (? + V— d and — ^ — V — c?. 17. Form a quadratic equation whose second member shall be 0, whose absolute term in the first member shall be — 4, and one of whose roots shall be — i. 18. One root of the equation 4a:;2__ig^_l_4_0 is 2 + V3 : find the second root. 19. Find, without solving, the sum and product of the roots of the equation 8x^ — lx— 5 = 0. 20. Form an equation whose roots shall be the recip- rocals of the roots of the equation 2 a;^ _ ^ _)_ j ._ q^ 21. Form an equation whose roots shall have the same absolute value as, but signs opposite to, the roots of x^+px + q=0. Cii. XVIII, §§ 232, 233] QUADRATIC EQUATIONS 295 GRAPHS OF EQUATIONS OF THE SECOND DEGREE 232. By the method employed in § 145 it is possible to construct the graph of any equation of the second degree ill two unknowns. Consider the equation y = ao? + hx + (?, the right hand member of which is evidently a part of the general form of complete quadratic equation in one unknown. The graphs of certain numerical forms of y = ao? -{-hx + c for various characters of the roots are interesting. 233. (1) When 52>4 qq^ and when Vi^— 4 ac is rational. 1. Plot the graph ofy=2? — 42:+8. Pi P2 P. A P. P. Pr Ps P9 x= —2 -1 1 2 3 4 5 6 y= 15 8 3 -1 3 8 15 In the table are found the coordi- nates of the various points. Locat- ing convenient points, and drawing a smooth curve through these points, the curve P,P,P,P,P,P,P,P,P,, Fig. 12, is the graph. The graph is Rpen to cut the X-axis at the points 1\ and P(;, whose coordinates are respectively (1, 0) and (3, 0). But, since the ^/-coordinates of the points where the graph crosses the X-axis are zero, the aj-coordinates of these -\ J\ Y ^^^ 1 \ \ '\ 4 1 \ / \ / \ / \ p, 1 \^' 1 \ x' \ P* n / X V J n Fig. 12. f: ints are the solutions of the equation, O/*^ — 4 a; + 3 = 0. 296 ELEMENTARY ALGEBRA [Ch. XVIII, § 234 In like manner, if the graph of any equation in the form y=ax'^+bx+c is plotted, the ir-coordinates of the points where the graph crosses the X-axis, will evidently be the solutions of aoc^ + bx + c = 0. The nature and approximate values of the solutions can therefore be determined from the graph. 234. (2) When P>4: ae^ and when Vfi^ ^ ^ ac i^ irra tional. Plot the graph of ^ = x^— ix }- 2. Fig. 13. V, Y pA p^ Y P. 1 \ i \ \ \ \ \ \ \ \ P. ^8, \ ^\ ^8 \ \ / \ / \ / \ / \ / \ / 1\ Pj / \ ^. / \ y y P-, a' X \ / r\ 1\ / \ i\ p. / \ ) x' \ / X P'. ^5 P. A n P, A Pa P, Ps A x= -2 -1 1 2 3 4 5 6 y= 14 7 2 -1 -2 -1 2 7 14 Fig. 14. The graph is constructed as shown in Fig. 13, and is seen to cut the X-axis at points whose o^-coordinates are between and 1, and between 3 and 4. By the usual method of solving the Cii. XVIII, §§235,236] QUADRATIC EQUATIONS 297 equation a^ — 4 a^ -4- 2 = 0, the roots are found to be 2 ± V2, or 0.26795+ and 3.73205 4- . These must therefore be the exact values of x where the graph crosses the X-axis. 235. (3) When Jfl = 4:ac, . 1. Plot the graph of y = x^-'4:X+4i. A Pi Pz P, A P. P7 P, A x= —2 -1 1 2 3 4 5 6 y= IG 9 4 1 1 4 9 16 A Here the equation, a^-4a; + 4 = 0, 1^ has equal roots, x=^2, and the graph. Fig. 14, touches the X-axis at the single point P5, whose coordinates are (2, 0). 236. (4) When P<4:ac. Plot the graph of Px Pz Ps P4 P5 A i'r P. A x= -2 -1 1 2 3 4 6 6 y= 17 10 5 2 1 2 5 10 17 If 2/=0, and the resulting equa- tion, i»2— 4 a; + 5 = 0, is solved, the roots are found tobeic=2±V — 1. Since these values are imaginary, they cannot represent any real dis- tance. Hence the graph. Fig. 15, does not cut the X-axis. The graphs of the equations which have been plotted have the same general shape, which will l)e found to be the same for all equa- tions of the form ?/ = ax^ + hx -f c. This curve is called the parabola. Y Y '1 \ \ \ \ \ ^2 P> 1 \ / \ / \ / ^, ^1 / \ 1 \ \ A J% 1 \ / x' h X Fig. 15. 298 ELEMENTARY ALGEBRA [Cii. XVIII, § 2;J: GRAPHS OF EQUATIONS CONTAINING / )' y < T\ II ^, ^ 1^ / U •JJ fioS s. ^ p ^ j ^1' r\ / \ a' Pj /;.. X \ \ \ // J ^ <: 7/ y K, s S^ I'l / ^/•,' < 4 li />; ^ ^ 'k L r' ^5 237. 1. Plot tne graph of .t2 + ^2^36. Solving a;2 + ?/2^S6, ■ a;- The na- '^ is sueh ?/ = ± V3(3^ tare of V36- that if a; takes any values less than — 6 or greater than o. (^ oe comes imagi- nary. Il IS necessary to constnict a table only for values of x between — 6 and +6. Fig. 1(3. a; = ±6 ±5 ±4 ±3 ±2 ±1 2/ = ±vn ±2V5 ±3V3'±4V2 ±V35 C The graph is constructed as shown in Eig. 16, using approxi- mations of the double values of the surd values of ?/. Points may be located closer together by taking fractional values of a;, as 1^, If, etc. The graph is seen to be a circle. : 2-2 — 6 r?T + 9. EXERCISE CXXII Plot the graphs of the following equations : 1. ?/ = 2^2 — 7 a; + 10. 6. y 2. 2/ = .T2-3a:+5. 7. a:2 + ^2^25. Z, y=:x^-1x-\-\. 8. x^^y'^^X^. ^, y = x^—7 x+4:. 9. y^ = 4:x. 5, y = x^-bx+6. 10. x^+y^-x-S^O. CHAPTER XIX SIMULTANEOUS EQUATIONS SOLVABLE BY QUADRATICS TWO UNKNOWN QUANTITIES 238. A system of two simultaneous quadratic equations involving two unknown quantities cannot, in general, be solved by quadratics. r. . . . (X^-f=S, (1) Solve the equations : \ ^ ^ Substituting in (2) the value of y in (1) and simplifying, x^ + 4ta^-lSx^-32x + 67 = 0. (3) Equation (3) cannot be solved by the method of quad- ratics ; and, in general, the solution of a pair of quadratic equations, chosen at random, will involve the solution of an equation of the fourth degree. There are, however, certain forms of simultaneous equa- tions which can be solved by means of quadratics. SIMULTANEOUS EQUATIONS SOLVABLE BY QUADRATICS 239. In § 151 it was shown that the coordinates of the point of intersection of two lines were the values of x and y in the solution of the two equations which the lines represent, since the coordinates of this point must satisfy both equations. For the same reason, if tlie graphs of two quadratic equations or a simple and a quadratic equation are plotted, the coordinates of the points of in- tersection of these graphs must be the solutions of the pair of equations. 299 300 ELEMENTAKY ALGEBRA [Ch. XIX, § 240 240. 1. Plot the m-aphs of the system : „ ' ^ "^ "" ^ "^ U2 + ^=5. (2) By § 145, construct the graph, AB, of x -{- y = 3. By § 237, construct the graph of x^ -\- y = 5. Pi P, P. P. ^3 P, Pr P. x = ±1 ±V2 ±vs ±2 ±V5 ±V6 ±V7 y = 5 4 3 2 1 -1 — 2 Locate suitable points and draw the smooth curve PgPiP'^. The intersections of the graphs AB and the smooth curve P'gPiPs, Fig. 17, will be points whose coordinates are solutions of the given system. If, in place of a; + ?/= 3, the graph of x-{-y = 6 is plotted, the graph will be found not to cut the parab- ola which is the graph of x^-]-y=5. Corresponding to this non-intersection of the two graphs are found imaginary values for x and y when the equations \ \ are solved simultaneously. ^ If the graph of 2 x-\-y—^ is plotted, the graph will be found just to touch the parabola at the point (1, 4). Corresponding to this fact, if the equations \ r \ {', \ / \ J\ / k B^ \^ A i \ \] \. k N i\ X' n A X // c / \ h \ ' \ n V ^ \ i \ / \ / \ 1 r r \ Fig. 17. l2x-\-y = ^ solution^ x=^l, 2/ = 4. ^.9^1 _ A ^^^ solved simultaneously, they have the single €h. XIX, § 241] SIMULTANEOUS EQUATIONS 801 241. 1. Plot the graphs of the system : By the same method nsed in the preceding paragraph the graphs of ■the two equations are plotted as shown in Fig. They intersect in the four points, P^ P2, P3, P4, whose coordinates are found by measurement to agree with the sohitions of the two equations, (4V2, 4), (4V2, -4), (-4V2,4),(-4V2,-4). ;c2 + 2 2/2=64,(l) x^-f=16. (2) ^' -y s y s z ^ z \^ ^^ M J^ 7 V 74-v --l A t A jq if X T V / V / vp' ^,y ^^ ^ y^- --^ \ Z X s / s ^ - s 31 Fig. 18. The graph of (1) is called an ellipse ; of (2), an hyper- bola. EXERCISE CXXIII Plot the graphs of the following systems and deter- mine by measurement the coordinates of their points oi intersection. 2. 3. xy = l. x + y = ^, ' a;2 + ^2 ^ 5^ xy==2, ' x^ — y'^=^ 16, . 2; + y = 8. p2-, \xy = ^2=5, 6. 6. 7. 8. .xy (a^--y^=^2i, l3:i:2_20 7/2 = 55. ( 4:X^ — xy = 6^ [S xy — y^ = 6, cx^+y^ = ll, Xx^ — y^= 15. 302 ELEMENTAEY ALGEBRA [Ch. XIX, § 242 CASE I 242. A simple equation and a quadratic equation. A system of simultaneous equations in which one equii- tion is simple and the other quadratic can always bo solved by substituting in the quadratic, equation the vahu: of one of the unknowns obtained from the simple equation. 1. Solve the equations : i „ ^ „' , Substituting in (2), x — l — y from (1), (7-?/)2 + 2/=34, . (3) simplifying in (?.), ^y--Uy + 15 = 0, (4) factoring in (4), (y - 3) (3 2/ - 5) = 0, (5) from (5), . 2/ = 3, or ?/ = -. (6) Substituting values of y from (6) in (1), 07 = 4, or a; = — . (7) o The given equations check if a; = 4 and y = 3 be substituted ; 16 5 and the given equations also check if a? = — and ?/ = - be sub- o o stituted. Such values of the unknowns which, taken together^ satisfy the given equations are called dependent values. Dependent values should always he found hy substituting the value of the unknoum first found in the simple equation., and never in the quadratic equation. It is to be noticed that the given equations are not verified by values which are not dependent. Cii. XIX, § 243] SIMULTx\NEOUS EQUATIONS 303 843. The use of the double signs, ±, read "plus or minus," and T, read "minus or plus," taken together are to be interpreted in the order in which the signs are read. Thus, I x=±l,, . fX = + l, (X IS equivalent to \ and <; [y = ±^, i?y = + ^^ is equivalent to \ J and /• /^ IT I Similarly J J i~ - . ^ -1. x = + l, 2/ = -2. EXERCISE CXXIV Solve the following systems of equations 1. ' xy — 5 X =: 1^ \2x^ + xy=S. ^ - 2/ = 4, 3^-2_^^^2. 5. 6. 7. 2 a; - 3 «/ = 2, 3x2-2«/2=115. . 9 a; + 7 z/ = 80. a^ + 22/ = 3, y £_+3j 2: + 2 4a;+3y = l, ^y + 3 X+1/ 5.y I 2y + 3 . 0. 10. 1 a;(J — a) ^(a + b) o? — b^ a ^ 2b Jj + ^b x-y' = 01 304 ELEMENTARY ALGEBRA [Ch. XIX, § 244 CASE II 244 When one of two simultaneous quadratic equations is homogeneous. A quadratic equation is said to be homogeneous when all j of the terms involved are of the second degree in the] unknown quantities. Thus, x^ — 3xy + 2y^ = is a homogeneous quadratic equa- tion. x'^-xi/-2f = 0, (1)1 (2)i 1. Solve the system : , _ Dividing (1) by y% factoring (3), 'x\^ fx^ yj \y. 2 = 0, ^ 2Y^ + 1] = 0, (3): (4) from (4), x=2y, or x=—y, (6) substituting x = 2y in (2), 4:y^ + y = 5, (6) solving (6), y = l, or 2/ = - i (^) substituting values of y from (7) in (5), x = 2, or x = — f, (8) substituting x = — y in (2), y^ -{-y = 5, (9) ■l±V2l solving (9), y- substituting y = "^ ^^ in (5), x = 2 lTV2i -, (10) (11) The solutions are : x = 2, rx=—^, X = ' 1 1 7/ — _ 5 2 I+V2I I+V2I -I-V2I Dh. XIX, § 244] SIMULTANEOUS EQUATIONS 305 EXERCISE CXXV Solve the following systems of equations : 3. 4. 5. 6. 7. .2:^2 + 3^2^11. ' 2 2)2 __ 3 ^^ + y2 ^ 0, ' a? + xy = 0^ 2x^-Sx-y = 4:. (2x'^ + xy-10y^=0, .x^+8xy + y = — l. ^x^-3xy+2y^=0, [xy — x + y = 4:. Sx'^ + x-y = 29. x^^ xy = 20 y\ x^ — x + y = 54. 8. 9. 10. 11. 12. 13. 14. cfi + y'^ = 2xy^ 2x^ — xy + y = 30. ' x^— xy — 2y'^: 0, Bx'^ + llxy + 2y^ = 0, x^ — xy -\- y = b. 'ix^ + 1xy ~ y'^= 0, x-2y + %y^=Z2. (lbx^-Uxy+\by^^O, ,a; + 2/-2«/2=_10. 8 a;2 + 2 xy - 3 «/2 = Q, a^ + a; + «/2 = 22. 3 x2 + 8 a;?/ + 5 2/2 = 0, 3x2 + 4a;^+^=_30. 15. 16. 17. 18. ^2a;2 + 9a;«/ = 35«/2, ,2x(x + «/)-ll?/ = 236. ' Qx'^=Wxy + 35 «/2^ . a;2 - 17 a;^/ - 180 y = -260. 9a:2_39 2.^4.22z/2 = 0, I3a^-7a; + y = 289. 10a^ + 23a;«/ + 12y2=o, 9a^ + 7a;«/ + 6jr = 132. 306 ELEMENTARY ALGEBRA [Ch. XL\, §§ 245, 246 CASE III 245. When each of two simultaneous quadratic equa- tions is homogeneous only in the unknowns involved. A sj^stem of two simultaneous quadratic equations which are homogeneous except in the absolute terms may be solved as in Case II, by combining such multiples of the two equations as will make equal the absolute terms. ^2 + ^y ^ 12, (1) .xy-'ly^ = \. (2) Multiplying (2) by 12, 12 xy -24.f-= 12, (3) subtracting (3) from (1), x^ - 11 xij + 24 t/- = 0. (4) Equation (4) may be solved as in Case II; or it may be solved by factoring. Factoring (4), (x-Sy) (x - 8 ?/) = 0, (5) from (5). x = 3 y, or x = ^y, (6) substituting x = 3y and x = Sy in (1), and solving the result- ing equations, ^ ^ ± 1^ 2, = ± ^V6, by substitution in (6), x = ±3, x = ± ^ V6. 1. Solve the system : \ ,x = 3, The solutions are : \ 2/ = l, x=—o, \^% ^•=— |V6, 2/=-l, b=iV6, l2/=-iV6, 246. An alternative method for solving equations of the class of Case III is called the vjc method. 1. Solve the system : |^ + ^y + 4/=6, (1) Let y = vx, and substitute in (1) and in (2), x^ -^x'v + i: x'v'' = 6, (3) 3 0^2 + 8 xV = 14, (4; factoring, a^{l+v + 4.v') = 6, (5) x"" (3 + S v') = U, (6) Ch. XIX, §246] SIMULTANEOUS EQUATIONS . 307 equating x^ in (7) and (8), ^_^^^_^^^, = g^^^, (9) clearing and simplifying in (9), 4 w^ + 7 y — 2 = 0, (10) from (10), v=\,ovv = -2, (] 1) substituting values of v from (11) in (7), 6 l + i + ¥f T 1 = 4, (12) ^-1_2 + 16~16' ^^^^ extracting square roots in (12) and in (13), (x==±2, _ (14) ia;=±iVlO. (15) When v = \, x=±2; substituting v=^ in y = vx, 2/ = i(±2)=:±f (16) When v = — 2, x = ± ^VlO ; substituting v=—2 in y=vx, .^ = q:2VIO. (17) The solutions are : 07 = 2, (x = -2, ra; = iVlO, ra; = -|VlO, b = i, U = -i U = -fVio, b = |Vio. The values of x inust alivays he substituted in y = vx. Since equations of the type of Case III may be reduced to a quadratic equation homogeneous in all its terms, and since such an equation may always be expressed as a quadratic in -, for - any quantity v may be^ substituted. If - = v, x=vy\ y y y y 308 ELEMENTARY ALGEBRA [Ch. XIX, § 247 1. EXERCISE CXXVI Solve the following systems of equations : xy = ^- (x^ + 3xy = 27, (x^ + y \x^ 3. 4. '2 ==20, xy= 8. r ^^ + 4 = 0, ' x^ — xy = 15, ^x^-f = 21. ['22;2 + :r7/ = 52, [2f-xy = S0. 6. 7. 8. 10. CASE IV ■f^ 2a;2-2/2=17. (2x^- 31/2 = 60, .3x2- 4 0:3/ + ^2 = 64. ' 6^2— 5a;?/ + 2 2/2=i2, 32;2+22:y-32/2=-3. [2a;2_2x?/-2/2=3^ U2 + 3a;?/+y2=ii. 3:r2_7^^_|_4^2__ _^ .2x^ + xy-Sf = 22. 247. When two simultaneous quadratic equations are each symmetric with respect to the unknowns involved. An equation is said to be symmetric with respect to the unknowns involved when the interchange of the unknowns does not change the form of that equation. Thus, x^ + xy + y'^ = 7, and xy + x + y = 5, are symmetric quadratic equations. A solution of a system of such equations may always be found by substituting x = u + v, and j^ '=^u — v^ in the given equations. (X^ + X7/-{-f=l, (1) .xy + x + i/=:5. (2) Let x = u + Vf and let y = u — v. Solve the system : Oil. XIX, § 247] SIMULTANEOUS EQUATIONS 309 Substituting x = u-\-v, and i/ = ^ — v, in (1) and in (2), (21 + vy + (u + v)(u-v)+(u- vf = 7, (3) {u + v) (u — v) + (it -\- v) + (u ^ v) = 5, (4) simplifying in (3) and in (4), 3u^ + v' = 7, ♦ (6) ^f2 + 2u-v' = 5, (6) transposing in (6) and in (6), v^ = 7 — 3u^, (7) v^ = u' + 2u-5, (8) equating v^ in (7) and in (8), 7 ---Su^^u^ + 2u- 5, (9) solving (9), u = ^, ov u = — 2, (10) Substituting i^ = |, in (5), v = ±^, x = u + v = ^±^ = 2, orl, And, ?/ = ^^-.'y = |-q:^ = l, or 2. Substituting w = — 2, in (5), 'y = ± V— 5, x = u + v = — 2±-\/—5, and, 2/ = i^ — 2; = — 2q: V— 5. The solutions are : x = 2, J/ = l, 'a; = l, fx = — 2 + -V—5, ra; = — 2 — V— 5, ^2/ = 2, \2/ = _2-V^ {2/ = -~2 + V^=^. Two simultaneous quadratic equations which are sym- metric, except in respect to signs, can often be solved by Case IV. The proof that equations of the type of Case IV can be solved by substituting x = u + Vy and y = u—v, is beyond the province of this book. 810 ELEMENTARY ALGEBRA [Cii. XIX, § 247 EXERCISE CXXVII Solving the following systems of equations : ^ + ^^y + 2/ = 29, r S(x^ + ^^) — 5xy = 15, 1^ + ^ 4. 5. 8. 10 11. 12 x^-^y^ — x — y = 22, ^ + y + ^y = — 1. ' ^y + ^(^ + 1 ) + y (2^ + 1) = 24, ^xy^^, xy-2x^-2y'^=^20, 4,xy + x + y=:^'l^. 3a;2 + 3/=8(:r + 2/)-l, xy — X — y = \. x'^ + y'^ + xy + x + y = 17, r^'2 + / - 3 a:^ + 2 :r + 2 ^ = 9 . 2:z: + 2?/ + ^^=16, ' ^^ + 2/^ + ^ + 2/ = 62, .5:^^ + 4(^2 + ^2)^328. r 2:2 + 2 :r2/ + ^2 + 5 2; + 5 ?/ = 84, . 0^ + ^2 + ^, _^ ^ 3^ 32. l + i- + i = 7, x^ xy 2/2 I :^:(2: -y)J^y(x + y^=%+ xy. x^ + y'^ + x + y = cfi^ xy + x + y = 3a Ch. XIX, §248] SIMULTANEOUS EQUATIONS 311 SPECIAL DEVICES 248. Special devices may be employed in finding solu- tions by shorter methods for some of the systems in the preceding cases, as well as for certain other systems whose equations are often of higher degree than the second. ^ + 2/ = 3, ' (1) 1. Solve the system : , „ « ^ Squaring (1) and subtracting from (2), -2xy== 20, (3) adding (3) and (2), o?-2xy + if = 49, (4) extracting square roots in (4), x — y = ±l, (5) -adding (5) and (1), a; = 5, or — 2, (6) subtracting (1) from (5), y = — 2, or 5. (7) r^r 1 . f x = 5, r x = — 2, The solutions are : .i ^ \ ^ [y = -2, l2/-=5. 2. Solve the system : :^;3 + ^3^1001, (1) U+2/=ll. (2) Dividing (1) by (2), x^-xy + y^ = 91, . (3) squaring (2) and subtracting from (3), -Sxy^-SOy (4) dividing (4) by — 3 and subtracting from (3), 1^ x'^2xy + f = Sl, (5) extracting square roots in (5), x — y= ±9, (6) combining (2) and (6), ic = 10 or 1, ?/ = 1 or 10. (7) 10, (x = l, The solutions are : \ ^ \ l2/=l. U = 10. 812 ELEMENTARY ALGEBRA [Ch. XLY, § 24l 1. 2. 3. 4. 5. 6. 8. 10. 11. 12. EXERCISE Solve the following systems ' a; + ^ = 6, xy = 5. x + y=20, xy = 51. . X2/ = 13. ^2+ 2/2 3^ 34, a;^/ = 15. x'^ + y'^=^ 25, 2 :z:?/ = 24. ' x + y^l2, ^x^ + y^=74. x-y = 2, 2^2.-2/2=20. :z;2 + ^2^34^ x + y = 8. X^ + y^==74:, X — y =2. xy = a, x^ + y^ = 6. 2: + 1/ = a, x^ — y^ = h, y^-\-xy=\b, x'^ + xy = 10. CXXVIII of equations : p2 4- ^2^436^ [ri; — 2/= 14. ' a;2 + o;^ = 15, 13. 21. 22. 23. 14. 15. 16. 17. 18. 19. 20. Vxy+y^ ( X' 13^ xy + y'^ lx-^y= — 2. x^ -i-Sxy^ 28, xy + 4: y^ = 8. x + xy + y=29, x^ -i-xy + y^= 61. ' x^ + xy + y^= 19, x^ — xy + y^= 7. 2 2:2 + 5 :ry = 33^ 2 y^ — xy= 12. ' a;2 + 5 2:y + ?/2 = 43. .x^ + 5 xy — y^= 25. I2: — y = 2. ,x + y = 5. x + y: X y -2 + -2 = 25. Ch. XIX, § 248] SlMtTLTANEOtrS EQUATIONS m 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. ^1 ^1 - + - = a, X y -^ + -^ = 6. 1+1 = 5, ' x^— xy =^ 153, :r + ^ = 3, r rz:^ + :r^ = 10, Uz/ - ^2 ^ - 3. ' ^ + y = - 3, 1+1 = 1 X y b 5 xy = 84: — x^y^^ X — y = 6, ^ + 2/ -f- 3 = 0, U2 + 22/2=8. '1 + 1=2, x^-7/ 2l' .2; + t/ = 7. 34. 35. 36. 38. 39. 40. y , 1 12 X 0^ + - = !, a;z/ - - = 2, y xy -U = l. I ha? + axy = a. a5 _ q xy [X y 41 42 0^— y^ =9, x + y + Vx+~y = 12, xy = 20. a;^^ — xy^ = 12, ofi — ^^z= 63. 314 ELEMENTARY ALGEBRA [Cii. XIX, § 240 THREE OR MORE UNKNOWN QUANTITIES 249.* Three simultaneous quadratic equations involving three unknown quantities cannot in general be solved by quadratic equations. The solutions of certain forms are illustrated in the following examples. U2_^2 2/2_^2^5^ (1) 1. Solve the system i i 2x-^ y + z = 6^ (2) [x + Ay-z = 5. (3) 11 — 5 V Eliminating z in (2) and (3), x= — ^, (4) o eliminating x in (2) and (3), z = -~ I' } (^) o substituting x and z from (4) and (5) in (1), simplifying and solving (6), 3/ = 1, or y = — 10. (7) {x = 2j Substituting values of y in (4) and (5), 2/ = l, z = l, 2/ = - 10, 2. Solve the system : '<2/+^) = -4, (1) "K^ + ^) = -10, (2) i<^ + 2/) = -54. (3) Dividing the sum of (1), (2), and (3) by 2, xy-\-yz-\-xz = — 34, (4) 30 subtracting (1) from (4), 2/ = ? (^) 24 subtracting (2) from (4), a;= --— , (6) Cii. XIX, § 240] SIMULTANKOUS EQUATIONS substituting y and x from (5) and (6) in (4), -30- -24 = -34, (J) solving (7), z = ±6, (8) substituting z from (8) in (G), x = Ti, (9) substituting z from (8) in (5), y = T5. (10) EXERCISE CXXIX* Solve the following systems of equations : 2. xy = — 42, a^2 = 48, yz = — 56. ' fc = 14, xz + yu=^ 11, XU + yz = 10, ^ + ^ + ^ + ^ = 10. S16 ELEMENTABY ALGEBRA [Ch.X1X,§249 1. REVIEW EXERCISE CXXX Solve the following equations : x^—1 __ 5r?; — 1 4. a?-^8x-^ ?. ■\/4:X-S'-Vx+l = l 5 2a;-3 5. a^x — 2 &2 = ab a^ + 1 x{ I x^'2 2(0;- 2) 7. 3 a; 3 6* 2x + 4: ^ x^+1 a + h . c 6. = -| 8. 3a: — 5 a — c X — a = 4 X c 2a:-3 « + S 3 h(x - c) 9. Va: + 3 + Va: + 6 — Va: + 11 = 0. 10. a;2 + 8a; + 6Va;2 + 8a;-8-3 = 0. 11. 12. 13. Solve the following systems of equations : 14.*J^a; = 15, r4 7_8a X y xy ■ a"" iofi — xy =^5^ x^ + xy^y==l, 3a; + 2i/-5 = 0. 15. 16. yz=20. ra:2 + 5^^ + 32/2==3, \ 3 a:2 + 7 a:?/ + 4 / = 5c 17. a^2/ == - 1, 4 a:2 + (2 ?/ - a;) (2 2/ + a;) = 7. Ch. XIX. § 249] SIMULTANEOUS EQUATIONS 317 18. 19. 20. ay hx a — b X + a y h -b a . -\/x + 1 — 'y/x — 1 = -Vy. 2 xy + 4: X — y — 2 = Q. 21. Construct the equation whose roots are — ^t ^nd 1-V5 ^ 22. What must be the value of c if the roots shall be equal in the equation, 'ix^ — 2x + c=Q'l 23. Determine the values of k if the roots of the equation, kx^ + 2fe — 3a; + 2 = 0, are real and equal, and verify the results. 24. Determine without solving the nature of the roots of 2^2-3:^ + 5 = 0; bx'^--Qx + l = 0. 25. Find the values of k in order that the equation, (x^ — Zx+2') + k(x^ — a;) = 0, may have equal roots. 26. The two distinct equations, x'^ + 2 px + q = {)^ x^ -^ 2 qx -\- p = 0^ are such that the roots of the first have tiie same difference as the roots of the second Prove that either j9 + g = — 1, ov p = q. CHAPTER XX PROBLEMS INVOLVING QUADRATIC EQUATIONS 250. Since the two roots of a quadratic equation can be rational, irrational, or imaginary, problems solved by means of such equations can have apparently such solutions. But because it is impossible to translate all the restrictions expressed or implied in the problem, into the equations formed from the conditions of the problem, solutions must always be verified by substitution in the problem itself. EXAMPLES 1. One of the two factors of 108 exceeds the other by 3. Find the factors. Let X = the first factor, and x + 3 = the second factor. By the conditions, x(x + 3) = 108. (1) Solving (1), cc = 9 or — 12; whence a; + 3 = 12, or — 9. Hence the factors of 108 are 9 and 12 ; or — 12 and — 9. Each of the above solutions satisfies (1) and the problem ; but if restrictions were imposed that both factors should be posi- tive, the second pair would be rejected; and if it were neces- sary that factors should be negative, the first pair would be rejected. 2. A company of 76 men and boys are seated in chairs arranged in such a way that the number of chairs in eacli row is 3 more than twice the number of boys; and thai 318 Ch. XX, § 250] QUADRATIC EQUATIONS 319 the number of rows is 4 less than the number of boys. Find the number of boys. Let X = the number of boys. By the conditions, (2 a; + 3) (a; - 4) = 76. (1) Solving (1), a; = 8, or — |. The restriction implied in the problem is that the solution shall be in positive integers, since it is absurd to speak of | of a boy. Hence the root— f must be rejected as a solution of the problem. In the following problems if possible use a single un- known, rather than several unknowns. EXERCISE CXXXI 1. The product of a number and its half is 18. Find the number. 2. The product of the third and seventh parts of a number is 21. Find the number. 3. What number is 2^ times its reciprocal ? 4. Find a number the sum of which and its reciprocal is 2. 5. Find a number the sum of which and 12 times its reciprocal is 8. 6. The sum of the squares of two consecutive integers is 145. Find the numbers. 7. One of two factors of a number exceeds the oth3r by 2. If the product of the factors is 80, find tlie numbers. 320 ELEMENTARY ALGEBRA [Ch. XX, § 250 8. The product of two factors of a number is 18|. Find these factors if one factor exceeds the other by 5. 9. The sum of two numbers is 9, and their product is 18. Find the numbers. 10. The sum of two numbers is 7, and the sum of their squares is 29o Find the numbers. 11. The difference of two numbers is 7, and their prod uct is 120. Find the numberso 12. The difference of two numbers is 4, and the dif- ference of their squares is 72. Find the numbers. 13. The sum of two numbers is 8, and the sum of their cubes is 152. Find the numbers. 14. Find two numbers such that the sum of the num- bers and the difference of their squares is 11. 15. Find two numbers such that tlieir sum is 15, and their product is 36. 16. If the length and breadth of a rectangle are each increased by 4 feet, the area is increased by 100 square feet ; but if the length and breadth are each diminished by 1 foot, the area is 88 square feet. Find the dimensions. 17. A rectangle whose area is 160 square inches is surrounded by a border 2 inches wide. The border contains 120 square inches. Find the dimensions of the rectangle. 18. The diagonal of a rectangle is 50 feet, and the perimeter is 140 feet. Find the area, 19. Find the length of a rectangle whose area is 11 Gl square feet, if the sum of its length and breadth is 70 feet. Ch. XX, §250] QUADRATIC EQUATIONS 321 20. A number of men each subscribed a certain amount to take up a deficit of $100; but 5 men failed to pay and thus increased the share of the others by |1 each. Find the share of each. 21. It took as many days to do a piece of work as there were men ; but if there had been 4 more men, these men could have done the work in 9 days. Find the number of men. 22. Divide 10 into two such parts that their product shall be 12 times their difference. 23. A number exceeds a second number by 4. Find these numbers if the sum of their reciprocals is -^^, 24. In a number of two digits the units' digit exceeds the tens' digit by 4, and the product of the number and the tens' digit is 192. Find the number. 25. A can do a piece of work in 3 more days than B; and both can do tlie work in 5^ days. How long will it take each alone? 26. Divide 10 into two such parts that the quotient of 10 and the greater part equals the quotient of the greater and less part. 27. The quotient of a number of two digits, divided by the sum of the digits, is 6 ; and if the sum of the squares of the digits be subtracted from the number, the remain der is 13, Find the number. 28. A sold goods for $56, and gained as many per cent is the goods cost. How much did the goods cost ? 29. A number exceeds a second number by 5 ; the differ- ence of their cubes is 665. Find the numbers. 322 ELEMENTARY ALGEBRA [Ch. XX, § 250 30. Separate 250 into two such numbers that the sum of their square roots shall be 22. 31. If A had sold 7 books less for $42, he would have received $1 a book more. Find the price of each book. 32. A sold a number of yards of cloth for $40. Had the price of a yard been 50 cents less he could have sold 4 more yards for the same money. Find the price per yard. 33. A bought two pieces of cloth, which together meas- ured 36 yards. Each piece cost as many dollars per yard as there were yards in the piece, and the cost of the first was 4 times the cost of the second piece. Find the number of yards in each piece. 34. A can row in still water 1| miles an hour faster than the current. It takes him 8 hours to make a round trip of 18 miles. Find the rate of the current, 35. A tap A can fill a cistern in 9 minutes less than a second tap B can empty it. If A and B are running, it takes 3 hours to fill the cistern. How long will it take B alone to empty it ? 36. In a number of two digits the tens' digit is double the units' digit ; and if the number be multiplied by the sum of the digits, the product is 567. Find the number. 37. Find two numbers whose difference multiplied by the greater produces 35, and whose sum multiplied by the less produces 18. 38. What is the price of eggs when 10 more for f 1 lowers the price 4 cents per dozen ? Ch. XX, §250] QUADRATIC EQUATIONS 323 39. A sum of money at simple interest for 1 year amounted to $20,800; if the rate were 1^ less, the amount would be $200 less. Find the principal and the rate per cent. 40. A party of friends went on a pleasure excursion, the expense of which they share equally. If the number of the party had been decreased by 7, and if the total expenses had been $150, the assessment for each person would have been $1 more than it was; but if the num- ber of the party had been increased by 8, and if the total expense had been $160, the assessment for each person would have been $1 less than it was. Find the number of the party, and the assessment for each person. 41. A and B had a money box containing $210, from which each drew a certain sum daily — this sum being fixed for each, but different for the two. After 6 weeks, tlie box was empty. Find the sum which eg-ch drew daily from the box, knowing that A alone would have emptied it 5 weeks earlier than B alone. 42. On a certain road the telegraph poles are placed at equal intervals, and their number per mile is such that if that number were less by 1, each interval between two poles would be increased by 2\^ yards. Find the number of poles, and the number of intervals in a mile. 43. A broker sells certain railroad shares for $3240. A few days later, the price having fallen $9 per share, he buys, for the same sum, 5 more shares than he had sold. Find the price and the number of shares transferred on each day. CHAPTER XXI RATIO, PROPORTION, VARIATION KATIO 251. The ratio of one number to another number is the quotient obtained by dividing the first by the second number. The quotient shows how the numbers compare. Thus, the ratio of 5 to T is indicated : 5 ^ 7, -f^, 5 : 7. The ratio of one quantity to another quantity of the same kind is the ratio of the numerical values of the quantities. Thus, the ratio of a dollars to h dollars is -. h The terms of a ratio are the terms of the fraction indi- cating the ratio ; the numerator is called the antecedent, and the denominator the consequent of the ratio. Thus, a and h are the terms, a is the antecedent, and h the consequent of the ratio ^. There is no ratio of one quantity to another of a different kind^ since it is impossible to compare such quantities. Thus, no ratio exists between a inches and h pounds. 252. If the ratio of two quantities can be expressed as a rational number, they are said to be commensurable ; if the ratio of two quantities is an irrational number, they are said to be incommensurable o 32^ Ch. XXI, § 253] HATIO S25 Thus, when - = -, a and h are commensurable; when h 4 2:= V2, a and h are incommensurable. The ratio of two commensurable quantities is called a commensurable ratio; the ratio of two incommensurable quantities is called an incommensurable ratio. Thus, when - = 5 and when - = V3, 5 and V3 are respec- b b tively commensurable and incommensurable ratios. 253.* An incommensurable ratio can always he expressed as a commensurable ratio whose value differs from the in- commensurable ratio by less than any assigned quantity^ however small. If a is a diagonal of a square of which J is a side, - = V2. b In § 185 it was shown that V2 may be determined to any equired degree of accuracy. In general, let a and b be any two incommensurable quantities. Let p be contained in b integrally (say) m imes, and let p be contained in a more than (say) n times, ind less than n + 1 times. That is, let, mp = 6, (1) np^, m. m (4) fividing (3) by (1), ?<- + -• (5) ' m m 326 ELEMENTARY ALGEBRA [Cii. XXI, § 25^ Since from (4), 7 >— , and from (5), -< — | — , - dif b m m m fers from — by less than — ; or, < — (6) m m m m Since it is always true that mp = S, by taking p smaller and smaller, m will increase : hence — will decrease and m may be made less than any assigned quantity. Therefore - can be made to differ from the commensurable ratio — . m by less than any assigned quantity, however small. Note. If p is very small, - is nearly equal to — ; but - :^ — . b m b m 254. The reciprocal of a given ratio is called an inverse ratio. Thus, f is the inverse ratio of -J. A ratio of equality is one in which the antecedent and consequent are equal ; a ratio of greater inequality is one in which the antecedent is greater than the consequent ; a ratio of less inequality is one in which the antecedent is less than the consequent. Thus, f, I, I, are respectively ratios of equality, greater inequality, and less inequality. The ratio found by squaring the terms of a given ratio is called a duplicate ratio ; the ratio found by cubing the terms of a given ratio is called a triplicate ratio. 2 3 Thus, — ^ and —^ are respectively the duplicate and the trip- licate ratios of - • h Oh. XXI, § 254] RATIO 327 EXERCISE CXXXII 1. Express the ratio of 5 to 7; 4| to 12; 6 to 1 ; 3j\ to 71. 2. Express the ratio of a cents to h cents ; m inches to n inches ; c dollars to a dollars ; m^ feet to i\^ inches. 3. Determine which of the following ratios are com- mensurable : 2 m 6V2 6V2 _5i (Ts/h 3' n V2' V3' 2^\' ^Vi' 4. Determine which of the following ratios are incom- mensurable : m 12 V5 1 11 16 V9 V:J + 1 - ' / — n V5 V8 V7 V16 V4 V3-1 5. Find both the duplicate and triplicate ratios of: c a/3 6 V3 ]V3 V2 Vm 1: T"' 7' v5' ^' Jn W 6. Determine which of the following ratios are those of jreater inequality and which are those of less inequality ; 2 6 9 a V^ V8 V5 + 4 4 4' 7' 8' 6' d' 2 ' VM ' 2 + V5" 7. Prove that a ratio of greater inequality is diminished f the same positive quantity is added to both terms. 8. Prove that a ratio of greater inequality is increased f the same positive quantity is subtracted from both arms. 9. Prove that a ratio of less inequality is increased if he same positive quantity is added to both terms. S28 Elementary algebea [Ch. xxi,§§255,25e PROPORTION 255. A proportion is an equation whose members are ratios. A proportion may be expressed thus: 7 = ^1 a : b = e : d^ a : b : : c : d. The terms of the equal ratios forming a proportion are called the terms of the proportion. The antecedents and consequents of the ratios are called the antecedents and ; consequents of the proportion. The first and fourth terms of a proportion are called the extremes and the second and third terms are called the means. The terms of a proportion are said to be proportional. The fourth term of a proportion is called a fourth proportional. When the second and third terms of a proportion are identical, it is called a mean proportional, and the consequent of the second ratio is called a third proportional. Thus, in the proportion, - = -, a,b, c, and x are proportional, b X a; is a fourth proportional; in the proportion, - = -, a? is a third proportional, and 6 is a mean proportional. A continued proportion is a series of equal ratios in which the consequent of each ratio is the antecedent of the next ratio. Thus, - = - = - is a continued proportion. bed 256.* If two incommensurable ratios^ - and — , are so related b d 71 n a n A- 7 to the commensurable ratio — , that — <-< — I , ivhen w. m b m m -^<£.< — H , however much n and m are increased^ then m d m m a _c b~d' Cii. XXI, § 2o7] PROPORTION 329 If 7^-, since both - and — lie between — and — | — , h d d m m m their difference must be some quantity less than — . But, ^ m since m can be made to increase, — can be made less than m any assigned quantity: hence - — - can be made as small d as is required; a fact which is true only when 7 = -. d Two incommensurable ratios are therefore equal under the conditions named above, and hence may form a proportion. PRINCIPLES OF PROPORTION 257. I. In any proportion the .product of the means equals the product of the extremes. If 2= J, (1) multiplying (1) by ScZ, ad = be. (2) II. If two products are each composed of two factors, thei^e factors form a proportion in which the factors of either product can be made the means^ and the other two factors the extremes. 0) (2) If ad^ = bc^ dividing (1) by bd, a b" _ c ''d' Similarly, h d a c a _ e h etc 330 ELEMENTARY ALGEBUA [Cii. XXT, § 257 III. The products of corresponding terms of two or more proportions are in p)ro port ion. If ^^ = |, 0) a and II _= ~, i^Z) n s 1 A ' o a m c r ^o\ by Axiom 3, t * — = "^ * ~' \^) ^ b n a s am cr or, rewntmg (3), - = -■ (4 IV. T/i6 quotients of the corresponding terms of two proportions are in proportion. If • f=!' O) and if ^ = ^, (2) 1 A • ^ a m c r ^q\ by Axiom 4, 7 -^ — = - — -9 v^>' •^ b n d s or, simplifying in (3), ^ = £* ("^^ V. If four quantities, a, b, c, d, are in proportion, they 7 . • ^7 J. ' b d are m proportion by inversion ; that is, - = — If ^=S' (^> b d by I9 ad = J^, (2) d b b d - = -, or : c a a dividing (2) by a^, _ = -, or =r ^- \ 3) Cn. XXT, § 257] PROPORTION 831 VI. If four quantities of the same Jcind^ a, J, ^, c?, are in proportion^ they are in proportion by alternation ; that is^ (1) (2) (3) a h d If a e b~d' by I, ad = be^ dividing (2) by cd, a _b c Note. ^2JH£^ ^ 10 pounds ^^^^^^ ^^ written by alternation, 3 inches 6 pounds 5 inches • • -u i since IS impossible. 10 pounds VII. If four quantities^ a, 5, c^ 6?, are in proportion^ they \ in proportic a + b c -\- d are in proportion by composition ; that is^ "; = -~ — , , b d a 4- (J 4- d or a c adding 1 to each member of (1), 2 + 1=1 + 1. (2) •x* /'Ci'\ a + b c A- d ^o\ or, rewriting (2), —^ = -=^. (3) a Similarly, (1), written first by V, and then by composi- .. . a + b c + d tion, IS — ' — = — = — r 332 ELEMKNTARY ALGEBRA [Cii. XXI, § 257 Vlil. If four quantities, «, 6, c, d^ are in proportion^ they I J- '^'^ J.T J. ' a — h c — d (1) (2) (3) (4) (5) (6) (7) IX. If four quantities^ a^ 5, c, c?, are in proportion^ they are in proportion by composition and division; that is^ o^-t- h _ c + d a — b c — d If |=|, (1) a a—b c—d b d a c If a b'^d' subtracting 1 from each member of (1), f->"j-^' Or, rewriting (2), a—b c—d b d ' \ writing (1) by V, b_d a e subtracting 1 from each member of (4), ^-1 = ^-1, a c or, rewriting (5), b — a d — c 5 a c multiplying (6) by • -1, a—b e—d Ch. XXI, § 257] PROPORTION 338 writing (1) by VII, ^ = ^, (2) writing (1) by VIII, ^ = "-^, (3) ;.;IV, a + h^c + d .^. a—bc—d X. Like powers^ or like roots of four quantities^ a, I\ (?, d^ which are in proportion^ are in proportion; or —- = — If 2 = 1 (1) raising each member of (1) to the nth. power, whetht^r n is integral or fractional, l'^ "" d"" XI. In a series of equal ratios the sum of the antecedents is to the sum of the consequents as any antecedent is to its own consequent. If cb __ c ^m __x ^-l^^ II — — — — , {^y d n y 1 i. a c m X ,o\ let - = r, - = r, - = r, - = r, (2) d n y clearing of fractions in (2), a = hr^ c = dr^ m^nr^ x=^yr^ (3) by Ax. 1, a + c + m + x = (J) + d + n + y^r, (4) dividing each member of (4) by {I + d •\-n + y), a-^ c-\- m-^-x _ _a _c__^m _x ^r, h -f d-\-n-{-y b ^ n y 334 ELEMENTARY ALGEBRA [Ch. XXI, § 257 EXAMPLES 1. Solve for V2a;+8-V2a:-5 1 Xs = -. (1^ V2a; + 3 + V2a;-6 2 V J By IX, 2V2a; + 3 3 -2V2a;-5 -i' (■/■ simplifying (2), V2a; + 3 = 3V2a;-5, (3J solving (3), x = 3. (4) 2 If*-^ " h~d' prove that ^^^_^^^^-^^. (1) Byx, 0? (? (2) since ^ = ^ = 1, il = i, P Q (3) by III, pa? _ qc' pb""" qif (i) by VI, pa? _ pb^ qc" ~ qcP' (5) by VIII, pa^ — qe^ _ pb^ — qd? q(? qrP ' (6) by VI, pa' —q(?_ qr;0 4. The volume of a sphere varies as the cube of the radius, and the volume of a sphere is 1^:37^ when the radius is 7. Find the volume of a sphere whose radius is 14. Let "F represent the volume and R the radius of the sphere. Then '^ = ^^^' ^^ V=mE% (1) substituting in (1) F= 14371 and B = 7, m = |^ ^ (2) hence volume = ^^^ • 14^ = ^^ • 8 = 11498|. (3) EXERCISE CXXXIV 1. If a: Qc ?/, and if x = 5 when ^ = 4, find x when y = 9. 2. If xcc —. and if 2: = 4 when v = 3, find y when x—2. 3. li xoc yz^ and if a; = 2 when ^ = 3 and 2 = 4, find x when 2/ = 2 and ^ = 6. 4. If X Qc -, and if :r = 16 when y = 3 and 2=8, find 2 when a: = 12 and t/ = 2. 5. If 2: X - + -, and if a; = 4 when y = 3 and 2 = 5, find ^ z y when :r = 3 and 2 = 2. 6. If :?; varies - directly as y and inversely as 2, and is equal to 4 when 2/ = 2 and 2=3, what is the value of x when 2/ = 35 and 2 = 15? 1 , \i y =. u — v^ a u varies as x^ and v as a;^, and if 1/ = 2 when :r = 1, and 2/ = 3 when x = 2, find the value of y in terms of x, 8. If a^ — 52 varies as c^, and if c= 2 when a= 5 and 6 = 3, find the equation between a, J, and (?. Cii. xxr, § 200] VARIA rioN 34S 9. If x^y^ and z xy, prove that x — zocy. 10. If 2; X 2/, prove that x^ + y'^oc xy, 11. li x + y OCX — y, prove that .x!^ +y^Qc xy, 12. If xyzy^ and xocz, and xocw^ when ^ and «^^, y and z^, 2/ iiiid z, are constants, prove that xccyzw. 13. The area of a circle varies as the square of the radius ; show that the area of a circle of 5 feet radius is equal to the sum of a circle of 3 feet radius and another of 4 feet radius. 14. Knowing that the volume, F, of a gas varies directly as the temperature, T, when 2^= 273° + the number of degrees in temperature (in the Centigrade System): if the volume of a certain gas is 400 c.c. when the tempera- ture is 27° C, find the volume of the gas at 127° C. 15. Find, under the law given in the preceding example, the volume of a gas at 0° C, if the volume is 250 c.c. at 18° C. 16. Knowing that the volume, F of a gas varies inversely as the pressure, P, upon it : if the volume of a gas is 100 c.c. when the pressure is 76 cm., find the volume when tlie pressure is 38 cm. 17. Under the conditions given in the preceding prob- lem, if the volume of a gas is 600 c.c. when the pressure is 60 cm., find the pressure if the volume is 150 c.c. 18. Knowing that the intensity of illumination, J, varies inversely as the square of tlie distance, D : if a candle throws a certain amount of light on a screen 2 feet dis- tant, what will be its relative illuminating power at a distance of 7 feet? 344 ELEMENTAKY ALGEBRA [Ch. XXI, § 260 19. Under the conditions given in the preceding prob- lem, if a candle and a gas flame are 12 feet apart, and if the gas flame is equivalent to 4 candles, where must a screen be placed on a line joining the candle and gas flame so that tlie screen may be equally illumined by each of them? V P V P 20. Knowing that -4r^= -\^ where F^ V^, etc., are as given in Problems 14 and 16 : if a mass of air at 0° C. has a volume of 600 c.c. at a pressure of 76 cm., find the volume when the temperature is 91° C. and the pressure is 190 cm. 21. Under the conditions given in the preceding prob- lem, if the volume of a certain mass of air at 27° C, and under a pressure of 225 cm. is 2000 c.c, find its volume at 127° C, under a pressure of 75 cm. 22. Knowing that the amount of bending, 5, of a rod varies jointly as the load, i, and the cube of the length, i', and inversely and jointly as the width, Tf, and the cube of the thickness, T, that is, B oc : if a rod 8 feet long, 4 inches wide, 1 inch thick, is bent 0.2 inch by a weight of 50 pounds, how much would a weight of 5 pounds bend a rod of like material, 24 feet long, 8 inches wide, and 2 inches thick ? 23. Under the conditions given in the preceding prob- lem, if a beam 16 feet long, 8 inches wide, 4 inches thick, is bent ^ inch by a weight of 1000 pounds, how much would a beam 10 feet long, 6 inches wide, 8 inches thick, be bent by the same weight ? CHAPTER XXII PROGRESSIONS ARITHMETICAL PROGRESSION 261. A succession of terms, each of which is obtained from the preceding term by the addition of the same posi- tive or negative quantity (the common difference), is called an arithmetical progression. Thus, 2, 5, 8, 11, etc., and — 1, — 2, — 3, etc., are arithmeti- cal progressions. The first term is usually represented by a, and the com- mon difference by d ; hence the progression is a, a + 6?, a + 2 c?, a + 3 c?, etc. The number of terms in a progres- sion is represented by n ; and the nth. term by /. Since each term is formed from the preceding term by the addition of 6?, the coefficient of c?, in any term, is one less than the number of the term in the progression. Thus, the third term is a + 2 d \ hence / = a +(/(,! -1). I. 1. Find the 10th term of the progression 2, 5, 8, etc. By the conditions, a = 2, d = 3, ^ = 10, by I, 10th term = 2 + 3 (10 - 1) = 29. 345 346 ELEMENTARY ALGEBRA [Ch. XXII, § 26] 2. Find the 10th term of the progression in which the 3d term is 11, and the 7th term is 27. By the conditions, a + 2d = ll, (1) and, a + 6d = 27. (2) subtracting (1) from (2), 4d! = 16, (3) or. d= 4, (4) substituting c^ = 4, in (1), a= 3, (5) by I, 10th term = = a + 9 d = 39. (6) EXERCISE CXXXV Find the last term of each of the following progressions: I. 2, 5, 8, ••• to 10 terms. 2. 8, 5, 2, •• to It terms. 3. 100, 95, 90, ... to 15 terms. 4. 5, 6 — (?, J — 2 (?, ... to 13 terms. Find the nth term of the following progressions in ^^^^^= 5. a = 3i,cZ=2f,^=10. 6. a = 76f, (^=-4f, n = 8. 7. a = h + c^ d = h — 0% n = p. 8. a=x — 7/^d= — t/^n = x^ — y^. Find the indicated terms in the following progressions : 9. 7th term ; the 3d being 10, and the 10th, — 5. 10. 6th term ; the 4th being 0, and the 9th, 15. II. 1st term ; the 7th being — 48, and the 13th, — 108. 12. 10th term ; the 5th being 28, and the 9th, 52. 13. 15th term ; the 31st being — 40, and the sum of th 3d and 11th, 4. tne i Ch. XXII, § 262] PROGRESSIONS 347 262. When three quantities are in arithmetical pro- gression, the middle term is called the arithmetical mean between the other two. If a, J, and c are in arithmetical progression, the arith- metical mean h can be found in terms of the other two. Since b — a = c — h^ i = ^ (a + c). Hence, the arithmetical mean between two quantities is one-half the sum of the quantities. In an arithmetical progression containing any number of terms, all the terms between the first and last are called arithmetical means between those terms. Insert 6 arithmetical means between 8 and 29. The progression evidently contains 8 terms ; a = Sj n = 8, Z = 29. By I, 29 = 8 + (^(8-^1), (1) solving (1), cZ=3. (2) Hence the progression is, 8, [11, 14, 17, 20, 23, 26,'] 29. EXERCISE CXXXVI 1. Insert 7 arithmetical means between 69 and 95. 2. Insert 13 arithmetical means between 13 and 209. 3. Insert 98 arithmetical means between 6 and — 489. 4. Insert 99 arithmetical means between — 5780 and 0. 5. Insert 4 arithmetical means between k and — 6. Insert 10 arithmetical means between w + V3 and m + V3 + 729. 7. Insert r arithmetical means between 1 and 3 r — 2. 348 ELEMENTARY ALGEBRA [Ch. XXII, § 263 263. If aS^ denotes tlie sum of n terms of an arithmetical progression, ^=a + (a + ^) + (a + 2(?)+... + (Z -d') + l, (1) cr, S^l +(l -d) + (l ^2d)+'"+Qa + d') + a, (2) adding (1) and (2), 2Ay=(a+Z) + (a + Z) + (a + 0+-+(^ + + (^ + 0. (3) or, 2S==n(a + l), (4) whence, 5==-[a + /)- II. Since, by I, Z= a + d(n — 1), substituting I in II, 5 = ? 2 a + (/(/?- 1) III. Equations I, II, and III are called the formulas of arithmetical progression. 1. Find the sum of 6 terms of the progression, 5, 3, 1, — 1, etc. By the conditions, a = 5, (^ = — 2, n = 6. Substituting a, d, and n in III, aS = f f 10 - 2 (5) | = 0. 2. How many terms of the progression, 4, 7, 10, ••• must be taken in order that the sum may be 69? By the conditions, a = 4, d = 3, S = 69o Substituting a, d, and S in III, 69 = - 1 8 + 3 (ri - 1) 1 , (1) reducing (1), 3n' + 5n^l3S = 0, (2) solving (2), n = 6, or - 2/. (3) Since n must always be a positive integer, n = 6 is the only solution. Ch. XXII, § 263] PROGRESSIONS 349 Problems of tlie class stated above will evidently always involve the solution of a quadratic equation, and it is therefore possible to obtain one, two, or no correct solutions according as one, two, or no solutions of the quadratic equation are positive integers. 3. In an arithmetical progression whose first term is 3, the sum of 7 terms is 105. Find the common difference. By the conditions, a = 3, n = 7, S = 105. Substituting a, n, and S in III, 105 = |-(6 + 6 d), (1) solving^), cZ = 4. EXERCISE CXXXVII Find the sum in each of the following progressions : 1. 1, 2, 3, 4, ... to 10 terms. 3. 7, 17, 27, ... to 8 terms. 2. |-, J, ^, ... to 12 terms. 4. 2, 2|, 3|-, ... to m terms. 5. 6|, 9|l 121^, ... to 13 terms. 6. 100, 90, 80, ... to 21 terms. 7. 178, 171, 164, ... to 11 terms. 8. 1, 1 + V2, I+2V2, ... to r terms. Find the number of terms in each of the following pro- gressions, so that the given sum may be obtained : 9. aS'=45; 15, 12, 9, .... 10. aS^=-1545; 50,43, 36, .... 11. /S^=1200; 31, 38,45, .... 12. ^=52i; |, |, 1, .... 13. jS = 80(1+W2); 3-V2, 3, 3 + V2, .... 350 ELEMENTARY ALGEBRA [Ch. XXII, § 263 In the following arithmetical progressions : 14. Find c?, and Z, if a = 3 and the sum of the first 13 terms is 351. 15. Find 6?, if the 12tli term is 38 and the sum of the first 13 terms is 351. 16. Find d, and Z, if a = 222, n = 223, and jS=^0. 17. Find a, and l,itd = Q,n = 10, and S= 310. 18. Find n, and c?, if ^ = 4, Z = — 22, and S== — 99. 19. Find ?^, and cZ, if a = ^, Z = 15|^, and S = 47. 20. Find 71, and a, if cZ = a; — 1, I = a^ + x^ + S x —1^ and S=5a^ + Sx^ + 6x. 21. The sum of the first 6 terms is 261, and the sum of the first 9 terms is 297. Find the first 9 terms. 22. The sum of the first 3 terms is 14, and the sum of the squares of these terms is 78. Find the terms. 23. The sum of the first half of the terms is 28, the sum of the second half is 222, the sum of the first and last terms is 50. Find the number of terms. 24. The sum of the last four terms is 20, the product of the second and fifth is 16. If the progression contains five terms, find the progression. 25. In a progression of eighteen terms the product of the two middle terms is 90, and the product of the first and eighteenth terms is 18. Find the first and last terms. Ch. XXII, § 264J PROGRESSIONS 351 GEOMETRICAL PROGRESSION 264. A succession of terms, each of which is obtained from the preceding term by multiplying it by the same positive or negative quantity (the common ratio), is called a geometrical progression. Thus, 2, 4, 8, 16, etc., and 1, — 3, 9, — 27, etc., are geomet- rical progressions. The first term is usually represented by a, and the common ratio by r; hence the progression is, a + ar + ar^ + ar^, etc. The number of terms in a progression is rep- resented by /?, and the ^th term by /. Since each term is formed from the preceding term by multiplying it by r, the exponent of r in any term is one less than the number of the term. Thus, the third term is ar^; and the 72-th term or / = ar^-i. I. 1. Find the 7th term of the progression 1, — 3, 9, •••. By the conditions, a = 1, r = — 3, n = 7, by I,' 7th term = 1(- 3)^ = 729. 2. If the 4th term of a geometrical progression is 1, and the 7th term is ^-, find the 1st term. By the conditions, m^ = 1, (1) and, ar^ = i, (2) dividing (2) by (1), 7^ = ^, (3) from (3), r = i, (4) substituting r = | in (1), a = 8. (5) 352 ELEMENTARY ALGEBRA [Ch. XXII, § 264 EXERCISE CXXXVIII J| Find the last term in each of the following geometrical progressions : 1. 2, 6, 18,-.. to 7 terms. 4. 27, 9, 3, ... to 8 terms. 2. 3, — 6, 12, ... to 6 terms. 5. 6, 3, |, ... to 10 terms. 3. 4, 8, 16, ... to 7 terms. 6. 1, -f, \^-, ... to 11 terms. In the following geometrical progressions : 7. Find the 7th term, the 2d term being 75, and the 5th, -f. 8. Find the 2d term, the 4th term being — 5, and the 7th, 625. 9. Find the 15th term, the 5th term being -2^, and the Q8 • 10th, -. 2^ 10. Find the 50th term, the 19th being 1200, and the 29th, 1200. 11. Find the 11th term, the 2d term being P — c^, and the 5th, (b + c)(b-cy. 12. Find the 10th term, the 3d term being h^^ and llie 13. Find the 7th term, the 2d term being 1, and th^. 4th, 17 - 12 V2. 14. Find the 8th term, the 4th term being 49 — 20 V6, and the 6th, 485 - 198 V6. 15. Find the 7th term, the 3d term being — 2, and the 8th, ~2i. Cii. XXII, § 265] PROGRESSIONS 353 265. When three quantities are in geometrical progres- sion, the middle term is called the geometrical mean between the other two. If a^ 5, and c are in geometrical progression, the geo- metrical mean, which is a mean proportional, can be found in terms of the other two. Since - = 7, a lfl=ac, (1) extracting square roots in (1), h = -Vae. (2) Hence, the geometrical mean between two quantities is the square root of the product of those quantities. In a geometrical progression containing any number of terms, all the terms between the first and last are called geometrical means between those terms. Insert 3 geometrical means between 6 and 486. The progression evidently contains 5 terms ; a = 6, n = 5, Z = 486. By I, 486 = 6 r\ (1) solving (1), r = 3. (2) Hence the progression is 6, [18, 64, 162,] 486. EXERCISE CXXXIX 1. Insert 2 geometric means between 1 and 64. 2. Insert 6 geometric means between — and ^• 3. Insert 11 geometric means between 1 and 2. 4. Insert 5 geometric means between 1875 and 3. 5. Insert 5 geometric means between 36 and — -^^-^c 354 ELEMENTARY ALGEBRA [Cii. XXII, § 2G6 266. If S denotes the sum of n terms of a geometrical progression, S= a + ar + ar^ + ar^ + ••• ar'*"^ + ar^'^. (1) Multiplying (1) by r, rS= ar + ar^ + ar^ -{- ar^ + ••• ar^~^ + ar% (2) subtracting (1) from (2), 8(r - 1) = ar'' - a, (3) from (3), 5 = ^'^^^^ = ^SllnSl. H. r — 1 /•— 1 Since Z = ar^~^^ rl = ar**, substituting rl for ar** in II, 5=^. III. r— 1 1. Find the sum of the progression, 2, 6, 18, ••• to 6 terms. By the conditions, a = 2, r =3, n = 6. By II, /y = ^(y7^ = 728. 2. The 3d term of a geometrical progression is 27, the 6th is 81. Find the sum of the- first 5 terms. By the conditions, ar = 27, (1) and, ar^ = 81, (2) dividing (2) by (1), t'= 3, (3) from (3), r = V3, (4) substituting in (1), a = 9, (5) substituting in II, S = ^(^^f-^ = 117 + 36-/3, (6) V3-1 Cii. XXII, § 266] PROGRESSIONS ' 355 EXERCISE CXL In the following geometrical progressions : 1. Find the sum of 3, — 6, 12, ••• to 6 terms. 2. Find the sum of 6, |, |, ••• to 10 terms. 3. Find the sum of |, |, |, ••• to 10 terms. 4. Find the sum of V2, 2, 2V2, ... to 8 terms. 5. Find the sum of V2 + 1, 1, V2 — 1, ... to 5 terms. 6. Find the sum of the first 7 terms, if the 2d term Ls 4, and the 5th, 256. 7. Find the sum of the first 5 terms, if the 3d term is 27, and the 5th, 48. 8. If a = 6, and r = — 2, find n^ if the sum of n termii is -30 9. The sum of the first 5 terms is 242, and the com- mon ratio is 3. Find the 5th term. 10. The sum of the first 4 terms is 9|^, and the common ratio is \, Find the 1st term. 11. Find the sum of the first 6 terms, if the 6th term is — ^Ys ^^^ ^^^ common ratio is — |. 12. Find the common ratio, and the sum of the first 5 terms, if the 1st term is \ and the 6th term is 864. 13. Find the sum of the first 10 terms of a geometric progression in which the 1st term is 243 and the common ratio is • V3 14. If the 4th term is y^g, and the 7th term is yj^, how many terms, beginning with the 1st, must be taken so that their sum is |||| ? 356 ELEMENTARY ALGEBRA [Ch. XXII, §§ 267,268 267. As in § 258, if a quantity retains the same value throughout a particular investigation, it is called a con- stant. If a quantity changes in value during a particu- lar investigation, it is called a variable. When the value of a variable can be made to approach the value of a constant in such a way that tPie difference of the variable and the constant can be made less than any assigned quantity, however small, the constant is called the limit of the variable. SUM OF AN INFINITE GEOMETRICAL PROGRESSION 268. If r>l, each term of a geometrical progression is larger than the preceding term, and the sum of n terms must increase indefinitely as n increases. If r = 1, the terms are all equal, and the sum of n terms must again increase indefinitely as n increases. If r < 1, and r > — 1, each term is less than the preceding term ; and it will be seen that the sum of n terms always remains less than some definite, finite quantity ; from which, however, by increasing n^ it can be made to differ by less than any assigned quantity, however small. As an illustration, consider the geometrical progression, 1 + 1 + 1 .... Applying III, S= ^~'^^ = 2-l, Hence, in this progression, the sum of any number of terras differs from 2 by just the last term. But, by increasing n the last term can be made as small as may be required. Evidently the sum of n terms can never be as large as 2, but it can be made to differ from 2 by a quantity less than any as- signed value. Hence 2 is the limit of the sum of n terms, as n increases indefinitely. Ch. XXII, § 269] PROGRESSION 357 269. When r <1, it is convenient to write II in the form, 1 — r 1 — rl — r Here f^ can be made as small as is required by increas- ing n. The second fraction, , can, therefore, be made 1 — r as small as is required by increasing the number of terms; and S can be made to differ from by less than any as- signed quantity. is, therefore, the limit approached by aS^ as ^ increases indefinitely. It is usually called the %nm of the infinite geometrical progression, but this must always be understood to mean the limit of the sum of the progression as n increases indefinitely. If S represents the limit of that sum, IV. 1-r 1. Find the sum of an infinite number of terms in the progression, l, 1 -^^, etc. By IV, ^=_J_=?. 2. Find the value of 0.4545 .... The decimal 0.4545 is evidently the geometric progression 45 I 4 5 _i 4 5 _i_ TTJU "T T Ty ~t" TTTTJ dTFTT "T * * *> I in which « = tVV ^' = Tk- t By IV, ^ = ^=5. 358 ELEMENTARY ALGEBRA [Ch. XXII, § 269 3. Find the value of 0.4555 .... The decimal 0.4555 is evidently y\ + the progressicn ron + rtTo + Tiro ¥"0" + •••; in which a = -^-^, r = -^^^ By IV, ^=^ = 1. (1) 1 - tV J-^ Hence 0.4555 ... =3-\ + J^ = |i. (2) EXERCISE CXLI In the following infinite geometrical progressions : 1. Sum to infinity, 2, — f, |, .••. 2. Sum to infinity, 5, 21, 1^, ••.. 3. Sum to infinity, 3|, — 2|^, 11, •••. 4. Sum to infinity, 4, — |, |, •••• 6. Find the value of 0.2544 .... 6. Find the value of 0.86464 .... 7. Find the value of 0.5124545 .... 8. Find the value of 0.2162525 .... 9. Find the value of 0.1248248 .... 10. Find the value of 0.18301830 .... 11. Find the sum to infinity, if the 4th term is 36 and the 7th is - 10|. 12. Find the 1st term, if the sum to infinity is — If and the 2d term is 2. 13. Find the 4th term, if the 1st term is 100 and the sum to infinity is lll^. Cii. XXIT, § 270] PROGRESSION * 359 270. A succession of quantities, whose successive terms are arranged in accordance witli some law, is called a series. Thus, arithmetical and geometrical progressions are series. If a series of quantity be given, it must be tested to determine the nature of the series. The abbreviations A. P. and G. P. indicate respectively arithmetical and geometrical progression. REVIEW EXERCISE CXLII 1. Show that 2a2(a + 3J), (a +5)3, and 2J2(j + 3^), are in A. P. 2. How many terms of the series 1, 8, 5, 7, ••• amount to 1,234,321 ? 3. The arithmetic mean between two quantities is •^^, and the geometric mean is 2. Find the quantities. 4. Find the sum of the terms in the series 1, 1 + J, 1 + 2 6, 1 + 3 J, ••• 1 + ^J, when 5 = 2, n^ll. 5. Sum the series — 3, 6, first as G. P., then as A. P., each to 5 terms. 6. If the arithmetic mean between a and b be double the geometric mean, find -• 7. How many terms of the series 42, 39, 36, ••• make 315 ? 8. Find the sum of 16 terms of the series 27 + 22| + 18+131 + .... 7h — • 1 9?i ■"" 2 9. Find the sum of k terms of the series 1, , -, g n n n— 3 , •••„ n 360 ELEMENTARY ALGEBRA [Ch. XXII, § 270 10. If a, 5, c?, and d are four quantities in G.P., show that 6 + c is the geometric mean between a + 6 and c -j- d. 11. Find the sum of all integral numbers between 1 and 207, which are divisible by 5. 12. Find the sum of all odd integral numbers between 74 and 692. 13. How many positive integral numbers of three digits are there which are divisible by 9 ? Find their sum. 14. Find four numbers in A. P., such that the sum of their squares shall be 120, and that the product of the first and last terms shall be less than the product of the other two by 8. 15. Find a G. P. in which the sum of the first two terms is 2, and the sum to infinity is 4. 16. The 1st term of an A. P. is 2, and d = ^. How many terms must be taken that their sum amounts to 192 ? 17. Find the G.P. whose sum to infinity is 4, and whose second term is J. 18. The sum of three numbers in A. P. is — 3, and their product is 8. Find the numbers. 19. Prove that in an A. P. of a limited number of terms, the sum of two terms, equally distant from the end terms, is equal to a constant. 20. Prove that if each term of an A. P. be multiplied by the same quantity, the resulting series will be in A. P. 21. Prove that in a G. P. of a limited number of terms, the product of two terms, equally distant from the end terms, is constant. N Cii. XXll, § 270] PROGRESSION 361 22. A body slides down an inclined plane 1290 feet long in 15 seconds. If it slides 9 feet the first second, and thereafter gains in distance traversed a fixed amount each second, find this gain. 23. A man deposits money in a bank every week-day for two weeks. Tlie first day he deposits $1.50, and on each succeeding day deposits three times as much as on the day previous. Find the amount to his credit at the end of the two weeks. 24. In starting an engine it was observed that the fly- wheel made f of a revolution the first second, 3| revolu- tions the second second, and 18| revolutions the third second. If it continued to gain speed at this rate, how many revolutions would it make in the eighth second ? If the wheel has a diameter of seven feet, how far would a point in its rim travel in nine seconds ? 25. During a freshet the overflow pipe of a reservoir discharged in a certain number of hours 1,562,496 gallons. If it discharged during the first hour 16 gallons and it continued to discharge on each succeeding hour five times as much as on the hour previous, find the number of hours the overflow continued to increase and the amount dis- charged the last hour. CHAPTER XXIII PERMUTATIONS AND COMBINATIONS 271. The various orders in which a number of things can be arranged are called their permutations. Thus, a and h can be arranged db, ha] while a, h, and c, can be arranged ahc, acb, hac, bca, cab, cba. 272. The various groups that can be selected out of a number of things, without reference to their order, are called their combinations. Thus, the groups of two things that can be selected from a, by and c, are ab, ac, and be. Unless the contrary is expressly stated, the things whose permutations or combinations are required will be understood as different things. Thus, the number of permutations of three different things, when taken two at a time, may be required. 273. // a single operation can be done in m different ways^ and when this operation has been done^ if a second operation can he done in n different ways^ the two operations can he done together in mn different ways. With the first way of performing the first operation there may be associated any one of the n ways of per- forming the second operation ; with the second way of performing the first operation there may be associated 362 Cii. XXILI, § 274] PERMUTATIONS AND COMBINATIONS 363 any one of the n ways of performing the second opera- tion, etc. That is, with each one of the m different ways of performing the first operation there may be associated n ways of performing the second operation. Therefore there are mn different ways of performing the two operations. Thus, the offices of president and vice-president can be filled from five candidates in 20 ways ; since any one of the five can be selected for president, the office of president can be filled in five different ways ; when the office of president has been filled, any one of the remaining four candidates can be selected for vice-president. Any one way of the five ways of filling the office of president can be associated with any one way of the four ways of filling the office of vice-president. Therefore the two offices may be filled in 5 • 4 = 20 different ways. Similarly, the above principle applies to more than two operations, each one of which can be performed in a definite number of ways. Thus, if a man has 5 coats, 3 waistcoats, and 6 pairs of trousers, he can dress himself in 5 • 3 • 6 = 90 different ways. PERMUTATIONS 274. The number of permutations of n different things taken r at a time is n(n — !)(/? — 2) ••• (n — r + 1). The problem of computing the number of permutations Df n different things taken r at a time is equivalent to bhe problem of filling r different places with n different things. 364 ELEMENTARY ALGEBRA [Cii. XXIII, § 275 The first place can evidently be filled with any one of the n different things. After the first place has been filled there remain n — 1 different things, any one of which can be put into the second place ; that is, the second place can be filled in n — 1 different ways for each way that the first can be filled. Hence the first two places can be filled in n(n — 1) different ways. After filling the second place, there remain n — 2 dif- ferent things, any one of which can be put into the third place ; that is, the third place can be filled in n—2 different ways. Hence the first three places can be filled in n(n— V)(n — 2) different ways, etc. Place 1st 2d 3d 4th ... rth Nurrber of ways . . n n-1 n~-2 n-3 ... n-(r-l) Continuing the process, it is evident that the number of ways in which each place can be filled is found by sub- tracting from n that number which is one less than the number of the place. Hence the rth place can be filled in n— (r — X) = n — r+\ different ways. Therefore the r different places can be filled by n different things in n(n — V)(n — 2^ ••• (n — r+V) different ways. The symbol for the number of permutations of n dif- ferent things taken r at a time is written „P^. Hence ./', = /7(/i-l)(/7-2)...(/7-r + 2)(/i-r + l). I. 275. The number of permutations of n different things taken n at a time can evidently be found by substituting n for r in I, Cj] XXIII, §275] PERMUTATIONS AND COMBINATIONS 365 „/'„ = /7(/;-l)(n-2) ••• (2)(1). 11. The product of the factors of „P„, that is, the product of the first n integral numbers, is called factorial /?, and is written \novn\ Formula II may therefore.be written „P„ = n! II. EXAMPLES 1. In how many ways can 8 different letters be inserted in 3 different letter-boxes, one and only one being placed in each box ? The first letter-box can be filled in 8 different ways; the second in 7 different ways ; the third in 6 different ways ; and the three in 8 • 7 • 6 = 336 different ways. That is, by I, 8P3 = 8(8-1)... (8-3-M), = 8.7.6 = 336. 2. In how many ways can the letters of the word Pingry be arranged ? Since there are 6 different letters, the 6 different letters may be arranged in the 6 different places occupied by the letters in 6 ! different ways ; or, by II, eP6 = 6!=720. 3. In how many different ways can 5 people be seated at a round table ? The order of arrangement cannot be that of position on a straight line, but on a closed curve. If one of the 5 be seated, so as to give a starting-point from which to reckon the order, the remaining 4 can be seated in the remaining 4 places in 4 ! different ways j or, by II, 4P, = 4!=24. 366 ELEMENTxVKY ALGEBRA [Ch. XXIII, §§ 276, 277 276. Tlie number of combinations of n different things . . /7(/7-l)(/7-2). .(//-r-f 1) taken r at a tune is — -- rl The symbol for the number of combinations of n dif- ferent things taken r at a time is written nC^- Each one of the combinations of JJ^ is a selection of r different things which can be arranged, by II, in rl different ways; hence the number of combinations of n different things taken r at a time, or ^(7^, when multiplied by rl equals the number of permutations of ^Prl that is, nC^' rl =7i(n — 1) ••• (n — r + 1), ^ n(n-l}(n-2)-'(n-r + l) or n^f— i • ^^^' p l The combinations of a, 5, and jine men are' and 1 ?*^^ '^^^^ted from 12 n,en-l IV, ^">en; ],ence,bjin . 12 • 1-1 ■ . ^/> OA a. Se J 6 • 5 TTT i-'^'^ ^' -^^^ ^^ chosen in gCs = = 15 ways. ajy § 273, the entire committee can be chosen in 56-15 = 840 different ways. 6. If letters in any order form a word, how many words can be formed from 8 consonants and 5 vowels, each word consisting of 4 consonants and 3 vowels ? By III, the selections of consonants and vowels are respec- tively gCi and sCg. ' ' 1.2.3.4 ' 5_,£^^,,^ ' ' 1.2.3 By § 273, the total number of selections of consonants and vowels is 70 . 10 = 700. Since each of the 700 combinations consists of 7 different letters, each combination can be per- muted in 7! = 5040 different ways. There are 700.5040 = 3,528,000 different words. Cii X:XI1I,§277] PERMUTATIONS AND COMBINATIONS 369 EXERCISE CXLIII bind the values of : ^' 10^3' 5. ,P,. 2- n^r 6. „Pa- ^' 12^6- 7. ,0,. 9. sO,. 10. 12(78. 11- 12^10- 4. ^Pg. 8. ^CY 12. i5(7i4. 13. In how many ways can 10 people sit in 4 chairs ? 14. In how many ways can the first 4 letters of the alphabet be arranged ? 15. How many numbers of 3 digits each, no digit being repeated, can be formed from the digits 1 to 9 inclusive ? 16. In how many different ways can 2310 be written as the product of its prime factors ? 17. A man has n different books, which he can place in 5040 different arrangements. Find the number of books. 18. How many combinations can be made of 10 differ- ent things taken in sets of 7 ? 19. On how many nights can a different guard of 5 men be selected from a body of 20 ? On how many of these ^"^iuld any one man serve ? 20. There are 20 things of one kind, and 10 of another. How many different sets can be made ea,ch containing 3 of the first kind and 2 of the second ? 21. In an examination paper of 10 questions any 3 can be omitted. Find the number of selections. 22. In how many ways can 5 people form a ring ? In how many ways a line ? 370 ELEMENTARY ALGEBRA [Ch. XXIII, § 277 23. How many different committees of 3 Republicans and 3 Democrats can be formed from 10 Republicans and 7 Democrats ? 24. How many even numbers of 4 digits each, no digit being repeated, can be formed from the digits 1 to 9 inclusive ? 25. In a boat's crew of 8 men one man can row only on stroke side. How many ways can the crew be seated ? 26. In how many different ways can a ball nine be arranged, the pitcher and catcher being always the same, but the others playing in any position ? 27. How many different sums of money can be formed with a cent piece, a nickel, a dime, a quarter, and a half- dollar ? 28. How many different quantities of anything ponder- able can be weighed with n different weights ? 29. How many changes can be rung with 3 bells out of 6 different bells ? How many with the whole peal ? 30. From 100 men how many juries of 12 men each can be selected if 25 men are excused and if A is always included ? ^ 31. If letters in any order form a word, how many words can be formed from 7 consonants and 5 vowels, each word containing 3 consonants and 3 vowels, and ending in a consonant ? 32. If each of n straight lines intersects all the others, not more than 2 lines intersecting in the same point, how many points of intersection will there be ? Cii. XXIII, §§ 278, 279] PERMUTATIONS, COMBINATIONS 871 278.* The mimher of permutations of n different things, taken r at a time^ when each of the n things can be repeated^ is /l^ After the first place has been filled by one of the n things, the second place can be filled by any one of the n things ; and the first two places can be filled in n^ ways, etc. Continuing the process, the first three places can be filled in n^ ways. The exponent of n is evidently the same as the number of places filled. Hence the first r places can be filled in n^ different ways. If x be the number of permutations of n different things, taken r at a time, when each of the n things can be repeated, X = n\ V. 279.* The number of permutations of n things^ taken n at a time, when p^ q, and r -" of the n things are respectively a, J, and c, -" is • p\ q\ rl ••• The proof will be best understood by taking a specific exam- ple : find the number of permutations of a^b^c=a - a- a-b -b- c. Place a distinguishing sign of each of the three letters a, and also upon the two letters b, thus : ai, a2, a^, bi, 62. Then ai, ag, ag, bi, 62, c, are 6 different things which can, by 11, be arranged in 6 ! different ways. Let X be the total number of permutations of a^b% in which 3 of the letters are a, 2 are b, and 1 is c. Since, by II, the 3 letters a, considered as ai, ag, ag, can be arranged in 3 ! ways, and the 2 letters b can be arranged in 2 ! ways, the total num- ber of permutations of the letters a%% considered as different letters, is a; • 3 ! 2 !, or 6 ! = a; • 3 ! 2 ! Hence x = -^ = 60- 372 ELEMENTARY ALGEBKA [Ch. XXIII, § 27?> Li general, let x represent the number of permutations of n things, taken n at a time, when p^ q^ r, ••• of the n things are respectively a^ 5, c?, •••. If in any one of the X permutations the p things a were different from each other and all the others, there will \}Q p I different permu- tations instead of a single permutation. Hence, if all the letters a were changed into p different letters, there would be in ^W x - p\ permutations. Similarly, if in any one ol the X'p\ permutations, if the q letters b were different from each other and all the others, there would hQ x - p\ q^ permutations. Continuing the process of changing the letters until they are all dift'erent, the total number of permutations will he x - p\ q\ r \ -". Since n ! also is the total number of permutations of n different letters, taken ^ at a time, n\=x - p\ q\ r\ ---^ or n\ jr = p\q\ r\ VI. EXAMPLES 1. Find the numher of ways in which a number of 3 digits can be formed of the 9 significant digits, repetitions being allowed. Each place can be filled in 9 different ways. Hence, by V, a: = 9^ =z 729. 2. Find the number of arrangements of the letters in the word Cincinnati, Of the 10 letters in the word Cincinnati, c is repeated twice, i is repeated three times, and 7i is repeated three times. Hence, by VI, .^, x= ^^' = 50,400. 213!3! Cii. XXIII, § 279] PERMUTATIONS AND COMBINATiUNS 373 EXERCISE CXLIV* 1. In how many ways can the following products be written as a different succession of factors : (1), ahcdef ; (2), a%c; (3), a%^c^ ; (4), a%^(^? 2. How many different arrangements can be made of the letters in the following words : (1), permutation ; (2), parallel ; (3), combination ; (4), Massachusetts; (5), in- commensurable ? 3. How many words, of 3 letters each, can be formed from a^ 6, (?, e^ z, (?, u^ if repetitions are allowed, and if any order of letters form a word ? 4. How many numbers of 3 digits each, repetitions being allowed, can be formed from the first 5 digits ? 5. How many odd numbers of 5 digits each, repetitions being allowed, can be formed from 0, 1, 2, ••• 9 ? 6. How many even numbers of 4 digits each, repetitions being allowed, can be formed from the digits 0, 1, ••• 9 ? 7. In how many ways can groups of 4 letters each, repe- titions being allowed, be formed from m, n^ r, s, u^ i\ w ? 8. In how many ways can groups of 3 letters each be formed from the word Illinois ? 9. In how many ways can groups of 3 books each be selected from 10 books, 3 of wliich are the same text in algebra, and 2 of which are the same text in geometry ? 10. How many different signals can be formed from 12 flags, 2 being red, 3 green, the rest yellow, if all the flags, placed in line, must be used to make a signal ? CHAPTER XXIV BINOMIAL THEOREM 280. The type forms given in § 172 when n = 2, 3, 4, 5, or 6 may be combined into the general form (a + by = a^ + na"-^b + "^"-^^ a/7-242 ^ ^ 1-2 n(n-\:)(n-2) ^_,l^ ^^^,_ 1.2-3 A proof — called the Binomial Theorem — that the laws governing the expansion of (a + by\ when n is any posi- tive integer, give the type form of I will now be given. 1. That I is true when n = 2, 3, 4, 5, or 6, may be seen by substituting in T, for example, n = 3. (a + 5)3= a3+ 3 a% + 3 ah'^+hK If 71 = 6, (a + 5)6 = ^6 + 6 a% + ^a^^ + ^'^''^a%^ ^ ^ 1-2 1.2.3 8>5.4.3^,^, ilA:ll3l2^j5 + 6^4.3.2.1^,^ ^1.2.3.4 ^1.2.3.4.5 ^1.2.3.4.5.6 2. If I is true, when n~h^ h being any positive integer, (a + 5)^ = a'' f ka'-'h + ^^]~^^ a'^'W J. • Jj • O 374 Ch. XXIV, § 281] BINOMIAL THEOREM 375 3. Multiplying both members of (2) by a + 5, 1 • A ^ k(k-l)(k-2') ^^_,^, + ... + a5* + a'b + ka'-W' + ^(^-'^') a'-^b' +-+ Jcab" + 5*+^ , ^ (^ + 1)(^)(7(;-1) ^,_2^3^ ... j^(k + l)ab' + ¥^\ (3) The right member of (3) has the same form as the right member of (2), (lc + 1^ taking the place of h. Hence if the theorem is true for any particuhir power, it is true for the next higher power. 4. The theorem was shown in 1 to be true for the 6th power; hence it is true for the 7th power: being now true for the 7th power, it is true for the 8th power, and so on for any power. 5. The theorem is true for (^a — by^ since {a — by = [« + (— 5)]% the signs of the successive terms being alternate^ plus and minus, the first term being plus. 281. Any required term can be written without com- pleting the expansion by observing the laws for the for- mation of particular terms. Thus, the fourth term of (^a + by is known to be ^0^-- ^X^- 2) ^^.3^3^ ^j^^ ^j^.^^ A. * A ' O term of (a-\-by+^ is known to be ^^ + ^^^ a^-'6^ etc. 1 • 2 87G ELEMENTARY ALGEBRA [Cii. XXIV, § 28'2 Similarly the rtli term of (a + 5)"^ is, n(n -V)(n- 2) «»» (n-r-\- 3X^ - r + 2) ^^-r+ijr-i . 1.2.8... (r-2)(r-l) and the (r + l)st term of (a + J)^ is, n(n — D(n — 2^ ■- Qi — r + 2)(7^ — r + 1) ^T^-rjr 1-2-8 ••• {r — l){r) 282. The number of terms in the expansion of (a + 5)% when 71 is a positive integer, is limited. Thus, by I, (a + J)4 = a^ + 4 aSJ + Il|a2t2 + i_L?JL|^J3 i . Z 1 • ^ • o . 4.3. 2. K, , 4.3.2.1.0 _i.5 1.2.3-4 1.2.3.4.5 Since the coefficients of all terms following the fifth contain a zero factor, all such terms disappear. In general, if n is a positive integer, the expansion of (a + by ends with the (n + l)st term. The coefficients of terms equally distant from the end terms are equal. It is evident that (^a + hy=(b + ay. (h + ay = h'' + 7ih''-^a^ '^^^^^^ b--'^(^- + -. nJa'^-l + a^ (4) (b + ay is merely the expansion ot (a + by written in de- scending powers of 6. The last term of I is the same as the first term of (4) ; the second term of I is the second from the last of (4), etc. Hence in the expansion of a binomial, terms after the middle term Qor teryns) take their coefficients in reverse order. Ch. XXIV, § 282] BINOMIAL THEOREM 377 EXAMPLES 1. Expand (3 a - 1/. By I, (3a-l/=(3a)^ + 5(3ay(-l)+f^(3a)X-l)2 5.4.3.2.1, ,,5 :(-!/ 1.2.3.4.6' = 243a'-405a<+270a8-90a2 + i5a-l, 2. Find the first 4 terms and the last 4 terms of (x — y)3i. By I, and § 282, . ■ 31 . 30 . 29 ,^,,, ..,2« , 31 . 30,..,,, .^ '.{xYi-yy^^-'-^l^ixfi-yf 1-2.3 ^ ' ^ "' 1-2 + 31(x)(-s,)«' + (-y)3i = a;'' - 31 a^y + 465 a;^^^ - 4495 a^/ ... + 4495 ar"^^ - 465 x'y"^ + 31 mj^ - f\ 8. Find the 6th term of (l-^^^. By § 281, the 6th term of 3 . 7 . 11 5V6 2< 231 6^V6 16 378 ELEMENTARY ALGEBRA [Ch. XXIV, § 281! EXEECISB CXLV Expand the following binomials : 2. (a -52)5. 8. (2x-yy. ' ^* ^^^ 3. (a2+52y. 9_ (a-2xy. 13. (\--l)*- 4. (l + x2y. 10. (3rr-2^)5. 14. (2Va-l)6. 5. CaJ — 1)". /I \6 / 1 1\6 6. (a;-i + 1/-2)5, Vx V \ bj 16 /2a-i aV^Y 19 ('^a-i aV^V „ /2«VJ lY 20 /^2aVF3 Viy \s /9 rt -r-v 18 '• \-^-^' \ b . 3V3^ Express in simplest form the indicated terms of the following binomials : 22. 4th term of {x — yy. 23. 2d term of {x — yy'\ 24. 11th term of (a - by\ ( 3 5"2\31 25. 5th term of ( o^b B b-^ 26. 6th term of f-^ - ^^y. \7 bVb VSaJ 27. 8thtermo£f^--Ji^^. \ b la) 28. 10th term of {— - — Y^- Ch. XXIV, § 282] BINOMIAL THEOREM 379 2V^ 6a/P\21 29. 6th terra of i — — — i • \ 6 a J 30. 8tli and 11th terms of f^^ - 6 VpY^ 31. 4th and 17th terms of {4^ ^^T* f 2 \* 33. (n — 2)d term of a ) 32. (r + l)st term of (2 a - hy 2 \*+^ 34. Find the first 4 and the last 4 terms of ( Va - 2-^)20. 35. Find the first 6 and the last 3 terms of (l - ^V^Vl 36. Find the terms that do not contain radicals in 37. Find the coefficient of x^^ in (a; + 2 x^y^. 38. Find the coefficient of a^ in [ a + - j • 39. Find the coefficient of a^^ in ( a? ) • 40. Find the term independent of h in [\\ — ^^ 41. Find the term independent of x in ( \^ — ^J • (2a :z: \' — _ _j CHAPTER XXV LOGARITHMS 283. The logarithm of any number is the exponent indi- cating the power to which a certain fixed number, called the base, must be raised in order to produce the given number. EXAMPLES 1. Find the logarithm of 25 if the base is 5. Since 26 = (5)^, the logarithm of 25 is 2. 2. Find the logarithm of 243 if the base is 9. Since 243= (3/= (32)^= (9)^, the logarithm of 243 is |==2.5. 3. Find the logarithm of 16 if the base is 8. Since 16 = (2)^ = (23)t = (8)*, the logarithm of 16 is 1 = 1.3333.-.. 4. Find the logarithm of ^j if the base is 3. Since — = -— = (3)"^, the logarithm of -^^ is — 3. 27 (3) EXERCISE CXLVI Find the logarithms of the following numbers : « 1. 8, 32, 2 V2, |, yl^, the base being 4. 2. 3, 27, 81 V3, 1 ^\, the base being 9. 3. 2, \, 232, _J_.^ the base being 16. 380 Ch. XXV, §§ 284-286] LOGARITHMS 381 284. In the common (or Briggs) System, the number 10 is always taken as the base. It may be shown that 100=1, 10<> = 1, 10-100, io- = i^,= o.oi, 103 = 1000, 10-^=105-0-001, 10^ = 10000. 10-^ = ^=0.0001. 285. Log 1 = is a short way of writing that, in the system in which the base is 10, the exponent of the power of 10, which produces 1, is 0. Hence, log 1 = 0, log 1 = 0, log 10 = 1, log 0.1 = -1, log 100 = 2, log 0.01 = -2, log 1000 = 3, log 0.001 = - 3, log 10000 = 4. log 0. 0001 = - 4. 286. It is evident that a number between 1 and 10 has a logarithm between and 1 ; a number between 10 and 100 has a logarithm between 1 and 2 ; a number between 100 and 1000 has a logarithm between 2 and 3 ; a number between 1 and 0.1 has a logarithm between and —1 ; a number between 0.1 and 0.01 has a logarithm between 1 and —2; a number between 0.01 and 0.001 has a logarithm between —2 and —3, etc. In general, the logarithm of a number greater than 1 is positive, and the logarithm of a number less than 1 is negative. 382 ELEMENTARY ALGEBRA [Ch. XXV, § 287 287. The logarithm of a number, not an exact power of 10, consists of two parts, — the characteristic, which is the integral part, and the mantissa, which is a fractional part expressed as a decimal. The characteristic of the logarithm of any number greater than 1 is always positive, and depends upon the number of significant digits in the number to the left of the decimal point. From the table in the preceding para- graph, it may be seen that any number containing two digits to the left of the decimal point has a characteristic of 1 ; that any number containing three digits to the left of the decimal point has a characteristic of 2, etc. Hence : The characteristic of the logarithm of any number greater than 1 is always one less than the number of digits preceding the decimal point. The characteristic of the logarithm of any number less than 1 is always negative, and depends upon the number of zeros between the decimal point and the first signifi- cant digit. From the table in the preceding paragraph, it may be seen that any number less than 1 and contain- ing no zeros between the decimal point and the first significant digit is — 1 ; that any number containing one zero between the decimal point and the first significant digit is — 2, etc. The characteristic of the logarithm of a number less than 1 is rarely written in a negative form, but thus : — 1 is written 9(+ decimal) — 10, — 2 is written 8(4- decimal) — 10, — 3 is written 7(+ decimal) — 10. Ch. XXV, §§ 288, 289] LOGARITHMS 383 The logarithm of a number less than 1 will have a characteristic which is the difference between 9 and the number of zeros between the decimal point and the first significant digit, minus 10. Hence : The characteristic of the logarithm of any number less titan 1 is negative^ and is the difference between 9 and the number of zeros between the decimal point and the first sig- nificant digits writing — 10 after the mantissa, 288. The mantissa of the logarithm of any number is given in the table on pages 394 and 395. PRINCIPLES OF LOGARITHMS 289. I. The logarithm of the product of two or more fac- tors is the sum of the logarithms of the factors. Let 10"^ = X, or log x = a^ (1) and let 10^ = ?/, or log «/ = 5, (2) multiplying (1) and (2), IQci+b __ ^y^ Qp i^g xg = a + b = log x + log y. (3) Similarly, I can be proved for the product of three or more factors. II. The logarithm of the quotient of two numbers is the logarithm of the dividend minus the logarithm of the divisor. Let lO"* = x^ or log x = a, (1) and let 10* = y, or log «/ = J, (2) dividing (1) by (2), 10« - ^ = -, or log - = a - 5 = log :r - log y. (3) i/ y 384 ELEMENTARY ALGEBRA [Ch. XXV, § 290 III. The logarithm of the power of a number is the prod- uct of the logarithm of the number by the exponent of the poiver. Let 10" = x^ or log a; = a, (1) raising both members of (1) to the 5th power, 10"^ = x^, or log x^=^ab^b log x, (2) IV. The logarithm of the root of a number is the quotient obtained by dividing the logarithm of the number by the index of the root. Let 10"* = x^ or log x =a^ , (1) extracting the 5th root of both members of (1), m=x\ or logx^ = j = ^-^ = \]ogx. (2) b b b Note. The above principles hold for any number whatever. 290. The mantissa of the logarithms of all numbers which have the same sequence of digits is the same. Let log 214.5 = 2.3314, then log 2145 = log(214.5x 10) =log 214.5-f log 10 = 2.3314 + 1 = 3.3334. Let log 214.6 = 2.3314, then log 0.002145 = log(214.5 -- 100,000) = log 214.5 -log 100,000 = 2.3314 - 5 = 7.3314 - 10. From the above examples, it is evident that changing the position of the decimal point is merely multiplying or dividing the given number by a power of 10. Ch. XXV, §§ 291, 292] LOGARITHMS 385 USE OF THE TABLE 291. To find the logarithm of a number consisting of three digits : On pages 394-395 find in the column under N the first two digits of the given number. The mantissa required will he foxmd at the intersection of the horizontal line containing the first tivo digits and the vertical column headed by the third digit. Prefix the proper characteristic, log 21.7 = 1.3365, log 0,429 = 9.6325 -10, log 970 = 2.9868, log 0.0211 = 8.3243 -10. Numbers containing less than three digits are similarly ^^^^^- log 0.27 = 9.4314 - 10, log 5 = 0.6990, log 0.0029 = 7.4624 - 10. 292. To find the logarithm of a number consisting of more than three digits: 1. Find the logarithm of 92.04. Mantissa of the log of the sequence 920 = 9638, mantissa of the log of the sequence 921 = 9643. An increase of one unit in the sequence gives an increase of 0.0005 in the mantissa; an increase of 0.4 of a unit in the sequence gives an increase of 0.4 x 0.0005 = 0.0002 in the mantissa. Therefore, * mantissa of the log of the sequence 9204 = 9640, prefixing required characteristic, log 92.04 = 1.9640. 386 ELEMENTARY ALGEBRA [Ch. XXV, § 293 2. Find the logarithm of 0.01238. Mantissa of the log of the sequence 123 = 0899, mantissa of the log of the sequence 124 ~ 0934. An increase of one unit in the sequence gives an increase of 0.0035 in the mantissa; an increase of 0.8 of a unit in the sequence gives an increase of 0.8 x 0.0035 = 0.0028 in the man- tissa. Therefore mantissa of the log of the sequence 1238 = 0927, prefixing required characteristic, log 0.01238 = 8.0927 — 10. 293. The process of making the proper correction in the logarithms of numbers of more than three digits is called Interpolation, and is based upon the hypothesis that adjacent mantissas increase proportionally with the corre- sponding numbers. Corrections made in this manner are not strictly accurate ; and even the mantissas given are only approximate, but are correct to 0.00005. If the cor- rection in the fifth decimal place be 5 or more, the fourth decimal place is increased by 1. In the table on pages 394-395 find the mantissa of the first three significant digits^ disregarding the position of the decimal point; subtract the mantissa thus found from the mantissa of the next higher number of three significant digits; multiply the difference thus found by the decimal represented by the remaining digits of the given number; add the product (to the fourth decimal^ to the mantissa of the first three digits. Prefix the proper characteristic. Ch. XXV, § 294] LOGARITHMS 387 294. To find the number corresponding to a given logarithm. 1. Find the number whose logarithm is 7.5521 — 10. From the table, 5514 is the mantissa of the sequence 356, and 5527 is the mantissa of the sequence 357 ; that is, a dif- ference of 0.0013 in the mantissa gives a difference of one unit in the sequence ; hence the mantissa 5521, being 0.0007 more than the mantissa 5514, gives a difference of -^^ of one unit (=0.5) in the sequence. Therefore, applying § 287, log 0.003565 = 7.5521 - 10. The number corresponding to a given logarithm is called the anii/ogarithm. EXERCISE CXLVII Find the logarithms of the following numbers : 1. 254. 7. 362. 13. 8.437. 2. 465. 8. 5685. 14. 0.003. 3. 200. 9. 6297. 15. 0.000569. 4. 908. 10. 1004. 16. 0.009186. 5. 2. 11. 0.8562. 17. 0.01089. 6. 20. 12. 0.008547. 18. 0.9989. Find the antiloga rithms of : 19. 0.3927. 25. 0.9821. 31. 0.0250. 20. 1.6395. 26. 1.6872. 32. 9.5299-10. 21. 8.7235. 27. 3.5689. 33. 8.7467-10. 22. 9.8420- -10. 28. 5.6372. 34. 2.8837. 23. 7.9069- -10. 29. 4.3204. 35. 8.9432-10. 24. 6.9903- -10. 30. 2.3974. 36. 7.0161-10. 388 ELEMENTARY ALGEBRA [Ch. XXV, § 291^ USE OF LOGARITHMS WHICH HAVE NEGATIVE CHARACTERISTICS 295. In finding the antilogarithm of< a negative logarithm^ — 10 should always appear at the end of the logarithm. EXAMPLES 1. Add the following logarithms : 9.6253 - 10 8.5145-10 18.1398 - 20 = 8.1398 - 10. 2. Subtract the logarithm 3.1461 from the logarithm 9 14^0 ^' ■*'^^' 2.1430 = 12.1430 - 10 3.1461= 3.1461 8.9969 - 10. 3. Subtract the logarithm 9.3141 — 10 from the loga- rithm 8.6537-10. 8.6537-10 = 18.6537-20 9.3141-10= 9.3141-10 9.3396-10. 4. Multiply the logarithm 8.1461 - 10 by 2. 8.1461 - 10 2 16.2922-20 = 6.2922 - 10. 6. Divide the logarithm 7.9101 - 10 by 3. 7.9101 - 10 = 27.9101 - 30 3 )27.9101 - 30 9.3034 - 10, Cii. XXV, § 295] LOGARITHMS 388 In multiplying a logarithm hy a fraction^ multiply the logarithm hy the numerator and divide this product by the denominator, in the order stated, taking care to simplify at each step. 6. Multiply the logarithm 8.3196 - 10 by f . 8.3196 - 10 2 16.6392 - 20 = 26.6392 - 30 3 )26.6392 - 30 8.8797 - 10. EXERCISE CXLVIII Perform the indicated operations in the following loga- rithms : 1. (9.7305 -10) + (9.3457 -10). 2. (8.5478 -10) + (9.8438 -10). 3. (0.6544) + (9. 7258 -10). 4. (0.8733) -(2. 7459). 5. (9.3476) -(9.5244). 8. (9.1436-10) x 4. 6. (8.2386 - 10) X 5. 9. (6.8433 -10) x|. 7. (8.8300 -10)-!- 3. 10. (9.8010- 10) -i-|. 11. (7.1431- 10) xf + (8.7153- 10). 12. (2.5157) xi- (9.9918- 10). 13. (6.5000) - (8.5431) x |. 14. (7. 2511 - 10) + (8.2190) x f . 15. (9.0909) X 5 - (8.1650) x |. 16. (2.0001) X f -(8.0999) x f 39G ELEMENTARY ALGEBRA [Ch. XXV, §290 COMPUTATIONS BY LOGARITHMS ooc i T7- ^ 4-1. 1 f 192.7 X 6.54 X 0.4683 296. 1. Find the value of ^^^^^^^^^^^^^ -^. log 192.7 = 2.2849 log 1624 = 3.2106 log 6.54 = 0.8156 log 0.0329 = 8.5172-10 log 0.4683 = 9.6705-10 log 1 .028 = 0.0120 log numerator = 2.7710 log denominator = 1.7398 log denominator = 1.7398 log fraction = 1.0312 fraction = 10.75 2. Find the value of V32.5 x 68.7 x 32.74. log 32.5 = 1.5119 log 68.7 = 1.8370 log 32.74 = 1.5151 log product = 4.8640 ^ log product = 2.4320 product = 270.4. 3. Findthe value of (5.235)3. log 5.235 = 0.7189 3 log 5.235 = 2.1567 (5.235)3 = 143.5. 4. Find the value of 0.763 x 62.8 + 8632 -^ 3.265. log 0.763 = 9.8825 - 10 log 8632 = 3.9361 log 62.8 = 1.7980 log 3.265 = 0.5139 log product = 1.6805 log quotient = 3.4222 product = 47.92 quotient = 2644. quotient = 2644. sum = 2691.92. Note. The last two digits are not accurate since a four-place table is used- Ch. XXV, § 296] LOGARITHMS 391 5. Find the value of — V8 x -^^. log 8 = 0.9031 log 1 = 10.0000 -10 ■^ log 8 = 0.4516 log 7 = 0.8451 -I log ^ = 9.7183 - 10 log I = 29.1549 - 30 log product = 0.1699 i log j- = 9.7183 - 10 product = - 0.1479. Note that the product is negative in accordance with the law of signs. 6. Solve the equation 3-^ = 4, by the use of logarithms. log 3^ = log 4, a; log 3 = log 4, log 3 0.4771 Notice that the above example is a case of an irrational number employed as exponent. EXERCISE CXLIX Compute by the use of logarithms : 1. 21.4x9.87. 11. 251.2 --0,785. 2. 6.92x58.4. 12. 0.09891 H- 0.001234. 3. 0.908x201. 13. 200.9-^10.01. 4. 65.31x0.319. 14. 8957^0.9081. 5. 0.8642x589.7. is. 0.7154 -i- 9.003. 6. 0.9034x0.00154. 16. 0.2167 h- .0.0375. 7. 698-5-20, 17. 0.04678-^892. 8. 0.583 -f- 2982. la 0.0001-5-894.5. 9. 0.9085-1-9.805. 19, 8.9x0.32x0.065. 10. 0.9651-1-0.8939. 20. 0.8x3x500. 392 ELEMENTARY ALGEBRA [Ch. XXV, §296 21. 0.3 X 0.09 X 0.1986. „„ 6456 x 0.6456 x 0.06456 22. 6.98x0.6851x0.32. 27x270x2700 23. 0.91x0.81x0.09. 29 0-4692 x 9231 x 64.82 24. 0.0061x3159^0.005468. ' 0.1492 x 0.8361 x 6987" „ 6.83x0.7816x0.9181 30. 0.5533x419.2x0.3265 . ^^- 9.2184 X 0.07436 ' 60.90x5.432x0.8406 „ 215.4 X 89.72 x 0.896 31 6384 x 0.0987 x 0.012 0.6671x19.2x88.32' ' 2007x0.3388x0.871 „„ 2.754x0.9803x2001 ,^ 0.7188x0.8159x0.0001 * 3721x0.1596x0.31 '0.01897x0.8963x0.3031* 33. (6.608)2. 39. V6479T. 45. -v^O.OOOS. 34. (2.755)2. 40. V9381. 46. -v/0.2756- 35. (1.01)25. 41. ^0.0182. 47. •v'0.1622- 36. (99.81)3. 42. ^6503. 48. A/85r2. 37. (49.73)*. 43. ■v'50. 49. ^f. 38. (0.9801)6. 44. ^0.1257. 50. -v/|. I 23 X 75 „ 3/0.152 X 0.025 51. -VI • 56. ■\/ • >'l3x0.85 ^ 25x0.085 y^. g2 , J.525 X 0.054 gy j/0.3756 x 0.265 351 X 0.062 * ^ 0.227 x 863 JO^ „ 1 0.768 X 0.0345 gg (0.03472)^ x -v/4011 2512.x 0.071 ' ■ (1.21) ,, J2. 01-6 X 0.06932 „ 5076 VO. 007109 54. \/ • 3"» „ — • ^ 0.1126x987 9834-V/0.045 3| 0.0435 X 3986 go (0.3143)^ ^ ' ^' 4534 X 0.087 ' ■l.63-V0.163 Ch. XXV, § 296J LOGARITHMS 393 61. (|)3v/36. 64. Vs+^T. 67. ■j^jV'Jf. 62. •v'o.aSVa. 65. ^384 + ^81. 68. -^mWM. 63. (11)6^8721. 66. -^^1 -7 7839 7701 7774 7846 61 62 63 7853 7924 7993 7860 7931 8000 7868 7938 8007 7875 8014 7882 7952 8021 7889 7959 8028 7896 7^)66 8035 7903 7973 8041 7910 7980 8048 7917 7987 8055 64 65 66 8002 8129 8195 8069 8136 8202 8075 8142 8209 8082 8149 8215 8089 8156 8222 80V)6 8162 8228 8102 8169 8235 8109 817() 8241 8116 8182 8248 8122 8189 8254 67 68 69 8261 8325 8388 8267 8331 8395 8274 8338 8401 8280 8:m 8407 8287 8351 8414 8293 8357 8420 8299 8363 8426 8306 8370 8432 8312 8376 8439 8319 8382 8445 70 71 72 8451 8513 8573 8457 8519 8579 8463 8525 8585 8470 8531 8591 8476 85.'57 8597 8482 8543 8()03 8488 8549 8609 8494 8555 ^15 8500 8561 8621 8506 8567 8627 73 74 75 8()33 8()92 8751 8639 8698 8756 8()45 8704 8762 8651 8710 8768 8657 8716 8774 86()3 8722 8779 86()9 8727 8785 8675 8733 8791 8681 8739 8797 8686 8745 8802 76 77 78 8808 8865 8921 8814 8871 8927 8820 8876 8932 8825 8882 8938 8831 8887 8943 8837 8893 8949 8842 8899 8954 8848 8^)04 8960 8854 8910 8965 8859 8915 8971 79 80 81 8970 9248 9149 9201 9253 9154 9206 9258 9159 9212 9263 9165 9217 9269 9170 9222 9274 9175 9227 9279 9180 9232 9284 9186 9238 9289 85 86 87 9294 9;M5 9395 9299 9350 9400 91304 9355 9405 9309 93()0 9410 9315 93(J5 9415 9320 9370 9420 9325 9375 9425 9330 9380 9430 9335 9385 9435 9340 93