IN MEMORIAM 
 FLORIAN CAJORl 
 
Digitized by tine Internet Arciiive 
 
 in 2008 witii funding from 
 
 IVIicrosoft Corporation 
 
 littp://www.arcliive.org/details/firstyearinalgebOOsomericli 
 
FIRST YEAR IN ALGEBRA 
 
 BY 
 
 FREDERICK H. SOMERVILLE 
 
 THE WILLIAM PENN CHARTER SCHOOL, PHILADELPHIA 
 
 NEW YORK •:• CINCINNATI .:• CHICAGO 
 
 AMERICAN BOOK COMPANY 
 
Copyright, 1905, by 
 FREDERICK H. SOMERVILLE. 
 
 Entered at Stationers' Hall, London. 
 
 FIRST year in ALG. 
 
 w. p. t 
 

 PREFACE 
 
 The aim of this little book is to provide an introduc- 
 tory course as a foundation to elementary algebra. A 
 minimum number of definitions, an early introduction of 
 the literal symbol in its simplest form, a clear conception 
 of the opposition of positive and negative quantity, and a 
 gradual introduction to the early processes are believed 
 to be the first essentials to successful later work. New 
 elements are introduced as the result of some natural 
 process, the exponent, for example, not being mentioned 
 or used until, in multiplication, the pupil meets the opera- 
 tion that produces it. 
 
 Certain important topics are given a more extended 
 treatment than is customary in most books prepared for 
 beginners. The application of the equation to the prob- 
 lem is made in a form that experience has shown to give 
 excellent results, and the reasoning powers developed by 
 the limited classifications have been equal to the demands 
 of the general problem. Substitution has a much more 
 important position than is usual in elementary teaching, 
 and its constant applications are designed to meet an 
 actual need felt by teachers in higher grades of work. 
 
4 PREFACE 
 
 The importance of factoring is emphasized by an unusual 
 amount of practice, and the arrangement of the type- 
 forms is 'one that has been given extended trial in the 
 class room. 
 
 Reviews are constant and consist of both classified 
 groups for topical review and miscellaneous exercises in 
 which the power of discrimination is first developed. 
 The frequent grouping of principles recently passed over 
 serves to keep them clearly in mind. The exercises have 
 been carefully graded throughout and are free from diffi- 
 cult and discouraging questions. 
 
 FEEDEEICK H. SOMERVILLE. 
 
 Philadelphia, Pa, 
 
■ CONTENTS 
 
 ' OHAPTEE PAGE 
 
 I. Symbols 7 
 
 Positive and Negative Quantity ... 10 
 
 n. Addition 18 
 
 Parentheses 22 
 
 Review .28 
 
 III. Subtraction 31 
 
 IV. Multiplication 36 
 
 V. Division 49 
 
 Review 58 
 
 VI. The Simple Equation 61 
 
 VII. Substitution 83 
 
 VIII. Special Cases in Multiplication and Division . 94 
 
 IX. Factoring 109 
 
 Review 131 
 
 X. The Highest Common Factor 140 
 
 XI. Fractions 143 
 
 Transformations 143 
 
 XII. The Lowest Common Multiple .... 151 
 
 The Lowest Common Denominator . . . 154 
 
 XIII. Fractions (Continued) 157 
 
 Addition and Subtraction of Fractions . 157 
 
 Multiplication and Division of Fractions . 161 
 
 Miscellaneous Fractions 165 
 
 Review 167 
 
 XIV. Fractional Equations 174 
 
 XV. Simultaneous Simple Equations .... 184 
 
 Review 197 
 
FIRST YEAR IN ALGEBRA 
 
 CHAPTER I 
 
 SYMBOLS. POSITIVE AND NEGATIVE 
 QUANTITY 
 
 1. In arithmetic we represent quantity by the use of 
 numbers^ and, in most cases, we work with units of a par- 
 ticular kind. 
 
 Thus, 4 pounds, 3 yards, and 7 dollars are units of a particular 
 kind. 
 
 2. In algebra we represent quantity by the use of the 
 letters of the alphabet^ and we do not, as a rule, assign a 
 particular value to these literal symbols. 
 
 Thus, 4 a, 3 m, 6 a:, and 7 y represent what we call general units, 
 and our operations with them are accomplished without reference to 
 any particular value. 
 
 ILLUSTRATIONS OF OPERATIONS WITH LITERAL SYMBOLS 
 
 At first the student may wonder that operations with 
 these new symbols of quantity are possible, but a com- 
 parison of an arithmetical addition with particular units 
 and an algebraic addition with general units will serve to 
 convince. 
 
 7 
 
8 SYMBOLS, POSITIVE AND NEGATIVE QUANTITY 
 
 By Arithmbtic By Algebra 
 7 apples 7 a 
 
 5 apples 5 a 
 
 3 apples 3 a 
 
 15 apples 15 a 
 
 No matter what the '''sl''' of the algebraic illustration stands 
 for, it is true that the sum of seven a's, five a's, and three 
 «'s is fifteen a's. 
 
 The principle is the same when units of dijBferent kinds 
 are considered together. 
 
 By Arithmetic 
 
 By Algebra 
 
 4 pounds + 3 ounces 
 
 4a+3& 
 
 6 pounds + 2 ounces 
 
 6a + 2& 
 
 10 pounds + 4 ounces 
 
 lOa + 46 
 
 20 pounds + 9 ounces 20 a + 9 6 
 
 Note that in both additions, units of the same kind are 
 added together, pounds added to pounds, ounces to ounces, 
 a's together, and 5's together. 
 
 SYMBOLS OF OPERATION 
 
 The principal signs of operation in algebra are identical 
 with those of arithmetic. 
 
 3. Addition is indicated by the " plus " sign, + . 
 
 Thus, a + b is the indicated sum of the quantity a and the quantity 
 b. Read " a plus J." 
 
 4. Subtraction is indicated by the " minus " sign, — . 
 
 Thus, a — 6 is the indicated difference between the quantity h and 
 the quantity a. Read " a minus b." 
 
SYMBOLS OF OPERATION 9 
 
 5. Multiplication is usually indicated by the absence of 
 sign between the quantities to be multiplied. 
 
 Thus, ab is the indicated product of the quantities a and 6. abx is 
 the indicated product of the quantities a, b, and x. 
 
 Sometimes a dot is used to indicate a multiplication. 
 Thus, a • b means the product of the quantities a and b. 
 
 An indicated product may be read by the use of the 
 word "times" or by the use of the literal symbols 
 only. 
 
 Thus, ab may be read " a times b." Or, simply, " ab.** 
 
 6. Division is indicated either by the sign of division, 
 -f-, or by writing in the fractional form. 
 
 Thus, a -H & is the indicated quotient of the quantity a divided by 
 the quantity b. 
 
 Also, - is another form for the same indicated operation. 
 b 
 
 7. Equality of quantities in algebra is expressed by the 
 sign of equality, = , which is read, " equals " or " is equal 
 to." 
 
 Thus, to express the fact that the sum of two quantities, a and b, 
 is the same as the sum of two other quantities, c and d, we write 
 a -{- b = c + d. 
 
 8. A coefficient is a known quantity showing how many 
 times another quantity, or group of quantities, is to be 
 added in an expression. 
 
 Thus, in 6 a + 8 xy "5" is the coefficient of «a"; «8" is the 
 coefficient of " xy." 
 
10 SYMBOLS. POSITIVE AND NEGATIVE QUANTITY 
 
 The student will see that coefficients are results of ad- 
 ditions, for, 
 
 5 a is an abbreviation ofa + a + a + a + a. 
 
 8 xy is an abbreviation of xy -\- xi/ + xy + xy -{- xy -{■ xy + xy + xy. 
 
 The advantage of the use of coefficients is evident. 
 
 Sometimes a coefficient is a literal quantity. 
 
 Thus, in abx, " a " may be considered the coefficient of bx, or 
 " ab " may be considered the coefficient of x. 
 
 When the coefficient of a quantity is " unity " or " 1," 
 it is not usually written or read. 
 
 Thus : a is the same as 1 a. xy is the same as 1 xy. 
 
 POSITIVE AND NEGATIVE QUANTITY. SIGNS OF QUALITY 
 
 9. In algebra we consider not only the amount of quan- 
 tity, but its nature or quality as well. 
 
 Illustration. Suppose a merchant has a credit in a bank 
 amounting to 1500, and suppose, also, that he owes the 
 sum of $500 for goods purchased. Clearly, the two con- 
 ditions are directly opposite to each other, for 
 
 the first $ 500 represents a possession, 
 the second $ 500 represents a debt. 
 
 Now possession is an addition to the man's property ; 
 debt, a subtraction from it : and the fact that these are 
 opposites of quality must be represented by symbols that 
 stand also for direct opposites. Hence, if we use 
 
 the plus sign to indicate possession, 
 
 we must, correspondingly, use 
 
 the minus sign to indicate debt. 
 
USE AND VALUE OF QUALITY SIGNS 11 
 
 So we will write + 500 representing his possession, 
 — 500 representing his debt. 
 
 From the foregoing we make this important conclusion : 
 
 10. As Signs of Quality the plus, +, and the minus, — , 
 symbols indicate absolute opposites in kind. 
 
 ILLUSTRATIONS OP THE USE AND VALUE OF QUALITY SIGNS 
 
 (a) If the distance in an easterly direction from Chicago 
 is considered "plus," the distance west from that city 
 would be considered as "minus." 
 
 west — Chicago 4- east 
 
 (5) We may consider the city of Memphis, Tennessee, 
 as in latitude about 35° north of the equator, or in lati- 
 tude -f- 35°. Correspondingly, Montevideo, about 35° in 
 latitude measured south from the equator, would be con- 
 sidered as in latitude — 35°. 
 
 (c) Suppose the temperature on a certain winter's morn- 
 ing was reported as 20° above the zero point in Philadel- 
 phia, and 20° below the zero point in Montreal. We would 
 consider the first as -}- 20°, and therefore the latter as —20°. 
 
 (d) If the force expended to lift a certain weight is 100, 
 and the force resisting the effort to lift it is 50, we would 
 express the conditions thus : 
 
 Force acting to raise it + 100. 
 
 Force acting against raising it — 50. 
 
 (e) Suppose two men weigh exactly 150 pounds. The 
 first man gains 10 pounds while the second loses 10 
 
12 SYMBOLS. POSITIVE AND NEGATIVE QUANTITY 
 
 pounds. The effects on the men are directly opposite, 
 hence the quality of the changes would be expressed thus : 
 
 First man's change in weight + 10 pounds. 
 Second man's change in weight — 10 pounds. 
 
 Clearly, therefore, 
 
 11. Whatever kind of quality we indicate hy the plus sign, 
 the opposite kind of quality will be indicated by the minus 
 sign. 
 
 THE USE OF OPPOSITE QUALITY SIGNS TOGETHER 
 
 12. The sum of a group of plus, or positive, units of the 
 same hind will be a positive quantity. 
 
 Thus, if A has three checks for f 100, $200, and $300 respectively, 
 we state his possession : 
 
 + 100 
 + 200 
 + 300 
 + 600 his possession in dollars. 
 
 13. The sum of a group of minus, or negative, units of the 
 same kind will' be a negative quantity. 
 
 Thus, if A owes three different men the sums of $100, $150, and 
 $ 200 respectively, we state his indebtedness : 
 
 - 100 
 -150 
 -200 
 
 — 450 his indebtedness in dollars. 
 
 Let us now consider the effect of bringing together into 
 one operation two or more quantities of different quality 
 sign ; that is, the combining of positive and negative quan- 
 tities in one group. 
 
USE OF OPPOSITE QUALITY SIGNS TOGETHER 13 
 
 Suppose a man has 5 dollars and owes 2 dollars. How 
 many dollars will he have after paying his debt ? 
 By arithmetic we obtain by ordinary subtraction : 
 
 5 
 2 
 3 • 
 
 But in algebra we consider the quality of the quantities, hence 
 the possession is given the + sign, consequently, the debt is given the 
 — sign. Then we write ^ 
 
 -2 
 + 3 
 
 In each case, clearly, there remain 3 dollars. But our algebraic 
 process has considered both amount and kind, and is based on this 
 reasoning : 
 
 His possession is +5 or +1 + 1 + 1 + 1 + 1. 
 
 His debt is — 2 or — 1 — 1. 
 
 Now it requires one + dollar to pay one — dollar, hence two + dol- 
 lars are used to pay the two — dollars. And three + dollars remain, 
 for the algebraic process has considered kind as well as amount in the 
 result. 
 
 Suppose, again, that the man has $ 4 but owes f 9. By 
 the same process : , ^ 
 
 -9 
 -5 
 
 For His possession is + 4 or +1 + 1 + 1 + 1. 
 
 His debt is - 9 or -l-l-l-l-l-l-l-l-l. 
 
 Here five negative dollars remain after the entire number of posi- 
 tive dollars has been used as far as possible to pay dollars of debt. 
 
 Again, the quality sign is evident and of as much importance as 
 the amount. 
 
14 SYMBOLS. POSITIVE AND NEGATIVE QUANTITY 
 
 From these illustrations we conclude : 
 
 14. The algebraic sum of two quantities with different 
 quality signs is positive or negative according as there are 
 more + or more — units considered. 
 
 With no more difficulty we may group several quantities 
 of different quality in one operation. 
 
 Suppose that the line AB represents the distance be- 
 tween two towns, and that (7 is a point midway between A 
 and B. Suppose, also, that distances to the right of O are 
 considered as -f- . It follows, therefore, that distances to 
 the left of Q will be considered as — . 
 
 A O B 
 
 + 10 
 
 
 -7 - 
 
 
 + 12 
 
 > 
 
 A man starts at (7, goes 10 miles towards B^ turns back 
 7 miles towards A^ and finally turns again and goes 12 miles 
 towards B. We express his journey thus : 
 
 + 10-7 + 12 miles. 
 
 He traveled 10 miles and 12 miles respectively towards B; also 
 7 miles towards A . Hence, his distance from C will be 
 
 + 10 + 12 - 7 = + 15 miles. 
 
 The result indicates not only that he is 15 miles from C at the end 
 of his journey, but, by its + sign, it shows him to be at the right 
 of C. 
 
USE OF OPPOSITE QUALITY SIGNS TOGETHER 15 
 Suppose a different journey. Thus : 
 
 A C B 
 
 + 10 
 
 15 
 
 + 8 
 
 -7 
 
 The several distances and directions are expressed thus : 
 
 + 10-15 + 8-7. 
 
 From which + 10 + 8 = + 18 his journey towards B. 
 and — 15 — 7 = — 22 his journey towards A. 
 
 Result — 4 his final distance at the left of C. 
 
 Using the same line, AB, let the student determine the 
 positions of the traveler at the end of the journeys indi- 
 cated in the following, not forgetting to make use of the 
 proper signs for locating him at either the right or the left 
 of his starting-point. 
 
 1. From C 8 miles towards B and 6 miles back towards A, 
 
 2. From C 9 miles towards B, 16 miles towards A, and 
 back 8 miles towards B. 
 
 3. From O 6 miles towards B, 10 back towards A, and 4 
 towards B. 
 
 4. From 17 miles towards A, 11 towards jB, 6 towards 
 A, and 21 towards B. 
 
 5. From C 7 miles towards A, 3 more towards A, 15 
 back towards B^ and 7 towards A. 
 
16 SYMBOLS. POSITIVE AND NEGATIVE QUANTITY 
 
 Exercise 1 
 
 In each of the following determine the required numeri- 
 cal result, together with its quality, whether positive or 
 negative. 
 
 1. A merchant has a credit of $1000 in a bank and gives 
 a check for the payment of a bill of |400. What is the 
 amount and nature of his balance ? 
 
 2. The temperature on a certain morning was 58° above 
 zero. At noon it was 72° above zero. What quality change 
 had taken place and how much ? 
 
 3. On a winter's day at noon the temperature was 12° 
 above zero. At 5 o'clock in the afternoon the temperature 
 had fallen 19°. What was the reading of the thermometer 
 at 5 o'clock ? 
 
 4. A has a cash balance of $45. He has B's note for 
 115 and C's check for 1 30, and he owes D |100. How 
 will A stand when he balances his account ? 
 
 5. A has $100. B has no cash whatever. C also has 
 no cash, and yet owes 8 100. How much better off than 
 B is A ? How much better off than C is B ? How much 
 better off than C is A ? 
 
 ALGEBRAIC EXPRESSIONS 
 
 15. An Algebraic Expression is an algebraic symbol, or 
 group of algebraic symbols, representing some quantity. 
 
 An expression is called numerical when made up wholly 
 of numerical symbols. 
 
 Thus, 20 + 10 — 13 is a numerical expression. 
 
ALGEBRAIC EXPRESSIONS 17 
 
 An expression is called literal when made up wholly or 
 in part of literal symbols. 
 
 Thus, ab + mn — xy is a literal expression. 
 5a + 3c — 17 is a literal expression. 
 
 16. The Terms of an algebraic expression are its parts 
 connected by the + or — signs. 
 
 Thus, in the expression ab + mn — xy, 
 
 ab, +7nn and — xy are terms. 
 It will be noted that the sign between two terms belongs to the term 
 following it. Also that the sign of the first term, when +, is not 
 usually written. 
 
 Thus, in the expression ab-\-mn — xy, 
 the first term is + ab. 
 
 Y7. The common expressions of algebra are named as 
 follows : 
 
 Monomial. One term. 
 
 3 a, 10 mn, 27 xyz are monomials. 
 Binomial. Two terms. 
 
 a + b,^m — x,10 — lcd are binomials. 
 
 Trinomial. Three terms. 
 
 3w + 7n — Sxyisa trinomial. 
 
 Polynomial. A name usually given to expressions of 
 more than three terms. 
 
 p. H. S. FIRST YEAR ALG. —2 
 
CHAPTER II 
 ADDITION. PARENTHESES. REVIEW 
 
 18. Addition in algebra, as in arithmetic, is the process 
 of combining two or more expressions into an equivalent 
 expression called the sum. 
 
 In algebraic addition we consider : 
 
 (1) Like quantities having the same signs, either all + 
 or all — . 
 
 (2) Like quantities having different signs, some + and 
 some — . 
 
 When the quantities to be added are like in sign, the 
 addition is in no way different from that of arithmetic. 
 Thus : 
 
 By Arithmetic By Algebra 
 
 7 books + 7 a -7a 
 
 10 books + 10 a - 10 a 
 
 15 books + log —15a 
 
 32 books + 32 a - 32 a 
 
 From which we conclude : 
 
 19. 77ie coefficient of the sum of similar terms having like 
 signs is the sum of the coefficients of the terms with the com- 
 mon sign. 
 
 When the quantities to be added are unlike in sign, the 
 resulting coefficients depend for their sign upon the excess 
 of one quality sign over the other. (Art. 14.) 
 
 18 
 
GENERAL METHOD 19 
 
 Applying this principle to the coefficients of like terms, 
 we will obtain the sum of the following : 
 
 5a4- 7& + 2c 
 2a-3&-5c 
 
 7a + 46-3c Result. 
 
 The result is obtained as follows : 
 
 fThe coefficient " + 7 " in the 1st term is the sum of the given 
 coefficients, + 5 and + 2. 
 
 The coefficient " + 4 " in the 2d term is the sum of the given coeffi- 
 cients, + 7 and — 3. 
 
 By Art. 14 this coefficient is the actual difference between 7 and 3, 
 with the sign of the greater. 
 
 The coefficient " — 3 " in the 3d term is the sum of the given coeffi- 
 cients, + 2 and — 5. 
 
 Here, as above, the coefficient "—3" is an actual difference with 
 the proper sign. 
 
 From this illustration we conclude : 
 
 20. The coefficient of the sum of similar terms having un- 
 like signs is the actual difference between the sum of the 4- 
 coefficients and the sum of the — coefficients^ with the sign 
 of the greater. 
 
 GENERAL METHOD 
 
 The following method is in general use for the addition 
 of expressions made up of similar terms. 
 
 Ex. 1. Add ^a + lh-2c, 2a-35 + 8(?, and -3a-t- 
 
 5a + 7& - 2c 
 
 2a-3& + 8c 
 
 ~3a + 2ft-10c 
 
 4a + 66- 4c Result. 
 
20 ADDITION. PABENTHESES. REVIEW 
 
 The coefficients of the result are obtained as follows : 
 
 Sum of the + coefficients of a, + 7 ; sum of the — coefficients of 
 a, — 3. Difference, + 4. 
 
 Sum of the + coefficients of &, + 9 ; sum of the — coefficients of 
 ft, - 3. Difference, + 6. 
 
 Sum of the + coefficients of c, + 8; sum of the — coefficients of 
 c, — 12. Difference, — 4. 
 
 Remember that the signs of the first terms, 5 a and 2 a, are " plus 
 understood." • 
 
 If not all of the terms considered are represented in each 
 given expression, it is well' to arrange in alphabetic order, 
 leaving space for such terms as an examination of the 
 problem shows need for. 
 
 Ex. 2. Add4a + 35 + 3w, 26 + 3(?-6?, 2a+3(^+2w-a;, 
 and 5b — 5m—Sx, 
 
 4a + 3& + 3m 
 
 + 2b + Sc- d 
 2a -\-Sd + 2m- X 
 
 + 5& — 5m — 3a? 
 
 6a + 10b + Sc + 2d -4a: Result. 
 
 Note that the sum of the coefficients of " m " is zero, and, there- 
 fore, the m term disappears in the result. 
 
 We are now ready to state the general process for addi- 
 tion of algebraic expressions. 
 
 21. Place like terms in vertical columns. 
 
 Add separately in each column the positive coefficients and 
 the negative coefficients^ and to the arithmetical difference of 
 their sums give the proper sign. 
 
GENERAL METHOD 21 
 
 22. Collecting terms in algebra is another expression 
 for addition. It is frequently used in cases where like 
 terms are scattered throughout an expression and in no 
 particular order. 
 
 Exercise 2 
 
 Find the sum of : 
 
 1. 2. 3. 4. 5. 6. 7. 8. 
 
 a -a 10a -10a -Sa -Ua 87a; -29y 
 la -la -6a 6a 15a -14a -18a; 13^ 
 
 9. 10. 11. 12. 13. 
 
 2a + 4h Za-\-lx —Im—lxz —8a;— 13 —aed-\- ^yz 
 ba-2h a-6x 4m-\-^xz lla;+ 1 ^acd-llyz 
 
 14. 15. 16. 
 
 bxy-\-'^xz— yz — 2nx—10 p —ah —x-\-l 
 
 2>xy 4-2?/2 Smy-\-llnx bab—2cd-\-x 
 
 17. 3a+26, 4a-35, -2a+75, and 5a-6h, 
 
 18. 2a— 3<?+a;, 3a + 4(? + 3a;, and a — 3 c — 2 a;. 
 
 19. 5a;+3y-10, -6a; + 2«/-9, and x-2y^l, 
 
 20. 4m — 37^^-2a?, —2n — x^ Sm — n, and 3n—Sx. 
 
 21. 2a-35— 4c— 5, a + 7h+^c—2, and -7a+35-c. 
 
 22. — 3a + 35-3c + 6, c — 25 + 5 — a, and 
 — 5 — 254-5flj — <?. 
 
 23. a-2y + h-Sx+10, 2h -S y -\-2 a-7, 
 3 a; — 2 a + y, and 45— 3a;— 2a— 11. 
 
22 ADDITION. PARENTHESES. REVIEW 
 
 Collect similar terms in each of the following : 
 
 24. 6 ah -\- She — 2 mn —Zx-\-2hc — cd — ah-\-% mn 
 
 — ah — S cd -{- X. 
 
 25. 3w+7 — a — a; + 8 — m+lla; — 15a — 4w 
 
 — 2n-^Sm — Sx. 
 
 26. 2ahc—5 hed -i- 2 cdx -11-^ 15 hcd —2abc 
 
 — cdx — 2 + ahc. 
 
 27. —2a + Qac-\-ab—^ac—a—bah-\-^a—2ac-\-2>ah. 
 
 28. Zd-[-a-d-\-c-'^d-2c+4:h-\-c-^h + lQa-2a. 
 
 29. — a — m + c— x-\-c — m-\-a — y — h-{-m — c — a 
 + m + h-\-y + a — c + h-{-x, 
 
 THE PARENTHESIS 
 
 23. The Parenthesis is used in algebra to indicate that 
 two or more quantities are to be treated as a single quan- 
 tity. Thus : 
 
 a + (ft — c) means that to a we are to add the difference of h and c. 
 a — (b — c) means that from a we are to subtract the difference of 
 b and c. 
 
 24. A sign hefore a parenthesis indicates an operation. 
 A + sign hefore a parenthesis is an indicated addition. 
 A — sign hefore a parenthesis is an indicated suhtraction. 
 
 We will first consider 
 
 THE PARENTHESIS PRECEDED BT THE PLUS SIGN 
 
 Suppose we add to 20 the sum of 10 and 5. 
 We write 20 +(10 + 5). 
 
PARENTHESIS PRECEDED BY MINUS SIGN 23 
 
 From this indicated addition we obtain 
 
 20 + (10 + 5)= 20 + 15 20 +(10 + 5)= 20 + 10 + 5 
 = 35 ^^' = 35 
 
 The result is the same whether the 10 and 5 are added before or 
 after removing the ( ). 
 
 Suppose, again, that we add to 20 the difference between 
 10 and 5. 
 
 We write 20 +(10 -5). 
 
 And, as before, 
 
 20 +(10 - 5)= 20 + 5 20 + (10 - 5)= 20 + 10-5 
 = 25 "^' =25 
 
 The same result in either case. Hence, we conclude : 
 
 25. if a + ( ) ^s removed^ none of the signs of its terms is 
 changed. 
 
 "The parenthesis preceded by the minus sign 
 Suppose we subtract from 20 the sum of 10 and 5. 
 We write and obtain 
 
 20 - (10 + 5) = 20 - 15 20 - (10 + 5) = 20 - 10 - 5 
 = 5 ^^' =5 
 
 The same result in either case. 
 
 Suppose, again, that we subtract from 20 the difference 
 between 10 and 5. 
 
 We write 
 
 20 -(10 - 5)= 20 - 5 20 -(10 - 5) = 20 - 10 + 5 
 ^15 "'' =15 
 
 Again, the same result in either case. Clearly, in the last operation 
 we are not to take all of 10 from 20 ; only the difference between 10 
 and 5 is to be taken. So if 10 is taken away, 5 must be added, and 
 we have 20 - 10 + 5, or 15. 
 
24 ADDITION. PARENTHESES. BEVIEW 
 
 We may conclude, therefore, that 
 
 26. If a — (^^ is removed^ the sign of every term in it must 
 he changed. 
 
 It must be remembered that 
 
 {a) The sign before a parenthesis indicates an operation 
 of either addition or subtraction. 
 
 (5) The sign of a parenthesis disappears when the paren- 
 thesis is removed. 
 
 The effect of all possible cases of + and — signs before 
 parentheses is clearly illustrated as follows : 
 
 + (+«) = + « —(+«)=—« 
 
 PARENTHESES WITHIN PARENTHESES 
 
 It is frequently necessary that we inclose in parentheses 
 parts of an expression already in another parenthesis. In 
 order to avoid confusion in such cases, we employ different 
 forms of the parenthesis. The forms are : 
 
 (a) The Bracket [ ]. (6) The Brace \\. (js) The 
 Vinculum " . Each has the same significance as the 
 ordinary parenthesis. Thus: 
 
 (a + 6) = [a + 5] = [a + ^1 = a-\-h. 
 
 In removing one or more of these signs of aggregation, 
 we observe the following : 
 
 27. Beginning with the innermost., remove the parentheses 
 one at a time. 
 
 If a parenthesis preceded by a minus sign is removed, the 
 signs of its terms are changed* 
 
PARENTHESES WITHIN PARENTHESES 25 
 
 After removing all parentheses, collect the terms of the 
 
 result. 
 
 Exercise 3 
 
 Observing the effect of the + and — signs preceding 
 parentheses that you remove, simplify the following : 
 1- (+5)+ (+2) = 5 + 2 
 
 . = 7. Result. 
 
 2. (+5) -(+2) = 5- 2 
 
 = 3. Result. 
 
 3. (+5) + (-2). 8. (+16) + (-17). 
 
 4. (+5) -(-2). 9. (-14) -(-13). 
 
 5. (-7) + ( + 6). 10. (-7a)-(+5a). 
 
 6. (+5)-(-9). 11. _(-5a)-(+7a). 
 
 7. (_10)-(_19). 12. -(+6w) + (-6m). 
 
 13. (-13:r?/) + (-llrry). 
 
 14. — (— 19mn) — (+20 wn). 
 
 15. 3m-(2m) + (-m) + (2m) = 3m-2w-wj + 2m 
 
 = 2m. Result. 
 
 16. (+6) -(+3) +(-2). 
 
 17. (-5) -(-2) -(-3). 
 
 18. a— (4a) + (—«) — (—4a). 
 
 19. -5x+(-2x')-(-x}-{Sx'). 
 
 20. a - [2 6 + (3 c - 2 6) - c] = a - [2 6 + 3 c - 2 6 - c] 
 
 = a-2b -Sc + 2b + c 
 ' = a — 2 c. Result. 
 
 21. 5a + (3a + ll). 26. - 3a;-(-rr + «/-9). 
 
 22. 4a-(3a + 6). 27. 4:r+ [22: + (2: + l)]. 
 
 23. 3a;+(2a;-3)+4. 28. 4a^- [2 2;-(a:- 1)]. 
 
 24. 5:r-(3a;-^ + l). 29. 3 ^/ - jy + (2 y-10) j. 
 
 25. 6c-(2c? + 3)-10. 30. 2a;?/- [a;2/ + 3-2;?/]. 
 
26 ADDITION, PARENTHESES. REVIEW 
 
 31. by+i\l-\2y-y + l\-], 
 
 32. -7-(-w+ [-3m- J2m + ^n)- 
 33. 
 
 4 ?n - [3 m - (w + 2) - 4] - {m + 3 n - (n - 1)} + n 
 
 4m- [3 w- n-2 -4]-{m + 3n- n + 1} +n = 
 4m- 3m+ n + 2 +4 -{m + 3n- n +'l} + n = 
 4m— 3m+ n + 2 +4 -m-3n+ n-1 +w = 5. Result. 
 
 34. a — [a + f a— (<x — a — 1)1] 
 
 35. — 3m+J — m + (— 2m — 6 — w)J. 
 
 36. 5tf-[3 + (-2c- [(?_(3(?+ J(?-70])] 
 
 37. (4 a - 1) - [2 a - (a + 1) - 3 a - 7] - (8 - a + 1) . 
 
 38. l_(_l)-j_l4-l_a4-(-l)|. 
 
 39. l+(-l) + ;-l-l_a-(-l){. 
 
 40. a-\-l—a + \-a-\-Q-a^a-V)\']. 
 
 41. a — [— a — J — a — (— a — a — l)j]. 
 
 42. «— 5(5 — c — a-c + l)— a — ft — 1)J. 
 
 43. a- [a-a-(a + l)- [a-(l-l-a + 2a)]5. 
 
 44. l_(_l)-5-(_l)j_j_(_;_(_i)_l|)|_l. 
 
 INCLOSING TERMS IN PARENTHESES 
 
 It is often necessary to group certain terms of an alge- 
 braic expression so as to treat the group as one term. 
 This operation is the exact opposite of the removal of 
 parentheses, hence we invert the statements of Arts. 25 
 and 26, and obtain the following : 
 
INCLOSING TERMS IN PARENTHESES 27 
 
 28. An^ number of terms may he inclosed in a -\- (^^ with- 
 out ehanging the signs of the terms inclosed. 
 
 29. Any number of terms may be inclosed in a — (^) pro- 
 vided the sign of each term inclosed is changed. 
 
 Exercise 4 
 
 Insert, in each of the following, a parenthesis inclosing 
 the last two terms, each parenthesis to be preceded by 
 the plus sign. 
 
 1. a + 2 6 + 7c. 4. l%ab + ^bc-2cd-M. 
 
 2. 5 a + 3 w + 2 w— 7. 5. Zm — -\:n — bp — xyz. 
 
 3. ^xy-\-l xz-\-^yz—^, 6. -^ a + lbb-10-\-yz. 
 
 Insert, in each of the following, a parenthesis inclosing 
 the last three terms, each parenthesis to be preceded by a 
 minus sign. 
 
 7. 3a- 5n + 7;)-19 = 3a- (5n-7j9 + 19). Result. 
 
 8. a6c — 4 cc? + wn — njo — 3 = a6c — 4 cc?— ( — mn + n/? + 3). Result. 
 
 9. x — y-{-z — \. 12. — 10 — 7;? + 8n — 5w. 
 
 10. 3^- 2 5-4 c + ^. 13. 4:a-\-b-\-c-^d. 
 
 11. ^ab-1bc + Qcd-^0. 14. 16-W + 3 w-3jt? + a;2. 
 
 15. a + b-c-\-d-e+f 
 
 16. — 3 — w + mn ~np — xyz, 
 
 17. x—b a — y ■\'Z — 1, 
 
 18. 5aH-2^— 3a;— 5. 
 
 19. ac — be — ah + ax — ay. 
 
28 ADDITION. PARENTHESES. REVIEW 
 
 GENERAL REVIEW — GROUP I 
 
 SIGNS OF OPERATION 
 
 The + sign. An addition or increase. 
 The — sign. A subtraction or decrease. 
 
 SIGNS OF QUALITY 
 
 sign. 
 
 ^, . r Absolute opposites in kind. 
 
 The — siqn. J 
 
 ADDITION 
 
 The sum of a group of all + units of the same kind is a 
 + quantity. 
 
 The sum of a group of all — units of the same kind is a 
 — quantity. 
 
 The sum of a group of both + and — units of the same 
 kind is either -\- or — in sign according as there are more 
 + or more — units considered. 
 
 PARENTHESES 
 
 The signs before paren- 1. . ^ . ._. _,,.. 
 
 ^, . A A -\- ( ) IS an indicated addition, 
 
 theses are signs of \ . ;r. ._. __ 
 
 \ A — ( ) IS an indicated subtraction, 
 operation. J ^ 
 
 (a) Removal 
 
 If a-\-(^)is removed., none of the signs of its terms is changed. 
 If a— (^} is removed^ all of the signs of its terms are changed. 
 The sign before aparenthesis disappears with the parenthesis. 
 
 (b) Insertion 
 
 Inclose an expression m a 4- ( ) and change no signs of 
 terms inclosed. 
 
 Inclose an expression in a — (^^ and change the sign of 
 each term inclosed. 
 
GENERAL REVIEW 29 
 
 Exercise 5 
 
 1. A merchant has 1500 and f 700 in two banks and 
 owes $ 800 for goods purchased. What is the expression 
 for his actual financial condition ? 
 
 2. If the same merchant had a dollars in the first bank, 
 b dollars in the second bank, and owed c dollars, what would 
 be the expression of his possession ? 
 
 3. On a certain day the temperature rose 20° from the 
 zero point, but at sunset it fell 13°. Express the change 
 and the result with proper signs. 
 
 4. If a thermometer indicates a rise of a° and, later, a 
 fall of x°, how shall we express the final result ? 
 
 5. A man weighs 150 pounds, but loses m pounds. 
 Later he gains n pounds. What is his final weight ? 
 
 '* 6. Express with proper sign the result of 
 
 (+3)-(+2)-(-3) + (-5) + (+2). 
 
 7. What is the sum of 
 
 _(_2)_(-3) + (-2)-(+3)? 
 
 8. Is the sum of 
 
 [_3 + (_2)-(-l)]-[_(-2) + C-l)] 
 positive or negative ? 
 
 9. Prove that 
 
 [3-(-2)-(l)] + [-2-(-3) + (-5)]=0. 
 
 10. A boy claims that 
 
 [(- 2) - (+ 3) - (- 5)] is the same as 
 [-(-2) -(-3) -(5)]. Is he right? 
 
 11. Find the sum of5a + 3& — m, 25 — 3<? + a;, — 4a — 
 5 6 + 5 c + Wj and a — 2c — x. 
 
30 ADDITION. PARENTHESES, REVIEW 
 
 12. Collect -2a + 3c-4(^, 4(?-a;+7, 4a— 6c-h3(?— 4, 
 
 and —a-\-2d + 1x — ^, 
 
 13. Simplify and collect 
 
 a + [ - 2 (7 - 3 a + (5 - c + a) + 5] . 
 
 14. Simplify and collect 
 
 1-!1 + (1-[1 + 1]-1)}. 
 
 15. Simplify and collect 
 
 10 _ [9 + 8 - (7 + 6 - 55 -f 4 - 3T2S - 1)]. 
 
 16. In a ball game the Bostons made — [2 — (3 + 2)] 
 runs, and the New Yorks made —3 — [—(2 + 3)] runs. 
 How many runs were made in all ? 
 
 17. A boy was asked to write his age on a slip of paper, 
 and he wrote " — (+[— 12 + (— 3)] ) years of age." How 
 old was he ? 
 
 18. Simplify and collect 
 
 - [2 - ^"^] - S - (-a - [- a - r=^]){. 
 
 19. Inclose the last four terms of the following in a 
 parenthesis preceded by a minus sign : 
 
 a — 3J4-4m — 5/1 + 27. 
 
 20. Inclose the last three terms of the following in a 
 parenthesis preceded by a plus sign : 
 
 ax — am — mx + mt/ — ay, 
 
 21. Beginning at the left, inclose each pair of terms in a 
 parenthesis, the first two parentheses to be preceded by 
 plus signs and the last two preceded by minus signs. 
 
 <i — Z'b — ^c-\-d — m — n-\-x — y, 
 
CHAPTER III 
 SUBTRACTION 
 
 30. Subtraction is the process of taking one quantity 
 from another quantity. 
 
 The quantity taken away is called the Subtrahend. 
 
 The quantity from which the subtrahend is taken is 
 called the Minuend. 
 
 The result obtained by taking the subtrahend from the 
 minuend is called the Difference. 
 
 31. The general process of subtraction depends upon 
 the principles already explained in Art. 26. For, 
 
 The quantity taken away may he considered as inclosed in 
 a parenthesis preceded hy a minus sign. 
 
 Thus, suppose we write : 
 
 From 10a + 36 + 7c take 6 a + 5 + <?. 
 
 The first expression is our minuend and the second expression 
 our subtrahend. 
 
 Consider each expression a single quantity and inclose each in a 
 parenthesis. 
 
 Then, using the minus sign of operation for the word " take," we 
 
 ^^^® (10 a + 3 & + 7 c) -(6 a + & + c). 
 
 Removing both parentheses, 
 
 10a + 3&+7c-6a-&-c. 
 
 Collecting, 4a + 2& + 6c. Result, or Difference. 
 
 31 
 
32 SUBTRACTION 
 
 We have, therefore, changed the signs of the terms of the original sub- 
 trahend and collected all terms. The same operation might have been 
 
 written thus: m^ . qt. • '7 -Mf- j 
 
 10a + «3o+7c Minuend. 
 
 6 a + ft + c Subtrahend. 
 
 4a + 26 + 6c Diif erence. 
 
 The student must bear in mind that the signs of the 
 subtrahend are not actually changed. They are only con- 
 ceived to he changed. At first the student will be greatly 
 assisted by indicating the conceived changes in his work, 
 but he should immediately form the habit of making the 
 change mentally. The given signs of the subtrahend should 
 never be altered in any way. 
 
 From the foregoing we will now state the general pro- 
 cess for subtraction. 
 
 32. To subtract algebraic expressions : 
 
 Place similar terms in vertical columns. 
 Consider the sign of each term of the subtrahend to he 
 changed^ and proceed as in addition. 
 
 Before taking up the practice of algebraic subtraction 
 the student should carefully note the following : 
 
 (a) Subtracting a positive quantity is the same as adding 
 a negative quantity. 
 
 For, (+5)-(+3) = +5-3 = 2, 
 
 and ( + 5) + ( - 3) = + 5 - 3 = 2. 
 
 (6) Subtracting a negative quantity is the same as adding 
 a positive quantity. 
 
 For, (+5)-(-3)= + 5 + 3 = 8, 
 
 and (+ 5) + (+ 3)= + 5 + 3 = 8. 
 
SUBTRACTION 33 
 
 These results follow directly from the simple removal 
 of parentheses. 
 
 Exercise 6 
 
 
 1. 
 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 From 
 
 + 10 
 
 
 -\-12a 
 
 -6a 
 
 -3a 
 
 + Sah 
 
 + 5a; 
 
 Take 
 
 4- 4 
 
 
 + 5a 
 
 -2a 
 
 -2a 
 
 -lab 
 
 -Sx 
 
 
 + 6 
 
 
 + 7a 
 
 -\a 
 
 — a 
 
 + 10 a6 
 
 + Sx 
 
 
 7. 
 
 
 8. 
 
 9. 
 
 10. 
 
 11- 
 
 12. 
 
 From 
 
 10 m 
 
 
 - le 
 
 6xy 
 
 -%ac 
 
 — 4mw 
 
 — 5a:0 
 
 Take 
 
 12m 
 -2m 
 
 
 4:0 
 -lie? 
 
 -\xy 
 V^xy 
 
 -2ae 
 
 — 9mn 
 
 19 xz 
 
 
 — a<? 
 
 -24:XZ 
 
 
 
 
 13. 
 
 
 14. 
 
 
 15. 
 
 From 
 
 
 4 
 
 a-\-6 
 
 7 
 
 m-13 
 
 — 
 
 13a:-7 
 
 Take 
 
 
 2 
 
 a-3 
 
 5 
 
 m-18 
 
 - 
 
 -6^ + 4 
 
 16. 17. 18. 19. 
 
 Sa + ^h + lO lm-4:n -2b +6 
 
 25 + 7 3m-2n + 7 2a-25 + 3m + 6 a + b-{-c 
 
 5a+ 5 + 3 4m — 2n— 7 —2a —3m —a—b — e 
 
 20. From 5 a + 4 m — 7 take 3 a — m + 3. 
 
 21. From 2 a + 4 a; — ^ take a — 5 a; — 3 y. 
 
 22. From 3 a; + 7 ?/ — 10 take —x—y + 6, 
 
 23. From 4iX — By—Zz take 2 «/ + 8 3. 
 
 24. From -3a + 4c-2(Z4-7 take 3a+7<7-2. 
 
 25. From 4 a + 11 m — a: take 3 a + 7 a: — ?/. 
 
 26. From 35 + 2c — 4m take — a— 5 + 2c — m. 
 
 F. H. S. FIRST YEAR ALG. — 3 
 
34 SUBTRACTION 
 
 27. Subtract 12 ah -2 be -10 from 11 ah - 4 ho -{- S. 
 
 28. Subtract 10-\-Sx-2i/ + z from 15 — 4:x-22/ + z. 
 
 29. Subtract 5m-^S n — 2 y -{-z from 2m—n+22/ — z. 
 
 30. Subtract 6a + 4J-3(? from 6a-46+3c-12i. 
 
 31. From the sum of 2m-\-2n — p and 3w — Sri + Sjt? 
 subtract the sum of m-^n—2p and Sm— 2n —p. 
 
 2m + 2n— j9 wi+ n — 2^? 5 m — n + 2j9 
 
 3 m — 3n+3jo 3 m — 2n— p 4:7n — n—Sp 
 5m— n + 2 ji? 4:771 — n — 3p m + 5jt> Result. 
 
 32. From the sum of 2 a + ih — c and 3a — 45 + 3 c 
 take the sum of S a + 5h—l c and 2a — 4:h-\-5c. 
 
 33. From the sum of 3a— 55 + 10 and ih + l c—7 
 take the sum of 6 + 8 c — 12 a and 16 5 — 19 + 6 a — 4 (?. 
 
 34. Take the sum of 4w + 37i — 5?/, Sn + 4:i/ — z, and 
 w+^ — 14 2 from the sum of 2m -{-Si/, ^n — x, 5p-\-z, 
 and — 12 w — 9. 
 
 ADDITION Ain) SUBTRACTION WITH DISSIMILAR COEFFICIENTS * 
 
 33. Thus far our additions and subtractions have been 
 made when the coefficients of like terms were all numeri- 
 cal and easily combined. Thus : 
 
 5x 15 ar 
 
 3a: 6a; 
 
 8 a; Sum. 9 a; DifEerence. 
 
 In later work it will often happen that coefficients will 
 be literal and not possible of combination into a simple 
 
 ♦ At the teacher's discretion this subject may be omitted until Exer- 
 cise 47 is reached. 
 
ADDITION AND SUBTRACTION 35 
 
 form like a numerical sum. In such cases our addition is 
 indicated and the process is as follows : 
 
 ax 
 
 bx 
 
 mx 
 nx 
 
 
 2 ax 
 ^cx 
 
 5 am 
 — 7 cm 
 
 (a + b)x 
 
 (m -\- n)x 
 
 (2a- 
 
 f 3c)a; 
 
 (5 a - 7 c) m 
 
 Add the 
 1. 
 
 Exercise 7 
 following : 
 2. 
 
 3. 
 
 4. 
 
 ay 
 
 cy 
 
 Sx 
 
 mx 
 
 ( 
 
 xyz 
 
 — axyz 
 )xyz 
 
 cm 
 — ahem 
 
 ( )^ 
 
 ( > 
 
 ( ^cm 
 
 ax 
 nx 
 
 5. 
 
 + So + 
 - he - 
 
 fan 
 an 
 
 
 6. 
 
 5771+ w 4- X 
 -am-\-en—S ax 
 
 (a + n) a; -h (3 — 5) <? + (w — a) w ( ) wi + etc. 
 
 7. ah + mn, 4 5 — en, and h -{- n. 
 
 8. 5 m -{- Sn -{- ax and 2 am — 4: en — x. 
 
 9. 4 6c + 5 5c^ + 3 /wTi, 2 (? — hmn, and — ac — ahd. 
 
 Subtract the following : 
 
 
 
 10. 11. 
 
 12. 
 
 
 mx hay 
 
 ma + 
 
 a; 
 
 nx —Zey 
 
 3a - 
 
 ax 
 
 {m — n)x (ba-\-^c)y ( )«+( )^ 
 
 13. Subtract 4 a; — 3 ^ from ax + %. 
 
 14. Subtract 5 m^i — 7 aa; from am/i + hx. 
 
 15. Subtract 3 my -{-2nx—Sz from a?/ + aa; + a;?. 
 
CHAPTER IV 
 
 MULTIPLICATION 
 
 34. Multiplication is the process of taking one quantity 
 as many times as there are units in another quantity. 
 Multiplication is an abbreviated form of addition, for 
 3x4 = 4 + 44 4. 5a = a + a + a + a-{-a. 
 
 The Multiplicand is the quantity to be multiplied. 
 
 The Multiplier is the quantity showing how many times 
 the multiplicand is to be taken. 
 
 The multiplier shows also, by its sign, how the multi- 
 plicand is to be taken; that is, whether to be added or 
 subtracted. Thus, 
 
 5 multiplied by + 3 indicates a positive or additive result, and 5 
 multiplied by — 3 indicates a negative or subtractive result. 
 
 The Product is the result of a multiplication. 
 
 SIGNS m IMULTIPLICATION 
 
 It has been stated that the multiplier shows not only 
 how many times a quantity is to be taken, but how it is to 
 be taken as well. 
 
 The following will illustrate : 
 
 (I) Positive Multiplier. A product to be added. 
 
 8 times 5 expressed with all signs gives 
 
 + 3x + 5=+5 + 5 + 5 = + 15. 
 3 times — 5 expressed with all signs gives 
 
 + 3x-5 = -5-5-5 = -15. 
 
EXPONENTS IN MULTIPLICATION 37 
 
 (II) Negative Multiplier. A product to be subtracted. 
 
 — 3 times 5 expressed with all signs gives 
 
 _ 3x + 5 = - ( + 5+5+5) = _ (+ 15) = - 15. 
 
 - 3 times — 5 expressed with all signs gives 
 
 _ 3 X - 5 = - (- 5 - 5 - 5) = - (- 15) = + 15. 
 
 By comparing these four possible cases we have 
 
 + 5 -5 +5 -5 
 
 + 3 + 3 - 3 - 3 
 
 + 15 - 15 - 15 +15 
 
 From which we make these important conclusions : 
 
 35. Like signs in multiplication give -f . 
 
 36. Unlike signs in multiplication give — . 
 
 COEFFICIENTS IN MULTIPLICATION 
 
 37. The coefficient of a term in a product of two alge- 
 braic expressions is the product of the coefficients of the 
 given multiplier and multiplicand. 
 
 Thus, 7 X 8a = (7 X 8)a = 56a. 
 
 ab X cdm = (ah x cd)m = abcdm. 
 
 EXPONENTS IN MULTIPLICATION 
 
 Suppose we obtain, arithmetically, the product of two 
 numbers, 16 and 8. We write : 
 
 16 
 __8 
 128 
 
 16 and 8 are factors of 128. 
 
88 - MULTIPLICATION- 
 
 38. The Factors of a quantity are the quantities that, 
 multiplied together, will produce it. 
 
 Let us now consider the 16 and the 8 as themselves 
 composed of factors. 
 
 Then 16 = 2x2x2x2 
 
 and 8 = 2x2x2 
 
 Hence, 128 = 2x2x2x2x2x2x2 
 
 In the same manner : 
 27 X 81 = (3 X 3 X 3) X (3 X 3 X 3 X 3). 
 36 X 9 X 16 = (2 X 2 X 3 X 3) X (3 X 3) X (2 X 2 X 2 X 2) ; etc. 
 
 It is clear, therefore, that all the factors of a multipli- 
 cand and a multiplier are included in a product. But this 
 method of expressing the fact is cumbersome, and we have 
 need for a briefer form. Hence, we use a small number 
 at the right and slightly above a factor that occurs twice 
 or more in a product, and our process is Written thus : 
 
 16 2* 
 
 8 or 28 
 
 128 2^ 
 
 The product indicates in a new way the number of factors 
 composing it. 
 
 39. The product of two or more equal factors is called 
 a Power. 
 
 40. The number showing how many times a factor 
 occurs in a power is called an Exponent. 
 
 In common practice we say : 
 
 16 = 2*, or, 16 is the fourth power of 2. 
 8 = 2^, or, 8 is the third power of 2. 
 128 = 2\ or, 128 is the seventh power of 2. 
 
MULTIPLICATION OF MONOMIALS 39 
 
 Similarly, with literal factors : 
 
 a X a^ = 0^+^ = a^, which we read " a ••• to the sixth power." 
 m^ X m^ = m^+^ = m^, which we read " a ••• to the ninth power." 
 
 We conclude, therefore, that : 
 
 41. The exponent of any factor in a product is the sum 
 of the exponents of that factor in the multiplicand and 
 multiplier. 
 
 The difference between a coefficient of a letter and an 
 exponent of a letter must be carefully noted. A numeri- 
 cal illustration will emphasize the difference : 
 
 5x2 = 2 + 2 + 2 + 2 + 2 = 10. Here 5 is a coefficient. 
 25 = 2x2x2x2x2 = 32. Here 5 is an exponent. 
 
 The difference is obvious. 
 
 The principle is the same no matter how many literal 
 factors occur in a product. 
 
 Thus : a%^ x a%^ = a%\ a%^c^ x a^h^c"^ = a^Wc^- 
 
 x^yz^ X xyH'^ = x^y^. xyz^ x xyz^ = x^yh^. 
 
 Note that when no exponent is written on a given letter, the 
 exponent is understood to be " 1." In common practice a^ is read 
 "a square"; a^ is read "a cube." Exponents of parentheses have 
 the same meaning. Thus, (a + &)3 = (a + &)(a + b)(a + &). 
 
 I. MULTIPLICATION OF A MONOMIAL BY A MONOMIAL 
 
 The following illustrates the general process : 
 Ex. 1. Multiply 4 a%^ by 3 aPc^ 
 
 ^^2^3 Explanation. We first determine the sign 
 
 3 ^^2^2 of the result. Since the signs of both monomials 
 
 • are like, the sign of the product is positive. We 
 
 • 12 a%^c^ Result. ^y^^^ determine the coefficient, which will be 
 
40 
 
 MULTIPLICATION 
 
 3 X 4 = 12. Then taking the literal factors one at a time, "we obtain 
 for the a factor, „ ,^- 
 
 ax a^ = a^-^^ = a% 
 b^xb^- 62+3 = b^ 
 
 ,2 
 
 for the b factor, 
 
 for the c factor, c^ (since this factor, occurring in the mul- 
 
 tiplier only, is not changed in multiplying). 
 
 Combining the sign, the coefficient, and the several powers of the 
 given factors, we have the product, 
 
 + 12 a%^c^ 
 
 It is good practice to determine the several parts of a 
 product in this order : 
 
 1. The sign of the product. 
 
 2. The coefficient of the product. 
 
 3. The exponents of the literal factors. 
 
 Exercise 8 
 
 Find the products of the following : 
 
 1. 
 
 2. 
 
 3. 4. 
 
 5. 6. 
 
 7. 
 
 2a 
 
 ba 
 
 ^x -4a 
 
 bm —5x 
 
 -Sx 
 
 3 
 
 -3 
 
 2x a 
 
 -m -3 
 
 20 X 
 
 8. 
 
 
 9. 10. 
 
 11. 
 
 12. 
 
 -5,y 
 
 
 — 3 a52 4 TY^n 
 
 5a^y 
 
 -%xf 
 
 -14^ 
 
 
 2a% - 3 mn 
 
 -xy 
 15. 
 
 2xz 
 
 13. 
 
 14. 
 
 16. 
 
 bxyz^ 
 
 
 — 6 rri^xy 
 
 — 3 mnx 
 
 4.yz^ 
 
 -Zy 
 
 
 4 c^:i^y 
 
 — 6 nxy 
 
 -12xz^ 
 
MULTIPLICATION OF POLYNOMIALS 41 
 
 17. Sab by 2ac. 23. —mVi/^hy—3mV^, 
 
 18. 17aho by - S aho. 24. -lOa^A by 3 a253^. 
 
 19. ^253^ by ^252^, 25. 3 a^Jcc? by 12 5c%. 
 
 20. 4 3^1/ by — 7?'y^^. 26. — 11 cmn^y by 5 mVa^. 
 
 21. 4w^i by —Smny. 27. — ISaJ^^^^ljy 5^^^?^, 
 
 22. 12 a2;^3 i^y _2m%. 28. 15am7i2yby — Sam^a;^^. 
 
 II. MULTIPLICATION OF A POLYNOMIAL BY A POLYNOMIAL 
 
 To illustrate the general principles of multiplication 
 when either multiplier or multiplicand, or both, are made 
 up of several terms, examine the following : 
 
 If, arithmetically, 12 is to be multiplied by 4, we obtain 
 
 12 
 _4 
 
 48 
 
 Now, if we consider the 12 as made up of a ten and two units, that 
 is, two terms, we may make our multiplication in this way ; 
 
 10 + 2 
 4 
 
 40 + 8 = 48 
 
 In the second multiplication the separation of the multiplicand 
 produced two products, whose sum is the same as the product of a 
 multiplication in the usual form. Clearly, therefore, we have only 
 changed the form of the arithmetical process, and the result is unchanged. 
 In like manner : 
 
 (z+ 7 
 4 
 
 4m -10 
 3m 
 12 m2 - 30 m 
 
 5 a6 +15 ac 
 3a 
 loa2& + 45a2c 
 
 lOxy -3 
 - 3x 
 
 4a + 28 
 
 - 30 a:2y + 9 a; 
 
 Each term of the multiplicand is separately multiplied by the 
 multiplier. 
 
By Separation 
 
 10 + 3 
 
 10 + 2 
 
 100 + 30 
 
 + 20 + 6 
 
 42 MULTIPLICATION 
 
 Suppose, again, that we multiply 13 by 12. 
 
 By^ the Arithmetical Process 
 
 13 
 
 12 
 
 26 
 13 
 156 100 + 50 + 6 = 156 
 
 In the separated process the result is the same as in the first pro- 
 cess. The product of (10 + 3) by 10 is added to the product of 
 (10 + 3) by 2. The only new step is the arrangement. By beginning 
 at the left we have brought the hundreds, tens, and units into proper 
 order for the addition of the partial products. 
 
 And here, as in the simpler case above. 
 
 Each term of the multiplicand is separately multiplied hy 
 each term of the multiplier. 
 
 The partial products are added as in ordinary multipli- 
 cation. 
 
 Applying the same principle to expressions containing literal terms, 
 we have : 
 
 a+4 m — 5 x +7 
 
 a+3 m-3 a;-4 
 
 a2 + 4a m^ - 5m. x^ + Ta; 
 
 + 3 g + 12 - 3 m + 15 - 4 a: - 28 
 
 a2 + 7 a + 12 m^ - 8 w + 15 x^ + 3 x - 28 
 
 Explanation of the First Multiplication 
 
 For the first partial product : +ax + a = + a2, +ax+4 = + 4a. 
 
 Hence, the first partial product, d^ -^ 4:a. 
 
 For the second partial product : +3x+a= + 3a, +3x+4= + 12. 
 
 Hence, the second partial product, + 3 a + 12. 
 
 The first term of the second partial product is written under the 
 second term of the multiplier, thus bringing like powers of a in the 
 same column. The addition of the partial products gives the result, 
 a2 + 7 a + 12. 
 
MULTIPLICATION OF POLYNOMIALS 43 
 
 In this multiplication the signs were all plus because of like signs 
 in every multiplication of coefficients. In tlie other two examples 
 both plus and minus signs occur, and the solution of each should be 
 carefully gone over before the student attempts to apply the principles 
 involved. 
 
 42. Arrangement of the Terms of Polynomials. In mul- 
 tiplying polynomials it is convenient to arrange the terms 
 of both multiplicand and multiplier so that like terms of 
 the partial products shall fall in the same column. This 
 arrangement is made by writing both expressions in ac- 
 cordance with the powers of some letter occurring in each 
 of them. 
 
 When the powers of a letter in an expression increase 
 from left to right, as in 
 
 a + a^ + a^-ha^ + a^, 
 
 the expression is said to be arranged in the ascending 
 powers of a. 
 
 When, on the other hand, the powers of the selected 
 letter decrease from left to right, as in 
 
 a"^ + a5 + aS + a2 4- a, 
 
 the expression is said to be arranged in the descending 
 powers of a. 
 
 43. The Degree of an Algebraic Expression. That par- 
 ticular term whose power is the highest of any in a given 
 expression determines the degree of the expression. Thus : 
 
 a^ + a + 1 is an expression of the second degree. 
 x^ — x^ — x^ is an expression of the fourth degree. 
 x'^y — x^y^ + 7 is an expression of the fifth degree, since its highest 
 power is a product of a first and a fourth power. 
 
44 MULTIPLICATION 
 
 A numerical factor in a term is not considered in naming 
 its degree. 
 
 From the principles of exponents, Arte 41, the degree 
 of the product of two algebraic expressions is equal to the 
 sum of the degrees of the given expressions. 
 
 We will now state the general process for multiplication 
 of polynomials. 
 
 44. To multiply a polynomial by a polynomial. 
 
 Arrange the terms of each polynomial according to the 
 ascending or descending powers of the same letter. 
 
 Multiply all the terms of the multiplicand hy each term of 
 the multiplier^ observing the principles that govern the signs. 
 
 Add the partial products thus formed. 
 
 Exercise 9 
 
 
 Find the products of the following : 
 
 1. 2. 3. 
 
 a + 55 2a2_3a bah-^c 
 a 4ta — 2a 
 
 4. 
 
 a2_2a + 3 
 a2 
 
 5. 3 a + 17 by 2 a. 
 
 6. VI ah -10 by 3a5. 
 
 7. 8 a^m — 3 a2^ by 5 amn, 
 
 8. 16 xy —10 xz—1 yz by — 2 xyz. 
 
 9. _l0a8_3^2+2a-4 by -6a2. 
 10. a8-3a26 + 3a62-53 by -a%\ 
 
MULTIPLICATION OF POLYNOMIALS 46 
 
 11. 
 
 12. 
 
 
 13. 
 
 2a + 3 5a2 + 
 
 7 
 
 
 3a + 4a: 
 
 3a - 7 2a2- 
 
 11 
 
 
 3a - 4a: 
 
 6a2+ 9a ■ 10a* + 14 
 
 a2 
 
 9 a2 + 12 ax 
 
 -14a -21 
 
 ■55 
 
 ab- 
 
 77 - 12 ax - 16 a;2 
 
 6a2- 5a-21 10 a* - 
 
 -41 
 
 as- 
 
 77 9 a2 - 16 x^ 
 
 14. 2a + S hj a + 1. 
 
 
 ia. 
 
 2m- S by Sm + d, 
 
 15. Sx-2 by 2a; + l. 
 
 
 20. 
 
 4^/2 + 5 by 5^2_i2. 
 
 16. 2a:+3 by ir-3. 
 
 
 21. 
 
 5ah — c by 4 a6 — 3 <?. 
 
 17. 42;-l by 2 a; + 3. 
 
 
 22. 
 
 5a;3_l by 11 a;^ - 2. 
 
 18. 3a- 7 by 2a- 6. 
 
 
 23. 
 
 Sab^-2hc by 2ah'^-5hc. 
 
 24. 16a^-10 by 5a^ + ll. 
 
 25. 4 mwa;2 _ 3 ji^^ by 2 mwa;^ + 5 tio;^. 
 
 26. a3 + 3a24-3a + l by a^+2a + l. 
 
 a8+3a2+ 3a + 1 
 a2 + 2a + 1 
 
 a6 + 3a*+ 3a8+ a^ 
 
 2a*+ 6a8+ 6a2 + 2a 
 
 a8+ 3aS + 3a+l 
 
 o5 + 5 a* + 10 a8 + 10 a2 + 5 a + 1 
 
 27. a2 + 2a + l by a2 4-2a + l. 
 
 28. w3-3m24- 3m-l by m^-2m-^l. 
 
 29. 2ic2+3a: + l by a^-x + l. 
 
 30. 5^3_3^2_|.2y_4 by «/2_4y + 2. 
 
 31. a3 - a25 + a62 _ 53 by a24-^^' + ^'2. 
 
 32. m^ + m^ + m^ + m + l by w — 1. 
 
46 MULTIPLICATION 
 
 33. 27 -{-lSa + 12a^ + Sa^ hj S-2a. 
 
 34. l-2x-h4:x'^-Sx^-\-163^ by l + 2rc. 
 
 35. 2a-{- a^ -Sa^ -1 hy 2- a + a^, ■ 
 
 First arrange the terms so as to have the same order in both mul- 
 tiplicand and m.ultiplier. In descending order we write : 
 
 a^-Sa'^-\-2a-\ 
 a^- g +2 
 
 (The student will complete the multiplication.) 
 
 36. a + l — a^ + a^ hy a-'2-\-a^. 
 
 37. a^-x^-\-6-Sa^-x by x^-{-4:-^a^-2x. 
 
 38. 'lO-5x-\-Sx^-x^ by x - 5x^- 2a^ -{-a^. 
 
 39. 4a2_ 8^3- 3^ + 16 4- a* by 2a- 8-6^2 + a3. 
 
 40. a + b + 2 by a + b-2. 
 
 a + 6 + 2 
 a + b-2 
 
 a2 + a& + 2 a 
 
 + a6 + 62 + 2 6 
 
 -2a -26-4 
 
 a2 + 2 a6 + 6^ - 4 
 
 41. a2 _|_ 52 _^ ^ _,_ 2 «5 _ ^c _ j^ by a + b-\-c. 
 
 First arrange terms according to the descending powers of a. 
 
 a^-h 2ab- ac+ W- bc-{- c^ 
 
 a + b+ c 
 
 a8 + 2 a% - a^c + ab^ - abc + ac^ 
 
 + a% +2a62- abc +¥-bh+hc'^ 
 
 +aV -f 2 q6c - ac^ + 6^0 - bc^ + c^ 
 
 a« + 3a26 + 3 n62 + 6» +0^ 
 
MULTIPLICATION OF POLYNOMIALS 47 
 
 42. 7W + w -t- 4 by w + 7^ — 4. 
 
 43. X -\-y -{-z by x-i- ^ + z. 
 
 44. m^+2 mn -^n^ -\-x by m^ + 2 w/i + w^ — a?. 
 
 45. x^ -\- y^ -\- a^ -\- 2 xy — ax — ay by a; + 3/ + a. 
 
 46. aH-6 + (?H-c?bya — 6— <? — c?. 
 
 Multiplication of polynomials is frequently indicated 
 by the use of parentheses. In such indicated operations 
 any two of the given expressions are multiplied and their 
 product is multiplied by the third, etc. An exponent upon 
 a parenthesis indicates the multiplication of the given 
 expression by itself as many times as the number of the 
 exponent. 
 
 47. (a -2)3. 48. (a + 5)(a4-6)(a-3). 
 
 a -2 
 a -2 
 
 I- 
 
 a2-2a 
 
 
 
 -2a + 
 
 4 
 
 
 a2-4a + 
 
 4 
 
 
 a -2 
 
 
 
 a8-4a2 + 
 
 4a 
 
 
 -2a2 + 
 
 8a- 
 
 -8 
 
 a + 5 
 
 
 a + 6 
 
 
 a2-i- 5 a 
 
 
 + 6a +30 
 
 
 a2+lla +30 
 
 
 a - 3 
 
 
 a8 + 11 a2 + 30 a 
 
 
 - 3a2-33a- 
 
 -90 
 
 a8_6a2 + 12a-8 0^+ 8a2- 3a-90 
 
 49. (a 4- 2)2. 51. (2 m- 3^1)3. 53. (Za^-bxy^. 
 
 50. (a-3:r)2. 52. (4^2-5 2^)2. 54. (b aV^ -1 ahcf. 
 
 55. (m2 -f- m — l)(m2 — m — l)(w2— 1). 
 
 56. (^x'^J^x + V)(x^-x + V)Qi^-x'^-\-V), 
 
 57. (^+-5)(a?-3)(a;-5)(:r + 3). 
 
48 MULTIPLICATION 
 
 When the terms of the multiplicand and multiplier are 
 dissimilar, the partial products cannot be added. In such 
 cases the additions of the partial products can only be 
 indicated. 
 
 58. (a + a;)(a + ^). 
 
 a + X 
 a -h y 
 a^ + ax 
 
 + ay-\-xy 
 a^ -^ax^ ay + xy 
 
 59. (a + 5)(a + c). 63. (^a + 7n)(h-{-n). 
 
 60. (2rz; + m)(3a^+2w). 64. (2a + 3 J)(3c? - 4(^). 
 
 61. (ah-c){ah-d). 65. (a^ ^-V)(jj(P'- x). 
 
 62. (2m-\-n')(bm — y'), 66. (a-\-h-{-V)(^a — x), 
 
 Perform indicated operations and simplify : 
 
 67. 3a2-(a-2)(3a + l)-5a. 
 
 68. (5 -1)2 +(6 + 1)2 -2(52+1). 
 
 69. (4a-l)(a + 2)-(2a-3)(2a + 5). 
 
 70. (a + 3)2-2a(a + 4) + (a-2)(a + 4). 
 
 71. rc(a;-l)+2:(2: + l)(rc+2)-ic2(a; + 4). 
 
 72. m(jn-\-n—l)—n(m—n + l) — m(m--V), 
 
 73. c(^ + l)_[(e-l)2_(l-e)]. 
 
 74. (a_5_l)2 + a(5 + l-a) + 5(a-6-l). 
 
CHAPTER V 
 DIVISION. REVIEW 
 
 45. Division is the process of finding one of two factors 
 when their product and the other factor are given. 
 
 The Dividend is the given product. 
 
 The Divisor is the given factor. 
 
 The Quotient is the factor to be found. 
 
 Division is the inverse of multiplication, and upon this 
 fact we base the principles by which we obtain the signs, 
 coefficients, and exponents of our quotients. 
 
 SIGNS IN DIVISION 
 
 « 
 
 From Arts. 35 and 36 we obtain : 
 
 (+a)x(+^)= + «^ +ah-^-\-a = + b (1) 
 
 Hence, 
 (+a)x(-^>)=-a5 inversely, -ab-t- + a=-h (2) 
 
 (_a)x(+^>)=-a5 ^y -ah-^-a= + h (3) 
 
 division, 
 (-a)x(-5)= + a6 -{-ah^ —a=—h (4) 
 
 Therefore, we conclude : 
 From (1) and (3): 
 
 46. Like signs in division give + . 
 
 From (2) and (4): 
 
 47. Unlike signs in division give — • 
 
 p. H. S. FIRST TEAR ALG. 4 49 
 
50 DIVISION. BE VIEW 
 
 COEFFICIENTS IN DIVISION 
 
 Since two factors multiplied together give a product, it 
 follows that this product divided by either of the factors 
 will give the other factor. Hence : 
 
 48. The coefficient of a quotient is obtained by dividing 
 the coefficient of the dividend by the coefficient of the divisor, 
 
 EXPONENTS IN DIVISION 
 
 By Art. 41, 
 
 This might be written, 
 
 aaaa x aa = aaaaaa. 
 
 Now, if the product of six factors be divided by the group of two 
 factors, we obtain by cancellation. 
 
 Or, subtracting the number of a factors in the divisor from the 
 number of a factors in the dividend, we obtain the number of a factors 
 in the quotient. Hence : 
 
 Similarly, a^^a'^ = a^-^ = a*. 
 
 x^^x* = x^-^ = x^. 
 
 From which we state the general principle for the expo- 
 nents of quotients : 
 
 49. The exponent of a factor in a quotient is the differ- 
 ence between the exponents of that factor in the dividend and 
 divisor. 
 
DIVISION OF MONOMIALS 51 
 
 It is good practice to obtain the several parts of a quo- 
 tient in, this order ; 
 
 1. The sign of the quotient. 
 
 2. The coefficient of the quotient. 
 
 3. The exponents of the literal factors. 
 
 I. DIVISION OF A MONOMIAL BY A MONOMIAL 
 
 The general process of dividing a monomial by a mono- 
 mial is as follows : 
 
 Ex. 1. Divide 10 a^ by 5 aK 
 
 5 a8 )10a5 Explanation. ,(1) Since the signs of the coefficients 
 2 a^ are like, the sign of the quotient will be + . (2) Divid- 
 ing the given coefficients, we have 10-4-5 = 2, the coefficient 
 of the quotient. (3) For the exponent of the literal factor, a^ -^ a^= 
 a^-s = a^. Hence, the complete quotient, 2 a^. 
 
 Ex. 2. Divide 36 a^a^ by - 12 aV. 
 
 — 12 a%8)36 a^x^ Explanation. (1) Since the signs of the coeffi- 
 3 a^x cients are unlike, the sign of the quotient will be 
 — . (2) Dividing the given coefficients, we have 
 36 -4- 12 = 3, the coefficient of the quotient. (3) For the exponents of 
 the literal factors, a^ ^ a^ = a^ and x^ -^ x^ = x. Hence, the complete 
 quotient, — 3 a^x. From these illustrations we state the general pro- 
 cess for the division of monomials. 
 
 50. To divide a monomial by a monomial : 
 
 Determine the sign of the quotient^ + or — according as 
 the signs of the dividend and divisor are like or unlike. 
 
 Divide the coefficient of the dividend hy the coefficient of 
 the divisor and affix the sign obtained. 
 
 Annex the literal factors, giving to each an exponent equal 
 to the difference between its exponent in the dividend and its 
 exponent in the divisor. 
 
52 DIVISION. REVIEW 
 
 E2:ercise 10 
 
 Obtain the quotients in the following : 
 
 1. 2. 3. 4. 5. 
 
 6. 7. 8. 
 
 5 a)15a3 -8^2 )24^8 Ibxy)-^^ 
 
 9. 10. 
 
 -8^25 ) -16^352 Zm^y )-'nrY&y^ 
 
 11. 12. 13. 14. 
 
 5:i;2^ -18a;2«/2a -25m2 -3a26c?(^ 
 
 II. DIVISION OP A POLYNOMIAL BY A MONOMIAL 
 
 A simple numerical example will illustrate the principle 
 of division upon which we base our work when the divi- 
 dend is a polynomial and the divisor a monomial. 
 
 Suppose we divide 48 by 4. 
 
 ^4? or, by separation, ^M + l ^„ 
 12 ' J' ^ 10 + 2 = 12. 
 
 Simil rl ^ a2^15a^-_12af ~ 5 a )10a - 15a2+ 20 gS 
 
 imiary, 5a2- 4a - 2 + 3 a - 4 a2 
 
 Hence, the general process. 
 
 51. To divide a polynomial by a monomial : 
 
 Divide each term of the dividend hy the divisor and con- 
 nect the several quotients hy the proper signs. 
 
DIVISION OF POLYNOMIALS 53 
 
 Exercise 11 
 
 Obtain the quotients in the following : 
 
 1. 2. 3. 
 
 a)a±5ab 4 a )Sa^-12a^ - 2 a )10 a^ + 6 ac 
 
 4. 5. 
 
 6. 
 
 7. 7a3_21a2 by la. 
 
 8. 4 m^o: — 12 wa;2 by 4 mx, 
 
 9. — 25 w%2 + 20 mW — 15 mhi^ by 5 wV. 
 
 10. 5 a%- 10 aH^ + 15 ah^ hy -bah. 
 
 11. -18^6-24^5 4-30m*-36m34-42w2 by -6w. 
 
 III. DIVISION OP A POLYNOMIAL BY A POLYNOMIAL 
 
 When both dividend and divisor are polynomials, the 
 principles already used are somewhat extended. We will 
 illustrate by a numerical example. Suppose we divide 
 156 by 12. 
 
 12)156(13 , . 100 + 50 + 6(10 + 2 
 
 j2 or, separating both dm- loo + 20 10 + 3 = 13 
 
 ~^ dend and divisor as on , e 
 
 og in former divisions, SO 4- 6 
 
 In the second process the divisor is written at the right of the divi- 
 dend. The process in its successive steps is as follows : 
 
 First. 100 -f- 10 = 10, the first term of the quotient. 
 
64 DIVISION. REVIEW 
 
 Second. Multiplying (10 + 2) by 10, we obtain (100 + 20), which 
 product is subtracted from the given dividend. The remainder, 
 (30 + 6), is the new dividend. 
 
 Third. 30 -=- 10 = 3, the second term of the quotient. Repeating 
 the process of multiplication and subtraction, the remainder is zero. 
 Hence, the quotient is (10 + 3). 
 
 Similarly, Dividend a^ -f 5 a + 6 ( a + 2 Divisor. 
 a2 + 2 a a + 3 Quotient. 
 
 + 3a+ 6 
 + 3a + 6 
 
 First, a^ -^ a = a, the first term of the quotient. 
 
 Second. Multiplying (a + 2) by a, we obtain (a^ + 2a), which 
 product is subtracted from the given dividend. The remainder, 
 (3 a + 6), is the new dividend. 
 
 Third. 3 a -^ a = 3, the second term of the quotient. 
 
 Fourth. Multiplying (a + 2) by 3, we obtain (3 a + 6), which, 
 subtracted from the last dividend, gives a remainder of zero, and 
 the division is complete. 
 
 And, in a more difficult case, the process is identical : 
 
 Dividend a* + a^ _ 5 ^^2 _,. 13 ^ _ 6 ( gg - 2 a + 3 Divisor. 
 a4 _ 2 a8 + 3 a2 a^ + 3 a ^ 2 Quotient. 
 
 + 3a8 
 + 3a8 
 
 -8a2 
 -6a2 
 
 + 13 a 
 + 9a 
 
 
 
 -2a2 
 -2a2 
 
 
 4a- 
 4a- 
 
 -6 
 -6 
 
 Hence, the general process for division of polynomials. 
 
 52. To divide a polynomial by a polynomial : 
 
 Arrange the dividend and the divisor according to the 
 ascending or descending powers of some common letter. 
 
 Divide the first term of the dividend hy the first term of 
 the divisor. 
 
DIVISION OF POLYNOMIALS 55 
 
 Write the result for the first term of the quotient. 
 
 Multiply all the terms of the divisor hy the first term of 
 the quotient and subtract the product from the dividend. 
 
 The remainder^ if any, is a new dividend, and the opera- 
 tion is repeated with successive dividends until the remainder 
 becomes zero. 
 
 Exercise 12 
 
 Divide : 
 
 1. 2m2+llm-f5 by 2^ + 1. 
 
 2. 2x^-\-x-Q by 2x-Z. 
 
 3. 3 c2 + 5 (? - 12 by (? + 3. 
 
 4. 3m2 + 14w + 15 by 3m + 5. 
 
 5. 16 + 8a: + rz;2 by 4 + 2;. 
 
 6. x^-^x-12 by x-%. 
 
 7. 7y2 + i23«/-54 by 7?/ -3. 
 
 8. ?>a%''--\-l\ab-20 by 3a6-4. 
 
 9. 21 m%* + 26 m2/?,2 _ 15 by 7mV-S. 
 
 10. 10a* + 23a2^-21a;* by lOa^-la^. 
 
 11. a^-\-Sa^-\-Sx-^l by x+1. 
 
 12. m^ + m2 — 3 m — 6 by m — 2. 
 
 13. a3 - 6 a2 + 8 a - 15 by a - 5. 
 
 14. mV — ■m^/i* — 7 ^2^2 + 3 by mV — 3. 
 
 15. 1 + a - 3 ^2 4- 9 ^3 by 1 + 3 a. 
 
 16. a* + 2 a3 + 2 a2 + a - 6 by a2 + a - 2. 
 
 17. 2x^-a^-Sx^-\-5x-2 by 22;2-3a; + 2. 
 
56 DIVISION. REVIEW 
 
 18. *9a+10-22a2 + 2a* + a3 by 2a2-l-a. 
 
 19. 9a^-6x+llx^+6a^-10 by -2 + 32^. 
 
 20. 2 m^ - 7 + 8 w + 8 w* + 9 m2 by - 1 + 2 m^ + m. 
 
 21. 6a^-{-2a^-5a^-6-\-ba-lla^ by 2 + 2a2_a. 
 
 22. - a* + a2 _ 2 a3 - 1 4- a^ + 2 a^ by a2 + 1 - a + a3. 
 
 23. a;2- y\xj-j_ 
 x^ — xy X -\- y 
 + xy -y^ 
 
 -\-xy -y'^ 
 
 24. x8 - ?/8(x 
 
 a;8 — x^y x^ + xy -\- y^ 
 
 + a:V 
 
 + x'^y - xy^ 
 
 25. a* — m\a -\- m 
 
 a* + ahn 
 
 a8- 
 
 a2m4- 
 
 am^- 
 
 — a^m 
 
 
 
 
 — ahn 
 
 -ahn^ 
 
 
 
 
 + ahn^ 
 
 
 
 + ahn^ + am* 
 
 
 
 
 — am^ ■ 
 
 -m* 
 
 
 
 — am^ ■ 
 
 -m* 
 
 26. a2 _ 52 ijy ^ ^ 5, 28. a^ - 53 by a - 6. 
 
 27. a^ — h^ by a— h, 29. a* — 5* by a + ^. 
 * Arrange terms of dividend and divisor in descending order. 
 
DIVISION OF POL TN0MIAL8 57 
 
 30. a3 - 8 by a - 2. 32. 8 a^ + 27 5^ by 2 a + 3 h. 
 
 31. m^ + 32 by m + 2. 33. 16 m* - 1 by 2 w + 1. 
 
 34. 125 a656 + 27^3 by 5^262 + 3^. 
 
 35. 81 a:*/ - 16 ^4 by 3 a:?/ + 2 2. 
 
 36. a8-3a&c+ S^ + c8 (a + 5+ c 
 
 a3 + ^2^ ^ a% (a2 _ a6 - ac + &2 _ &c + c« 
 
 - a26-a2c.-3a6c+ b^-\- c^ 
 
 - a%-ab^- abc 
 
 -ah + 
 -ah 
 
 ab^- 
 
 2abc+ b^ + c^ 
 abc - ac'^ 
 
 
 + 
 + 
 
 ab^- 
 ab^ 
 
 abc + ac"^ -^^ b^ + c^ 
 + b^ + bh 
 
 
 : 
 
 abc + ac"^ - 
 abc — 
 
 bh+ c« 
 
 62c - &c2 
 
 
 
 ^ac"^ 
 + ac^ 
 
 
 In this division the student will note that in all new dividends the 
 letter a is given precedence, with powers of b and c following in their 
 original order. 
 
 37. a^ + ^ah + y^-c^ hj a-\-h + c, 
 
 38. a;2 — 4 a; H- 4 — ?/2 by a: — 2 — y. 
 
 39. c^ + 2cd + d?-4:x^ hy c-\-d—2x, 
 
 40. 4 2^ + 12 a: + 9 - ^2 by 2 a; + 3 + ^. 
 
 41. ^2 + &2 4. c2 + 2 ^6 + 2 a(? + 2 5c? by a 4- J + <?. 
 
 42. ofi -{- y^ -{■ z^ — 2> xyz hj x -{- y + z. 
 
 43. w^ — 2 m^ + 1 by w2 _ 2 m + 1. 
 
58 DIVISION, BEVIEW 
 
 GENERAL REVIEW — GROUP II 
 
 The following group of principles presented in an 
 abbreviated form covers the new work since the preced- 
 ing group. It is suggested that the student review again 
 the principles covered in the first group, for the elementary 
 processes that have been thus far presented cannot be too 
 well fixed before more advanced work is attempted. 
 
 SUBTRACTION 
 
 The subtrahend^ with all signs changed^ is added to the 
 minuend. 
 
 MULTIPLICATION 
 
 Signs. Like signs in multiplicand and multiplier produce 
 a + product. Unlike signs in multiplicand and multiplier 
 produce a — product. 
 
 Exponents. The exponent of any factor in a product is 
 the sum of the exponents of that factor in the multiplicand 
 and multiplier. 
 
 Arrangement of Terms. Multiplicand and multiplier 
 should he arranged in the same order^ both ascending or both 
 descending. 
 
 DIVISION 
 
 Signs. Like signs in dividend and divisor produce a + 
 quotient. Unlike signs in dividend and divisor produce 
 a — quotient. 
 
 Exponents. The exponent of a factor iyi a quotient is the 
 difference between the exponents of that factor in the dividend 
 and the divisor. 
 
 Arrangement of Terms. Dividend and divisor should 
 be arranged in the same order^ both ascending or both de- 
 scending. 
 
REVIEW 59 
 
 Exercise 13 
 REVIEW 
 
 1. Add (a + 2)(a + 3) and (a + 1)2. 
 
 2. What is the sum of 3(a2 _ ^ + 2) - 2(a - 3 + a2) - 
 («-!)(« + 3)? 
 
 3. Subtract (a + 2)2 from (a + 1)^. 
 
 4. Subtract (a + 2)(a — 3) from a^ 4. ^ _ 5. 
 
 5. By how much does 4(a — 3) — 3(5 + 1) exceed 
 2(a + 6 - 2) ? 
 
 6. Subtract (a + 3) from (a — 2)2 and multiply the 
 result by (a— 1). 
 
 7. From (a2+ 3)2 take (a+ 3)3. 
 
 8. By how much does (a+ 6 + (?)2 exceed 2(aJ + «c4-5(?)? 
 
 9. What must be added to (a + a;+ 1)^ that the sum 
 may be (^a-\-x— 1)2 ? 
 
 10. Subtract (a + 3) from (a2 — 2 a) and from the re- 
 mainder take «2 _ 4 ^ _ 3. 
 
 11. Prove that w(a — rr) + a(rc — m) + a;(m — a) = 0. 
 
 12. Multiply the sum of a^ -j. 2 and a^-\-a — l by d^ — 1. 
 
 13. Multiply the sum of (a + 2) and («2 _ 3 ^) by 
 [«(a + l)_(2a-2)]. 
 
 14. Simplify a + 5 - [(3 a - 5) + (a - 3 6) - a] + [a - 
 (5 — 2 a)] and multiply the result by a — 4 5. 
 
 15. Simplify (2; + l)(a:+3)(rr + 4)-(rr-l)(a;-3)(a:-4). 
 
 16. Simplify (c + l)c- [(c- 1)2- (1 - c)]. 
 
 17. Simplify (a + a;) (a + y) — ((X — a:) (<^ — y). 
 
60 DIVISION. REVIEW 
 
 18. Divide a^ + 3 ^2 + 3 a + 1 by (a + 1)(« + 1). 
 
 19. Find the continued product of (2a + 3)(3a— 2) 
 (a-1). 
 
 20. Simplify 3a-[55-(2aH-J35-(a-[5-^^=^2T])l)], 
 and multiply the result by (a + 3 6). 
 
 21. Find the dividend if the quotient is a — 2 and the 
 divisor a^ — a-\-Q, 
 
 22. The product of two expressions is a* + 11 a^— 12 a 
 — 5 a^ + 6, and one of them is 3 — 3 a + a^. Find the 
 other expression. 
 
 23. The difference between two expressions is a^—^x 
 + 2, and the greater expression is 4iX^ — 2x-{-l. Find 
 the smaller expression. 
 
 24. Divide the product of (a^— 4)(a2— 9) by a^-f-a— 6. 
 
 25. Subtract a^— 4a; from (x^-\-x-{-l')(x-\-2~) and divide 
 the result by (3 a; + 1). 
 
 26. Add a^-\-a-2> to the quotient of Qa^-\-^a^-a-\-12') 
 ^(a + 4). 
 
 27. Divide [(a2 - 2 a - l)(a - 1) + 2 a2 _ 2 a] by [(a 
 + 2)(a + l)-(a2+2a + 3)]. 
 
 28. What is the sum of [(a + 1)^— «(a + 3)] and 
 
 29. Subtract a— [2a — ( — a— l-f-a)] from 1+ { — a — 
 [-(-2a + l)]i. 
 
 30. Add the quotients of [(a^ — 1) -?- (^ — 1)] and [(a^ 
 _2a; + l)-j-(a;-l)]. 
 
CHAPTER VI 
 THE SIMPLE EQUATION 
 
 53. An Equation is an expression of equality between 
 two quantities. 
 
 Illustration. If five books of exactly the same size and 
 kind weigh all together ten pounds, we have a known and 
 an unknown quantity. For, 
 
 The weight of the whole number is known. 
 
 The weight of one book is unknown. 
 
 But the conditions given permit us to write the fact 
 
 that, in weight, i- ^ ^ ^r^ i 
 
 ^ o books = 10 pounds. 
 
 Or, if " h " stand for the weight of one book, we abbre- 
 viate thus : r r -if. 
 
 = lU. 
 
 And we have an equation between two quantities, — a 
 certain number of books on one hand, and a certain number 
 of pounds on the other. 
 
 Now if 5 books weigh ten pounds, 1 book weighs (10 h- 5) 
 pounds, or 2 pounds. Or, from our equation, 
 
 5 = 2. 
 Hence, our conclusion. 
 
 54. From an equation we obtain the value of an unknown 
 quantity/ in terms of a given known quantity/. 
 
 61 
 
62 THE SIMPLE EQUATION 
 
 55. The expression obtained for the value of an unknown 
 quantity/ in an equation is called a Moot. 
 
 Thus : 2 is a root of the equation 5 & = 10. 
 
 56. The Left Member of an equation is the expression at 
 the left of the sign of equality. The Right Member is the 
 expression at the right of the sign of equality. 
 
 57. A Simple Equation is an equation which, when re- 
 duced to its simplest form, has no power of the unknown 
 quantity higher than the first power. The following are 
 examples of simple equations : 
 
 5 a? = 15 is an equation in x. 
 7 y = 35 is an equation in y. 
 5 2 = 40 is an equation in z. 
 
 3 a; + 2 = 2 a; -i- 7 is an equation in x, but is not reduced. 
 (x + 5)2 = a:^ -f 7 a; + 6 is an equation in x that, when simplified, 
 will contain only the first power of x. 
 
 The final letters of the alphabet are commonly used for 
 unknown quantities in equations. 
 
 58. The principles upon which the solution of equa- 
 tions depends are the truths called 
 
 AXIOMS 
 
 1. If equals are added to equals^ the sums are equal. 
 
 2. If equals are subtracted from equals, the remainders 
 are equal. 
 
 3. If equals are multiplied by equals, the products are 
 equal. 
 
 4. If equals are divided by equals, the quotients are equal. 
 
 Applications of these axioms to processes with equations 
 may be illustrated by the following numerical operations : 
 
TRANSPOSITION OF TERMS IN EQUATIONS 63 
 
 12 = 10 + 2 12 = 10 + 2 
 
 By Ax. 1 add 2= 2 By Ax. 2 subtract 2= 2 
 
 14 = 10 + 4 (1) 10= 10 (2) 
 
 12 = 10 + 2 12 = 10 + 2 
 
 By Ax. 3 multiply _2 2 By Ax. 4 divide ^ = \^- + f 
 
 24 = 20 + 4 (3) 6 = 5 + 1 (4) 
 
 From which we conclude : 
 
 From (1) and (2). 77ie same quantity/ may he added to, or 
 
 subtracted from^ both members of an 
 
 equation. 
 From (3) and (4). Both members of an equation may he 
 
 multiplied or divided by the same 
 
 quantity. 
 
 Or, taking the generral equation, A = B. 
 
 From (1) A+C = B+C. From (3) AC = BC. 
 
 From (2) A-C = B-C. From (4) A---C = B^C. 
 
 TRANSPOSITION OF TERMS IN EQUATIONS 
 
 Frequently we are required to find the root of, or solve^ 
 equations in which the known quantity occurs in both 
 members. Thus : 5 a; + 2 = 17 
 
 We want only the x term in the left member, so we will 
 subtract 2 from that member. But if we subtract a num- 
 ber from one member, we must subtract the same number 
 from the other member also. That is, 
 
 5 re + 2 = 17 
 
 By Ax. 2, Subtracting : 2=2 
 
 5 a; = 17 - 2 
 
64 THE SIMPLE EQUATION 
 
 This is the same result that would have followed from 
 transposing the 2 to the right member and changing its 
 sign.. Hence, 
 
 59. A term may he transposed from one member of an 
 equation to the other member by changing its sign from + to 
 — , or from — to -\-, 
 
 We are now ready for the 
 
 GENERAL METHOD FOR SOLVING SIMPLE EQUATIONS 
 
 60. To solve a simple equation : 
 
 Perform all indicated multiplications and remove all 
 parentheses. 
 
 Transpose the terms containing the unknown quantity to 
 the left member^ and all other terms to the right member of 
 the equation. 
 
 Collect the terms in each member. 
 
 Divide both members by the coefficient of the unknown 
 quantity. 
 
 In the solution of equations the following will frequently 
 save much labor: 
 
 (1) The same term with the same sign in both members of 
 an equation may be canceled. 
 
 (2) All the signs of the terms of an equation may be 
 changed without destroying the equality. 
 
 Exercise 14 
 
 Solve ths following equations : 
 1. 5a;=35. 2. 7a;=-56, 3. -4a:=20, 4. -3a:=-27, 
 
METHOD FOR SOLVING SIMPLE EQUATIONS ^6 
 
 5. 3^ = 5, 6. -6a: = 19, 7. 13a:=0, 
 
 x = ^. X— — ^g^. 2; = 0. 
 
 8. 3a; = 18. 14. -18a; = 54. 20. -5^=16. 
 
 9. 7a: = 42. 15. -52:=-20. 21. 3a:=-13. 
 
 10. 5:c=90. 16. -3 a; =-39. 22. -2 2; = -9. 
 
 11. 32; = -18. 17. 42^=1. 23. 3a; = 0. 
 
 12. 62;= -30. 18. 92^=5. 24. 72;= 0. 
 
 13. -3a:=12. 19. 5rr=13. 25. -8a:=0. 
 
 26. Solve 5:^ + 2=17, 27. Solve 7a:- 3 = 3a:+ 13, 
 5x=17-2, 7ar-3a:=13 + 3, 
 
 5a; = 15, 4x= 16, - 
 
 a; = 3. Result. a; = 4. Result. 
 
 28. 32:+ 7 = 19. 37. 3a;+4 = 2a;+5. 
 
 29. 2a:-5 = ll. 38. 62;+ 8 = 5 a:+ H. 
 
 30. 32:-5 = 0. » 39. 82:-3-5a:-ll = 0. 
 
 31. 5a;-8 = -13. 40. 62;- 12 = 3^;- 3. 
 
 32. 3a;-7 = -9. 41. 42^-5=28+^. 
 
 33. 3-22^=6. 42. 17-32^=45-102;. 
 
 34. 5 4- 3 2; = 5. 43. 52^ + 1- 42;=— 2. 
 
 35. 4— 32; = 4. 44. 32^+1 = 62:— 1. 
 
 36. -18 2:4- 5 = -10. 45. -5 2: -10 = -10 + 3 2:. 
 
 F. H. 8. FIRST YEAR ALG. — 6 
 
66 TBE SIMPLE EQUATIOn 
 
 46. Solve 5a;-[2a;-(a^-2a;-l)] = 5. 
 
 5 a; - [2 x - (x - 2 a: + 1)] = 5, 
 
 5a;-[2a;-a: + 2a;-l] =5, 
 bx-2x-\-x-2x + l = b, 
 
 bx-2x-\-x-2x = b-l, 
 2x = i, 
 
 X = 2. Result. 
 
 47. 4:X-(2x-\-l)=15. 
 
 48. (5a:-3)-18 = 3a:-(2-a;). 
 
 49. 12-(3a:+7) = 3-(2:r-5). 
 
 50. 4 a:- [2ri:-a; + l] = 0. 
 
 51. 4a;-5(2; + l)-[2rz;-(a; + l)]=l. 
 
 52. - i2rr-(a; + l)]=13-(2:r-l)-3a;. 
 
 53. Solve (ix-\-S')(2x-5) = 2(x-2y~2(x+l) 
 
 2 x^ + X - 15 = 2(x^ -^ X -\- i)-(2 X + 2) 
 •^^^2 + a: _ 1 5 =>2^2 _ 8 a; + 8 - 2 x - 2 , 
 + a: + 8a: + 2a: = 8-2 + 15, 
 11 a: = 21, 
 
 X = i\. Result. 
 
 54. Cx-l)(x-\-2) = (x-S)(x-2y 
 
 55. (2rr-5)(2^-3) = (a;-l)(2: + l)+a:2. 
 
 56. (x-5y-^S(x-h2y=^(a^-l)-S, 
 
 57. 2(2; - 1)2 - 3(0; - 2)2 = 6 - (a; - 1)2. 
 
 58. 3[2aj-4(3rir2+l)] = lT-4(3a:-l)2. 
 
 59. - [2(a:-4)(2:-5)-(rr-3)(2:-4)] = (5-a:)(a;-2> 
 
 60. (a;-2)3-(2;-l)3+(3a;-2)(a;-2) = 0. 
 
THE SOLUTION OF PROBLEMS 67 
 
 THE SOLUTION OF PROBLEMS 
 
 61. A Problem is a question to be solved. 
 
 In a problem certain conditions are given in which an 
 unknown number is involved. It is the value of this un- 
 known number that we seek. And, assisted by the state- 
 ment of the conditions that relate to that number, we 
 form an equation and obtain its value. 
 
 LITERAL SYMBOLS FOR UNKNOWN QUANTITIES 
 
 62. If a tank holds a certain number of gallons, five 
 tanks of the same kind will hold just five times as many- 
 gallons. That is, 
 
 If 1 tank holds 10 gallons, 5 tanks hold (5 x 10) gallons = 50 gallons. 
 
 If 1 tank holds 20 gallons, 4 tanks hold (4 x 20) gallons = 80 gallons. 
 
 If 1 tank holds x gallons, 5 tanks hold (5 • x) gallons = 5 a: gallons. 
 
 If 1 tank holds y gallons, m tanks hold (m • y) gallons = my gallons. 
 
 Also : 
 
 If A has $100 and B has $50 more than A, 
 
 B has (100 + 50) dollars. 
 If A has X dollars and B has f 50 more than A, 
 
 B has (a; + 50) dollars. 
 If A has X dollars and B has y dollars more than A, 
 
 B has (x + y) dollars. 
 
 Exercise 15 
 Oral 
 
 1. If X denotes a certain number, write an expression 
 for 10 more than x. 
 
 2. If y denotes a certain number, write an expression 
 for 12 less than y. 
 
68 THE SIMPLE EQUATION 
 
 3. If X denotes the number of bushels of apples that a 
 certain barrel will hold, how many bushels of apples will 
 there be in 8 similar barrels ? 
 
 4. John solved x examples and William solved y ex- 
 amples. How many did both together solve ? 
 
 5. A boy had m marbles and lost n of them. How 
 many had he left ? 
 
 6. A boy earned x cents and found twice as many. 
 How many cents had he in all ? 
 
 7. A boy solved a examples and his sister solved five 
 more than that number. How many did she solve ? 
 
 8. John piled x cords of wood and Charles piled 3 
 cords less than John. How many cords did both to- 
 gether pile ? 
 
 9. John solved x examples and William solved the same 
 number less seven. How many examples did William 
 solve ? 
 
 10. Three men together pay a bill. B pays twice as 
 much as A. C pays three times as much as A. How 
 much does each pay if A pays x dollars? 
 
 11. A library chair cost y dollars, a bookcase x dollars, 
 and a table as much as the chair and bookcase together. 
 How much did all together cost ? 
 
 12. A line is 10 inches long and x inches are added to 
 it. What is the increased length ? 
 
 13. From a line x inches long y inches are cut off. 
 How much of the line remains? 
 
THE SOLUTION OF PROBLEMS 69 
 
 14. Three lines of lengths, a, 5, and c, respectively, are 
 placed in one straight line ; and c? inches are cut from 
 the whole. What is the length left ? 
 
 15. The sum of two numbers is 40 and the smaller, 
 number is x. What is the greater number? 
 
 16. The sum of three numbers is 50. One of them is 
 X, another 20. What is the expression for the third 
 number ? 
 
 17. The smaller of two numbers is i/ and the larger is 
 X. What is the difference between them ? 
 
 18. Write three consecutive numbers if the least of the 
 three is x. 
 
 19. Write four consecutive numbers if the least of the 
 four is m. 
 
 20. Write five consecutive numbers, the greatest being a. 
 
 21. Write five consecutive numbers, the middle one of 
 the five being n, 
 
 22. What is the next odd number above m when m it- 
 self is even ? 
 
 23. What is the next odd number above m when m 
 is odd ? 
 
 24. Write the three consecutive odd numbers below c. 
 
 25. Write the five consecutive even numbers above d, 
 d being even. 
 
 26. What is the sum of the three consecutive even 
 numbers next below k^ k being odd ? 
 
 27. Write five consecutive odd numbers so that the 
 middle one shall be m. 
 
70 THE SIMPLE EQUATION 
 
 28. Write the product of three consecutive even num- 
 bers, the middle one being x. 
 
 29. If a man is x years old now, how old will he be in 
 10 .years ? 
 
 30. If a man is y years old now, how ol^ was he twelve 
 years ago ? 
 
 31. A man is twice as old as his son whose age is x 
 years. What is the sum of their ages ? 
 
 32. A man is three times as old as his son and six 
 times as old as his daughter. What is the sum of their 
 ages if the daughter's age is x years ? 
 
 33. How old was a man m years ago if his age at pres- 
 ent is a years ? 
 
 34. A man earns x dollars a year for a period of y years. 
 In that time he has spent z dollars. Write the expression 
 for his savings. 
 
 35. X and y are two numbers and x is the larger of the 
 two. Express the fact that four times the difference of 
 the two is equal to their sum. 
 
 36. William has a wallet containing x dimes. How 
 many cents has he ? 
 
 37. Express the condition that h dimes shall equal m 
 nickels. 
 
 38. A has X dimes and y nickels in his pocket and he 
 spends 10 cents. Write the expression for the amount he 
 has left. 
 
 39. A man travels a miles an hour for a period of m 
 hours. How many miles does he travel in all ? 
 
 40. A man travels m miles in a period of x hours. 
 What is his rate of traveling in miles per hour ? 
 
THE SOLUTION OF WRITTEN PROBLEMS 71 
 
 41. A boy rides x miles in a boat. Then he walks 
 2 miles. Then by train he goes twice as far as he has 
 already traveled. Write the expression for the total 
 number of miles in his journey. 
 
 42. Write the expression for the statement that the 
 square of the sum of a and 6 is 3 less than the square of 
 the difference between 3 m and x, 
 
 SUGGESTIONS FOR THE SOLUTION OF WRITTEN^ PROBLEMS 
 
 From the foregoing oral exercise it will be clear to the 
 student that no rule can be given that will cover all cases 
 of problems. The following suggestions, however, will 
 give a general outline as to the method by which we 
 reach a solution. 
 
 63. In solving a problem : 
 
 1. Study the problem to find that number whose value is 
 required. 
 
 2. Represent this unknown number or quantity by x. 
 
 3. The problem will state certain existing conditions or 
 relations. Express those conditions in terms of x. 
 
 4. Some statement in the problem will furnish a verbal 
 equation. Express this equation algebraically by the aid of 
 your own written statements. 
 
 To aid the beginner a classification of four common 
 types of problems has been made, and solutions for each 
 group are given wherever new elements are introduced. 
 After these four groups a collection of miscellaneous, un- 
 classified problems gives abundant opportunity for the 
 application of principles already learned. 
 
72 THE SIMPLE EQUATION 
 
 The four groups are as follows : 
 
 1. Problems involving one number. 
 
 2. Problems involving two or more numbers. 
 
 3. Problems involving the element of time. 
 
 4. Problems involving the element of value. 
 
 Exercise 16 
 
 Problems involving One Number 
 
 * 
 
 1. Four times a certain number is 36. Find the number. 
 We will represent the unknown number by x, and we write : 
 Let X = the required number. 
 Then, from the given condition, 
 
 4 a; = four times that number. 
 
 But the problem states that 
 
 36 = four times that number. 
 
 Hence, from our assumed condition and fiom the given condition, 
 we have 4 x and 36 representing the same quantity, 
 From which, our equation, 
 
 4 a: = 36. 
 
 x=9. I 
 
 Therefore, the required number is 9. 
 
 2. A certain number increased by 10 equals 25. Find 
 the number. 
 
 3. If 5 is taken from a certain number, the remainder 
 is 13. Find the number. 
 
 4. Three times a certain number is diminished by 7 
 and the remainder is 17. What is the number ? 
 
THE SOLUTION OF WRIT! J]N PROBLEMS 73 
 
 5. John has three times as i: ,ny books as William. 
 Together they have 20 books -low many books has 
 William ? How many has Jo \m I 
 
 6. If three times a certain number is added to five 
 times the same number, the sum is 72. Find the number. 
 
 7. Four times a certain niimbeY is subtracted from 
 seven times the same number und the remainder is 36. 
 Find the number. 
 
 8. By adding 12 to a cei tain number I get a result 
 five times the original numbji*. What was the original 
 number ? 
 
 9. I double a certain number and subtract 9 from my 
 result. My remainder is 3 more than my original number. 
 What was the number ? 
 
 Let . X = the required number. 
 
 2 X = double the number. 
 2 a: — 9 = 9 less than double the number. 
 Also, X -\- S = S more than the original number. 
 
 From the conditions of the problem the last two expressions of the 
 statement are equal, hence our equation : 
 2x-9 = x-[-3. 
 2x-x = S + 9. 
 
 a: = 12, the required number. 
 
 10. If 5 is added to a certain number, the sum is equal 
 to 3 less than three times the original number. Find the 
 number. 
 
 11. Three is subtracted from 5 times a certain number, 
 and the remainder is 5 more than the original number. 
 What was the number ? 
 
 12. Find that number which, if doubled, exceeds 60 by 
 as much as the number itself is less than 60. 
 
74 THE SIMPLE EQUATION 
 
 Exercise 17 
 Problems involving Two or More Numbers 
 
 1. The sum of two numbers is 21, and the greater num- 
 ber is twice the smaller one. What are the numbers ? 
 
 Let X = the smaller number. 
 
 Then 2x = the larger number. 
 
 {x -}- 2 x) or 3 X = their sum. 
 From the problem, 21 = their sum. 
 
 Hence, 3 a; = 21. 
 
 X = 7, the smaller number. 
 2 X = 14, the larger number. 
 
 2. There are two numbers, one of which is 7 more than 
 the other and their sum is 31. Find them. 
 
 Let X = the smaller number. 
 
 Then a; + 7 = the larger number. 
 
 2 a; + 7 = the sum of the numbers. 
 From the problem, 31 = the sum of the numbers. 
 
 Hence, 2 a: + 7 = 31. 
 
 2a; = 31 -7. 
 2a: = 24. 
 X = 12, the smaller number, 
 a; + 7 = 12 + 7 = 19, the larger number. 
 
 3. There are two numbers, one of which is four times 
 the other. Their sum is 12 more than twice the smaller 
 number. Find the numbers. 
 
 4. There are two numbers whose difference is 24, and 
 the greater number is 3 more than twice the smaller num- 
 ber* What are the numbers ? 
 
THE SOLUTION OF WRITTEN PROBLEMS 75 
 
 5. The sum of three numbers is 45. The second num- 
 ber is twice the first number, and the third is equal to 
 twice the sum of the first and second. Find the three 
 numbers. 
 
 6. The sum of three numbers is 25. The third num- 
 ber is twice the first number, and the second is 5 Ifess than 
 the third. Find the numbers. 
 
 7. Find three consecutive numbers whose sum is 45. 
 Let X = the smallest number. 
 
 Then a: + 1 = the next larger number. 
 
 X -\-2 = the largest number. 
 3 a: + 3 = their sum. 
 Hence, 3 a: + 3 = 45. 
 
 3a; = 45 -3. 
 3a: = 42. 
 X = 14, the smallest number, 
 a; + 1 = 15, the next larger number, 
 a; + 2 = 16, the largest number. 
 
 8. Find three consecutive odd numbers whose sum 
 shall be 21. 
 
 (Let x,x + 2,x + 4:, represent the numbers, and state like Problem 7.) 
 
 9. Find the five consecutive even numbers whose sum 
 is equal to seven times the least number. 
 
 10. Find four consecutive odd numbers such that the 
 sum of the three smallest shall be 23 less than four times 
 the greatest one. 
 
76 THE SIMPLE EQUATION 
 
 11. Divide 19 into two parts such that the smaller part 
 plus twice the larger part shall be 29. 
 
 Let X = the smaller part. 
 
 Then 19 — x = the larger part. 
 
 2(19 — x) = twice the larger part. 
 Hence, from the given conditions, 
 
 x + 2(19 -x) = 29. 
 Solving, X = 9, the smaller part. 
 
 And 19 - a: = 19 — 9 = 10, the larger part. 
 
 12. Divide 72 into parts such that 3 times the larger 
 part added to 5 times the smaller part shall be 274. 
 
 13. Divide 100 into two parts such that twice the larger 
 part shall be 4 more than five times the smaller part. 
 
 14. Divide 59 into two parts such that twice the smaller 
 
 part shall be 2 less than twice the larger part. 
 
 15. Divide 28 into parts so that 14 less than three times 
 the larger part shall be equal to three times the small part 
 increased by 4. 
 
 Exercise 18 
 
 Problems involving the Element of Time 
 
 1. A boy is 5 years older than his sister. In 4 years 
 the sum of their ages will be 21 years. J'ind the present 
 age of each. 
 
 Let X = the present age of the sister. 
 
 Then x + 5 9= the present age of the boy. 
 
 a; + 4 = the sister's age 4 years from now. 
 X -\- 9 = the boy's age 4 years from now. 
 From which 2 a: + 13 = the sum of their ages 4 years from now. 
 
THE SOLUTION OF WRITTEN PROBLEMS 
 
 11 
 
 Hence, our equation, 
 
 2 a; + 13 = 21. 
 Solving, X = 4:, the sister's age now. 
 
 a; + 5 = 9, the boy's age now. 
 
 2. A man is twice as old as his son and the sum of 
 their ages after 10 more years will be 92 years. Find 
 the present age of each. 
 
 3. Five years ago the sum of the ages of A and B 
 was 40 years. B is now four times as old as A. What 
 is the present age of each ? 
 
 4. In seven years the sum of A's age and B's age will 
 be 8 years less than five times A's present age. At the 
 present time A is three times as old as B. Find the age 
 of each after seven years. 
 
 5. A man is twice as old as his brother. Five years 
 ago he was three times as old. Find the present age 
 of each. 
 
 Let X = the brother's age now. 
 
 2 X = the man's age now. 
 X — 6 = the brother's age five years ago. 
 2x — 6 = the man's age five years ago. 
 Hence, d(x — 5) = 2 a; — 5. 
 
 Solving, X = 10, the brother's age now. 
 
 2 a: = 20, the man's age now. 
 
 6. In seven years a man will be twice as old as his 
 brother. The sum of their present ages is 31 years. Find 
 the present age of each. 
 
78 THE SIMPLE EQUATION 
 
 7. A man is five times as old as his sister but in four 
 years he will be only three times as old. What is the 
 present age of each ? 
 
 8. A man 50 years of age has a son 15 years old. In 
 how many years will the father be twice as old as the son ? 
 
 9. One boy is 16 years old and another boy is 8 years 
 old. How^ many years ago was the oldest boy three times 
 as old as the youngest ? I 
 
 10. The sum of the present ages of a man and his son 
 is 52 years. In two years the man will be three times as 
 old as the son. What will be the age of each when the 
 sum of their ages is 100 years ? 
 
 Exercise 19 
 Problems involving the Element of Value ^ 
 
 1. Divide 1100 among A, B, and C, so that B shall 
 have twice as much as A, and C $ 10 more than what A 
 and B together receive. 
 
 Let X = the number of dollars A receives. 
 
 Then 2x = the number of dollars B receives. 
 
 3 a; + 10 = the number of dollars C receives. 
 Hence, 6 a; + 10 = the total received by aU three. 
 
 6a: + 10 = 100. 
 Solving, a; = 15, the number of dollars A receives. 
 
 2 a: = 2 • 15 = 30, the number of dollars B receives. 
 3 a: + 10 = 3 • 15 + 10 = 55, the number of dollars C receives. 
 
 2. A has $ 3 less than B. C has as many dollars as A 
 and B together. All three have 1 26. How many dollars 
 has each ? 
 
THE SOLUTIOJ^ OF WRITTEN PROBLEMS 79 
 
 3. A certain number of yards of cloth cost $2 per 
 yard, and the same number of yards of silk cost % 5 per 
 yard. The total cost of both lots was % 70. How many 
 yards of cloth are there in each lot ? 
 
 Let X = the number of yards in each lot. 
 
 Then 2 x = the value of the cloth in dollars. 
 
 5 X = the value of the silk in dollars. 
 Hence, 7 x = the total value of both lots in dollars. 
 
 From which we write our equation, 
 
 7 a; = 70. 
 X = 10, the number of yards in each lot. 
 
 4. 12 men receive f 31 for a day's work. A part of 
 the men work at the rate of f 2 per day, and the other 
 part receive $ 3 each per day. How many men worked at 
 each rate ? 
 
 Let X = the number of men working at $ 2 per day. 
 
 Then 12 — x = the number of men working at f 3 per day. 
 2x = the total dollars paid to the first lot. 
 3(12 — x) = the total dollars paid to the second lot. 
 Hence, the equation, 
 2 a; + 3(12 -a:) = 3L 
 
 X = 5, the number of men working at $ 2 per day. 
 12 — a: = 12 — 5 = 7, the number of men working at $ 3 per day. 
 
 5. A man pays a bill of 1 49 with five-dollar and two- 
 dollar bills, and uses the same number of each kind. 
 How many bills of each kind were used ? 
 
 6. A boy has $ 14 in two-dollar bills and half-dollars. 
 He has three times as many coins as he has bills. How 
 many has he of each ? 
 
80 THE SIMPLE EQUATION 
 
 7. Eight oranges cost a fruit dealer 11.90. A portion 
 of the lot cost him 20 cents per dozen, and the remainder 
 cost him 30 cents per dozen. How many dozen were there 
 of each kind? 
 
 8. A man bought 90 postage stamps, the lot being 
 made up of the five-cent and two-cent denominations. 
 Twice the value of the two-cent stamps was the same as 
 the value of the five-cent stamps. How many of each 
 kind were purchased, and what was the total amount paid 
 for them? 
 
 Exercise 20 
 
 Miscellaneous Problems 
 
 1. Find two numbers whose sum is 80 and whose dif- 
 ference is 10. 
 
 2. A and B together have 190, but A has 110 more 
 than B. How many dollars has each ? 
 
 3. A certain number when multiplied by 4 exceeds 15 
 by as much as the original number is less than 15. What 
 is the number ? 
 
 4. One of two numbers is 4 more than the other, and 
 the difference of their squares is 72. What are the 
 numbers ? 
 
 5. The sum of three consecutive numbers is 26 more 
 than the second number. What are the three numbers ? 
 
 6. The difference between the ages of a father and son 
 is 36 years, and the father is four times as old as the son. 
 Find the age of each. 
 
 7. There are two numbers whose sum is 30, and whose 
 difference increased by 6 equals the smaller number. 
 What are the numbers? 
 
THE SOLUTION OF WRITTEN PROBLEMS 81 
 
 8. Divide 70 into two parts such that twice the larger 
 part shall equal three times the smaller part. 
 
 9. A man divides $700 among his four sons. Each 
 gets 1 50 more than the next oldest. How much does each 
 get? 
 
 10. Divide 23 into two parts such that 1 less than four 
 times the smaller part shall equal 1 more than twice the 
 greater part. 
 
 11. Ten times a certain number is as much above 13 as 
 19 is above 6 times the number. What is the number ? 
 
 12. The difference between the squares of two consecu- 
 tive even numbers is 28. Find the numbers. 
 
 13. The sum of two numbers is 13 and the difference 
 between their squares is 13. What are the numbers ? 
 
 14. In a family of four children the oldest is twice the age 
 of the youngest, and the sum of the ages of the oldest and 
 youngest equals the sum of the ages of the other two. 
 All together their ages amount to 54 years. Find the age 
 of each of the four, the third child being 15 years old. 
 
 15. One number is three times another. Subtract the 
 smaller number from 15 and the larger from 23, and the 
 remainders are equal. Find the numbers. 
 
 16. At a certain election a total vote of 1248 was cast. 
 The successful candidate received a majority of 64. How 
 many votes were cast for each candidate ? 
 
 17. A man has four hours of time at his disposal and he 
 walks out into the country at a rate of 4 miles per hour. 
 How many miles can he walk so that, by returning on a 
 trolley car at the rate of fifteen miles per hour, he will 
 return at the end of his time ? 
 
 F. H. S. FIRST YEAR ALG. 6 
 
82 THE SIMPLE EQUATION 
 
 18. A walks over a certain road at the rate of 4 miles an 
 hour. Three hours after he left his home, B starts after 
 him at a rate of 5 miles per hour. How many miles will 
 A have gone when B overtakes him ? 
 
 19. A and B are 48 miles apart and start at the same 
 time to travel toward each other. A goes at a rate of 3 
 miles an hour and B goes at a rate of 5 miles an hour. 
 How many hours will pass before they meet, and how far 
 will each have traveled ? 
 
 20. A man paid $178 for a horse and a harness. The 
 cost of the horse was f 10 more than the cost of five simi- 
 lar harnesses. What was the cost of the horse and the 
 harness ? 
 
 21. A man walked the first 10 miles of a journey, then 
 rode a certain distance in a train, and finally traveled in 
 an automobile twice as far as he had already come. He 
 traveled 45 miles in all. How many miles did he go by 
 train and in the automobile ? 
 
 22. Ten yards of cloth and 5 yards of silk cost in all 
 $60. The cost of the silk was twice the cost of the cloth. 
 What was the cost of each per yard ? 
 
 23. How can you pay a bill of $3.50 so that you will 
 use the same number each of dimes and quarter dollars, 
 and no other coins? 
 
 24. The sum of the ages of a father and son is 92 years ; 
 but if the son's age is doubled, it will be 4 years more 
 than his father's age. Find the age of each. 
 
 25. A had $7 more than three times B's money. A 
 gave B $8 and now he has only $1 more than B. How 
 much had each at first ? How much has each now ? 
 
CHAPTER VII 
 SUBSTITUTION 
 
 64. Substitution is the process of replacing literal factors 
 in algebraic terms by numerical or by other literal values. 
 
 Thus : If a = 5 and 6 = 7 : 
 
 (1) (a + 6) = (5 + 7) 
 
 = 12. Kesult. 
 
 (2) (2a-a&) = (2.5-5.7) 
 
 = (10 - 35) 
 
 = - 25. Result. 
 
 If x=:2m, y = dm, and z = 6m: 
 
 (3) (a: + ?/ + 2) = (2 m + 3 m + 5 m) 
 
 = 10m. Result. 
 (4:) (x + y)(2x-y-\-Sz) = (2m + dm)(2-2m-Sm + S-5m) 
 = (5 m) (4 m — 3 m + 15 m) 
 = (5 m) (16 m) 
 = 80 m2. Result. 
 
 GENERAL METHOD 
 
 65. To substitute numerical or literal values in a given 
 expression : 
 
 Replace the literal factors of the terms of the given expres- 
 sion hy their respective given values. 
 
 Perform all indicated operations^ and simplify the result. 
 
 83 
 
84 SUBSTITUTION 
 
 Occasional problems will permit the substitution of given 
 values either before or after indicated operations are per- 
 formed ; but, in general, the better plan is to make substi- 
 tution the first step, for in practical applications this order 
 is almost universal. 
 
 I. Substitution of Numerical Values 
 Exercise 21 
 
 Find the numerical value of the following, when a = 4, 
 6=3, c = 2, and c? = 1. 
 
 1. 
 
 a-{-h -\- c. 
 
 2. a—^h + Be. 
 
 a 
 
 +&+c=4+3+2 
 
 a-2& + 5c = 4-2(3)+5(2) 
 
 
 = 9. Result. 
 
 = 4 _ 6 + 10 
 = 8. Result. 
 
 3. 
 
 4 6 _ 3 « 4. 2 ^. 
 
 5. lOa-{-h-CSc-d), 
 
 4. 
 
 1 a-d+^c-h. 
 
 6. ib-lSc-C2a-^d)'], 
 
 7. 
 
 ah — ^hc^b ad. 
 
 
 
 ab-dbc^5ad = (4) (3) - 3 (3) (2) + 5(4) (1) 
 
 
 = 12- 
 
 -18 + 20 
 
 
 = 14. 
 
 Result. 
 
 8. 2ab-Sed-{-2bd. 
 
 9. Bad-^bd-hScd. 
 
 10. Sabc — b bed + 2 acd — 12 abd, 
 
 11. ab — (be — cd}. 
 
 12. abc — \_acd — (bcd—abd)'\. 
 
 13. 2ab-(Bcd-\-2a- ab + ed}. 
 
 14. bc-\-[bd-(iab-cd)]+(iab-bc}l 
 
GENERAL METHOD 85 
 
 15. Simplify a{a -\- e) — c(c — a) when a = 4 and c = 3. 
 
 a(a -f c) - c{c - a) = 4(4 + 3) - 3(3- 4) 
 = 4(7)-3(-l) 
 
 = 28 + 3 
 
 = 31. Result. 
 
 16. Simplify (a + 6)2 _ (<^ + 6) (a - 5) - (« - 5)2 when 
 a = 5 and 5 = 2. 
 
 (a-f6)2-(a + />)(a-5)-(a-6)2= (5 + 2)2- (5 + 2) (5-2) -(5-2)2 
 
 = (7)2 -(7) (3) -(3)2 
 = 49-21-9 
 = 19. Result. 
 
 17. (a + 2)2 -(a -1)2 when a = 3. 
 
 18. 10(6f+-4)2-(3a + 2)2 when a =10. 
 
 19. 4(2a;+-?/)2— 8(a:+-y)(a:— ?/) when x=2 and ^ = 0. 
 
 20. 3(m — 2a;)2— 2(w + 2a;)(m — 2a;)+-4a; when m = 5 
 and x=l. 
 
 21. 2(c + (^)-S2(6' + ^)((?-(?)-3(c-(^)S when c=l 
 and c? = — 1. 
 
 22. a(a + 6)-[6(a-6)2 + (a+-5)3] when «=-3 and 
 ^.= -4. 
 
 23. (3 a+-m)(3 « - w) + [9 a2 _ j^^ _ ^(2 w- 9a)j] 
 when a = and m = 2. 
 
 24. 5a(a+-l)-(2-a)(3<x + 5)-2a(a+-l) when a = 0. 
 
 25. (3 w - 2)2 + (m + w + 1)2 - (2 m - l)(m + 1) when 
 
 7W = 71 = — 1. 
 
86 SUBSTITUTION 
 
 II. Substitution of Literal Values 
 
 When the values given for substitution are literal 
 quantities, the process is identical with that of numerical 
 substitution. The results, however, will be in terms of the 
 literal quantities substituted. 
 
 Exercise 22 
 
 Substitute the given literal values and reduce the fol- 
 lowing: 
 
 1. (a + xy — (a — xy^ when a=2x. ♦; 
 
 (a + xy -(jci-xy = {2x-{- xy -(2x- xy 
 =(dxy-(xy 
 
 = 8 a;2. Result. 
 
 2. (a — 3 a; — x^^ when x= — a. 
 
 3. (a 4- a?) (a — x)^ — (a + 3 xy when a = — x. 
 
 4. (x + m}(x-\-m — 7i) when m = x and n= — x. 
 
 5. (2 a — "^ c -\- x)(x — a) — acx when a = x and c = 0. 
 
 6. 5 ac — (a — 3 <?) (a + 10 c) — c + ((? — a)^ when <? = 0. 
 
 7. 2(mH-a;) — 2(w + a;)(77i — 2:) — 3 771a; when m=a and 
 a;= 3«. 
 
 8. (2 a; + 1)2 -3(a:- 1)2 when a: =4 a. 
 
 9. (3a;2-4a;+l)- (2 a: + 1)2 whena:=-«5. 
 
 10. (a 4- a:) (a — a:) — (a — a;) + 2 a; when x= — ^ a. 
 
 11. (a + 6+l)2-(a+6)2+2(a + 5)-l when a=-x 
 and 5 = — 3 a;. 
 
 12. (Qx^-\\xy-10y'^^-Qx(x-2y') + l^y'^ when 
 x = m and y — — m. 
 
APPLICATIONS OF SUBSTITUTION 87 
 
 APPLICATIONS OF SUBSTITUTION 
 I. To VERIFY AN EQUATION 
 
 66. To verify an equation is to prove the truth of the 
 result obtained for its root. 
 
 When we verify an equation we show that our result, 
 when substituted for x in the given equation, reduces both 
 members to the same quantity. 
 
 In such a case a root is said to satisfy an equation. 
 
 Exercise 23 
 
 1. Show that X = 1 satisfies the equation 3 a; + 4 
 
 = 2 2; + 5. 
 
 dx-\-i = 2x + 6. 
 
 3(1) + 4 = 2(1) + 5. 
 
 3 + 4 = 2 + 5. 
 
 7 = 7. 
 
 2. Show that X = S satisfies the equation 3 a; — 4 
 = 2:+- 12. 
 
 3. Show that x= — 4: satisfies the equation 2{x + 3) 
 = 4 + (r^^-2). 
 
 4. Show that x=0 satisfies the equation 5 (rr — 2) 
 = 3(a;+l)_13. 
 
 5. Show that x= — l satisfies the equation (a; + 1) 
 (a;+2)=<a;+l). 
 
 6. Is 2 a root of the equation 2(^3p 4- 2 a; + 1) — (a; + 2) 
 = 2a^ + 6? 
 
 7. Prove that — 4 will satisfy the equation (a; + 4)^ 
 + (a: + l)2=(a; + 3)2+2a;(a;+l)-a^. 
 
88 SUBSTITUTION 
 
 8. Show that x = satisfies (4a; - l)(a; + 3) - da? 
 
 _(_10:r+3) + 6 = 0. 
 
 9. Prove 2 c a root of the equation 2cx-\-d = 4:C^-{-d, 
 
 10. Show that x=l satisfies 3(<x + b}x — 2 (a — b)x = 
 a + 5b. 
 
 II. The Use of Formulas 
 
 67. A Formula is an algebraic expression for a general 
 principle. 
 
 Thus, if the area of a triangle is 
 represented by S, the base by 5, 
 and the altitude by a, the formula 
 
 S = —■ gives us an expression for 
 
 the area of the triangle in terms of its base and altitude. 
 Then if we know the values for the base and the altitude, 
 we obtain the area by the substitution of those values and 
 the reduction of the formula. 
 
 We will now make application of this principle to a few 
 formulas in common use. 
 
 (a) The Circumference of a Circle 
 
 Let C = the circumference of a circle. 
 
 R = the radius of the same circle. 
 
 3.1416 = the constant relation between the circumference 
 and the diameter of a circle. (This constant is 
 usually represented in formulas by the Greek 
 letter ir, called " pi.") 
 
 Then C = 2 7ri2 is the formula for the circumference of a circle 
 
 whose radius is known. 
 
APPLICATIONS OF SUBSTITUTION 89 
 
 Exercise 24 
 
 1. Find the circumference of a circle whose radius is 
 10 feet. 
 
 Given R = 10, 
 
 TT = 3.1416. 
 Then C = 2 7rR 
 
 = 2 X 3.1416 X 10 
 
 = 62.832 feet. Result. 
 
 2. Find the circumference of a circle whose radius is 
 14 inches. 
 
 3. Find the circumference of a circle whose diameter 
 is 18 feet. (The diameter of a circle is equal to twice its 
 radius.) 
 
 4. The circumference of a circle is 12.5664 feet. Find 
 the radius of the circle. (Substitute the known values 
 and solve the formula for B.) 
 
 5. The circumference of a circle is 31.416 feet. Find 
 its diameter. 
 
 6. In going one mile a carriage wheel turns 440 times. 
 Find the radius of the wheel. " 
 
 (b) The Area of a Circle 
 
 Let S = the area of a circle. 
 
 R — the radius of the same circle. 
 
 TT = the constant, 3.1416. 
 
 Then S = irR^ is the formula for the area of a circle 
 
 whose radius is known. 
 
90 SUBSTITUTION 
 
 Exercise 25 
 
 1. Find the area of a circle whose radius is 10 feet. 
 Given R = 10. 
 
 TT = 3.1416. 
 Then S = irR^ 
 
 = 3.1416 X 102 
 = 3.1416 X 100 
 = 314.16 square feet. Result. 
 
 2. Find the area of a circle whose radius is 14 inches. 
 
 3. Find the area of a circle whose diameter is 18 feet. '! 
 
 4. The area of a circle is 28.2744 square feet. Find 
 the radius. 
 
 5. The area of a circle is 12.5664 square feet. Find 
 the diameter. 
 
 6. What is the area of a circular ring whose two diame- 
 ters are 10 and 12 feet respectively ? 
 
 7. How many rods of fence will be needed to inclose a ; 
 circular park whose area is 176.715 square rods ? 
 
 8. A man has a circular pond whose diameter is 100 
 feet. He builds around it a walk 6 feet wide. What is 
 the area of the walk ? 
 
 (c) TVie Sum of a Series of Numbers or Quantities 
 
 68. A Series is a succession of numbers formed accord- 
 ing to some fixed law. Thus : 
 
 1, 3, 5, 7, 9, 11, 13, etc., 
 
 is a series in which each term is greater by 2 than the 
 preceding term. 
 
APPLICATIONS OF JSUBSTlTUTtOK 91 
 
 A series formed in this way is called an Arithmetical 
 series. 
 
 When certain elements of a series of this kind are 
 known, others can be found by the use of formulas. 
 Thus : 
 
 s... 
 
 Let a = the first term of a series. 
 
 . I = the last term of a series. 
 
 n = the number of terms in the series. 
 
 aS = the sum of the terms in the series. 
 
 Then S = ^(a -{■ I) is the formula for finding the sum 
 
 of an arithmetical series when the first 
 term, the last term, and the number of 
 terms are known. 
 
 Exercise 26 
 
 1. The first term of an arithmetical series is 2, the last 
 term 18, and the number of terms 9. Find the sum of 
 the terms in the series. 
 
 From the statement, a = 2, I = 18, n = 9. 
 
 Hence, 5 = ^(a + Z) 
 
 = |(2 + 18) 
 = 90. Result. 
 
 2. There are 10 terms in an arithmetical series. The 
 first term is 5 and the last term 15. What is the sum of 
 the terms in the series ? 
 
 3. The sum of a series of sixteen terms is 136, and the 
 last term of the series is 16. What is the first term of 
 the series ? 
 
92 SUBSTITUTION 
 
 4. A certain arithmetical series has a first term 12, 
 and a last term 18. The sum of the terms is 105. How 
 many terms are there in the series ? 
 
 5. The first term of an arithmetical series of nine terms 
 is — 35, the last term — 3. Find the sum of the terms. 
 
 6. The first term of a series of six terms is — 5 x^ and 
 the second term is 5 x. Find the sum of the series. 
 
 7. The sum of the terms of an arithmetical series is 
 21 x + 33. The first of the six terms in the series is 2 a; 
 + 3. Find the last term. 
 
 8. The sum of a succession of odd numbers that began 
 with 7 is 176. How many numbers are there in the series 
 if the last number is 37 ? 
 
 (d) The Formula for Falling Bodies 
 
 When a body falls to the earth, it gains a velocity of 32 
 feet per second for each second of its fall. The following 
 formulas are in common use for experiments with falling 
 bodies. 
 
 Let a = the gain in velocity per second. 
 
 t = the number of seconds during which the body falls. 
 S = the space through which the body falls. 
 Then, (1) S = lat"^ is the formula for finding the space passed 
 
 over by a falling body in a given time. 
 Again, let a = the gain in velocity per second. 
 
 t — the number of seconds during which the body falls. 
 V — the velocity of the body at the end of any given 
 second. 
 Then, (2) v = a< is the formula for finding the velocity of a fall- 
 ing body at the end of any given second. 
 For a we use 32 feet, or, in the Metric System, 980 centimeters. 
 
APPLICATIONS OF SUBSTITUTION 93 
 
 Exercise 27 
 
 1. In 2 seconds a ball fell from the top of a flag pole 
 to the ground. How many feet did the ball fall ? 
 
 Given a = 32 feet, 
 
 t = 2 seconds. 
 Hence, 5 = -| at^ 
 
 = i X 32 X 4 
 
 = 64 feet. Result. 
 
 2. A ball dropped from the top of a tower struck the 
 ground in 3 seconds. How high was the tower ? 
 
 3. A coin was dropped from the edge of a cliff and it 
 struck the water in 6 seconds. How far did the coin fall ? 
 
 4. What is the velocity of a ball at the end of the 3d 
 second of its fall ? 
 
 Given 
 
 a = 32 feet, 
 
 
 < = 3, the number of the second considered 
 
 Hence, 
 
 V = at 
 
 
 = 32 X 3 
 
 
 = 96 feet. Result. 
 
 That is, the ball is falling at a rate of 96 feet per second at the 
 end of the third second. 
 
 5. A stone fell from a roof and fell for 5 seconds before 
 striking the ground. How far did the stone fall ? What 
 was its velocity at the end of the 4th second? At the 
 end of the 5th second ? 
 
 6. The velocity of a ball when it struck the ground 
 was found to be 288 feet per second. For how many 
 seconds had the ball been falling ? What was its velocity 
 at the end of the 7th second ? 
 
CHAPTER VIII 
 
 SPECIAL CASES IN MULTIPLICATION AND 
 DIVISION 
 
 MULTIPLICATION 
 
 The product of vsimple forms of binomials may often be 
 obtained without actual multiplication. The following 
 examples will show the principles upon which the abbre- 
 viated processes depend. 
 
 The square of the 
 sum of two quanti- 
 ties 
 
 II. 
 
 The square of the 
 
 difference of two 
 
 quantities 
 
 III. 
 
 The product of the 
 
 sum and difference 
 
 of two quantities 
 
 a + 
 a + 
 
 a + 
 
 a — 
 
 a2 + 
 
 ab 
 
 a2- 
 
 ab 
 
 ab + b^ 
 
 a^ + ab 
 
 a^ + 2ab-hb^ 
 
 a^-2ab + b^ 
 
 - ab-b^ 
 -b^ 
 
 From these actual multiplications we make the follow- 
 ing conclusions : 
 
 From I. : *• 
 
 69. The square of the sum of two quantities equals the 
 square of the firsts plus twice the product of the first hy the 
 second^ plus the square of the second. 
 
 94 
 
 I 
 
MULTIPLICATION 95 
 
 L From II. : 
 
 70. The square of the difference of two quantities equals 
 the square of the first, minus twice the product of the first 
 hy the second, plus the square of the second. 
 
 I From III. : 
 
 71. The product of the sum and difference of two quan- 
 tities equals the difference of their squares. 
 
 Exercise 28 
 Write by inspection the following indicated products : 
 
 1. {a + x)\ 4. (« + 2)2. 7. (2a+3)2. 
 
 2. (^x + yy. 5. (a 4- 2^)2. 8. (3 a + 5)2. 
 
 3. (2:4-2)2. 6. (a + 5c)2. 9. (4 a; +3)2. 
 
 10. (m-w)2. 13. (a -5)2. 16. (a6-7)2. 
 
 11. (2; -5)2. 14. (2 a -3)2. 17. (^252- 5)2. 
 
 12. (a- 7)2. 15. (5 a- 2)2. 18. (3am-2ma;)2. 
 
 19. (a + 6)(a-6). 23. (4a + 10)(4a-10). 
 
 20. {m -\- x) (^m — x') . 24. (5 w + 7?z)(5?w— 7w). 
 
 21. (a+5)(a-5). 25. (2 ^2 + 9)(2 a2- 9). 
 
 22. (2a + 3)(2a-3). 26. (6 a5 + 5)(6 a5- 5). 
 
 27. (2a5 + 3 5c)2. 31. (^2x^ + ll}(2a^-ll). 
 
 28. (4 a;2?/2 — 7)2. 32. (ax^ + xy^Qaoc^ — xy). 
 
 29. (3m3-10)(3m3 + 10). 33. (2^5 + 11)2. 
 \ 30. (bxy-\-^xzy. 34. (5m7-8)2. 
 
96 SPECIAL CASES 
 
 IV. The Difference of Two Squares obtained from 
 Trinomials 
 
 Expressions that contain three terms may be grouped 
 in many cases so as to come under the principles that 
 govern the binomial cases already considered. The pro- 
 cess depends upon the grouping of two of the three given 
 terms in a parenthesis^ this parenthesis being treated as one 
 term. 
 
 Three different groupings are possible, as follows : 
 
 I. II. 
 
 (a + 5 4- c)(a + h -c) (a + 5 -f- c)(« - ^ + c) 
 
 III. 
 
 (a + h — c){a — h -\- c) 
 
 The terms inclosed in parentheses must always he like in 
 sign. 
 
 From I. (a + & + c) (a + J - c) = [(a + &) + c] [(a + i) -c] 
 
 = (a + hy - c2 
 = a2 + 2 a& + &2 _ c2. Result. 
 
 From II. {a-^l) + c){a-h+c) = [(a + c) + &][(a + c) - ft] 
 
 = (a + c)2 - 6-2 
 = a2 + 2 ac + c2 - h\ Result. 
 
 From III. In this case only one term has the same sign in each 
 expression, — the a term. Hence, the last two terms of each expression 
 must be inclosed in parentheses. The sign of the second term of 
 each expression becomes the sign of the parenthesis. 
 
 (a + & - c) (a - & + c) = [a + (& - c)] [a - (6 - c)] 
 = a^- (h-cY 
 = a^-(h'^-2bc-i-c'^) 
 = a^-b^-\-2bc- c\ Result. 
 
MULTIPLICATION 97 
 
 Exercise 29 
 
 Write by inspection the following indicated products : 
 
 1. [(a-i-c}-\-x']l(a + c)-x']. 9. (x^-\-x-^lX^ + ^-'^)' 
 
 2. [(a+5) + 2] [(a+^)-2]. 10. (a;-a + 5)(a;-a-2>). 
 
 3. [(a-2)-{-x'][(a-2)-x]. 11. (m-x+lXm-^-x+iy 
 
 4. l(x^-^l)+x]l(a^ + l)-x']. 12. (c-x-\-2Xc-\-x + 2'), 
 
 5. [a+(tf+tZ)] [a-((?+^)]. 13. (a + w-l)(a-m4-l). 
 
 6. [m + (a;-^)] [^-(a;-?/)]. 14. (a4-^-5)(a — ^ + 5). 
 
 7. (fl^ + c + a;)C«4-c-:K). 15. (c2-2-t?)(c2_2 + <?). 
 
 8. (a4-5 + 3)(a + &-3). 16. (tf2_^^_2)(c2_c_|.2). 
 
 17. [(a + a:) + C^ + l)][(a + a:)- (2/4-1)]. 
 
 18. [(a + ^)-(c?-c?)] [(a + 5)+(c-(^)]. 
 
 19. (a-b + c-l)(a-^h + c-hl}. 
 
 20. (a3-a2_^_l)(^3_^^2^_^_l), 
 
 V. The Product of Binomials having a Common Literal Term 
 
 The actual multiplication of the binomials (a + 7) and 
 (a + 5) gives ^ ^ 7 
 
 a + o 
 a2+ 7 a 
 
 + 5 a + 35 
 a2 + 12 a + 35 
 
 The terms of the product are made up as follows r 
 a"^ = a ' a, the product of the given first terms. 
 
 + 12 a = (+ 7 + 5) a, the product of the common literal term by the 
 sum of the given last terms. 
 + 35 = (+ 5) (+ 7), the product of the given last terms. 
 
 F. H. S. FIRST TEAR ALG. — 7 
 
98 SPECIAL CASES 
 
 In the same manner we may obtain the product of the 
 binomials (x — 4) and (x — 9). 
 
 x:x = x^, (_4-9)a; = -13a:, (-4) (-9) = + 36. 
 
 Hence, (x - i) (x - 9) = x^ -13 x -\- 36. Result. 
 
 Or, in a case where the signs of the second terms are unlike, 
 (x -\- 9) (X - S). 
 
 x-x = x% (+ 9 - 3) a; = + 6 a:, (+ 9 ) ( - 3) = - 27. 
 
 Hence, (ar + 9) (x - 3) = x^ + 6 a: - 27. Result. 
 
 And, again, the product of (m — 15) (m + 7). 
 
 m'm = m% (- 15 + 7) wi = - 8 w, (_ 15) (+ 7) = - 105. 
 
 Hence, (m - 15) (?« + 7) = m^ - 8 m - 105. Result. 
 
 From these illustrations we may state the general 
 process. 
 
 72. The product of two binomials having a common 
 literal term is obtained as follows : 
 
 The first term is the product of the given first terms. 
 The second term is the product of the common literal term 
 hy the algebraic sum of the given second terms. 
 
 The third term is the product of the given second terms. 
 
 Exercise 30 
 Write by inspection the following products : 
 
 1. (a + 2) (a + 3). 6. (a + 12)(a + l). 
 
 2. (a + 5)(« + 3). 7. (a +11) (a +10). 
 
 3. (c+6)(c?+7). 8. (a: + 7)(rr + 20). 
 
 4. (w + 5)(m + 9). 9. (re + 16) (a; + 9). 
 
 5. (a: + 8) (a; +10). 10. (2: + 14) (2; + 15). 
 
MUL TIP Lie A TION 99 
 
 11. (a -4) (a -5). 16. (a-12)(a-10). 
 
 12. (a;-3)(rr-4). 17. (a5 - 7) (a6- 30). 
 
 13. (a; -8) (37-1). 18. (rr^ - 15) (a;?/ - 3). 
 
 14. (m-7)(w-4). 19. (c2_i0)(c2_20). 
 
 15. (3/ -10) (2/ -3). 20. (c2c22_7)(^^_ll). 
 
 21. (2; + 5) (a: -3). 28. (2:2 + 6)(a;2_ii). 
 
 22. (a;+3)(2:-5). 29. (a2_ 5) (^2 + 13). 
 
 23. (6J_4)(a + 7). 30. (w3+7)(^3_4). 
 
 24. (a + 4) (a -7). 31. (a52_9)(^52 + 20). 
 
 25. (w-9)(7yi + 10). 32. (a;2?^2 + io)(^^2_i9), 
 
 26. (2; + 5) (2; -9). 33. ((?3(^3_i5)(^^ + 2). 
 
 27. ((?-12)(c+ll). 34. (ax^-S0x)(a2^-{-2x), 
 
 VI. The Product of Any Two Binomials 
 
 An application of the principle already used in the fore- 
 going exercise enables us to obtain the product of any two 
 binomials. 
 
 By actual multiplication : 
 
 2a + 5 
 3a + 4 
 
 + 8 g + 20 
 6 a2 + 23 a + 20 
 
 The two multiplications that result in the terms -f- 15 a 
 and +S a are called cross-products. 
 
 In multiplication by inspection the student will be 
 helped if he will imagine that the terms entering into the 
 
100 SPECIAL CASES 
 
 cross-products are connected as indicated below. Then 
 in the example above we would have 
 
 (2 a + ^)(3 a + 4) 
 
 From which form we immediately write the products 
 that together make the middle term. 
 
 Thus, (+ 2 a) (+ 4) + (+ 3 a) (+ 5) = + 8 a + 15 a = + 23 a. 
 
 Similarly : 
 
 In (4 a - 7) (3 a - 5) the middle term is 
 ( + 4a)(-5) + ( + 3a)(-7) = (-20a)+(-21a)=-20a-21a=:-41a. 
 
 In (2 a + 5) (3 a — 4) the middle term is 
 ( + 2a)(-4) + ( + 3a)( + 5) = (-8a) + ( + 15a) = -8a + 15a=+7a. 
 
 In (3 a -2) (9 a + 4) the middle term is 
 (+3a)( + 4) + ( + 9a)(-2) = ( + 12a) + (-18a) = +12a-18a=-6a. 
 
 From these illustrations we may state the general process. 
 
 73. The product of any two binomials is obtained as 
 follows : 
 
 The first term is the product of the given first terms. 
 The second term is the algebraic sum of the cross-products 
 of the given terms. 
 
 The third term is the product of the given second terms. 
 
 Exercise 31 
 
 Write by inspection the following products : 
 
 1. (2a-f-3)(a + l). 5. (4a;4-l)(2?:+3). 
 
 2. (3a:+2)(2rr + l). 6. (3 m4- 1)(2 tw + 5). 
 
 3. (2 a: + 3) (a; + 3). 7. (5 a: + 2) (3 a; -hi). 
 
 4. (2a + l)(3a + l). 8. (7 a: + l)(a;+ 9). 
 
MULTIPLICATION 101 
 
 9. (2a-7)(3«-4). 13. (5m-7)(8m-5). 
 
 10. (3a-l)(a-6). 14. (10 a- 17) (8 a - 3). 
 
 11. (4c-l)(3(?-5). 15. (7m-5y)(8w-3«^). 
 
 12. (3^-4)(2?/-3). 16. (2a-56)(7a-26). 
 
 17. (4 a -f- 7 m) (5 a - 9 m). 21. (11 «+ 10 5) (9 a - 7 5). 
 
 18. (7rc-4«/)(8a; + 9?/). 22. (7 rr- 13 2/) (5 a; + 10?/). 
 
 19. («6-3 2/)(5a5 + 2^). 23. (a5tf-9)(7a5c + ll). 
 
 20. (a;2/+7 2)(4a:^-2). 24. (12 a- 5) (5 a + 12). 
 
 VII. The Square of Any Polynomial 
 
 By actual multiplication : 
 (a + 6 + c)2 = a2 + &2 + c2 + 2 a6 + 2 ac 4- 2 &c. 
 
 (a + & + c + c?)2 = a2 + &2 + c2 + ^2^. 2 aft + 9 ac + 2 ac? + 2 5c + 2 &rf+2 erf. 
 (a-6-c-rf)2 = a2 + 62 + c2 + d2_2a&-2ac-2arf + 2&c+2M+2crf. 
 
 It will be noted that 
 
 (1) All the squares in the product are positive. 
 
 (2) The other terms are positive or negative according 
 as the signs of the factors composing them are like or 
 unlike in sign. 
 
 (3) The coefficient of each product of dissimilar terms 
 is 2. 
 
 Applying these principles we have 
 
 (a-2& + 3c)2 = (a)2+ (_2&)2+ (3 c)2 + 2(a)(- 2 &) +2(a)(3c) 
 + 2(-2 6)(3c) 
 
 = a2 + 4 62 + 9 c2 _ 4 a& +6 ac - 12 &c. 
 
102 SPECIAL CASm 
 
 Hence, the process of writing the square of any poly- 
 nomial is as follows : 
 
 74. The square of any polynomial is the sum of the 
 squares of the several terms together with twice the product 
 of each term into each of the terms that follow it. 
 
 Exercise 32 
 
 Write by inspection the following products : 
 
 1. (a + 5 + a:)2. 9. Qa-{-h^-m-\-ny, 
 
 2. (m + n-{-xy. 10. (x + a + m + yy. 
 
 3. (x + y + zy, 11. (^a-h^-c-df, 
 
 4. (^a + h~xy, 12. (a- 2 w + 3 a;- 2)2. 
 
 5. Qm-n-py 13. (2a-3 6 + 4(?-5c?)2. 
 
 6. (a + 2 5 + 3 0^. 14. *(\-x + x^-a?y. 
 
 7. (2a-35 + (?)2. 15. '^(a^^x^-x-\-iy, 
 
 8. (3m-47i-5)2. 16. *(a3-3a2_4a4.5)X 
 
 DIVISION 
 
 In certain cases where both dividend and divisor are 
 binomials we are able to write the quotients without 
 actual division. Such divisions are possible only in cases 
 where the powers of the terms of the dividend are like ; 
 that is, both terms of the dividend must be squares, both 
 cubes, both fourth powers, etc. We will consider 
 
 * After squaring, the terms should be collected. 
 
DIVISION 103 
 
 I. The Difference of Two Squares 
 Since by Art. 71, 
 
 it follows that 
 
 _ = a — b and =a-\-b, 
 
 a-{-o a — b 
 
 From which we state the general principles : 
 
 75. The difference of the squares of two quantities may be 
 divided by either the sum or the difference of the quantities. 
 
 If the divisor is the sum of the quantities^ the quotient will 
 be the difference of the quantities. 
 
 If the divisor is the difference of the quantities, the quo- 
 tient will be the sum of the quantities. 
 
 Exercise 33 
 Divide by inspection : 
 
 1. a^-x^ hj a-x. 8. 4 a2 _ 1 by 2 a - 1. 
 
 2. a2 _ ^ by a-c. 9. 4 a^ _ 25 by 2 « - 5. 
 
 3. a^ — m^ by a-{^ m. 10. 9 m^ — 16 by 3 m + 4. 
 
 4. x^-y^ by X- y. 11. 25 w^ - 81 by 5 m -f 9. 
 
 5. ^_lbya; + l. 12. 16 m*- 36 by 4 ^2 - 6. 
 
 6. c2-4byc-2. 13. 100 - 81 a:^ by 10-9^:2. 
 
 7. m2 - 25 by m + 5. 14. x^y^ - 144 by xy + 12. 
 
 15. 81 a%^ - 169 c by 9 a6 - 13 c. 
 ie. 196w*-49c6 by 14m2+7<?3. 
 
104 SPECIAL CASES 
 
 n. The Difference op Two Cubes 
 By actual division 
 
 a — 
 
 From the form of the quotient and its relation to the 
 divisor we state : 
 
 76. The difference of the cubes of two quantities may be i 
 divided by the difference of the quantities. 
 
 The quotient is the square of the first quantity., plus the 
 product of the two quantities^ plus the square of the second 
 quantity. 
 
 Exercise 34 
 
 Divide by inspection : 
 
 1. a^-7^ hj a-x. 9. 27 - (?3 by 3 - <?. 
 
 2. a^ — (^ by a — c. 10. a%^ — 1 by ab — 1. 
 
 3. a^ — m^ by a — m. 11. 'j^y^ — 125 by xy — 5. 
 
 4. a? — y^ by x — y. 12. 27 a^ — 64 c^ by 3 a — 4 c. 
 
 5. a^-S hy a-2. 13. 8 a^ -21 b^ by 2 a- Sb. 
 
 6. 0^-21 by x-S. 14. 125a3_8 by 5a-2. 
 
 7. a3 - 1 by a-1. 15. 1000 - c^d^ by 10 - cd. 
 
 8. 8 -a^ by 2-2:. 16. 729-512^:3 by 9-8:r. 
 
 III. The Sum of Two Cubes 
 By actual division 
 
 ■ — = a^ — ab + o^ • 
 
 a -hb 
 
DIVISION 105 
 
 From the form of the quotient and its relation to the 
 divisor we state : 
 
 77. The sum of the cubes of two quantities may he divided 
 hy the sum of the quantities. 
 
 The quotient is the square of the first quantity^ minus the 
 product of the two quantities^ plus the square of the second 
 quantity. 
 
 Exercise 35 
 
 Divide by inspection : 
 
 1. a^-\-h^ by a + h. 8. m^ + 125 by m + 5. 
 
 2. m^ + n^ by m + n. 9. 27 + a^ by 3 + a. 
 
 3. e^-\-m^ by (? + m. 10. 27 <?3 + 64 by 3 c + 4. 
 
 4. d^ + 7^hjd^-x. 11. 64 2:3 + 125 by 4 :rH- 5. 
 
 5. 0^ + ^ by 2: + 2. 12. 272^3^4-125 by 3 2:?/ + 5. 
 
 6. a;3 + 27 by a: + 3. 13. 125 a^ + 512 by 5 a + 8. 
 
 7. 2j3^_64 by2; + 4. 14. 512^3 + 72963 by 8a + 96. 
 
 15. 1 + 1000^:3 by 1 + 10 a;. 
 
 16. 1000 a%^ + 64 c by 10 a5 + 4 e, 
 
 IV. The Sum and Difference of Any Two Like Powers 
 
 By actual division we may prove that the following 
 binomials may be divided by the corresponding binomial 
 divisors. The different groups are arranged according to 
 the exponents of the powers, whether odd or even num- 
 bers, and, also, according to the signs between the terms. 
 
106 
 
 SPECIAL CASES 
 
 The Difference of Even Powers, 
 
 etc. 
 
 may each he divided by either a -\- b or a — b. 
 
 The Difference of Odd Powers, 
 
 - 
 
 . may each be divided by a — b only. 
 
 a-b ^ 
 a^-¥ 
 a^-h^ 
 a'' -b'^ 
 etc. 
 
 The Sum of Odd Powers 
 
 a+b 
 a^ + b^ 
 a^ + 6^ 
 a7 + 67 
 
 etc. 
 
 may each be divided by a -\- b only. 
 
 The Sum of Even Powers. 
 
 a^ + J2, a* + &^, a® + &®, etc., are not divisible by either a + b or a — b. 
 
 The following divisions will assist us in deriving the 
 method for writing the quotients of any binomial divi- 
 sions by inspection. 
 
 ' =a^ + a^b + ab^ + b\ 
 
 = a* + a^b + a%^ + ab» + b*. 
 
 = a* - a^b + a^b^ - ab^ + &*. 
 a-\-b 
 
 a* 
 
 -M 
 
 a 
 
 -b 
 
 a* 
 
 -b^ 
 
 a 
 
 + b 
 
 a6 
 
 -66 
 
 a 
 
 -b 
 
 a6 
 
 + 66 
 
hiviston 107 
 
 From the form of the quotients and their relations to 
 their divisors we state : 
 
 78. 1. The number of terms in the quotient is the same 
 as the exponent of the powers of the dividend. 
 
 2. The exponent of a in the first term of the quotient is the 
 difference between the given exponents of a in the dividend 
 and divisor^ and this exponent decreases by 1 in each succes- 
 sive term. 
 
 3. The exponent of h is 1 in the second term of the quo- 
 tient, and this exponent increases by 1 in each successive 
 term until equal to the difference of the given exponents of b 
 in the dividend and divisor. 
 
 4. If the sign of b in the divisor is +, the signs of the 
 quotient are alternately 4- and —; if the sign of b is — , 
 the quotient signs are all + . 
 
 Exercise 36 
 Divide by inspection : 
 , a^-b^ ^ x^ + m^ „ //8_i 
 
 ■L. --• 6. • 11. 
 
 a-b 
 
 a^ — a^ 
 
 a — x 
 
 a^ + y^ 
 
 x + y 
 
 m^-a^ 
 
 m-\-x 
 
 (?7 + C^7 
 
 2. 7. 12. 
 
 13. 
 
 4. 9. '-—' 14 
 
 x-^m 
 
 a^-c^ 
 
 a— c 
 
 a%^-c^ 
 
 ab -{- c 
 
 ^+1 
 
 x + l 
 
 s" + /I 
 
 a-1 
 
 1-x^^ 
 1 + x ' 
 
 x^y^ — z^ 
 xy — z 
 
 rn) — Yi^ 
 m — n 
 
 5. .-^. xo. :— . 15. ^±|2. 
 
 c-\- d x + y w + 2 
 
108 SPECIAL CASES 
 
 Exercise 37 
 Oral Drill in Multiplication and Division by Inspection 
 Give the fallowing products : 
 
 1. (a + 5)2. 11. (7w + 3?/;3)(7m-3«/2). 
 
 2. (a -7)2. 12. (a + 5)(a + 3). 
 
 3. (a + Sxy, 13. (a;-9)(a;-7). 
 
 4. (w-5n)2. 14. (a6-6)(a5 + 4). 
 
 5. (3a;-2«/)2. 15. (m-14)(w + 7). 
 
 6. (bax-^hyf, 16. (c- ll)(c + 15). 
 
 7. (7?ww-ll)2. 17. (3a + l)(2a + l). 
 
 8. (« + 7)(a-7). 18. (4 2-7)(50-2). 
 
 9. (3a + 5)(3a-5). 19. (3a:+10)(2 2;-4). 
 10. (5a;-12)(5rc + 12). 20. (5 a - 7)(7 a + 10). 
 
 Give the quotients of the following : 
 o, - - oa w^-27 ,, 64^:3 + 216 
 
 21. 27. 33. ■ 
 
 8rr + 6 
 125a;3 + 343 
 
 5a;-i-7 
 
 125a;3_^729 
 
 52; + 9 
 
 22. -• 28. —• 34. 
 
 23. -. 29. — 35. 
 
 aP' 
 
 -62 
 
 a 
 
 -h 
 
 a2 
 
 -y^ 
 
 a 
 
 + h 
 
 aP 
 
 -16 
 
 a 
 
 + 4 
 
 m^ 
 
 -36 
 
 m 
 
 + 6 
 
 977Z2-64 
 
 3 
 
 w + 8 
 
 7^ 
 
 -8 
 
 24. 30. — 36. 
 
 w + 6 ^x — o 
 
 m-3 
 
 (73-125 
 
 ^-5 
 
 8-27^:3 
 
 2-3a: 
 
 64:r3-125 
 
 4:X-5 
 
 8^34.1 
 
 2a + l 
 
 27 w3 + 125 
 
 a — x 
 
 m^ — n^ 
 
 m— n 
 
 <^ + d^ 
 
 32. ^ ' 38. 
 
 x-2 3m + 5 c? + (^ 
 
CHAPTER IX 
 
 FACTORING. REVIEW 
 
 '• 79. When an algebraic expression is the product of two 
 or more expressions, each of these latter quantities is called 
 a Factor of it. 
 
 Thus, a% = a X a X b. 
 
 And the factors of a% are a, a, and b. 
 
 80. Factoring is the process of separating an expression 
 or quantity into its factors. 
 
 Under separate groups we will now consider the types 
 of expressions whose factors are readily obtained. 
 
 WHEN EACH TERM OF AN EXPRESSION HAS THE SAME MONOMIAL 
 FACTOR 
 
 Let US compare the following expressions : 
 
 (1) 10 -f 15 + 20 = 5(2 H- 3 + 4). A factor " 5 " in each 
 term. 
 
 (2) a3+2a2 + 4a=:a(a2 + 2a + 4). A factor "a" in 
 each term. 
 
 In both expressions we have divided each term by the 
 factor common to the terms. 
 
 The factors that make up the expression are, therefore, 
 
 The common factor and the quotient obtained hy dividing 
 the expression hy it. 
 
 109 
 
110 FACTORING. REVIEW 
 
 Applying the principle we have the following : 
 a2 + a=rt(a + l). 
 5a;8 + 10a;= 6x(x^ + 2). 
 3 ms + 9 m2 - 12 m = 3 m(m^ + 3 m - 4). 
 da%-6 am + 9 a&3 = 3 ah^cP' - 2 a6 + 3 ft^). 
 
 Exercise 38 
 Factor : 
 
 1. 5a+10. 6. lOir-15^ + 20. 
 
 2. 7^-35. 7. 16m-247i + 8. 
 
 3. 12 a -15. 8. 11 am + 22 an -11, 
 
 4. 14m -26. 9. lax — liaz-\-21a. 
 
 5. 4a-85 + 16. 10. lS + 9n-21m, 
 
 11. 5 a6<? + 10 abx + 15 abi/. 
 
 12. 5 2; — 20 a:^ -\-SOxz — 40 a:?/2. 
 
 13. a2 + a. 19. 15m^n-20mV-S6mnK 
 
 14. a3_^2_^. 20. na^x + 51a^x^-d^a2^. 
 
 15. m% + M7i2. 21. 27 w%— 9m%2 4.l8w%3. 
 
 16. 0^1/ — xy-\- xy^. 22. 13 a* — 39 a^^ — ^b a^n. 
 
 17. a^i^ — ^252 _|_ ^253^ 23. a^x^y — aaPy^ — ax^y^. 
 
 18. 4a«-12«2+l6a. 24. a^mH^ - ahnhi^ - a^whiK 
 
 25. a^J* - 2 ^352 ^ 3 ^253 _ 4 ^455. 
 
 26. rn^x — 3 w^a: + 2 m^a; — 5 mx, 
 
 27. 4a46 + 5a3j2_8^253^2a54. 
 
 28. a^m — 2 a^Jrw + aJ^^ _ ^^3 _ 2 ^77^2 _ ^^^ 
 
TRINOMIAL EXPRESSION 111 
 
 TRINOMIAL EXPRESSIONS 
 
 I. The Perfect Trinomial Square 
 
 81. A Perfect Square is a product of two equal factors. 
 
 82. A Square Root of a perfect square is one of its two 
 
 equal factors. (Square root is indicated by the sign y'. 
 Thus, V16 = 4.) 
 
 By Arts. 69 and 70 : 
 
 (a - 6)2 = a2 - 2 a& + h^. 
 
 Each resulting expression is a perfect trinomial square. 
 In each product 
 
 The first term is a perfect square. 1 ^^ -, 
 
 The last term is a perfect square. J 
 
 The middle term is twice the product of the square roots 
 
 of the square terms. 
 The sign of the middle term is + or — according as the 
 
 sign of the second term of the given binomial was + 
 
 or — . 
 
 These four elements are the conditions necessary for a 
 perfect trinomial square. Thus : 
 
 a2 + 10 a + 25. x^ -2^x + 144. x'^y^ + 8 xy + 16. 
 
 ,^^2 _ 18 m + 81. * 4 a2 + 12 a& + 9 h\ 9 c^ - 42 c^ + 49. 
 
 are perfect trinomial squares, for each trinomial has two 
 positive square terms with a second term twice the product 
 of the square roots of the square terms. 
 
112 FACTORING. REVIEW 
 
 To factor a perfect trinomial square : 
 
 If necessary^ arrange the terms in ascending or descending 
 order. 
 
 Take the square roots of the first and last terms^ and con- 
 nect them with the sign of the middle term. 
 
 The resulting binomial is one of the two required equal 
 factors. 
 
 Examples : Factor a'^ -\- 12 a + S6, 
 
 The square root oi a^ = a. 
 
 The square root of 36 = 6. 
 
 The sign of the middle term is + • 
 
 Therefore, a + 6 is one of the two required factors. 
 
 Or, a2 + 12 a + 36 = (a + 6)2. 
 
 Factor 9 a'^ -{- 25 2^ - SO ax. 
 
 Rearrange thus, 9 a^ — 30 ax + 25 x^. 
 
 The square roots of the first and last terms are 3 a and 5 x. 
 
 The sign of the middle term is — . 
 
 Hence, 9 a^ _ 30 ax + 25 a:2 = (3 a - 5 xy. 
 
 Exercise 39 
 
 1. a^ + 6a-{-9. 9. a^^-{-Uah -^4:9. 
 
 2. a2+8«4-16. 10. hV-24:bx-\-lU, 
 
 3. a2 + 10a+25. M.. 64. -\- c^cP - IQ cd. 
 
 4. a2_i0a + 25. 12. mV -\- SQ mn + S24:. 
 
 5. ic2-14a;+49. 13. x^yh'^ -40 xgz + ^00. 
 
 6. m2+20m + 100. 14. a^^e^ - 12 ahc+S6. 
 
 7. a2+81-18a. 15. a^mV -[-SS amn -{- S61, 
 
 8. c2-30(? + 225. 16. a^-6ah-]-9h^. 
 
TRINOMIAL EXPRESSION 113 
 
 17. m^ - 12 mn-\-S6n^. 24. 16^2 -24 m + 9. 
 
 18. 2^-^lSxt/ + Slf. 25. 9c2-42c + 49. 
 
 19. 4a2_4ttw + m2. 26. 25 a2 + 60 a6 + 36 52. 
 
 20. dx^-\-QxZ + z\ 27. 16 m2- 72 7710^+81 2^2. 
 
 21. SCX + X^-^IQC^, 28. 36 2^ + 1082:2^2 4. 81 ^4. 
 
 22. 4:X^-12x+9, 29. 49^252 4. 140 a6c+ 100 (?2. 
 
 23. 30 a + 25 + 9^2. 30. 81 a*-198 a2a^+121a^. 
 
 II. The Trinomial whose Highest Power has the 
 Coefficient Unity 
 
 By Art. 72 : 
 
 (a + 2)(a + 3) = a2 + 5a + 6. 
 
 The terms of the product are obtained as follows : 
 
 a^ is the product of a by a. 
 + 5 a is the product of (2 + 3) by a. 
 + 6 is the product of 2 by 3. 
 
 From which we conclude : 
 
 The first and last terms of the trinomial are products of 
 the first terms and the second terms, respectively, in 
 two required binomials. The middle term of the tri- 
 nomial is the product of the common literal term of 
 the binomials by the sum of the numerical terms. 
 
 It remains therefore, in determining the factors of a 
 trinomial in this form, to write 
 
 a2 + 5a + 6=:(a+ )(a + ), 
 
 the missing terms being the two factors of 6 whose sum is 
 5 (i.e. 2 and 3). 
 
 F. H. S. FIRST TEAR ALG. — 8 
 
114 FACTORING. REVIEW 
 
 Similarly : 
 a2 + 10 a + 16 = (a + ?)(a + ?). (The factors of 16 whose sum is 10.) 
 x2 - 9 a; + 20 = (a; - V) (x - ?) . (The factors of + 20 whose sum is - 9.) 
 
 In the following exercise the W possible cases of ex- ' 
 pressions formed by binomials having like or unlike signs 
 in their second terms are separately considered. ] 
 
 Exercise 40 
 
 (1) The third term + . The second term + . 
 
 The signs of the last terms of both factors +. *• j 
 
 Illustration : x^ -h S x + 15 = (a: + 5) (x + 3). 
 
 Factor : 
 
 1. ^2 + 3 ^ 4. 2. €. 2:2 + 12 rr + 32. 
 
 2. a2 + 5 a + 6. 7. a^ + 19 a + 48. 
 
 3. a^-\-l a + 12. 8. c2 + 20 c + 36. 
 
 4. 52 _^ 8 5 _^ 15. 9. m^-^Slm-h 58. 
 
 5. w2 + 12 w + 20. 10. x^-{-14:x + 48. 
 
 (2) The third term + . 77ie second term — . 
 The signs of the last terms of both factors — . 
 
 Illustration : a;^ - 8 a: + 15 = (a: - 5)(a: - 3). 
 
 11. a2 - 4 a + 3. 16. ^2 _ 14 ^ _^ 24. 
 
 12. w2 - 5 m + 4. 17. c^ -lie -^ 24. 
 
 13. 2^2 _ 13 ^ 4. 12. 18. 2:2 _ 10 a; + 24. 
 
 14. a^-Sa-^ 15. 19. m^-15m-h 36. 
 
 15. 52 _ 11 5 + 28. 20. c2 - 15 c 4- 44. 
 
TRINOMIAL EXPRESSIONS 115 
 
 (3) The third term — . The second term either + or —. 
 
 The signs of the last terms of the factors unlike^ the 
 
 greater last term having the same sign as the second term of 
 the given trinomial. 
 
 Illustration : a:^ + 2 a; - 15 = (a: + 5) (a: - 3). 
 
 a;2-2a;-15= (a; - 5) (a: + 3). 
 
 21. x^ -\- X - 6, 28. m^ -11m- 60. 
 
 22. m^ + m- 20. 29. c^-15c - 34. 
 
 23. a^- a- 30. 30. m^+11 m- 60. 
 
 24. x^+Zx- 10. 31. ^2 _ 23 a - 78. 
 
 25. 52 _ 5 5 _ g, 32. ^2 _ 9 ^ _ 112. 
 
 26. d^ -ba- 14. 33. z2 _^ 13 ^ _ 140. 
 
 27. a2 + 8 a - 20. 34. y'^ - (d y - 135. 
 
 35. a2 4- 14 a + 24. 45. :r2 - 11 a: + 24. 
 
 36. 2^2 _ 9 a; _ 22. 46. x^- bx- 24. 
 
 37. c2 + 7 (? - 60. 47. a2 -f 17 « + 60. 
 
 38. x^-l\x- 26. 48. 39 - 16 ?/ + y2. 
 
 39. 10 + 7 :r + a:2. 49. m^ - 15 m - 76. 
 •40. 28 - 16 w + w2. 50. c2 - 72 + c. 
 
 41. :r2 - 2; - 132. 51. x^ -\-SSx- 70. 
 
 42. 110 - 53 ^ - x^ 52. m2 - 8 m - 84. 
 
 43. y^ -15y - 54. 53. :?:2 _,_ 2 :z; _ 120. 
 
 44. c^ + 21c- 46. 54. ip'-y- 210. 
 
116 FACTORING. BEVIEW 
 
 III. The Trinomial whose Highest Power has a Coefficient 
 Greater than Unity 
 By Art. 73 : 
 
 (a + 3)(2 a + 1) = 2 a2 + 7 a + 3. 
 
 The terms of the product are made up as follows : 
 
 2 a^ is the product of a by 2 a. 
 + 3 is the product of + 1 by +3. 
 
 + 7 a is the sum of the cross-products of a by 1 and 2 a by 3, 
 or, a + (3 X 2 a) = a + 6 a = 7 a. 
 
 Or, applying the principle to any pair of binomials, we 
 may state the general conditions governing their product 
 thus : 
 
 The first term is the product of the two first terms of 
 the binomials. 
 
 The last term is the product of the two second terms of 
 the binomials. 
 
 The middle term is the algebraic sum of the cross-prod- 
 ucts of the first and second terms of the given binomials. 
 
 Knowing, therefore, the process by which each term was 
 obtained in multiplication, we reverse the reasoning and 
 find the numbers from which the terms were formed. 
 
 Thus, to factor 2 a^ + 7 a + 3, 
 
 we seek, 1st, Two numbers whose product is 2 a^. 
 
 2d, Two other numbers whose product is + 3. 
 3d, That particular arrangement of these selected numbers 
 the sum of whose cross-products is + 7 a. 
 
 The following discussion will show that a particular 
 arrangement of the selected term-factors is necessary to 
 obtain the middle term of the given trinomial. 
 
 I 
 
TRINOMIAL EXPRESSIONS 117 
 
 I Factor 6^2 + 19^ + 15. 
 
 Since all the signs of the trinomial are +, the signs of the terms of 
 the binomial factors will be + . 
 
 Factors of 6 : Factors of 15 ; 
 
 2 3 
 
 3 5 
 
 The sum of the cross-products, (3 x 3) + (5 x 2) = 19. 
 
 Hence the factors of 6 a^ + 19 a + 15 = (2 a + 3) and (3 a + 5). 
 
 To show that this one pair of factors and this particular arrangement 
 of the numerical coefficients are the only possible selections that will 
 give by multiplication both end terms and the middle term of the 
 given trinomial, let us examine the results from other selections. 
 
 Suppose we had selected 
 
 Factors of 6 : Factors of 15 : 
 6 15 
 
 1 1 
 
 The sum of the cross-products, (1 x 15) + (1x6)= 21. 
 Hence, this arrangement is manifestly wrong. 
 
 Again, Factors of 6 : Factors of 15 : 
 
 6 1 
 
 1 15 
 
 The sum of the cross-products, (1 x 1) -f (15 x 6) = 1 + 90 = 91. 
 And the arrangement fails to produce the required middle term. 
 
 Therefore, we conclude : 
 
 In every case we seek the particular pairs of factors of the 
 first and last terms of the given trinomial^ the algebraic sum 
 of whose cross-products shall he the middle term. 
 
 The following exercise is so arranged as to assist the 
 beginner in determining the different possibilities that may 
 arise from like and unlike signs in the binomial factors 
 involved. 
 
118 FACTORING. REVIEW 
 
 Exercise 41 
 
 (1) The third term +. The second term +. 
 The signs of the last terms of both factors +. 
 
 Factor : 
 
 1. 2 «2 + 3 ^ + 1. 5. 8 a2 + 22 a + 15. 
 
 2. 3 ^2 + 5 a 4- 2. 6. 6 ^2 +23 a + 20. 
 
 3. 4 a2 + 8 a + 3. 7. 9 ^2 + 21 a + 10. 
 
 4. 6 a2 + 17 a + 12. 8. 25 a2 + 35 a + 12. 
 
 (2) The third term + • The second term — . 
 The signs of the last terms of both factors — . 
 
 9. 6a^-lx-h2, 13. 12 a;2 _ 23 re + 10. 
 
 10. 4 2^2 - 16 2; + 15. 14. Qx^-Ux-\- 35. 
 
 11. 6x^-l^x + 6. 15. 28 2^2 _ 45 ^ + 18. 
 
 12. 10 2^2 _ 41 ^ _^ 21. 16. 15 2^2 _ 59 2; + 56. 
 
 (3) The third term — . The second term either -^ or —. 
 The signs of the last terms of the factors unlike^ the 
 
 greater cross-product having the same sign as the second term 
 of the given trinomial. 
 
 17. 
 
 6^2+7^_5. 
 
 24. 
 
 30c2-ll(?-30. 
 
 18. 
 
 102:2_3^_13. 
 
 25. 
 
 9^2-15^-50. 
 
 19. 
 
 15m2-7m-4. 
 
 26. 
 
 12 2:2_40a,_63. 
 
 20. 
 
 122;2_i3^_14, 
 
 27. 
 
 1022 4.;2_24. 
 
 21. 
 
 16 a2 4. 18 a -9. 
 
 28. 
 
 21a2+65a-44. 
 
 22. 
 
 20^>2_276-14. 
 
 29. 
 
 42a2-25a-60. 
 
 23. 
 
 21a2_23a-20. 
 
 30. 
 
 20m2-37?w-18. 
 
TRINOMIAL EXPRESSIONS 119 
 
 Exercise 42 
 Factor: Miscellaneous Factoring 
 
 1. a2_ 14^^49. ' 15. 2b-10x + 3^. 
 
 2. 4a24_7a + 3. 16. a;2 + 49a; + 48. 
 
 3. 5tt2_35^. 17. 128-24c + c2. 
 
 4. 2a2_7^_^6. 18. 15a^-a:-28. 
 
 5. 2^2 _|. 13 ^ + 30. 19. a;2-16a: + 48. 
 
 6. 9:r3_36^2_|_io8a^. 20. 6'2t^ - 8 (?d; + 16. 
 
 7. 36-182: + a;2. 21. m27i2-8m7i-48. 
 
 8. a;2_i4^4.48. 22. a%'^-10a%'^-lba%K 
 
 9. 4(?2-f.9c + 2. . 23. 2^ -19 2: +48. 
 
 10. 6?2-19a + 60. 24. 3m2-25w-18. 
 
 11. Uac + l^hc-22cd. 25. 49^4-112^2/1 + 64^2. 
 
 12. 36^2-60^ + 25. 26. 12 -"Imn + mH^ 
 
 13. a:2_iia;_180. 27. 24 a6 + 9 ^2 + 16 62. 
 
 14. a^-mx-nO. 28. a:2_26:r+48. 
 
 29. 11 m^n^ + 22 ^2^^ - 77 m^yi^. 
 
 30. a2_i2a-28. 
 
 31. 52 wV + 39 ^271 — 26 mn^ — 13 mhi^. 
 
 32. -18 0^ + 2:2 + 81. 35. ^m^-50mn + 7n^. 
 
 33. a;2_i3^_48. 35. 2:2-222:-48. 
 
 34. 4 2;2+17a: + 4. 37. 4:n^i-9x^-l'2nx. 
 
120 FACTORING. REVIEW 
 
 BINOMIAL EXPRESSIONS 
 
 I. The Difference of Two Squares 
 
 By Art. 71 : 
 
 (a + b)(a-b) = a^-b^ 
 
 Hence, the factors of a^ — b^ are (a + b) and (a — 5). 
 In like manner : 
 
 a^ - x^ = (a -\- x)(a - x). 
 
 a2-9 = (a + 3)(a-3). 
 
 4a2-25 = (2a + 5)(2a-5). 
 
 Hence, in general, to factor the di£Perence of two 
 squares : 
 
 Extract the square roots of the square terms. 
 
 One factor is the sum of their square roots. 
 
 The other factor is the difference of their square roots. 
 
 Factor : Exercise 43 
 
 1. a^-b^. 11. 4a2_9. 21. *a^ - xK 
 
 2. a^-x\ 12. 9^2-25. 22. *a:*-/. 
 
 3. a^-1. 13. 16^2-9. 23. *a^-ie. 
 
 4. m2-l. 14. 4a;2-25. 24. *m* - 81. 
 
 5. a2_4. 15. 9^2-64. 25. *16m4-8l2;*. 
 
 6. 2^-9. 16. a2_9 52. 26. 9a262_49^^2. 
 
 7. ?/2_i6. 17. 4a2-49a;2 27. 16a2^,2_9^^2,,2, 
 
 8. ^2-25. 18. 16 2^2 _ 25 ^2, 28. 121 a^b^-c\ 
 
 9. 2^-49. 19. 36 0:2-81^2. 29. 25a;4/-862io. 
 10. w2-81. 20. 49^2-100^2. 30. 144^10-169. 
 
 * Three factors. 
 
BINOMIAL EXPBESSIONS 121 
 
 31. (a + hy-c^= (a-\-h-hc)(a + b-c). Result. 
 
 32. (m-ny-x^ 34. (a-2f-x^. 36. 16(« + 2)2-9. 
 
 33. ((?4-t?)2-l. 35. iQa + hy-c^ 37. 36((? + 2)2-49. 
 
 38. a2_(5_|.^)2. 
 
 = (^a-\-b-\-c)(a-b-c). Result. 
 
 39. m^-(n + iy. 43. 25-9(a+J)2. 
 
 40. a2_(^_i)2. 44, 36a2_25(a + l)2. 
 
 41. 16-(a + ^)2. 45. 49w2-9(w-l)2. 
 
 42. 25-(3 + 2;)2. 46. 64ic2«/2_25(ajy+l)2. 
 
 II. The Difference of Two Cubes 
 
 83. A Perfect Cube is a product of three equal factors. 
 
 84. A Cube Root of a perfect cube is one of its three 
 equal factors. 
 
 By Art. 76: 2i,i^ = „. + <,j + j,. 
 
 a — b 
 
 From which the factors of a^ — b^ are (a — b) and (a^ + ab + b^). 
 
 Similarly ; a^ - x^ = (a - x) (a^ + ax + x^). 
 c3_8 = (c-2)(c2+2c + 4). 
 27^3 - 64 =. (3 m - 4) (9 m^ + 12 m + 16). 
 
 Hence, in general, to factor the difference of two cubes : 
 
 One factor is the difference of the cube roots of the 
 quantities. 
 
 The other factor is the sum of the squares of the cube 
 roots of the quantities plus their product. 
 
u 
 
 52 
 
 FAC 
 
 'TOEING, REVIEW 
 
 
 
 Factor : 
 
 
 Exercise 44 
 
 
 
 1. 
 
 a^ - ¥. 
 
 8. 
 
 2^3-125. 
 
 15. 
 
 8^3-27. 
 
 2. 
 
 a^ — a^. 
 
 9. 
 
 27 -m3. 
 
 16. 
 
 27 - 64 m3. 
 
 3. 
 
 m^ — n^. 
 
 10. 
 
 64-2:3. 
 
 17. 
 
 125^3-64^3. 
 
 4. 
 
 e^-1. 
 
 11. 
 
 a%^-U. 
 
 18. 
 
 216 2^3-27. 
 
 5. 
 
 ^3-8. 
 
 12. 
 
 mV - 125. 
 
 19. 
 
 27^363-512. 
 
 6. 
 
 0^-21. 
 
 13. 
 
 27 - a^y\ 
 
 20. 
 
 8 m3- 343^13. 
 
 7. 
 
 m3-64. 
 
 14. 
 
 64 — mV. 
 
 21. 
 
 125 2:3_ 729^^3. 
 
 III. The Sum of Two Cubes 
 
 By Art. 77: ?^=a^-ah + 
 
 From which the factors of a^ + ^3 are (a + h) and 
 ia^-ab + h^). 
 
 Similarly : a^ + x^=(a + x) (a^ - ax + x^). 
 c8 + 8=(c + 2)(c2-2c + 4). 
 125 + a:3 =(5 + a;)(25 - lOx + x^). 
 27 m3 + 64 = (3 m + 4)(9 m^ - 12 m + 16). 
 
 Hence, in general, to factor the sum of two cubes : 
 
 One factor is the sum of the cube roots of the quantities. 
 The other factor is the sum of the squares of the cube roots 
 of the quantities minus their product. 
 
 Factor: Exercise 45 
 
 1. 2^3 + ^3. 4. ^3+8. 7. (^ + 125. 
 
 2. a^+m^ 5. (?3_^27. 8. 27 + 2^8. 
 
 3. 2:3^1. ^. a3_|_64, 9. 64-1-^3. 
 
EXPRESSIONS HAVING FOUR TERMS 123 
 
 10. wV+64. 13. M-^M^ 16. 8^3+343^. 
 
 11. A3 + 125. 14. 8^3+27 63. 17. 512^3+27^^. 
 
 12. 27H-a353. 15. 216^3 4- 27. 18. 1000 2^3^729. 
 
 Exercise 46 
 Factor • Miscellaneous Factoring 
 
 1. a^-9x^. 11. 64^3+125. 21. Q4:c^-}'1, 
 
 2. a3-8m3. 12. 25-64m2. 22. 81 c*^^*- 36 ic^. 
 
 3. a3+27. 13. 216-27(?3. 23. 729a^-Sc^ 
 
 4. c3 - 27 a;3. 14. 4 m* - 9 a;*. 24. 64 a;3 - 125 ^\ 
 
 5. 2:3 4_ 125. 15. 8w3n3-125. 25. 64a:6_i. 
 
 6. 4 77i2_9. 16. 25 2:6 _ 9. 26. 64 2:8_i. 
 
 7. 1-64 2:3. 17. 272:6-8/. 27. 512 2:6 _ 27. 
 
 8. 27 2:3 + 8. 18. 49m6-9 7i6. 28. 1292^-125^. 
 
 9. 125^3-8. 19. 8^6 -125 53. 29. 1000 2:3 _ 27 2/3. 
 10. 25 a2- 36 2:2. 20. 64 m9- 27 7^9. 30. 1728 2:3_ 1331. 
 
 EXPRESSIONS HAVING FOUR TERMS 
 
 Certain common expressions having four terms can be 
 so grouped as to come under types that have already 
 been considered. 
 
 I. The Grouping of Terms to show a Common 
 Binomial Factor 
 
 In the expression ax + bx + ex, the factor " x " is common to the 
 terms. Factoring, 
 
 ax + bx -{• ex =(a + b + c)x. 
 
124 FACTORING. BEVIEW 
 
 Similarly, in a(x -\- y) -\- h{x + ?/)+ c{x + y), the binomial factor 
 " (a; + y) " is common to the terms. Factoring, 
 
 a{x + y)+ h{x + y)-\-c{x + ij) = {a + & + c)(a;+ ?/). 
 
 In the above illustration the common binomial factor is 
 readily seen, but in many expressions it is necessary to 
 make a careful examination of the given terms before a , 
 particular binomial can be found. In such cases we find, I 
 by inspection, what terms hear to each other the same rela- 
 tion; and then we obtain a common binomial factor by the _, 
 necessary rearrangement and grouping. Thus : 
 
 Factor ax -{- ay -\- hx ■\- hy. 
 
 It will be noted that " a " is common to the first two terms, and 
 that "&" is common to the last two terms. Furthermore, if the 
 common factor, a, is taken out of the first two terms, the same expres- 
 sion results as when the common factor, 5, is taken from the last two 
 terms. 
 
 Hence, the process : 
 
 ax + ay -\- hx ■{ hy = (ax + ay) + (bx + hy) 
 = a(x-{-y) + b(x-^y) 
 Collecting coeflacients, =(a + h)(x + y). Result. 
 
 Sometimes we make use of the parenthesis preceded by the minus 
 sign in order to group the terms so as to show a common binomial 
 factor. Thus : 
 
 Factor ax + ay — bx — hy. 
 
 ax -\- ay — bx — by = (ax + ay) — (bx + by) 
 
 = a(x + y)-b(x + y) 
 
 Collecting coefficients, =(a — b)(x + y). Result. 
 
 Many examples of this type admit of more than one way of group- 
 ing terms. But the final result is unchanged, no matter what may 
 have been the grouping for a common binomial factor. The change 
 is in the order of the resulting factors. 
 
EXPliESSIONS HAVING FOUR TERMS 125 
 
 Factor: Exercise 47 
 
 1. ac -\- be -\- ad -\- bd. 11. m^— 2m-\- mx — 2 x. 
 
 2. am + mx -\-an + nx. 12. 2c^ — 6c-{-cd — Sd. 
 
 3. ac — bc+ ad— bd. 13. a^ -\- a^ + a -^ 1. 
 
 4. ayn — mx -{-an — nx. 14. a?^ + a;^ + a; + 1. 
 
 5. a^ + ab -\- ac -\- be. 15. ^3_^ 3 ^2_|_ 2 ^ + 6. 
 
 6. e^ -\- ex -\- e?/ -i- xy. 16. 2:^ + a:^ — re — 1. 
 
 7. a2 4. ^5 _^ <^ 4. 5. • 17. ^4 _|_ ^ _ ^ _ 1, 
 
 8. ^2 4- aa; + « + a:. 18. m^ 4- 2 m^ — 9 m — 18. 
 ^. a^-ax^a-x. \^. e'^-\-2(f^-\-e-\- 2. 
 
 10. a^-ab^-2a-2b. 20. a* + 3 a^ - 27 a - 81. 
 
 II. The Grouping of Terms to form the Difference of 
 Two Squares 
 
 By multiplication : 
 
 (a + & + c) (a + & - c) = «2 + 2 a& + Z>2 _ c2. 
 (x + a + 2) (x + a - 2) = x2 + 2 aa: + a2 _ 4. 
 {x - y ^ z){x - y - z)= x^ - 1 xy -^ y'^ - z^. 
 
 Each result has four terms. Three of the terms are per- 
 fect squares and the fourth term is twice the product of the 
 square roots of two of the square terms. With these facts 
 in mind, the student will carefully factor the following 
 expressions of similar form. 
 
 Factor a^^-2 ab+b'^- x^. 
 
 a2 + 2 a6 + 62 - a;2 = (a2 + 2 a& + &2) _ x"^ 
 = (a + by-x^ 
 = (a-\-b + x)(a + h — x). Result. 
 
126 FACTOBING, REVIEW 
 
 Factor a:^ -h'^-2hc- (^. 
 
 = [a + (6 + c)][a-(6 + c)] 
 
 = (a + & + c)(a-6-c). Result. 
 
 Factor a;^ _^ ^2 _ ^2 _ 2 ^a?. 
 
 a;2 + a2 _ ^2 _ 2 ax = (x2 - 2 ax + a2) - ^2 
 = (a;-a)2-z2 
 
 = (x — a + 2) (x — a — z). Result. 
 
 It will be clear to the student that the grouping is' 
 simply a matter of determining what three terms make up 
 the desired trinomial square. The term that is not a square 
 is the hey to the grouping arrangement, for its literal factors 
 show the square terms with which it must be grouped. 
 
 It may be of help to the student to familiarize himself 
 with still another method by which the proper grouping 
 may be obtained. From the signs of the square terms we 
 derive the following : 
 
 (1) When only one given square term is plus it is 
 written firsts and the other three terms are inclosed in a 
 pareiithesis preceded hy a minus sign. 
 
 (2) When only one given square term is minus it is 
 written last., and the other three terms are written first in 
 a plus parenthesis. 
 
 Factor : ExerclBe 48 
 
 1. a^^^ab-{-h^-m\ 4. 4:<^- ^cd-\- d? -1. 
 
 2. m^ + ^mn+n^-x^. 5. 4:a^-12ax-\-9 x^-25. 
 
 3. x^-2xy-^y^-z^ 6. a^ + b^- (^-{.2aL 
 
GENERAL AIDS TO FACTORING 127 
 
 8. m2 — 7i2+2m + l. 15. 4 2:2-^2 — 2a — 1. 
 
 9. a^ + 2/2 — ^2 _ 2 a^y. 16. m^ -\- 10 xi/ — x^ — 25 1/\ 
 
 10. a:4 + a:2-4-2a;3. 17. e^ _ ^6 - d^-12d, 
 
 11. a2-62_25c-c2. 18. 4a2 + l-4a:2_4^. 
 
 12. w2-ri2-27i-l. 19. -2rz; + l + 2;2_a:4. 
 
 13. a;2-/ + 2y-l. 20. 6<? + 9a2-l-9tf2. 
 
 REPEATED FACTORING 
 
 The process of factoring must frequently be applied 
 more than once before an expression is resolved into its 
 prime factors. In the following review it is first neces- 
 sary that each example be carefully examined to determine 
 under which type it belongs. Then, having determined 
 that a particular type will apply, it remains to carefully 
 examine resulting factors to see whether or not they are 
 themselves to be refactored. From the different type- 
 forms that have been presented we are able to make the 
 following classification of expressions that we have learned 
 to factor. Basing our conclusions directly upon the num- 
 ber of terms in a given expression, we are able to make 
 the following helpful hints for the factoring of miscel- 
 laneous and unclassified forms. 
 
 GENERAL AIDS TO FACTORING 
 
 First. Determine if each term has a common monomial 
 factor. If so, remove it, and apply the further ex- 
 aminations suggested below. 
 
128 FACTORING. REVIEW 
 
 Second. What is the number of terms in the expression to he 
 factored I* 
 
 I. If two terms, place it under the proper binomial 
 form. 
 
 (a) If the exponents of the terms are even, 
 the coefficients perfect squares, and the 
 second term negative in sign, the expres- 
 sion is the difference of two squares. 
 (6) If the exponents of the terms are divisible 
 by three and the coefficients of the terms 
 are perfect cubes, the expression is the 
 sum or difference of two cubes, according 
 to the sign of the second term. 
 II. If three terms, place it under the proper tri- 
 nomial form. It is well to apply at once the 
 test for a perfect trinomial square, which test 
 failing, we know that it belongs under one of the 
 two remaining trinomial forms. 
 III. If four terms, look for the proper grouping that 
 jvill place it under one of the forms already 
 given. 
 
 (a) If the expression has three perfect square 
 terms, apply the grouping test for the dif- 
 ference of two squares, one a trinomial 
 expression, the other a monomial. 
 (5) If no terms, only two terms, or all four 
 terms are squares, look for a grouping in 
 pairs that shall give a common binomial 
 factor. 
 Third. Continue the processes until the resulting factors 
 cannot be refactored. 
 
GENERAL AIDS TO FACTORING l20 
 
 Typical Cases of Repeated Factoring 
 
 Factor a3 + 10 ^2 _^ 25 a. 
 
 a8 + 10a2 + 25a = a{cfi + 10a + 25) 
 = a{a + 5)2. Result. 
 Factor a^ — x^. 
 a^ - x6 = (a3 + x^) (a3 _ x^) 
 
 = (a + a:)(a2 - ax + x'^){a - x)(cfi + aa; + x^). Result. 
 Factor a^- 13 2^2 ^36. 
 
 a;4 _ i3a;2 + 36 = (a:2 _ 4)(a;2 - 9) 
 
 = (x + 2)(a: - 2) (a: + 3)(a: - 3). Result. 
 
 Factor a:^ — a;* + a;^ — ^. 
 a;^ — x* + x^ — a: = a;[a;^ — x^ + x^ — 1] 
 
 = a;[(a;6-a;3) + (a;2-l)] 
 
 = a:[x8(a:2-l)+(a;2-l)] 
 
 = a:(a;8+l)(a:2-l) 
 
 = x{x + l)(a:2 - a: + l)(x + l)(a: - 1). Result. 
 
 Exercise 49 
 
 Review. Miscellaneous Factoring 
 ^ Factor : 
 
 1. a2_i0a + 25. 8. a^-:^-Qx. 15. m2-3w-70. 
 
 2. a2_ 10^.^24. 9. ^3-64. 16. 4a2_i2a + 9. 
 
 3. a^ — a. 10. m2 — 64. 17. a;* — 8 a;. 
 
 4. 2:2_42;4.4. 11. x^j^\2x+20, 18. 3a2_i2^4.12. 
 
 5. 2;2 + 5a; + 4. 12. a;2 + 9a;+20. 19. 272:5_^2. 
 
 6. m^-4m. 13. a;24-2l2:4-20. 20. 4a:2_|_4^_,_l^ 
 
 7. a;2_^5^_g4^ 14^ a;3_92;. 21. m^+21, 
 
 F. H. 8. FIKST YEAR ALG. — 9 
 
130 FACTORING. REVIEW 
 
 22. 5^2-24^-5. 49. c3-c2_c+l. 
 
 23. Sa^-lOx+S. 50. a^-da. 
 
 24. Sm^-Sm, 51. c3_7^_60<7. 
 
 25. 81-18a^+a^. 52. ^(a-^by-cf^. 
 
 26. 642^-27. 53. c2_^_42. 
 
 27. 5a5-5a. 54. Qa^-lx-S. 
 
 28. a2-lla-26. 55. 27 a;* -a:. 
 
 29. a3+10a2-24a. 56. a* -16 J*. 
 
 30. a* -81. 57. 9(a + l)2-4. 
 
 31. a* +a. 58. c'^- 5c + 4. 
 
 32. ^3 + 4 a2 + 4 a. 59. a^ — S a -\- ax — S x. 
 
 33. m2-<i2 4.2a-l. 60. 2x^-^x-55. 
 
 34. c2_^4^_21. 61. l-a-a2 + a3. 
 
 35. x^-1. 62. 125^3- 8. 
 
 36. b^-b^ + b — 1, 63. a7_^. 
 
 37. a^ + 7 a^ + 10 a. 64. m^ — n^ ^ ^ _ ^, 
 
 38. a^-3a;3_p7^2. gg, 80-5w*. 
 
 39. ^2-2^+1-0^2. 66. a^^Sa^^Sa + 1. 
 
 40. Qx^-X-12. 67. (?5_81(?. 
 
 41. ab-{-be — ad — cd. 68. a;^ — 13 a;2 _|_ 4 ^. 
 
 42. (^-IQc. 69. 4a;4_92;2_62;-l. 
 
 43. a;3_2a; + a;2_2. 70. 36 m%* - 13 ^2/12 4. 1. 
 
 44. m^ -\- m^ -\- m + 1. 71. 343 a:^ — 1. 
 
 45. 2^Jr2a^-x-2, 72. 64a:3+729?/3. 
 
 46. m5-18m3+81m. 73. 1- c^ - c^-\- (^, 
 
 47. a2_3^_4. 74. 20 a: + 100 + a:2. 
 
 48. 8a;2-24a;«/ + 18t/2. 75. 6m2-8m-8. 
 
GENERAL REVIEW 131 
 
 GENERAL REVIEW — GROUP III 
 
 THE SIMPLE EQUATION - AXIOMS 
 
 The same quantity may he added to or subtracted from both 
 members of an equation. 
 
 Both members of an equation may be multiplied or divided 
 by the same quantity. 
 
 TRANSPOSITIONS 
 
 A term may be transposed from one member of an equation 
 to the other member provided its sign be changed. 
 
 PROBLEMS 
 
 To solve : 
 
 (a) Represent the desired unknown value by x. 
 
 (b) State the given conditions of the problem in terms of x. 
 
 (c) Write the equation that fulfills the conditions and solve 
 for X. 
 
 SUBSTITUTION 
 
 In finding Numerical or Literal Values 
 
 Replace the given literal factors in the terms by the given 
 values. Perform all operations and simplify the result. 
 
 In USING Formulas 
 
 Substitute the known values and the necessary constant 
 values. Solve the formula for the required element. 
 
 In VERIFYING Equations 
 
 Substitute in the original equation the value obtained for 
 the root. If simplification makes each member the same^ the 
 root found is correct. 
 
132 FACTORING. REVIEW 
 
 SPECIAL CASES IN MULTIPLICATION - TYPE-FORMS 
 
 The Product of Binomials having Like Terms 
 
 (a + &)2 = a2 + 2 a& + h\ (« _ hy = a^ - 2 ah -\- b^. 
 
 (a + b)(a-b) = a^-b^ 
 
 Trinomials grouped for Binomial Forms 
 
 (a + ft + c)(a + 6 - c) = [(a + 6) + c] [(a + 6) - c] 
 (a + b + c)(a-b + c)= [(a + c) + 6] [(a + c) - ft] 
 (a + ft - c)(a - & + c) = [a + (& - c)] [a - (6 - c)] 
 
 Binomials having Unlike Terms 
 
 (a + b)(a + c) = a'^ + (b + c) a -\- be. 
 
 (ax + b) (ex ■}■ d) = acx^ + (ad + &c) x -f bd. 
 
 The Square of Any Polynomial 
 
 (a + & + c)2 = a2 + &2 + c2 + 2 a& + 2 ac + 2 &c. 
 
 SPECIAL CASES IN DIVISION. - TYPE-FORMS 
 
 The Difference of Two Squares 
 
 = a -\- 0. = a — 0. 
 
 a — b a + b 
 
 The Difference of Two Cubes 
 
 a — b 
 The Sum of Two Cubes 
 
 «i±A'=a2-aft + &^ 
 a + b 
 
GENERAL REVIEW 133 
 
 The difference of equal even powers of a and h is divisible 
 by both a-\-b and a — b. 
 
 The difference of equal odd powers of a and b is divisible 
 by a — b only. 
 
 The sum of equal odd powers of a and b is divisible by 
 a + b only. 
 
 FACTORING - TYPE-FORMS 
 
 A common monomial factor may occur with any of the later 
 forms. 
 
 The Binomial Forms 
 
 The difference of two squares. 
 Illustration : a^ - b\ 
 
 The difference of two cubes. 
 Illustration : a^ — h^. 
 
 The sum of two cubes. 
 Illustration : a^ + h\ 
 
 The Trinomial Forms 
 The perfect trinomial square. 
 Illustration : a^ + 2 aft + ft^ or a^ -2ah + bK 
 
 The trinomial having the coefficient of x unity. 
 Illustration : x^ + bx -\- c. (Where b and c may be any numbers.) 
 
 The trinomial having the coefficient of x greater than unity. 
 Illustration : ax^ + bx-\- c. (Where a, &, and c may be any numbers.) 
 
134 FACTORING. REVIEW 
 
 Expressions having Four Terms 
 
 The polynomial having a common binomial factor. 
 Illustration : ax ■{■ hx -{■ ay ■\- by. 
 
 The polynomial containing both trinomial and monomial 
 squares. 
 
 Illustration : a"^ -h 2 ab + b^ - c\ 
 
 General Suggestions for Factoring 
 
 First. Examine the given expression for a common mono- 
 mial factor. 
 
 Second. The number of terms in the expression and the 
 powers and coefficients will serve to indicate the form under 
 which it comes. 
 
 Third. Apply the proper separation and note carefully 
 whether the factors found can be ref adored. 
 
 Exercise 50 
 
 Review 
 Solve and verify : 
 
 1. 2x(x-\-V)-(x-iy={x-\-^\ 
 
 2. x-l2x-(^x-l-x^)'] = (x-2y-l, 
 
 5. 6(a;-l)2-3(2:+2)2-(3rr-l)(a; + 4) = 0. 
 
 State and solve : 
 
 6. What two numbers are those whose sum is 90, and 
 whose difference is 26 ? 
 
GENERAL REVIEW 135 
 
 7. Divide 84 into two parts, one of which shall be 12 
 more than double the other. 
 
 8. Find the three consecutive odd numbers whose sum 
 is 129. 
 
 9. The difference between the squares of two consecu- 
 tive numbers is 23. Find the numbers. 
 
 10. A's age is three times B' s age, but in twelve years 
 A will be only twice as old as B. Find the present age 
 of each. 
 
 Make the following substitutions : 
 
 11. If a: = 3 and y = — 1, what is the value of (x — y)^ 
 
 12. What is the value of (x-^a-\- 1)^ — 3 (a; — a)^ when 
 a: = — 3 and « = — 7 ? 
 
 13. Find the value of (m + a;) (w + 5 a;) — (3 m + 2 a: + mxy 
 when x= — m. 
 
 14. Find the value of S in the formula S=-(a -f V) 
 when a = J, ^ = i\ and n = 21. 
 
 15. What is the value of the expression (^x—a')(^x-\-a—y') 
 — (x—y)(x-\-a)a— (x — af^ when x = — y and a = ? 
 
 Obtain the following required values by substitution in 
 the proper formulas : 
 
 16. What is the area of a circle whose radius is 10 
 inches ? 
 
 17. In going one mile how many turns are made by a 
 wheel 4 feet in diameter ? 
 
136 FACTOBING. BEVIEW 
 
 18. How many feet will a body fall in 7 seconds ? 
 
 19. What is the sum of the first ten even numbers ? 
 
 20. Show that there are 75.3984+ inches in the circum- 
 ference of a circle whose radius is 1 foot. 
 
 Multiply by inspection : 
 
 21. (a + 3) (a -4). 26. (a; + 4)(:r-9). 
 
 22. (2a-l)(3a + 2). 27. (a^-x^J^x-iy, 
 
 23. (5a;-3^)(2a;-«/). 28. (2;- 14) (2 a; -f 5). 
 
 24. (a+2 6-2(?)2. 29. (3 a -4) (11 a + 15). 
 
 25. (a;-l)(a;-5). 30. (2x + b y^(p x - 12 y^. 
 
 Divide by inspection, supplying omissions by possible 
 divisors : 
 
 31. tziA. 36. 64 + 729^3 
 
 32. ^^ 37. 
 
 33. 64^. 
 
 a-2 
 
 27 + a^ 
 
 ? 
 
 64-a:3 
 
 ^-x 
 
 27 m3 - 125 
 
 ? 
 
 343 - a3 
 
 34. 39. 
 
 35. 40. 
 
 ? 
 
 
 8l2;2- 
 
 25 
 
 ? 
 
 
 27a:3_ 
 
 343 
 
 ? 
 
 
 729 a;3. 
 
 -512 
 
 ? 
 
 
 1000- 
 
 27:1^ 
 
GENERAL BEVIEW • 137 
 
 Factor : 
 
 41. m^-16m. 49. -12^2+^3 + 36 a. 
 
 42. S2^-15a^ + 9x. 50. c^-Slc. 
 
 43. a^-lSa^ + Sla. 51. :?^ - 4 ^2 - 8 rr + 32. 
 
 44. m3+7m2-18w. 52. 3^2_9^^2_22:^. 
 
 45. 8-2a2-4a + a3. 53. x^-21x. 
 
 46. 3 a;5 _|_ 17 ^ + 10 a;. 54. ^^j* + a:2 _ 2 a;3 _ ^2^. 
 
 47. 36(:?;-l)2-25. 55. 9 (a + 1)2 -4. 
 
 48. 30 a;3 + 65 a;2 _ 140 2:. 56. 2;^ _ 4 ^_,_^_4. 
 
 57. What is the quotient when the sum of x^ — x + 7 
 and 3a:2_|_4a;_6is divided by 2; + 1 ? 
 
 58. Solve and verify (a;+ 1)(2 2:-3) = (2;-3)(2a:-l). 
 
 59. Prove that (a -^ x') (a — x -{- 1) = (a — :r) ( a + ic) 
 -\- a + X. 
 
 60. Solve and verify (2; - 5) (3 2: + 7) = (2 a: - 1) (a; - 3) 
 
 + x(x-^l). 
 
 61. Find the value of S x — x {x — ly — 5 x(x -{- 4) 
 — (x — 1)2 when a; = 3. 
 
 62. What must be added to 3a;— [— 2a^+(^ — 5 — a^)] 
 to make x— (^a — h — x^? 
 
 63. If a = 2, 1= 10, and r = — 2, find the value of S in 
 : the expression : rl — a 
 
 r— I 
 
 64. Show that — I is a root of the equation, 3 (a; — 1) 
 (a; + l) = a;(3a;+4). 
 
138 • . FACTORING. REVIEW 
 
 65. Find the value of (a — 5) (a + ^ — <?) — («— ^ + 0^ 
 — ahc when a = 1, ^ = — 1, c = 0. 
 
 66. T he score of a football game was — (3— [12— (3 
 -2-4)]to-4-(-3 + 3-4). What was the actual 
 result of the game ? 
 
 67. Through what distance will a body fall in 3 seconds 
 if a in the formula S = ^ at'^ is 980 centimeters ? 
 
 68. What are the exact binomial divisors of the binomial 
 2:6-1? 
 
 69. Write the square of (a — 3 5 + 5 c). 
 
 70. Find the value of (3 re + 2) (2 2: - 3) + (2 :r - 5)2 
 when x= ^. 
 
 71. What are the factors oi m^ — n^ — jj^— 2np? 
 
 72. If X equals 3, which is the greater expression, 
 
 2x-\Sx-(-5x-l)\ or 5x- [2 x- (^-Sx -{-!)']? 
 
 73. Multiply by inspection (a — 3 bx) (7 a + 4 hx^. 
 
 74. Find the value oi (Sx-4: a}(2x+7 a^-^Sx-bay 
 — aV when x=2a. 
 
 75. If a = 3, 71 = 7, and c? = — 2, what is the value of I 
 in the formula l = a+(n—l)d? 
 
 76. What are the factors oi 2 xy — 14: y — ax + 7 a? 
 
 77. The divisor in a certain example was 2x^ — x— 3, 
 the quotient a^-\-x— 5. What was the dividend ? 
 
 78. A man's weight increases by 80 pounds, and he 
 then weighs 110 pounds less than twice his original 
 weight. What was his original weight? 
 
GENEBAL BEVIEW 139 
 
 79. What are the factors oi b a^ — 2i3^— 5a^? 
 
 80. Simplify aH-[— a— { — a -\- a — {a— 1) \ — a -^ l"]. 
 
 81. From what expression must a^— S x-^1 be sub- 
 tracted to give a remainder of x^ ? 
 
 82. What is the sum of the factors oix^— 1 — 2 3:^ +2x7 
 
 83. What is the result when (5 a: -}- 1) (2 a; — 3) is taken 
 from (ic + 2) (:?: + 1) + (:r - 2)2 ? 
 
 84. To what expression must S m^ — 2'm^ -[-Ihe added 
 to produce ? 
 
 85. What are the factors oi x^ - 21 x^ - 4. a^ -\- lOS ? 
 
 86. A man receives his month's salary in five-dollar 
 and ten-dollar bills. He receives one more bill of the 
 smaller denomination than of the larger. If he was paid 
 •f 110, how many bills of each denomination did he receive ? 
 
 87. How much greater than a; + 1 is the quotient of 
 
 88. What are the factors oi2a^-2f-4:i/-2? 
 
 89. John is 8 years older than William, and John's age 
 is as much above 6 as William's age is below 38. Find 
 the age of each. 
 
 90. Two men were paid the same sum for a week's 
 work. If one of them had received 18 more and the other 
 $S less, the one would have received just twice as much 
 as the other. What amount did each receive ? 
 
CHAPTER X 
 HIGHEST COMMON FACTOR 
 
 85. A common factor of two or more algebraic expres- 
 sions is an expression that can be divided into each without 
 a remainder. 
 
 Thus, a%^, a%\ a'^b^, and a%^ can each be divided by ab. 
 Hence, ab is a common factor of a%'\ a^b*, a%^, and a^6^. 
 
 86. The highest common factor of two or more algebraic 
 expressions is the expression of highest degree that can be 
 divided into each of them without a remainder. 
 
 Thus: 
 
 Given, 
 
 g^ aV)^ is the expression of highest degree that 
 
 will divide each of the four expressions without 
 -,, a remainder. 
 
 From this we make the following conclusion : 
 
 87. The Highest Common Factor of two or more expres- 
 sions is the product of the lowest powers of the factors common 
 to the given expressions. 
 
 The abbreviation "H. C. F." is commonly used in 
 practice. 
 
 THE H.C.F. OF MONOMIALS 
 
 In the simple forms of monomial expressions, the 
 H. C. F. is readily found by inspection. 
 
 140 
 
THE n. C. F. OF POLYNOMIALS 141 
 
 Oral Exercise 
 Find the H. C. F. of : 
 
 1. a%^ and a%^. 7. 16 m% and 24: nh;. 
 
 2. mVx and mnV, 8. 15a^i/^ and 20^:223^ 
 
 3. 2m^x^ and 4m^2^. 9. 12 mV and ISm^/i^. 
 
 4. 3^3(^5 and 6c(^*. 10. IS c^d'^m and 2T<?3J%. 
 
 5. 3 a'^mx^ and 9 «wV. 11. 35 a%^c^ and 42 ^352^3. 
 
 6. 10 x^y^z^ and 15 x7/h^. 12. 51 a^^y^^ and 17 a^i/. 
 
 13. 12 ^25^, 16a63c?, and 20 abc^ 
 
 14. 18 mwa;, 24 mpx, 30 /wwp, and 36 npx. 
 
 15. 5m^nx, 10 mn'^x, 15mnx^, and 20w2?i2a;. 
 
 16. 33 abx, 44 «<?a7, 55 aJ^?, and 66 obex. 
 
 THE H. C. F. OF POLYNOMIALS 
 
 When the given expressions are polynomials, we ob- 
 tain the H. C. F. by factoring. Hence 
 
 THE GENERAL METHOD FOR FINDING THE H.C.F. OF POLYNOMIALS 
 
 88. Factor the given expressions. 
 
 The product of the lowest powers of the common factors is 
 the required H. Q. F. 
 
 Illustrations : 
 
 Ex. 1. Find the H. C. F. of a^- 3 a^- 10 a, a^- 8 ^2+ 15 a, 
 and a^ — 25 a. 
 
 a3 - 3 a2 _ 10 a = a(a - 5)(a + 2) 
 
 a3_25a = a(a + 5)(a-5) 
 H. C. F. = a(a - 5). Result. 
 
142 HIGHEST COMMON FACTOR 
 
 Ex. 2. Find the H. C. F. of a* + 27 a, 2a^ + a^- 15 a\ 
 
 9 a + 6 ^2 + ^3, and a^ - 81 a. 
 
 a^ + 27a = a(a + 3)(a2 - 3 a + 9) 
 2 aS + a2 - 15 a = a(a + 3)(2 a - 5) 
 9 a + 6 «*^ + a8 = a(3 + a)(3 + a) 
 
 a^-Sla = a(rt2+ 9)(a + 3)(a - 3) 
 
 H.C.F. = a(a + 3). Result. 
 
 Exercise 51 
 Find the H. C. F. of : 
 
 1. ah + J, ae -{- c. 11. a^ — ay^^ a^—2 c^y + a?/^^ 
 
 2. ax-\-hx^ ay-\-hy. 12. Saa:^— 3a, aa;2_ ;[o«2; + 25a. 
 
 3. aa; 4- a, ax — a. 13. or^ — 5 a;^ — 14 a;, 2 a::^ + 4 a;^. 
 .4. m^ + m, m^ — m. 14. a^ -f 27 a, a^ — 3a2 + 9«. 
 
 5. c^ + c, c^-1. 15. a;2-3a: + 2, a:2_l^ x^ - x. 
 
 6. a2_9^ a2_^3a. 16. m^-9, 2m^-6m, m^-m-6. 
 
 7. a5 + ^ a%-\-ab. 17. 5m2-10mw+5w2, lOm^-lOw^. 
 
 8. m2-l, (m-l)2. 18. a^ _ i^ ^3 _ ^^ ^4_i^ ^3 _ i. 
 
 9. rcHl, (a: + l)2. 19. 1-a:^ 1 + 5 a^ + 4 a^, x-\-a^. 
 10. a^ — 1, 3^-\-x-\-l. 20. a2_^^^^^^2;, a* + a, a* + a^. 
 
 21. 2a2+5a-3, 3a2 + 5a_12. 
 
 22. a2 + 5a + 6, a3 4-"3«2+2a, a2+7^ + io. 
 
 23. a2-(^»-02^ ^_(a_5)2. 
 
 24. lO-lc + e^, 16-10(?+c2, 6-c-c2. 
 
 25. ar2+4a:-21, 3ar2-8a;-3, 3a^-14a;+15. 
 
 26. 12a^-12b^ 9a2_962, 13a^-15b\ 
 
 27. 47w24-17w-15, 25 + 10m + m2, 3m3-75m. 
 
 28. a^-a^c-Sx^-\-Sex, x^-\-ax^-^a^-Sax, 
 
 i 
 
CHAPTER XI 
 FRACTIONS 
 
 TRANSFORMATIONS 
 
 89. An Algebraic Fraction is an indicated quotient of 
 two expressions. 
 
 Thus, 2, ^±*, 5i±l!, «i±«*±*' are fractions. 
 h a — b x^ — y^ a ■\- h 
 
 90. The Numerator of a fraction is the indicated divi- 
 dend; the Denominator of a fraction is the indicated 
 divisor. 
 
 Thus, in the fraction, "^ "^ ^ ^ 
 
 a^ — 2x 
 
 a^ + 2 a; is the numerator, a^ — 2 a: is the denominator. 
 
 91. An Integral Expression is an expression that does 
 not contain a fraction. 
 
 THE SIGNS OF FRACTIONS 
 
 92. The Sign of a Fraction is independent of the signs 
 of its numerator and denominator. 
 
 Thus, +^, +-, H — ^ are positive fractions. 
 
 , — , ^^- are negative fractions. 
 
 n 5 a: — 1 
 
 143 
 
144 FRACTIOJ^S 
 
 The sign of a fraction may he affected hy the sign of its 
 numerator or denominator or by the signs of both together. 
 
 Thus, +.ll^=-«, +ZlE=+«, _Jz£=+«, _ZL^=_« 
 '6 5'-6&' b 6' -b b 
 
 Again, the signs of the factors in the numerator and 
 denominator of a fraction may affect the sign before the 
 fraction. 
 
 Thus, 
 
 a 
 
 (1) (+«) .^+±, (3) 
 
 (+6)(+c) be ^' (_6)(_c) 'be 
 
 -« =^^. (2) (-«) =-±. (4) 
 
 From which we make the following important con- 
 clusions : 
 
 From (1) and (4): 
 
 93. An odd number of negative factors in numerator and 
 denominator together may he changed in sign provided the 
 sign before the fraction is also changed. 
 
 From (2) and (3): 
 
 94. An even number of negative factors in numerator 
 and denominator together may he changed in sign without 
 affecting the sign before the fraction. 
 
 The following important principle underlies many pro- 
 cesses with fractions : 
 
 95. Multiplying or dividing both numerator and denomina- 
 tor of a fraction by the same quantity does not change the 
 value of the fraction. 
 
BEDUCTION TO LOWEST TERMS 
 
 145 
 
 REDUCTION TO LOWEST TERMS 
 
 96. To reduce a fraction to its lowest terms is to change 
 its form without changing its value. 
 
 A fraction is in its lowest terms when the numerator and 
 denominator have no common factor . 
 
 45 a%^c ^ 3x3x5 a%^c ^ 3 ab^ 
 
 4 
 
 Ex. 1. 
 Ex. 2. 
 
 60 a%^c 4x3x5 aWc 
 
 a^-\- ax _ a(a-{-x) 
 a^ — x^ 
 
 Result. 
 
 a — x 
 
 Result. 
 
 Result. 
 
 {a -\- x^{a — x') 
 
 Ex 3 a;2 + 5a: + 4 ^ {x^V){x-\-\^ ^ x^-\ 
 a:2 + 8a: + 16 (ic+4)(a; + 4) a; + 4 
 
 Hence, the general principle : 
 
 97. To reduce a fraction to its lowest terms : 
 
 Factor both numerator and denominator. 
 Cancel the factors common to both. 
 
 Factors common to the numerator and denominator may 
 be canceled. 
 
 Terms cannot be canceled under any condition. 
 
 Exercise 52 
 
 Reduce to lowest terms : 
 
 3. 
 
 ia^ 
 12 ab 
 
 15 x^^ 
 
 35 xY^ 
 
 2Sa^f 
 
 4. 
 
 6. 
 
 F. H. 8. FIRST YEAR ALG. 
 
 39 m^n 
 65 mn^ 
 
 76 a^y^z 
 
 23 a^c^ 
 69a^cd' 
 10 
 
 8. 
 
 9. 
 
 48 m%y 
 
 72 m%% ■ 
 
 135 a^mn^x^z^ 
 81 a^nxh^ 
 
 a^-\- ax 
 a^-{-ac 
 
c^ — a 
 
 ax-\-a 
 
 bx^^l^x 
 
 dx + 2a 
 
 4:X^-SX 
 
 4:X^+20X 
 
 a^ + la^ 
 
 a^-Sa^ 
 
 a^ + x^ + x 
 
 a^-1 
 
 
 m^ + Sm- 
 
 -4 
 
 m^— 5m + 4: 
 
 a3_^^2_6 
 
 a 
 
 a^-da 
 
 
 2a^-la- 
 
 -15 
 
 2a^-lla 
 
 +5 
 
 2a^-2a^ 
 
 
 146 FRACTIONS 
 
 10. r^. 15. ^4+4=4^- 20. «^ + 8« + T 
 
 a*^ — a^ + 5 a 
 
 ^^ --- ■ --- . 16 ^^=^- 21. 
 
 rr3 — 1 
 
 12. ^^^=: ;f^- 17. — 22. 
 
 X^ — X 
 
 -.«".•" , « a^ — 9 7^2 
 
 13. „ ^ „ • 18. -• 23. 
 
 (« - 3 w)2 
 
 14. !^ ! 19. 24. 
 
 x^ 4- a; H- 1 m^ 4- w^2?^ 4 a^ — 4 aj^ 
 
 25 4tz3 + 5a2-21a 28 ^-(^-0' 
 
 a* + 2Ta * (3-c)2-a2 
 
 26^ a2-(6 + 02 ^^^ 2^ + (^ + ^'):r + a3 
 
 (a — 5)2— ^2 a;2_|_^^ _[_ 5^^_j_^j 
 
 27. ^^= ■ ^' 30. ■ • 
 
 (a: + l)2_2;2 a;2_i4.2a-a2 
 
 REDUCTION TO MIXED EXPRESSIONS 
 
 98. A Mixed Expression is one that is made up of both 
 integral and fractional terms. 
 
 1 1 
 Thus, a + - and x + 1 are mixed expressions. 
 
 a a; + 1 
 
 When the numerator of a fraction is of a degree higher 
 than the degree of the denominator, it is possible to change 
 the fraction to a mixed expression by the process of divi- 
 sion. Thus, 
 
REDUCTION TO MIXED EXPRESSIONS 147 
 
 Ex. 1. Reduce the fraction 
 
 ^ "^ to a mixed expression, 
 a — 1 
 
 By division we obtain : 
 
 a3 + 1 (g- 1 
 
 gs- g^ g2 + g + 1 
 
 g2 + l 
 
 g^ — g 
 
 g + 1 
 g-1 
 
 + 2 
 
 The quotient is g^ + g + 1 ; the remainder, + 2. Hence the 
 remainder is the numerator of a fraction, of which the divisor, 
 g — 1, is the denominator. 
 
 Hence, ^-±1 = a'^^a + l^- — ^. Kesult. 
 
 g — 1 g — 1 
 
 From which we state the general principle : 
 99. To reduce a fraction to a mixed expression ; 
 
 Divide the numerator by the denominator. 
 
 Write the remainder over the denominator^ and annex the 
 resulting fraction to the quotient ; remembering that the sign 
 of the remainder becomes the sign of the fraction. 
 
 
 
 Exercise 53 
 
 Reduce to mixed 
 
 expressions : 
 
 1. 
 
 x-^l 
 
 ^' a-\ ' 
 
 2. 
 
 a* 4-1 
 
 x'-Sx-2 
 
 g 2^-2x^-hx-\-l 
 x^—x—1 
 
 ^ a:* + 4 a; + 1 
 a + l X — 5 x^-\- x — 1 
 
148 
 
 FB ACTION 8 
 
 ^ s^ + x^~2 
 
 ' x^-hl 
 
 x^ + x^-2 
 
 ^'+1 riS 
 
 x^ + x 
 
 ^+1 ^ + i-^v4- A^^s^i*- 
 
 x^-x-2 -^ ' ^ 
 
 x^ +1 
 
 -x-S 
 
 
 w2-4 ' a2^a-l' • x^ + 1 
 
 REDUCTION OF MIXED EXPRESSIONS 
 
 By Art. 99, 
 
 
 -^+l=.a.l. V. 
 
 a — 1 a — 1 
 
 Reversing the process, 
 
 a + l+_2_^ (« + l)(a-l)+2 
 a — 1 a — 1 
 
 ^ 0^^-1 + 2 
 
 = ^^i±i. Result. 
 a-1 
 
 Hence, 
 
 100. To reduce a mixed expression to an improper 
 fraction ; 
 
 Multiply the integral expression hy the denominator of the 
 fraction and add the product to the numerator^ remembering 
 
REDUCTION OF MIXED EXPRESSIONS 149 
 
 that the sign of the fraction becomes the sign of the 
 numerator. 
 
 Write the given denominator under the result. 
 
 Exercise 54 
 
 Reduce to improper fractions : 
 
 1. 
 
 a 
 
 4. 
 
 , h 
 ax -\ — . 
 
 X 
 
 7. ""-^x. 
 
 X 
 
 2. 
 
 X 
 
 5. 
 
 1 
 
 X . 
 
 X 
 
 8. -+a, 
 z 
 
 3. 
 
 c 
 
 6. 
 
 1 
 
 a . 
 
 a 
 
 9. ?-a. 
 a 
 
 10. x-^l-\ . \Z. 01^— x+l-\- 
 
 x-\-l x-\-\ 
 
 11. <. + 2+-5_. 14. ^+2+^^?L±l). 
 
 a + 2 X — 6 
 
 x—^ 4a2_j.2a4-l 
 
 16. -— ^ + « + 1. 
 a — 1 
 
 Suggestion : Inclose the integral expression in a parenthesis. 
 
 Then, ^+l + a+l = t+l±Sl±M«^Z^ 
 
 a — 1 a — 1 
 
 ^ "'^ Result. 
 
 a-1 
 
150 FRACTIONS 
 
 17. --]-x+b. 19. — — — =i— + a — a; — 1. 
 
 X -\- z a -\- X 
 
 18. ?!^+4-:r2. 20. <^~^) +a:2 + a; + l. 
 X + z x^ + x—1 
 
 21.. + !-^^ + ^. 
 
 The sign before the fraction becomes the sign of the numerator. 
 
 Hence, . + 1 _3^+j^ (^ + DC^ + 2) - (3. + 2) 
 
 x + 2 .r + 2 
 
 a;2+3:i; + 2-3a:-2 
 
 x + 2 
 
 = — ^— . Result. 
 
 23.1-x + :fi--^. 25. a2+2- ^"-"' + "' . 
 
 1 + a; a — 1 
 
 26. a^ + a — 2 — 
 
 a-1 
 
 27. x^- x-4:-h 
 
 x-\- 
 
 i)- 
 
 oo 2 To fi 2-8wM 
 28. m'^ — 2 m — 1 — 1 • 
 
 29 2a (^-l)(^+'^) + (^-B) 1 
 a + 1 
 
CHAPTER XII 
 
 LOWEST COMMON MULTIPLE. LOWEST 
 COMMON DENOMINATOR 
 
 THE LOWEST COMMON MULTIPLE 
 
 101. A Common Multiple of two or more algebraic 
 expressions is an expression that will contain each of 
 them without a remainder. 
 
 Thus : a%^ will contain a%'^, a%\ a'^b^ and a%^ 
 Hence, a%^ is a multiple of a%% a%^, a^b^, and a^b^. 
 
 102. The Lowest Common Multiple of two or more alge- 
 braic expressions is the expression of lowest degree that 
 will contain each of them without a remainder. 
 
 Thus: 
 
 Given, ■ 
 
 a^b^ is the expression of lowest degree that will 
 contain each of the four expressions without a 
 „, remainder. 
 
 From this we make the following conclusion : 
 
 103. The Lowest Common Multiple of two or more expres- 
 sions is the product of the highest powers of all the different 
 factors in the given expressions. 
 
 The abbreviation "L. C. M." is commonly used in 
 practice. 
 
 151 
 
152 COMMON MULTIPLES AND DENOMINATORS 
 
 THE L.C.M. OF MONOMIALS 
 Oral Exercise 
 Find by inspection the L. C. M. of : 
 
 1. a?h and aV^. 7. IQmH and 24^2^. 
 
 2. mV and mV. 8. 9 ah'^c^ and 12 a^5c. 
 
 3. 2 m^a; and 3 m^a;. 9. 10 (P'dx and 8 ed?y. 
 
 4. 3 a52 and 6 a^. 10. 15 a;^^ and 20 A. 
 
 5. 8 mx and 12 na;. 11. 35 a%h^ and 42 w^Jw. 
 
 6. lOx^y'^z and ^xy^^. 12. 51 am^ and 34 a/i^. 
 
 THE L.C.M. OF POLYNOMIALS 
 
 In finding the L. C. M. of polynomials, we must first 
 separate the expressions into their factors, as in the case 
 of finding the H. C. F. Hence, 
 
 The General Method for finding the L. C. M. of Polynomials 
 
 104. Tractor the given expressions. 
 
 The product of the highest powers of all the different 
 factors is the required L. Q. M, 
 
 Illustrations : 
 
 Ex. 1. Find the L. C. M. of a^ - ah and ah - h\ 
 
 a^ — ah = a(a — 6) 
 a&-&2 = 6(a-6) 
 
 L. C. M. = a& (a - 6) . Result. 
 
THE L. C. M. OF POLYNOMIALS 153 
 
 Ex.2. Find the L. C. Mo of x^ + x-2, a^-x-Q, and 
 
 x^ -{- X -2 =(x - l)(x + 2) 
 
 x^-x-Q = (x-{-2)(x-d) 
 
 x^-4:X + S = (x-l)(x-S) 
 
 L.C.M.=(a;-l)(a; + 2)(a;-3). Result. 
 
 Ex.3. Find the L.C.M. of a^ -\- 4. a^ + 4 a, aH-4:x, 
 
 and 5 a2 - 20 a + 20. 
 
 a8 + 4 a2 + 4 a = a(a + 2) (a + 2) 
 
 a^x-^x = x(a-\-2){a-2) 
 
 5 a2 - 20 a + 20 = 5(a - 2) (a - 2) 
 
 L,C.M. = 5ax(a + 2)2(a-2)2. Result. 
 
 Exercise 55 
 Find the L. C. M. of : 
 
 1. a^ -\- a^ ac -\- c. 9. a^^ — 9, a:^ — 6 a: + 9. 
 
 2. ax — a^ x^ ~ X. 10. rrfi-\-2m-\- 1, m^ — m. 
 
 3. m^ + m, m^ — m, 11. (P' —\^ c^ — 1. 
 
 4. c2 + 2 c, c2 - 2 (?. 12. a^ - 4 a; + 4, 2:3 _ 8. 
 
 5. a:2_^^ (^2-1). 13. 4:a^-12a + 9, 4^2-9. 
 
 6. a2 + 3a, (a -3)2. 14. a^-6a;2 + 9a:, a;5_8l2;. 
 
 7. 2<a: + X), 33:24.3^. 15. a;2 + 3a; + 2, a^ + 4a; + 3. 
 
 8. 5a:2-15a:, 3a;3-27a:. 16. x(x^- x -\-l'), x^+1. 
 
 17. 5a3_l25a, 2a3-20a2 + 50a. 
 
 18. (m + ^)^, (w — ^)^ w^ — ^^. 
 
 19. c\c-d}, c(ic^-d^), c + d. 
 
 20. 4(62 + 5m), 6(5m-m2), 8(^2 _;^2) 
 
154 COMMON MULTIPLES AND DENOMINATORS 
 
 21. 2m^ — m^ — m^ 2 m^ — S m^ — 2 m. 
 
 22. Sa^-^x^ 5a^-10x^+5x. 
 
 23. 2a^-2a, 3 a(a - 1)2, 4 a(a + 1)2. 
 
 24. 2^ + 1, 2 + Sx + a^, (a;+l)(l + a;). 
 
 25. a^-9, a3-27, a* -81. 
 
 26. ax -{- a -\- X -{- 1^ ax^ -\- ax — x^ — x. 
 
 27. ax+Sa-2x-6, a(a-2y, a\a^- 
 
 8). 
 
 REDUCTION TO EQUIVALENT FRACTIONS WITH A COMMON 
 DENOMINATOR 
 
 105. The Lowest Common Denominator of two or more 
 fractions is the lowest common multiple of their denomina- 
 tors. (Usually abbreviated thus, "L. C. D.") 
 
 The process of reducing fractions to their lowest com- 
 mon denominator is illustrated as follows : 
 
 Reduce 
 
 5x Sx 
 
 , and - to equivalent fractions hav- 
 4 6 
 
 ing a common denominator. 
 
 The L. C. M. of tlie denominators, 3, 4, and 6, is 12. 
 Hence, 12 is the lowest common denominator of the three denomi- 
 nators, and the required denominator of the equivalent fractions. 
 
 12 
 
 12 
 
 4, 1st quotient. Hence, — ;— = 
 
 4 x5a: 
 4x3 
 
 20 a: 
 12 
 
 1st Resul 
 
 3x3^ 
 3x4 
 
 _9x 
 12' 
 
 2d Result. 
 
 2 X a: 
 2x6 
 
 _2x 
 12* 
 
 3d Result. 
 
 — = 3, 2d quotient. Hence, 
 4 
 
 12 
 
 — = 2, 3d quotient. Hence, 
 
 20 X 9 X 2 X 
 
 Therefore, , — , and — are the required fractions having 
 
 12 12 12 ^ ^ 
 
 the same denominator, each equivalent, respectively, to the given 
 
 fractions. 
 
nEDUCTtON TO EQUIVALENT FRACTIONS 155 
 2. Reduce -^^^ — and ^^^ — to equivalent fractions 
 
 X^ + X X^ — X 
 
 having a common denominator. 
 
 The L. C. M. of the denominators x^ -\- x and x^ — xis, x (x^ — 1). 
 Hence, x (x^ - 1) is the L. C. D. 
 
 ^i^!:::ll = x-l, 1st quotient. (^-1)(^-1) _felili. Ist Result. 
 
 ^^^-1)^,^1, 2d quotient. (^ + 1) (^ + 1) i^±iy 2d Result. 
 
 Therefore, A^-^ — i— and ^^"^ ^ — are the required equivalent 
 x(x^ — 1) x(x^ — 1) 
 
 fractions having the same denominator. 
 
 From these processes we state the general method. 
 
 106. To reduce two or more fractions to equivalent 
 fractions having a common denominator. 
 
 If necessary/, reduce the fractions to their lowest terms. 
 
 Find the lowest common multiple of the given denominators^ 
 for the lowest common denominator. 
 
 Divide each of the given denominators into the common 
 denominator. 
 
 Multiply both numerator and denominator of each fraction 
 by the respective quotients obtained. 
 
 Exercise 56 
 
 Reduce to equivalent fractions having a common de- 
 nominator : 
 
 3^ 2^ 3 A A 1 
 
 4 ' 3 ' ' ^x' 2x X 
 
 5a 2a 5 3 4 
 
 6 ' 9 * ' 2x' 4.x 3a;" 
 
156 COMMON MULTIPLES AND DENOMINATORS 
 5 2 2,^- 13. ' ' 
 
 ah he ac ' xQc — T)' a;(a;— 3) 
 
 6. A, J_ A_. 14 ^ ^ 
 
 ^y'' x^y^^ xy^ ' 2 a; + 3' 4a;2— 9 
 
 a%^ a%^^ ah"^ 
 
 a b c 
 
 TT^nx miP^'x m^nx^ 
 
 x-{-l x — 1 
 
 a + 6 a— 6 
 
 11. — ^, ^ 19. 
 
 ^2_l' (a;- 1)2 
 
 2 2 
 
 12. — , -— . 20. 
 
 a3- 
 
 ■1' a3 + a2 + a 
 
 
 
 3 
 
 1 
 
 
 a^- 
 
 ■x-Q' 
 
 a^2+2a;- 
 
 15 
 
 
 1 
 
 2 
 
 
 2x^ 
 
 -a^-1 
 
 .' Qx^-x- 
 
 -2 
 
 
 2 
 
 3 
 
 a* 4- a 
 
 
 a2- 
 
 ■a+1' 
 
 
 
 X 
 
 a;2 — re— 6' 
 
 1 
 
 x^j^^x+2' 
 
 X 
 
 1 
 
 
 2 3 
 
 
 aQa + iy (a + iy ' (a-\-hy(a-i-hy\a+hy 
 
 X X X . • 
 
 21. 
 
 22. 
 
 23. 
 
 2(0^4-1)' 3(a;-iy 2(rr2-l) . 
 2x 4x Sx 
 
 5(a: + ly 3(a;-iy 2(a:2_2a; + l) 
 
 X X X 
 
 a^ — x-^-V x^ -\-x x'^ + x 
 X a^ + x^ -\-x X 
 
 25. 
 
 1 1 i 
 
 8^4-15' 5w8-125w' 7w-35' 6m4-30 
 
CHAPTER XIII 
 FRACTIONS (Continued). REVIEW 
 
 I. ADDITION AND SUBTRACTION 
 
 If two or more fractions have the same denominators, 
 their numerators may be added and a single fraction will 
 
 result. 
 
 „, 2,3 5 
 
 Thus : = + = = =• 
 
 7 7 7 
 
 a b c a -\- b + c 
 
 XXX X 
 
 X y z _x — y — z 
 a a' a a 
 
 By Art. 106 we may reduce any given fractions to 
 equivalent fractions having a common denominator ; hence 
 it follows that any given fractions may be added. 
 
 The following illustrations will serve to show the 
 process : 
 
 Ex. 1. Simplifv -^+-^- 
 
 The L. C. D. is a^ - b^ 
 
 Dividing each denominator into the L. C. D. and multiplying the 
 corresponding numerators by the quotients, we obtain 
 
 a . b _ a(a + b) + b(a — b) 
 
 b a + b a^-b'^ 
 
 _ a^-\-db-\-db-b'^ 
 a'^-b'^ 
 
 _ ai^2ab-b'^ 
 a^-b^ 
 157 
 
 Result. 
 
158 FRACTIONS 
 
 Ex. 2. Simplify -^ - , ^ , + ^f~^ • 
 
 TheL.C.D. isa:8+l. 
 
 Dividing each denominator into the L. C. D., multiplying the cor- 
 responding numerators by the respective quotients, and writing the 
 result over the L. C. D., we obtain 
 
 X X . 2x^-1 _ x(x^-x + 1) - x(x-\-l) + 2x^-l 
 
 4- 
 
 x+lx^~x+l x^ + l x^+1 
 
 x^ -x^ + X - x^ - x-^2x^-l 
 
 X3+ 1 
 
 x^-l 
 
 a;8+ 1 
 
 Result. 
 
 From which we make the statement for the general 
 process : 
 
 107. To add algebraic fractions : 
 
 Reduce the given fractions to equivalent fractions having a 
 common denominator. 
 
 Write the sum of the numerators of these fractions over 
 the common denominator., remembering that the sign of each 
 fraction becomes the sign of its numerator in the addition. 
 
 Simplify : 
 
 1. 
 
 « + *. 
 
 X x 
 
 2. 
 
 c Ze 
 
 a^2a 
 
 3. 
 
 xy xz 
 
 Exercise 57 
 
 
 4. 
 
 A^A. 
 
 ax bx 
 
 5. 
 
 \*h\- 
 
 6. 
 
 1 1 1 
 m 71 p 
 
7. 
 
 2 2 3^ 
 
 
 a X y 
 2 2 4 
 
 8. 
 
 ah ac he 
 
 9. 
 
 5 3 2 
 
 mn mp np 
 
 10. 
 
 111 
 
 xij xz yz 
 
 ADDITION AND SUBTRACTION 159 
 
 S ah b ah ah 
 ' 4 ~6~'^T 
 
 14. 
 
 ,_ a — '^ + x. X — a-\-\ 
 J-5' 7i 1 : 
 
 12. ^+i+^=i. 
 
 ^^' ~^^^2" 
 x-Z x-1 
 
 14 
 
 5x — 5a-\-c c-\-Sa-\-Sx 
 
 17. 
 
 5 3 
 
 4a-3c + l Sa-\-c-l 
 
 18. a+2 g-l ^ g + l 
 
 19. 
 
 6. 3 2 
 
 2a-l 7a+3a-3 
 5 10 2 
 
 a;- 5 a;+ 3 2a:- 5 
 12 6 9 ' 
 
 21. 1+-1-. 24. ^ 
 
 aa + (j 7m3m — 1 
 
 oo ^ J 2 o« 2 1 
 
 22. 1 -. 25. — - — 
 
 2a;a:4-l ah a — h 
 
 23. -.,^ + i. 26. -1^+ 1 
 
 a^— 4a a;-f-la:— 1 
 
160 FBACTIONS 
 
 27. -^ ?-. 36. ^±1 + 2^. 
 
 m — n m -\- n a -{- 2 a -{- S 
 
 28. -^ ?-. 37. ^±l + ^-Zl. 
 
 29. ^ 2^. 38. "+1 .+ «-l 
 
 a-2 a + 3 a2 + a + l a^-a+l 
 
 5 5 „^ a — 2 a; , 4 «a; 
 30. . 39. h 
 
 ^^2 a-3 6 
 
 31. — -H -— . 40. 
 
 a(^a + 1) a(a - 1) a + 3 a^ - 9 
 
 32. — ^ ?__ ^^ ^ + I_£^. 
 
 a(a - 1) a(a + 1) ic - 7 a; + 7 
 
 33. J—^-J-, 42. .^-A. + ^ + ' 
 
 ^2 _^ ^ ' ^2 _ ^ (a: + 1)2 (a; - 1)^ 
 
 34 _JL__l£_ 43 3a; + 2 32;-2 
 
 • l_^ l_a;2- • 32^-2^3:^:4-2* 
 
 Sx 2x ^1^7^ + 4 7w-4 
 
 35. —» 44. — 7 — — J. 
 
 rr-1 x-\-l 7w-4 7m + 4 
 
 w — 2 m + 2 
 
 * ^2— 2m + 4 m2 + 2w + 4* 
 
 46. , .A ., + ^ 
 
 47. 
 
 (^ + l)(^+2) (a; + l)(a;-2) 
 
 5 2_ 
 
 (a:-l)(2: + l) cx-iy 
 
MULTIPLICATION AND DIVISION 161 
 
 4 2 
 
 48. 
 
 49. 
 
 a2+3a + 2 a^ + Ba + Q 
 1 1 
 
 2^-{-lx-{-12 x^-}-Sx+15 
 
 ah ah , 1 
 50. -f- 
 
 a-{-h a — h a^ — h' 
 51. 2^-^^+ ^ 
 
 52. 
 
 ^2 — 1 m-\-l m — 1 
 a-2 a + 2 a2-4* 
 
 53. . ^ ■ + 1 1 
 
 54. „ 1 ,-^i-+ 1 
 
 m^— w + l m* + m w + 1 
 55. .JU+^J-.+ 1 
 
 a;2 — a^ {x + a)2 (x — a)^ 
 
 56. ^Il1-^^iJ+ ^(^^-^) , 
 x-{-% X — 4: X^ — X — 12 
 
 II. MULTIPLICATION AWD DIVISION 
 
 The processes of multiplication and division of algebraic 
 fractions are not unlike the same processes with arithmeti- 
 cal fractions. 
 
 (1) Multiplication. 
 
 The product of two or more fractions is the product of 
 the numerators divided by the product of the denominators. 
 
 F. H. S. FIRST YEAR ALG. — 11 
 
162 FRACTIONS 
 
 Thus: «x-^x^" = ^. 
 
 X y z xyz 
 
 ah cm nx _ abcmnx 
 mn bx ay mnhxay 
 
 _ c 
 
 ~ y 
 
 Canceling, 
 
 (2) Division. ^ 
 
 The quotient of two fractions is obtained by inverting 
 the divisor and using it as a multiplier, whence the process 
 becomes a multiplication of the given fractions. 
 
 Thus: 
 
 
 a . d _a h _ah 
 c b c d cd 
 
 am 
 
 bm _ am xz __az 
 
 xy 
 
 xz xy bm by 
 
 The general process may be stated as follows : 
 
 108. To multiply a fraction by a fraction : 
 
 Multiply the numerators together for a new numerator^ and 
 the denominators for a new denominator. 
 
 By cancellation reduce the resulting fraction to its lowest 
 terms. 
 
 109. To divide a fraction by a fraction : 
 Invert the divisor and proceed as in multiplication. 
 The following examples will illustrate the processes : 
 
 Ex. 1. Multiply 1^^ by i^. 
 ^^ %mny ^ Z ax^ 
 
 \^€?-x 4 my _ 15 X 4 a^xmy 
 8mny dax^ 8 x dmnyax^ 
 
 2nx 
 
 I 
 
MULTIPLICATION AND DIVISION 163 
 
 Ex. 2. Multiply : 
 
 x-y ix-yy 
 Factoring numerator and denominator in the multiplier, we have 
 
 x + y ^ ^!_z_ii = ^±1 X (^ + y)(^ - y) 
 
 x-y {x -yy x-y {x - y){x - y) 
 
 Canceling, 
 
 Ex. 3. Simplify : 
 
 i^±y^. Result. 
 {x - yy 
 
 
 Factoring the terms of the second and third fractions, we have 
 
 g + l^^ (a-\y . a^-l ^a+1 ^^ (a-l){a-l) . (a + l)(a-l) 
 a-l a2-3a + 2 " a^-^ a-l (a-l)(a-2) ' (a + 2)(a-2) 
 
 Inverting the divisor, = ^^ x (^"^X^"^) x (« + 2)(«-2) 
 ^ ' a-l (a-l)(a-2) (a + l)(a-l) 
 
 Canceling, = ^^-^. Result, 
 
 a — 1 
 
 Simplify : 
 aJc 6 X 
 
 ^ 8 ca; 10 m2 
 m y a; 
 
 • 9i/ 2ac 
 
 Exercise 58 
 
 , 21^25 5a^ 
 10 mx 8 «S 
 
 89 a(?2 5 moi^ 
 
 6 m^ ^ 2 m^ 
 * 13^ ' 39^' 
 
164 FRACTIONS 
 
 ^ 16 ahc . 32 ao ^^ rn^-l 2m^+4:nv' 
 
 ' 27 mx ' 9x' 
 
 8 1^_21^ ^^ 
 
 32 m/i 40 WW 
 
 3£_ 4w^ 25^ 
 5 m 15 a;^ 12 w 
 
 15. 
 
 35 g3^ 18 aH^x 28 y 
 ' 12 a^«/ ' 49 c^ ' 45 a^Jc' * 
 
 or — X x^ + x 
 
 m2+ 2m 
 
 m^-\-m 
 
 2^2-1 
 
 2^3 + 1 
 
 ^-1 
 
 C?-l 
 
 ^+1 * C2 
 
 -c+1 
 
 a + 1 . a2 
 
 -1 
 
 a - 2 ' a2 
 
 -4 
 
 :?; -1 
 
 :i:2_l 
 
 2^3 + 1 • :^2 
 
 -a;+l 
 
 a^—xy x^ — 4:xy+^y'^ 
 
 a^-9 5a; + 10 i^-^y x^-y' 
 
 X X x^—1 
 
 19. 
 
 20. 
 
 21. 
 
 22. 
 
 23. 
 
 24. 
 
 25. 
 
 a; + 1 x—1 ax 
 
 2(a-{- 1)2 5 (g - 1)3 , 5 a 
 3(a-l)2 ' 6(a-l)2 * 8(a+l) 
 
 a2^>2g2 _ 4 ^252g2_^5g_6 
 
 a^^(P-9 ' a%'^(^ + ^ ahc + 4^ ■ 
 a2_3^_18 a3_27 
 
 (a: + l)(a;+2) (a;+3)(a:+5) 
 (aj + 3)(a; + l) (a:+2)(a:+7)* 
 
 a^-a:-2 rc2-3a;-4 
 a^_5^+6 * a;2_73,+ i2' 
 
 6-\- x — a^ 4 — 4 a;+ a^ 
 4-2^2 '9_eJa: + ar2" 
 
26. 
 
 27. 
 
 28. 
 
 29. 
 
 30. 
 
 MISCELLANEOUS FBACTIOKS 165 
 
 ^2-1 a2-2a-3 
 
 a2 - 5 a + 6 
 
 ' a?- 
 
 -3« 
 
 + 2 
 
 
 a;3 - 16 a; 
 
 
 2;2 
 
 + 3:r-4 
 
 
 aa:^ — 7 a:z: + 12 a 
 
 ' x^ 
 
 -4a; + 3 
 
 
 i?;3+l 
 
 ax^- 
 
 - a 
 
 
 
 x^ + 2x+l 
 
 a^- 
 
 -1 
 
 
 
 4a2_l 
 
 9a2 
 
 -4 
 
 6a^+a- 
 
 -2 
 
 6 a2 - a -^ 2 
 
 8^3 
 
 + 1 
 
 • 4a2_2a 
 
 + 1 
 
 «2 _^ 2 a5 4- ^ 
 
 -^ 
 
 a2- 
 
 -2atf + ^- 
 
 -62 
 
 a2_2a6 + 6 
 
 -(?2 
 
 
 (« + ^ + 0' 
 
 
 III. MISCELLANEOUS FRACTIONS 
 
 It frequently happens that an example in fractions in- 
 cludes addition or subtraction with multiplication and 
 division. The student should note carefully the indicated 
 order in such problems, remembering that parentheses 
 indicating addition must be simplified before further 
 operations are possible. 
 
 Exercise 59 
 
 Simplify : 
 
 ^ (i-i)(i-f} '■ e-)e-> 
 
166 FRACTIONS 
 
 •■'J 
 
 Va; a/\a; ay \a b eJ\ab + oc+ac. 
 
 11 
 
 12 
 
 /^i _ A^ -L- /'^ _ ^^ 17 / ^ — ^ m-{-n\f m-\-n \ 
 
 W f)\x 'yj ' V 2 3 Adm-nJ 
 
 /^_^y___a^___\ ^g^ /a + 1 x-l\ , fx ^ ^ 
 \a x) \^ — 2 ax -\- d^j \ a x J \a J 
 
 \x-]-a J\x — a J \Qo^—ay 
 
 V c ^ )\^a-\-xy-cJ 
 
 20. 
 
 21. 
 
 (x-^vf-(x-iy 
 
 x-l (^-1) 
 
 ' \x aj\ xj \ x) 
 
 - (--:i7)(lf?l)(f-i-'> 
 
 V. 1-xyJ \ 1-xy J 
 
GENERAL REVIEW 167 
 
 GENERAL REVIEW — GROUP IV 
 
 THE H.C.F. 
 
 The H. 0. F. of two or more expressions is made up of 
 those factors that are common to the expressions. 
 
 The prodvct of the lowest powers of the factors common to 
 the given expressions is the H. C. F. 
 
 THE L.C.M. 
 
 The L. C. M. of two or more expressions is made up of 
 all the different factors that occur in the expressions. 
 
 The product of the highest powers of all the different 
 factors occurring in the given expressions is the L. C. M. 
 
 An H. 0. F. divides. An L. C. M. contains. 
 
 FRACTIONS 
 
 Signs 
 
 The sign of or before, a fraction is independent of the 
 signs of the numerator and the denominator of the fraction. 
 
 The Signs of thk Factors of the Numerator and the 
 Denominator of a Fraction 
 
 Changing the signs of an even number of factors in both 
 numerator and denominator of a fraction does not change the 
 sign of tlie fraction. 
 
 Changing the signs of an odd number of factors in both 
 numerator and denominator of a fraction changes the sign 
 of the fraction. 
 
168 FRACTIONS 
 
 Introducing a Factor in Both Terms of a Fraction 
 
 Both numerator and denominator of a fraction may he 
 multiplied by the same quantity without affecting the value of 
 the fraction. 
 
 Removing a Factor from Both Terms of a Fraction 
 
 Both numerator and denominator of a fraction may he 
 divided hy the same quantity without affecting the value of 
 the fraction. 
 
 The Transformations of Fractions 
 
 I. To reduce a fraction to its lowest terms is to change its 
 form without changing its value. 
 
 II. To reduce an improper fraction to a mixed expression 
 is to change to a form in which a part of the terms are 
 integral; the remainder^ fractional. 
 
 III. To reduce a mixed expression to a fraction is to 
 change its form so that no integral terms remain. 
 
 IV. To reduce a given number of fractions to equivalent 
 fractions having a common denominator is to change the 
 form of each so that the denominators of the resulting frac- 
 tions shall be the same. The operation depends upon the 
 introducing of the same factor in both numerator and 
 denominator. 
 
 Addition and Subtraction of Fractions 
 
 Fractions having the same denominators may he added hy 
 adding their numerators. 
 
 Fractions having different denominators may he added 
 after they are changed to equivalent fractions having a 
 common denominator. 
 
GENERAL REVIEW 169 
 
 Signs 
 
 In addition and subtraction of fractions^ the sign of 
 each fraction becomes the sign of its numerator when the 
 numerators are added. 
 
 Multiplication and Division of Fractions 
 
 The product of two or more fractions is the product of the 
 numerators divided by the product of the denominators. 
 
 Cancellation^ or the removing of the same factor from both 
 numerator and denominator of a fraction^ reduces the result 
 to its simplest form. 
 
 The quotient of two fractions is obtained by inverting 
 the divisor and using it as a multiplier., whence the process 
 becomes the same as in the multiplication of fractions. 
 
 Bzercise 60 
 
 REVIEW 
 
 Find the H. C. F. of : 
 
 1. a^ + x^ x^ — X, x(x + 1)2. 
 
 2. a3_2a2 + a, (a-iy. 
 
 3. a2-l, a^-1, a^-1. 
 
 4. a^+a-6, 2a2-5a + 2, 3a2-4a-4. 
 
 Find the L. C. M. of : 
 
 5. ^2-6^-42, 2a2+6a-36. 
 
 6. (2:2-1), (^-1)2, x^-\-2x-^l. 
 
 7. a^-{-x^y^ xy^ — y^, x'y^ — 7^y^. 
 
 8. a2-l, a2-2a+l, a3-3a2+3«-l. 
 
170 FRACTIONS 
 
 Find the H. C. F. and the L. C. M. of : 
 
 9. c^-\-8cd^ 2(^-Sc(P, c^-i(^d + 4:(^€p, 
 
 10. 2a;2_2, 4ic3-4, 2a;2 + 2ic-4. 
 
 11. x^-{-x-2, a^+2x-S, x^-\-Sx-4:. 
 
 12. (a-\-bf-c^ P-(a + cy, (h + cy-a\ 
 
 Reduce to lowest terms : 
 
 13. 5^- 20a; ^^ 125-27^:3 
 
 lbx{x-2)^ 25-30ic4-9a;2 
 
 12 a -12 X ,„ a:* -2* 
 
 14- ^ — 7. ^ \ — ^- 18. 
 
 ^^ g2 - 9 g + 20 a:2-(a-l)2 
 
 c2_7^^12' * (x + iy-a^' 
 
 - g 15 a:^ 4- a:^ — 6 a; x^ -\- x -\- ax -\- a 
 
 21 a:;^ — a;^ — 10 a? a;^ + a; — ca; — c 
 
 Change to mixed expressions : 
 
 a3_2a2_5 a:3_^ + 7 
 
 21. 23. — ' 
 
 «2-l a;2 + a:+l 
 
 a:^ + a:2 - 7 2:g^-2^2_^_3 
 
 ' x^+1 ' ' x^-1 
 
 Change to improper fractions : 
 
 25. ^ + 1+^. 27. (:, + 2)'- ^(^-|^X 
 
 a:— 1 a: + 4 
 
 26. «2_a + i_«'(«-l). 28. ^-a-3. 
 
 a+1 a— 2 
 
GENERAL BEVIEW 171 
 
 29. ^+2:2 + 2 
 
 30 
 
 Find the sum of : 
 2 . 1 
 
 33. 
 
 34. 
 
 2:^—1 x^ + x + 1 x — l 
 1 1 3a 
 
 (a - 1)2 (a + 1)2 (^2 _ 1)2 
 
 35. - + 
 
 36. 
 
 37. 
 
 ^_2 ' ic-1 (a;-2)(a;-l) 
 
 2.2 4 
 
 -^^x-i-2x^-\-^x+S 2:2 + 52:4-6 
 3 2 
 
 (2;-l)(2:-2)(2: + 2) (x - 2) {x -{- 2) (x -\- 1) 
 
 38. ^ ^- + 
 
 (a-6)(a-c) {a-b)(h-c) {a — c^(h-c) 
 
 Simplify : 
 
 ^2_l ^_\ 
 
 39. 
 
 ecg-2 6?2 g2_g^ c2_ 3 g^ 4.2^^2 
 • e2 + tf(^ * (^-^)^ ' c2_^2 * 
 
 2 
 
 42 
 
 ^-)a->)-G-^^>Xra-'> 
 
172 FRACTIONS 
 
 43. 
 
 a2 + 2a-3 a^-Ga+S ' (a-l)(3 + a) 
 
 g^ — 2 am + m^ — x^ a + m — x 
 cfi-\-2 am -\-m^ — a^ a — m + x 
 
 45. 
 
 / g + l _ a-l\ , ( a + 1 . a — l\ 
 
 46. Find the H. C. F. of m^ + ac -\- am + cm and 
 m^ — ac -{- am— em, 
 
 47. Add ^-^ I ^ + <^ m^-a^ 
 
 a m 4: am 
 
 48. Reduce , r, -— 5, to equivalent 
 
 1-y 1-f ! + «/ + / 
 
 fractions having a common denominator. 
 
 49. Collect —4 ^^4- ^ 
 
 l-2a l + 2a 1-4^2 
 
 50. Simplify — — - • — — 7 X - 
 ^ -^ ab + l ab-1 
 
 51. Change ~ "^ to a mixed expression. 
 
 52. Simplify ^-^ . -.-^^ . -— ^. 
 
 53. Find the L.C.M. of 6a + 2, 27a2- 3, and 108 a^ + 4. 
 
 54. Find the sum of 
 
 x^-^Sx-10 x^-\'4tx-6 
 
 Jhange x^—2z 
 fraction 
 
 55. Change a^^- 2 a:- 1 + ^^i^ to an improper 
 
56. Collect 
 
 GENERAL REVIEW 173 
 
 a b 
 
 (l^aXa + b) (l-5)(a+5) 
 57. Find the L. C. M. of a^ + Sa^-\-lQ, a^-lG, and 
 
 58. 
 
 59. 
 
 Simplify (.-l + -A3).(.-3--A^) 
 
 60. Find the H. C. F. and the L. C. M. of 
 
 x^ -\- ax -{■ a + X, x^ — hx + X — h, and a^ -\- ex -{- x + c. 
 
 61. 
 
 ^^"p^^^^ ( <--9d^ H-^T^rr 
 
 62. Collect - + 
 
 x+2 a^+S x^-2x-{-4t 
 
 2 
 
 63 
 
 Simplify (^__f\^^^ 
 ^ ^ \x-l x-lj x-1 
 
 64. Simplify fab + 5 + ^]-^ fab -^4 + ^' 
 
 2 
 
 65. Change a^ — a^ + a — 1 -\ to an improper 
 
 fraction. ^ 
 
 r.r> \AA ^-"^ X-2 . 12 
 
 66. Add 
 
 x-^2 x-\-4: x^-\-6x-\-S 
 
 ^' I'f ^^ + 2 am a^—Sm^ a^-\- 4: am^ 
 a^ + 4 w^ 2 a% H- 4 am^ a — 2 m 
 
 68. Simplify (^+2^)-(4^-2^> 
 
CHAPTER XIV 
 
 SIMPLE FRACTIONAL EQUATIONS. PROBLEMS 
 
 110. To Clear an Equation of Fractions is to change its 
 form so that the fractions disappear. 
 
 Thus, in the fractional equation, 
 
 we may multiply both members by 5, whence we have 
 
 ^=15. 
 5 
 
 Or, reducing, x = 15. 
 
 Wlien two or more fractions occur in an equation, the 
 process of clearing is similar to that above, but the multi- 
 plier is the L. Q. M. of the given denominators. 
 
 Thus, to solve the fractional equation, — — ;; = s» 
 
 4 
 
 L. C. D. = 12. 
 
 3a; a: 5 
 4 3~6* 
 
 Multiplying by 12, 
 
 9a:-4x = 10. 
 
 
 5a: = 10. 
 
 
 a: = 2. 
 
 Hence, the general statement for solving fractional 
 equations. 
 
 174 
 
SIMPLE FRACTIONAL EQUATIONS 175 
 
 111. To solve an equation containing fractions : 
 
 Multiply both members of the equation by the L. C. D. of 
 all the fractions, remembering that the sign of each fraction 
 becomes the sign of its numerator. Complete the solution by 
 the methods already learned. 
 
 The following solutions will illustrate the general prac- 
 tice. The student should carefully work out each step 
 before undertaking the exercises. 
 
 Ex.1. Solve^+?^ = 5. 
 4 3 
 
 The L. C. D. is 12. Multiplying both members by 12, we have 
 
 3(a;+ l) + 4(2a;-5) =60. 
 
 3a; + 3 + 8a; -20 = 60. 
 
 lla: = 77. 
 
 a: = 7. Result. 
 
 Ex.2. Solve £^-^£±1=^. - 
 
 O D Z 
 
 The L. C. D. is 30. Multiplying both members by 30, we have 
 10(a: - 1) - 6(2 a; + 1) = 15(a; + 1). 
 lOx - 10 - 12a; - 6 = 15a; + 15. 
 -17 a; = 31. 
 
 a; = - f f Result. 
 
176 FRACTIONAL EQUATIONS. PROBLEMS 
 
 Ex.3. Solve 2^-^ = 2 ^"^ 
 
 X-\-\ X — 1 x^—1 
 
 The L. C. D. is x^ — 1. Multiplying both members by x^ — 1, we 
 have 
 
 (x-l)(x- l)-(x + l)(x + 2) = 2(x'^- l)-2a:2. 
 
 (a:2 _ 2 a: + 1) - (:r2 + 3 a: + 2) = (2 a;2 - 2) - 2 x2. 
 
 x^-2x+l- x^-Sx -2 = 2x^- 2-2a:2. 
 
 ~-5x = -l. 
 
 — \. Result. 
 
 Exercise 61 
 
 2^ 45_^^29 
 * 3 5 2~15' 
 
 8 ^+1 , x-1^2 
 
 ' 8 2 3 
 
 a; 4-1 2a:-l _o 
 
 10. ^ + ^ + 4 = 0. 
 
 a; + l a^-l _o 
 ^^* "5 3 ^• 
 
 2a:-8 8a; + l x-1 
 
 Solve : 
 
 
 
 
 1. 
 
 i^ 
 
 2x 
 3 
 
 7 
 = — • 
 
 2 
 
 
 2. 
 
 1+ 
 
 2^: 
 5 
 
 11 
 15* 
 
 
 3. 
 
 2a; 
 3 
 
 2 
 
 5 
 6* 
 
 
 4. 
 
 a; — 
 
 8a: 
 
 2 
 
 1 
 
 2* 
 
 
 5. 
 
 1+ 
 
 2a; 
 5 
 
 + ?? = 
 ^15 
 
 = 0. 
 
 6. 
 
 I* 
 
 8a; 
 2 
 
 +f- 
 
 25 
 12* 
 
 12. 
 
 4 2 6 
 
 ,^ 4a;-8 , 8a; + l ^-10 
 13. ! — = . 
 
 2 7 14 
 
 5a;-8 6a;-l 8a;-2 ^Q 
 2 3 4* 
 
SIMPLE FRACTIONAL EQUATIONS 177 
 
 15. 1(^ + 1) -1(^-1) =2. 
 
 16. |(2:.-l)-f(3:.+ 5) = J5. 
 
 18. Q^x}Q-x) + x^ = 0. 
 
 x-i-2 5-xx-^lS 
 
 19 
 
 8 
 
 Sx-^2 l-2x ^-6 ^Q 
 5 3 2 ' . 
 
 21. ^ + 3 = ^. 
 
 22. i(:.-2)-i(^+2)-i(^-l)=0. 
 4 a; - 3 a^ 4- 10 
 
 23. 4 
 
 2 
 
 24. Ax ^^ "2-2 
 
 o« (^ + 1)^ Ca: + 2)2 _ 2-^2 
 
 23. • 
 
 3 2 6 
 
 x + \ x—1 2a;— 1 3a: 
 
 
 
 ''' 3 
 
 X 
 
 -1 
 
 a;-3 
 
 X 
 
 + 2 
 
 a: + 5 
 
 X 
 
 -6 
 
 a:-7 
 
 X 
 
 + 3 
 
 a; + 5 
 
 X 
 
 + 1 
 
 a:4-2 
 
 X 
 
 -3 
 
 X-Q 
 
 2 
 
 a:-3 
 
 2x-{-5 
 
 2 4 6 
 
 „„^— J. u. — ^ „, 3a;— 1 2a; — 1 -, 
 
 27. :; = -' 31. = 1. 
 
 a; + 1 a; — 1 
 
 28. = 32. — — = 2. 
 
 a;— 1 a:— 2 
 
 23 ^_^^^^_^^ 33 5a;-l_3^2a: + l. 
 
 a;+2 a;-l 
 
 30. ----- ^ ---r- . 34 4a;-5 _ 5a;-l g^Q 
 
 3a; + 4 3a;+7 2a;-l a; + l 
 
 r. H. 8. FIRST YEAR ALG. — 12 
 
178 FRACTIONAL EQUATIONS. PROBLEMS 
 
 ,, X x^-Sx 4 ^^ x+S x-S 7 
 
 35. — — = — • 36. 
 
 5 &x-^l 5 x-S x+S x^-d 
 
 ^„ 2x-5 2x-\-5 8 
 
 37. 
 
 38. 
 
 2x + 5 2x-5 4tx^-2b 
 
 x+2 a;-4 ^12 
 
 X— 2 x + 4i X 
 
 _ 1/5^+3 2£-ll a; + 2 
 
 ^^* 2i""i 3~r-6- 
 
 3.2 1 
 
 40. 
 
 41. 
 
 42. 
 
 2(a; + l) 22^-1 3(a:+l) 
 
 Ifx o\ 2/:r A .10+£ 
 
 ^j^x'^-^ a^-x^ + 2 
 
 x-\-l x — 1 
 
 a;+l x + 2 
 
 Sx 2x X 5 2x 
 
 45. 
 
 46. 
 
 x+1 3(:r + l) 2(a;+l) 6 x-i-l 
 
 2 ^ 3 ^ 15 
 a:— 3 2^+1 3a;— 1 
 
 ^„ a;2+2: + l, X x^—x + l 
 
 47. h — 
 
 48. 
 
 a: + l a^-1 x-1 
 
 X a^-\-l X 
 
 2a;+2 ^^-^ %x-Z 
 
SIMPLE FRACTIONAL EQUATIONS 179 
 
 PROBLEMS PRODUCING SIMPLE FRACTIONAL EQUATIONS 
 
 In our first application of the equation to written prob- 
 lems we made only such statements as would give equa- 
 tions without fractions. In the following exercise we 
 shall be able to make use of fractional expressions in our 
 .statements, and, consequently, in our equations as well. 
 The method by which we reach our solution will differ in 
 no way from the method employed in our earlier equa- 
 tions. The suggestions for solving a problem (Art. 63) 
 are repeated as an aid to the solutions required. 
 
 In solving a problem : 
 
 1. Study the problem to find that numher whose value 
 is required. 
 
 2. Represent this unknown number or quantity by x. 
 
 3. The problem will state certain existing conditions or 
 relations. Express those conditions in terms of x. 
 
 4. Some statement in the problem ivill furnish a verbal 
 equation. Express this equation algebraically by means of 
 your own written statements. 
 
 No statements or solutions are given in the following 
 exercise. With the aid of a suggestion whenever a new 
 element is introduced, the student should make his own 
 statement with little or no difficulty. 
 
 Exercise 62 
 
 1. The sum of the fourth and the fifth parts of a number 
 is 9. Find the number. 
 
 2. Find that number, the sum of whose third and 
 fourth parts is 5 less than the number itself. 
 
180 FRACTIONAL EQUATIONS. PROBLEMS 
 
 3. The difference between the fourth and the ninth 
 parts of a certain number is 2 more than one twelfth of 
 the number. What is the number ? 
 
 4. There are three consecutive numbers such that when 
 the least is divided by 4, the next by 3, and the greatest 
 by 2, the sum of the quotients equals the largest number. 
 Find the numbers. 
 
 5. The sum of two numbers is 21, but if the greater 
 number is divided by the smaller number, the quotient is 6. 
 What are the numbers ? 
 
 6. The difference between two numbers is 14, and when 
 the greater number is divided by the smaller number the 
 quotient is 2 and the remainder is 3. Find the numbers. 
 
 (Hint : The dividend minus the remainder will exactly contain 
 the quotient.) 
 
 7. A man sold 10 acres more than one third of his 
 wood lot and had left 12 acres. How many acres were 
 there originally in the wood lot ? 
 
 8. Two numbers differ by 7 and one of them is five 
 fourths of the other. What are the numbers? 
 
 9. The denominator of a certain fraction is greater by 
 2 than the numerator. If 1 is added to both numerator 
 and denominator, the fraction becomes f. What was the 
 original fraction ? 
 
 10. A man sold a cow for $ 25 more than one third of 
 what she cost him, making $ 5 by the sale. What was the 
 original cost of the cow ? 
 
SIMPLE FRACTIONAL EQUATIONS 181 
 
 11. Out of a certain sum a man paid a bill of f 45, 
 loaned one fifth of the remainder, and found that he had 
 remaining f 44. How much had he originally ? 
 
 12. Find two consecutive numbers such that ^ of the 
 least is equal to | of the greater. 
 
 13. The sum of two numbers is 72. If the larger num- 
 ber is divided by the smaller number, the quotient is 5. 
 What are the numbers ? 
 
 14. The sum of two numbers is 75, and when the smaller 
 number is divided into the larger number, the quotient 
 is 3 and the remainder is 3. Find the two numbers. 
 
 15. The largest of three consecutive odd numbers is 
 divided into the sum of the other two, and the quotient is 
 1, the remainder 11. Find the numbers. 
 
 16. A boy's age is one third of that of his father, but in 
 4 years the boy's age will be two fifths of the father's age. 
 What is the age of each at the present time ? ^ 
 
 17. A certain boy is one and one third times as old as his 
 brother, but 4 years ago he was twice as old. Find the 
 present age of each. 
 
 18. The sum of the ages of a father and son is 39 years, 
 and if the son was one year older he would be one fourth 
 as old as his father. How old is each ? 
 
 19. A can do a piece of work in 3 days and B can do 
 the same work in 4 days. How many days will the work 
 require if both work together ? 
 
 (Hint : A does all in 3 days, therefore he does | of it in 1 day. 
 Let X = the number of days required when both work together. 
 Then how much of the work will both together do in 1 day ?) 
 
182 FRACTIONAL EQUATIONS. PBOBLEMS 
 
 20. A can do a piece of work in 4 days, B the same work 
 in 5 days, and C the same work in 6 days. How many 
 days will it take to perform the task if all three work 
 together ? 
 
 21. A can do a piece of work in 7 days and B can do the 
 same work in 9 days. C is called in to help, and all three 
 together do the work in 3 days. In how many days could 
 C alone have done the work ? 
 
 22. A certain flock of sheep contains 18 more than half 
 the number of sheep in a second flock. In both flocks 
 together there are 90 sheep. Find the number of sheep 
 in each flock. 
 
 23. A man spends one fourth of his salary for household 
 expenses, one seventh for rent, and one tenth for miscel- 
 laneous expenses. How much is his annual salary if, after 
 the above expenditures, he has left 1 1420 ? 
 
 24. A man left two thirds of his estate to his widow, 
 one twelfth to each of two sons, one twenty fourth to a 
 brother, and iSOOO to his church. What was the total 
 amount of his estate ? 
 
 25. A man being asked his age, replied, " If three times 
 my age be decreased by 14 years and the result be divided 
 by my age, the quotient will be J." What was the man's 
 age? 
 
 26. The total runs made in a baseball game was 11. If 
 the winning team had made 5 more runs and the losing 
 team 2 more runs, the quotient of the winning runs divided 
 by the losing runs would have been 2. How many runs 
 were made by each team? 
 
SIMPLE FRACTIONAL EQUATIONS 183 
 
 27. The distance around a rectangular field is 84 rods, 
 and the length of the field is | the width. What is the 
 length of each side of the field? What is the area of the 
 field ? 
 
 28. A man paid 13750 for two automobiles, paying 
 prices for each such that | the cost of the cheaper one was 
 $ 125 more than ^ the cost of the better one. What was 
 the cost of each ? 
 
 29. A farmer has his cattle housed in three barns. In 
 the first barn there are 3 more head than \ the whole 
 number ; in the second barn, 2 less than | the total ; and 
 the remainder, 19 head, are in the third barn. How many 
 head are there in all and how many in each of the first 
 two barns ? 
 
 30. A certain fielder played in seven games of baseball 
 and made three more hits than he made runs. If four 
 times the number of hits he made is divided by the number 
 of runs increased by 6, the quotient is 3. How many hits 
 and how many runs did he make in the seven games ? 
 
 31. An estate was divided among three heirs, A, B, and 
 C. A received 1500 more than one fourth of it; B, 1500 
 less than one half of it; and C, $500 more than one sixth 
 of it. What was the total amount of the estate ? What 
 amount did each receive ? 
 
 32. Find the three consecutive even numbers such that 
 1 less than one half the first, plus 2 less than one half the 
 second, plus 3 less than one half the third, equals 15. 
 
CHAPTER XV 
 
 SIMULTANEOUS SIMPLE EQUATIONS. 
 REVIEW 
 
 If X and y represent any two unknown quantities, then 
 
 X -\- y — their sum, 
 
 X — y = their difference. 
 
 Suppose we assume that x=10 and y=l. Then 
 
 a; + 2/ = 17, 
 x-y = ^. 
 
 It follows, therefore, that w^e have two equations in 
 which the same unknown quantity has the same value. 
 Hence, the definition: 
 
 112. Simultaneous Equations are equations in which the 
 same unknown quantity has the same value. 
 
 113. Simultaneous equations are solved by obtaining 
 from the given equations a single equation with but one 
 unknown quantity. This process is called Elimination. 
 
 TRANSPOSITIONS AND DIVISIONS 
 
 Before taking up the solution of simultaneous equations, 
 we must be able to change the form of an equation in two 
 
 184 
 
TBAN8P0SITI0NS AND DIVISIONS 186 
 
 unknowns so that we have an expression for one of the 
 unknown quantities in terms of the other. 
 
 (I) Whei^ the Coefficient of x is Unity 
 Given the equation x + 2 y = 7. 
 Transposing, a; = 7 - 2 y. 
 
 That is, 7 — 2 2/ is the expression for the value of x in terms of y. 
 Again, given 4 ?/ — x = 9. 
 
 Transposing, —x=9 — 4:y. 
 
 Changing all signs, a; = 4 y — 9. 
 
 Oral Exercise 
 
 In each of the following equations give the value of x in 
 terras of ^ : 
 
 1. x-{-i/=b. 5. i/-\-x = ll. 9. —x—l7/=—3. 
 
 2. :r-f By = 10. 6. — 4y + a?=13. 10. x— Si/— 2 = 0, 
 
 3. x—2'i/=7. 7. —2^ + 3^ = 4. 11. y — 4 — a;=0. 
 
 4. x — 5^=S. 8. 2i/ — x = — l. 12. ^ = — S~x. 
 
 (II) When the Coefficient of x is Any Number 
 Given the equation 3 x + 2 y = 5. 
 Transposing, Sx = 6 — 2y. 
 
 5 — 97/ 
 
 Dividing by 3, x= '- — -^^ • Result. 
 
186 SIMULTANEOUS SIMPLE EQUATIONS. REVIEW 
 Oral Exercise 
 
 In each of the following equations give the value of x in 
 terms of y : 
 
 1. 2a;+3?/ = 5. 7. 7a;-4?/ = 0. 
 
 2. ^x-y = ^. 8. Zy-2x = ^. 
 
 3. 2x + by = ^, 9. 2y-Zx-l = 0. 
 
 4. 3a;-22/=:-l. 10. 6a;-2i/ + l = 0. 
 
 5. 5a: + 3«/ = -2. 11. _2?/ + 6a; + l = 0. 
 
 6. 5a;-3 = 4y. 12. -?/-32;-5 = 0. 
 
 The student must not be misled by the fact that in both 
 of the foregoing exercises we have asked for the value of x 
 in terms of y rather than for the value of y in terms of x. 
 The value of either in terms of the other may be written 
 without difficulty. The x term was selected merely to 
 avoid confusion. In the following exercises the student 
 will learn to select whichever x ox y value will give the 
 probable best solution. 
 
 ELIMINATION BY SUBSTITUTION 
 
 Applying the principle already learned in the foregoing 
 oral exercises, we will now illustrate the solution of a pair 
 of simultaneous equations in x and y, 
 
 Ex. 1. Solve 32:4-2?/ = 7; 2:r + «/ = 4. 
 
 3^:4-2^ = 7. (1) 
 
 2 a; + 3/ = 4. (2) 
 
 From (2), y = 4 - 2 :r. 
 
ELIMINATION BY SUBSTITUTION 187 
 
 (The value of y is obtained from this equation because its selection 
 avoids a fractional result in transposing.) 
 Substituting in (1), 
 
 3 a; + 2(4 - 2 a:) = 7. 
 
 3x+8-4a: = 7. 
 
 - a: = - 1. 
 
 a:=:l. 
 
 For the value of y we substitute in one of the original equations. 
 Hence, substituting in (2), 
 
 2 a: 4- y = 4. 
 
 2(1) + y = 4. 
 
 y = 2. 
 
 X = l.) 
 
 Hence, [ Result. 
 
 2/ = 2.) 
 
 Ex. 2. Solve 5a;+2?/ = ll;3ic + 4^ = l. 
 
 5x + 2y = n. (1) 
 
 3a: + 4y= 1. (2) 
 
 From (2), x = ^ ~^^ . 
 
 o 
 
 Substituting in (1), 
 
 From which, y = — 2. 
 
 Substituting in (2), 3a: + 4(-2) = l, x = S. 
 From which, a: = 3. y 
 
 -2.> 
 
 Result. 
 
 From these illustrations we state the general process for 
 elimination by substitution. 
 
188 SIMULTANEOUS SIMPLE EQUATIONS. BEVIEW 
 
 114. From one of the given equations obtain the value of 
 one unknown quantity in terms of the other unknown 
 quantity/. 
 
 Substitute this value in other equation and solve. 
 
 Exercise 63 
 Solve and verify : 
 
 1. x-^i/ = 2, 10. 5 X- 7/ = -IS, 
 Sx-\-2y = 5. dx-}-2^ = 0. 
 
 2. a: + «/ = 3, 11. x — ^ = — 7, 
 2rr+3«/ = 8. 32:-2?/ = -18. 
 
 3. a; + 3/ = 5, 12. 5 ic + ^ = 15, 
 3a: + 2y=12. 22:-3^=6. 
 
 4. 5a;+T^=12, 13. 5^ -a: = 4, 
 3a: + y = 4. 2a;-^ = -8. 
 
 5. 4a;+2?/ = 22, 14. 4a;+3y=9, 
 5 x — 2^ = 5, X— 6 i/ = 0. 
 
 6. 4:i/-Sx=16, 15. 6x-2 7/=7, 
 Sx+2i/ = 26. 3a: + 4y = 12. 
 
 7. 4a;-9«/ = -5, 16. 6 re -3?/ =2, 
 5t/-2x=5. 2x-Sy = Q. 
 
 8. 6 a; + 5 2/ = 4, 17. - 5 a; + 3 ?/ = 2, 
 2a;-3?/ = -8. 6^ + 5a:=l. 
 
 9. Sx + 2 = -l y, 18. 8a^+3?/ + 5 = 0, 
 2a;=3^-9. 2a;=3y. 
 
ELIMINATION BY COMPABISON 189 
 
 ELIMINATION BY COMPARISON 
 
 The process of elimination by comparison is illustrated 
 in the following : 
 
 Ex.1. Solve 5a;+2y = 9; 2x-{-Si/ = S. 
 
 From each equation obtain the value of one unknown in terms of 
 the other, selecting the same unknown for each value. 
 
 5 a; 4- 2 y = 9. (1) 
 
 22:+3y = 8. (2) 
 
 From (1), x= ~ ^ . 
 
 5 
 
 From (2), x = ^~^^ ' 
 
 9-2w8-3w 
 
 Hence, Art. 58, 
 
 O J!i 
 
 Clearing, 2(9 - 2 t/) = 5(8 - 3 y; 
 
 18-4y = 40-15 ?/. 
 
 y = 2. 
 
 Substituting in (1), 
 
 5x4-2(2) = 9. 
 
 x = l. 
 
 x= 1 
 
 y 
 
 :;;) 
 
 Result. 
 
 From this illustration we state the general process for 
 elimination by comparison. 
 
 115. From each equation obtain the value of the same un- 
 known quantity/ in terms of the other unknown quantity. 
 Place these values equal to each other and solve » 
 
190 SIMULTANEOUS SIMPLE EQUATIONS. REVIEW 
 
 Exercise 64 
 
 Solve and verify : 
 
 1. 2x+Sy = lS, 9. 6a; + y=ll, 
 2x-{-^=7. ^-6x = -lS, 
 
 2. 3 2^4-2^ = 25, 10. 5 re + 2 3/ = 20, 
 2x-y = 5. Sx-4:i/ = 12, 
 
 3. 3rr + 5i/ = 29, 11. 4a:+3y = ll, 
 3rr+2?/=17. 2x + 5^ = 16. 
 
 4. 4:r-y = l, 12. 2a;+3^=9, 
 3a; + 2«/=20. 4a;+92/ = 19. 
 
 5. 3a;-2i/ = 13, 13. 6 a; -2?/ =3, 
 4a;+3y=6. 3r?; + 2^=l. 
 
 6. 5a;4-3y = — 5, 14. 3a; — 4«/= — 9, 
 7a:H-2?/ = 4. 3?/ + 3a;=-7. 
 
 7. 22; + 3?/=^18, 15. 8a;4-3y=-4, 
 3a;-2y=l. 4a;-5?/=-2. 
 
 8. 3rr + y=l, 16. 2 ?/ + 5 a; = 2, 
 Ua:+3y = l. 10a; + 32/-l = 0. 
 
ELIMINATION BY ADDITION OR SUBTB ACTION 191 
 
 ELIMINATION BY ADDITION OR SUBTRACTION 
 
 This method of elimination produces a solution free of 
 fractions, and is frequently useful when the given coeffi- 
 cients are small numbers. 
 
 (1) 
 
 (2) 
 
 Ex. 1. Solve 
 
 3a;+ 2^ = 11 
 4a;- 3i/= 9 
 
 Multiply (1) by 3, 
 Multiply (2) by 2, 
 Adding, 
 
 9a; 4- 62/ = 33 
 
 8a;- 6^ = 18 
 
 17a; =51 
 
 Substituting in (1), 
 
 a; = 
 
 ::) 
 
 Result. 
 
 Ex. 2. Solve 
 
 Multiply (1) by 5, 
 Multiply (2) by 2, 
 Subtracting, 
 
 Substituting in (1), 
 
 3a; + 
 2a; + 
 
 2^ = 10 
 5«/ = 14 
 
 15a; + 102/ = 50 
 4a; + 10^ = 28 
 
 11a; 
 
 22 
 x= 2. 
 
 y 
 
 ii] 
 
 (1) 
 
 (2) 
 
 Result. 
 
 Hence, for eliminating by addition or subtraction : 
 
 116. Multiply one or both given equations by the smallest 
 numbers that will make the coefficients of one unknown quan- 
 tity equal. 
 
 If the signs of the coefficients are unlilce^ add the equa- 
 tions ; if like., subtract the equations. 
 
 For practice the student should solve the equations in Exercise 63 
 by the method of addition and subtraction. 
 
192 SIMULTANEOUS SIMPLE EQUATIONS. BEVIEW 
 
 FRACTIONAL SIMULTANEOUS EQUATIONS 
 
 Clear the following simultaneous equations of fractions, 
 and solve the resulting equations by the method of substi- 
 tution or by the method of comparison : 
 
 
 
 
 Exercise 65 
 
 1. 
 
 2 3 
 
 
 7. 
 
 2.+1 y+1^ 
 3 ' 5 
 x+1 y+l_ . 
 5 2 
 
 2. 
 
 2 5 
 
 
 8. 
 
 3-+1 5 + 2^ = 2, 
 
 2* ^Y^ 4='>- 
 
 3. 
 
 2 3 ' 
 
 4^ 2 
 
 
 9. 
 
 y-J + 1 -+J-0. 
 
 b d 
 
 4. 
 
 2 4 
 
 
 10. 
 
 3v-5a; 3 
 2 ==8' 
 2a; + 4.y_4 
 a;-32/ 3 
 
 5. 
 
 2 3 
 
 = 8, 
 
 11. 
 
 rr + ^ + l 2 
 12 3' 
 
 
 ^ + 1 + 2^= 
 4 ' 5 
 
 = 2. 
 
 
 8 2 
 
 6. 
 
 x + 2 ,y-'4 
 4 ' 3 ~ 
 
 = 3, 
 
 12. 
 
 22:H-?/4-l_3 
 3rr-?/ + 2 6' 
 
 
 a:-3 .y-2_ 
 3 ' 5 " 
 
 = 2. 
 
 
 4a:-3«/ + 2_5 
 2a;-3^ + l 3 
 
SIMULTANEOUS EQUATIONS 193 
 
 PROBLEMS PRODUCING SIMULTANEOUS EQUATIONS 
 
 Problems involving two unknown quantities are readily 
 solved by the use of two different letters, x and ?/; and 
 the pair of simultaneous equations obtained from the con- 
 ditions of the problem is treated in accordance with either 
 of the methods already learned. The following problems 
 require the use of two unknown quantities and, in each 
 problem, two equations are necessary for a solution. 
 
 Exercise 66 
 
 1. The sum of two numbers is 15, and their difference 
 is 3. What are the numbers ? 
 
 Let X = the greater number, 
 
 y = the less number. 
 Then x-\-y = 15. (1) 
 
 x-y = 3. * (2) 
 
 From (1), y=15-x. 
 
 Substituting in (2), x — (15 — x) = S. 
 x-15 + x = 3. 
 2a:=18. 
 x = 9. 
 Substituting in (1), 9 + y = 15. 
 
 y = Q. 
 Hence, x = 9, the greater number. " 
 
 y = Q, the less number. 
 
 Result. 
 
 2. The sum of two numbers is 29, and their difference 
 is 5. What are the numbers ? 
 
 F. H. S. FIRST TEAR ALG. — 13 
 
194 SIMULTANEOUS SIMPLE EQUATIONS. REVIEW 
 
 3. The difference between two numbers is 15, and the 
 quotient of the larger divided by the smaller is 6. Find 
 the numbers. 
 
 4. The difference between two numbers is 40, and if 
 the larger number is divided by the smaller, the quotient 
 and the remainder are each 4. Find the two numbers. 
 
 5. One half the sum of two numbers is 14, and 4 
 times their difference is 16. Find the numbers. 
 
 6. One half the sum of two numbers is 2 more than 
 their difference. Five times the smaller number equals 
 twice the larger number. Find the numbers. 
 
 7. There are two numbers such that 3 times the first 
 added to 4 times the second equals 37, and 4 times the first 
 less 3 times the second equals — 9. Find the numbers. 
 
 8. If 3 is added to the greater of two numbers and 2 is 
 subtracted from the smaller, the sum of the results will be 
 14. If the greater number is divided by 2, the quotient is 
 1 less than the smaller number. Find the numbers. 
 
 9. Add the least of two numbers to one half the 
 greater and the sum is 12. Subtract the least number 
 from one fifth the greater and the difference is — 5. 
 What are the numbers? 
 
 10. If the greater of two numbers is divided by the 
 less, the quotient is 3 and the remainder 6. If 4 times 
 the less number is divided by the greater, the quotient is 
 1 and the remainder 6. Find the numbers. 
 
 11. The sum of two numbers is divided by their differ- 
 ence, and the quotient is 5. If the sum of the numbers is 
 divided by 12, the quotient is f . What are the numbers ? 
 
SIMULTANEOUS EQUATIONS 195 
 
 12. If 1 is added to both the numerator and denomi- 
 nator of a certain fraction, the resulting fraction is |. If 1 
 is subtracted from both the numerator and denominator of 
 the fraction, the resulting fraction is J. Find the fraction. 
 
 (Hint : Let - = the given fraction. Then ^ "^ = -, etc.) 
 y 2/4- 1 4 
 
 13. A certain fraction becomes ^ when 5 is added to 
 the denominator, but if 5 is added to the numerator, its 
 value is |. What is the fraction ? 
 
 14. If 1 is added to both numerator and denominator 
 of a certain fraction, the resulting fraction is ^. If the 
 original fraction is inverted and 3 is added to both numer- 
 ator and denominator, the resulting fraction is ^, What 
 is the given fraction ? 
 
 15. The numerator of a certain fraction is multiplied 
 by 3 and the denominator by 2, the resulting fraction 
 being f ; but when the numerator is increased by 2 and 
 the denominator increased by 2, the resulting fraction is ^. 
 What is the fraction ? 
 
 16. When the numerator of a certain fraction is doubled 
 and the denominator increased by 3, the result is J. If 
 the denominator is doubled and the numerator decreased 
 by 1, the result is -^q. What is the fraction ? 
 
 17. A father is 32 years older than his son, but 8 years 
 ago the father was 5 times as old. What are the present 
 ages of each ? 
 
 (Hint : Let x = the father's age now, y = the son's age now. Then 
 X — y = 32. And x — S = 5(y — 8). State and solve.) 
 
 18. Three years ago a certain boy was twice as old as 
 his sister, but at the present time the sum of their ages is 
 30 years. Find the ages of each at the present time. 
 
196 SIMULTANEOUS SIMPLE EQUATIONS. REVIEW 
 
 19. The sum of the ages of John and William is 42 
 years, but one third of John's age is 1 year less than 
 one half of William's age. What is the age of each ? 
 
 20. Four years ago A was twice as old as B, but in 
 6 years A's age will be nine sevenths that of B. What is 
 the age of each now ? 
 
 21. A lady bought 10 yards of cloth and 4 yards of 
 silk, paying f 32 for the whole. Later, at the same prices, 
 she bought 5 yards of cloth and 6 yards of silk for 128. 
 What was the cost of the cloth and of the silk per yard ? 
 
 (Hint : x = cost per yard of the cloth in dollars, y — cost per yard 
 of the silk in dollars. Then 10x + 4:y= .32, etc.) 
 
 22. A man bought 4 horses a^nd 3 wagons for #780, and 
 later, 5 more horses and 2 more wagons, paying $870 for 
 the second lot. The cost of both horses and wagons was 
 the same, respectively, in each lot. What was paid for 
 each ? [ 
 
 23. Five pounds of coffee and 3 pounds of rice cost 
 f 1.80, and 4 pounds of coffee and 7 pounds of rice cost, at 
 the same rates, $1.90. W^hat is the cost of the coffee and 
 of the rice per pound ? 
 
 24. A boy has 32 marbles in two pockets. One half of 
 the number in one pocket is 2 less than the number in the 
 other pocket. How many marbles has he in each ? 
 
 25. A said to B, " If my weight divided by 3 is added 
 to your weight divided by 4, the total is 90 pounds." 
 B replied, " If one eighth of my weight is diminished by 
 one sixth of your weight, the result is 5 pounds." What 
 was the weight of each ? 
 
GENERAL REVIEW 197 
 
 GENERAL REVIEW — GROUP V 
 
 FRACTIONAL EQUATIONS 
 
 Clearing of Fractions 
 
 Multiplying both members of a fractional equation by the 
 lowest common multiple of their denominators removes the 
 fractions from the equation. 
 
 Signs 
 
 The sign of a fraction becomes the sign of its numerator 
 when the equation is cleared of fractions. 
 
 Simultaneous Simple Equations 
 
 For each solution we must have as many equations as there 
 are unknown quantities. 
 
 elimination 
 
 We make use of two methods : 
 
 1. By Substitution. In which we replace an unknown 
 quantity in one equation by the value of that unknown ob- 
 tained from the other equation. 
 
 2. By Comparison. In which we form an equation with 
 the values of one of the unknowns in terms of the other^ these 
 values being obtained from the given equations. 
 
 FINAL REVIEW 
 Exercise 67 
 A 
 
 1. What is the sum of (a-\-b')x — {a—b^y-^2(a-^c)z 
 - 3 (a + 5) :c + 4 (a - 6) y -f 3 (a + 25 + 3 (a + ^>) a; - 
 
198 SIMULTANEOUS SIMPLE EQUATIONS. REVIEW 
 
 2. Take the sum of 2a^-a^ + a-2> and 2a^^Za^ + 
 ^ a + Q from the sum of 3 a — 4 a^ -f -i a^ + 2 and 1 + 3 a^ 
 - 3 a3 4- 4 a. 
 
 3. Take the result of [(^2- ^4- 1)- (2a- ^2 + 2)] 
 from the sum of a + 2 — a2 and a?—Za—l. 
 
 4. Take a;2_^ 2 rcH- 3 'from 2^ + 3 re +2 and multiply 
 the result by the sum of 4 a: — 10 — a;2 and 11 -\- x^ — ^ x. 
 
 5. Divide the sum of a;^ — 7 a;2 + 14 2; — 17 and 2 a^ — 
 32: + 2 by (2; - 3)2 + 4 (a: -1). 
 
 6. Subtract Q — x — ^x^ from the product of (— 3 + 
 2:2-2:) by (x-2 + x^). 
 
 7. Divide 4a*-9a2+6a-l by 2a2_3a + l and 
 multiply the quotient by 2 a — 3. 
 
 8. Simplify [(3 «« + a^ - 2 a) + (a^ _ a* + 1) - (2 a^ + 
 2 ^5 _ a* — 2 a + 2)] and divide the result by (a2 + a + 1) 
 (a2_a + l). 
 
 B 
 
 Simplify : 
 
 9. a:-[l + |2:-(l + 2:)S]. 
 
 10. _J_[_(^_^_1)]|. 
 
 11. a — [ — ( — ; — a J — a) — 4 a] . 
 
 12. c-(2d-(i-c-\l-2d-ic-T^^)\')-'V). 
 
 13. -[c+ Jc-(c-l)-(c + l)-ci -(?]. 
 
 14. 22:-[2: + l-S2: + l-a-(2;+l4-a-w)|]. 
 
 15. (a + l)2-(2a-l)2+3a(a-2). 
 
 16. a(a-l)2- a(a2 + a-l)-4a(l-a). 
 
 17. (a + l)(2;+l)- (a-l)(2:-l)-2(a + a;-l). 
 
GENERAL REVIEW 199 
 
 18. 2(a+l)(a-5) + 3(a-2)(a + 5)-(4a+5)(a-8). 
 
 19. (a-l)3-(a+l)3 + 7a2 4-2. 
 
 20. Qi-\-x+iy-a(x-\-l-^a)-x(a-\-x-V), 
 
 C 
 
 Solve and verify : 
 
 21. 5a;-3(2a:-ll)-(rr-l) = 0. 
 
 22. 4a;+ [2 a;- (a:H-l)] =a;-5. 
 
 23. 2a;2-(2; + l) = 22;(a;-3). 
 
 24. 3 n; + (:z^ - 1) (2 2^ + 1) = (2 a; - 1) (a; + 1) - 1. 
 
 25. (2a;4-l)2-3a;(ic+2)-(a;-5)2 = 0. 
 
 State and solve : 
 
 26. One number is 4 times another number, but if the 
 smaller number is increased by 17 and the larger num- 
 ber decreased by 22, the results are equal. Find the 
 numbers. 
 
 27. A is 26 years older than B, and A's age is as many 
 years more than 34 as B's age is years less than 34. Find 
 the age of each. 
 
 28. A man has four sons, each 3 years older than the 
 next youngest. The age of the oldest is 21 years less 
 than the sum of the ages of all four. Find the age of 
 each. 
 
 29. A man paid a bill of '$7.20, using dimes, quarters, 
 and dollars. He used twice as many quarters as dimes 
 and three times as many dollars as dimes. How many 
 coins of each kind did he use ? 
 
200 SIMULTANEOUS SIMPLE EQUATIONS. BEVIEW 
 
 30. Two automobilists start toward each other from 
 towns 88 miles apart. The first travels at a rate of 10 
 miles an hour and the second at a rate of 12 miles an hour. 
 In how many hours will they meet ? 
 
 31. A man rows downstream at a rate of 8 miles an 
 our, but upstream his rate is but 4 miles an hour. How 
 
 .ar downstream can he go and return in 6 hours ? 
 
 32. A boy agreed to work 50 days at 10.50 a day for 
 each day he worked. For each day he was idle he agreed 
 to forfeit fO.25. He received $22 when the settlement 
 was made. How many days did he work ? 
 
 D ' • 
 
 Find the value of : 
 
 33. (a-l)2-a(a + l) +^(a-l) when a =2. 
 
 34. (x-Sy-(x-\-S)(x-S)-(x-\-Sy whena;=-2. 
 
 35. 2 a (^a -{- X — 1} — (^a — x){2 a -\- x^ when a = 3 and x 
 = -3. 
 
 36. m (m + H + 1) — (w — n + 1) (m + w — 1) when m = 
 
 and n = -- 
 
 2 
 
 37. 3a;(a:+l)2-3(a;+l)3+3(a;+2)(a;-5) when x 
 
 2 
 
 — ~ 3- 
 
 Substitute in the following formulas and obtain required 
 values : 
 
 38. If a = 4, w = 6, c? = 2, find the value of I in 
 
GENERAL REVIEW 201 
 
 39. If a = -2, Z=-128, r = - 2, find the value of aS' in 
 
 a rl— a 
 
 40. If a = 3, 5 = 4, <? = — 7, find the value of x in 
 
 """ 2^ 
 
 41. If a = 6, 6 = — 1, c = — 35, find the value of x in 
 
 _5_V62-4ac 
 a; = 
 
 2a 
 
 42. If a = 1, <f = 2, w = 11, find the value of aS' in 
 
 43. If (? = 3, ? = 40, n = 13, find the value of a in 
 
 1 = a-\-(n — V)d. 
 
 44. Given a — — b^ Z = 15, /S'= 105, what is the value of 
 winAS'=^(a-|-Z)? 
 
 A 
 
 45. How far will a body fall in 6 seconds if g in the 
 formula aS'=-| ^^2 is 32 ft.? 
 
 46. How many turns are made by a wheel 4 feet in 
 diameter in going 3 miles ? 
 
 47. What is the area of a circular pool whose diameter 
 is 100 feet ? 
 
202 SIMULTANEOUS SIMPLE EQUATIONS, BEVIEW 
 
 E 
 
 Multiply by inspection: 
 
 48. (2a + 3a:)2. 54. (4a:- 1) (3 a;- 5). 
 
 49. (4a-7«/)2.* 55. (2x^-4:')(Sx^-\-l}. 
 
 50. (3« + 5a;)(3a-5a;). 56. (5a:- 8) (4 a; + 3). 
 
 51. (a;+T)(a:-10). 57. (2 a: + 7) (3 a:- 10). 
 
 52. (a^-4)(a^H-13). 58. (a + 2 a; -3)2. 
 
 53. (2a: + 3)(3a: + 4). 59. (2a3 + 3 ^2 + ^ + 2)2. 
 
 Divide by inspection : 
 
 60. 4 a2 _ 25 by 2 a - 5. 66. 27 a^ _ 1 by 3 « - 1. 
 
 61. a^ - 64 by a - 4. 67. a* - 1 by « - 1. 
 
 62. ^3 _|. 125 by a + 5. 68. a*-lbya + l. 
 
 63. 8 a^- 27 by 2 a - 3. 69. a* - a:^ by « - a:. 
 
 64. 125 4- a:^ by 5 + a;. 70. a^ — a:^ by a — a:. 
 
 65. 64 - a3 by 4 — a. 71. a:^ _|_ 32 by a; + 2. 
 
 F 
 
 Factor : 
 
 72. a^-ia. 76. a^-64a;. 
 
 73. 27 63(^3 + 1. 77. 12m'-m + x^-m\ 
 
 74. 16a2-40a+25. ' 78. ar^- lla:2__ 423,. 
 
 75. ^2 _ 19 ^_ 120. 79. a;2-a2_i0a_25. 
 
GENEBAL REVIEW 203 
 
 80. ^x^ + Qx-^b. 88. m3-8-3(w-2). 
 
 Q1. a^-x^ + x-l, 89. 4 2^ + 17 a: + 4. 
 
 82. 4(2; -1)2 -9. 90. ax-\-a-\-^x + ^. 
 
 83. 39-10c-c2. 91. 216 m3- 125. 
 
 84. 4:c*-32a;. 92. x-lbx^ + ZQa^. 
 
 85. 7c2_98c + 343. 93. 25 2:2_4(^_2)2. 
 
 86. b-10x-bc^ + bx\ 94. 10a:2-63 + 17a;. 
 
 87. 6r2-a;-2. 95. a:5_a;3_^2^1. 
 
 G 
 
 Find the H. C. F. and the L. C. M. of: 
 
 96. 9(c + (^)2, 30((?*-(?*)^ 45(c2-2c^ + (i2^. 
 
 97. x{x-iy, 0:2(2: +1)2, x^x^-l). 
 
 98. 3(:r + l)(2:+2), 4(:z: + l), 6(a:+2). 
 
 99. m2-m-20, m2+8m + 16, m2 + 10m+24. 
 
 100. 9(^:2-:?: -2), 6(^-2a:'-3), 12(^-5 2: + 6). 
 
 101. 3(^3 + 53), 6(a2-52), 9(^3-53). 
 
 102. 22:2-a:-l, 22^-3a;-2, ?>x^-^x+Q. 
 
 103. xy ■\- X -\- y + 1^ xy — X'\- y —\, xz -\- z -\- x -\- 1. 
 
 104. (^2-52)2, 7(<z + 5)3, 6(a*-54). 
 
 105. a^-(x-{-iy, (a-\-xy-h (^ + 1)2-2:2. 
 
204 SIMULTANEOUS SIMPLE EQUATIONS. REVIEW 
 
 H 
 
 Reduce : 
 
 "*'• Zi — 5 — I ^ o 5 to lowest terms. 
 
 ^4 . Q Q , o 9 to lowest terms. 
 
 "^* ~T"^ — 2r~; — m . iQ to lowest terms. 
 
 ^ _ ^2 _ 20 
 ^^^' "TT — TT" ^^ ^ mixed expression. 
 
 ~2 T" *^ ^ mixed expression. 
 
 111. rr^ — 07 4- 1 to an improper fraction. 
 
 37+1 i- x- 
 
 112. a — - — a + 2 to an improper fraction. 
 
 113. c-\- d—1 — -— to an improper fraction, 
 
 c+d+1 ^ ^ 
 
 I 
 
 Collect : 
 
 114. 1 • -1 
 
 115. 
 
 x—2a x^—5ax+6a^ x—Sa 
 
 g + 4 g-4 15g+2 
 e-4: c + 4. c^-ie' 
 
 116. -1-^+ 1 
 
 a2+3a + 2 a2 + 5a + 6 
 117. ^<^ I « 2 a 
 
 2a + 2 3a-3 6a2_6 
 
118. 
 
 119. 
 
 120. 
 
 121. 
 
 122. 
 
 GENERAL REVIEW 
 
 1 
 
 1 , 
 
 1 
 
 (a+1) 
 
 {a-\-iy ( 
 
 :«+i)3 
 
 x^-1 
 
 2x2 x-1 
 
 
 x-1 : 
 
 r-l x+1 
 
 
 2 
 
 2 
 
 4 
 
 a?-l 
 
 l + a2 (^2. 
 
 -IX^'^ + I) 
 
 7x 
 
 a:-2 
 
 1 
 
 ^-8 ' 
 
 a:2+2a; + 4 
 
 x-2 
 
 a^ 
 
 , 1 
 
 x 
 
 {x-{-iy 
 
 ' 37+1 (:r 
 
 + 1)^ 
 
 2x 
 
 3 
 
 2 
 
 205 
 
 123 
 
 ' 3(a;-2)2 2^ + 4 a:2-4 
 
 Simplify : 
 
 124. [a + 6' —J. 
 
 io« a2-4a + 3 a^-2a 
 
 a^ — 6 a + 5 a^ — a^ 
 
 127. r^«^Vfi-i+i\ 
 
 \am — ax ■{- mxj \a m xj 
 128. fl-lY-^V^^^^'. 
 
 U^ Ai + W i-:r2 
 
 3 2: +3 
 
 129. fx-i + _^y/|^ 1' 
 
 I ^r+iy V3 2 6, 
 
206 SIMULTANEOUS SIMPLE EQUATIONS. UEVIEW 
 2^2-3^^ + 2 2:2-16 
 
 130. 
 
 131. 
 
 2;2-5a;4-6 * a^ + ^-12' 
 c + d c^-d^ . c2+^ 
 
 132 jx-iy-a? ' x^-^a-iy 
 
 133. 
 
 ^2— 9 _^ r 6t9^ — 3 n aa; 4- ^ + 3 a; + 31 
 c2 — 4 * [_cm + 2 m c?a; + ^ — 2 a; — 2 J* 
 
 Solve and verify : 
 
 x-1 x-{-2 x-2 x + 1 
 
 134. 
 
 6 
 
 135. 2(£+3)^3_2^+l^ 
 
 5 2 
 
 136. i^-|(.-l)-£±l = 0. 
 
 137. 
 
 :r + l 2^-1 8 
 
 X --l x+1 x^—1 
 
 138. ^^-±1-= ^-^ + 
 
 2a + 6 3a-9 6a2_54 
 
 State and solve: 
 
 139. The difference between the numerator and denomi- 
 nator of a certain proper fraction is 5. If 4 is added to 
 both the numerator and the denominator, the resulting 
 fraction is -^2 • What is the original fraction ? 
 
GENERAL REVIEW 207 
 
 140. A certain number is decreased by 12 and the 
 remainder divided by 4. If the resulting quotient is 
 increased by 7, the sum is the same as if the original 
 number had been increased by 7 and then divided by 3. 
 What is the original number ? 
 
 141. A boy can mow a lawn in 7 hours if he works alone, 
 and a second boy can do the same work alone in 6 hours. 
 In how many hours can the boys together accomplish the 
 task ? 
 
 142. A gentleman orders a bill of books. Three less than 
 half the whole number are fiction, 2 more than a third of 
 the whole are history, and 3 less than a quarter of the 
 whole are biography. How many books are there of each 
 kind? 
 
 143. Divide $ 630 among A, B, and C, so that A shall 
 receive half as much as B, and C two thirds as much as A 
 and B together receive. 
 
 Solve by substitution : Solve by comparison : 
 
 144. 2x-\-i/ = 3, 147. 4:x + ^i/ = — 29, 
 Sx-2i/ = 4:, 5ic-4y=-13. 
 
 145. 4a;-5^ = 2, 148. ll:r+7?/4-29 = 0, 
 2 :r + 3 ^ = 12. 3 a; - 33 = - 11 y. 
 
 146. 22; + 5?/-4 = 0, 149. -10 + 5x=S7/, 
 Sx-\-2i/-{-5 = 0. 1 2/-4: = -2x. 
 
208 SIMULTANEOUS SIMPLE EQUATIONS. REVIEW 
 Solve by either method : 
 
 153. ^-y+^ =2, 
 
 4 
 
 154. ^-1=^, 
 5 2 
 
 3 "^ 4 ' 
 
 150. 
 
 I* 
 
 1-^ 
 
 
 X 
 
 10" 
 
 -!=->■ 
 
 151. 
 
 3 
 
 -¥=-*• 
 
 
 3a: 
 2 
 
 -I-"- 
 
 152. 
 
 a: — 
 
 2 
 
 i.^=., 
 
 155. 
 
 ^ + .y + 3 ^2 
 
 x-y-2 3' 
 
 a;+l . .y + 1 ^3 ^_-^jf2^4 
 
 2 5 ' a^ + y-3 3* 
 
 State and solve : 
 
 156. Find two numbers whose sum is 37. and whose 
 difference is 13. 
 
 157. One fourth of a number added to one third of a 
 second number gives a sum of 8. One third of the first 
 number increased by one fifth of the second gives a total 
 of 7. What are the numbers ? 
 
 158. If the sum of two numbers is divided by 7, the 
 remainder is 4 and the quotient, 6 ; but if the difference 
 between the numbers is divided by 7, the remainder is 2 
 and the quotient, 4. Find the numbers. 
 
UNIVERSITY OF CALIFORNIA LIBRARY 
 This book is DUE on the last date stamped below. 
 
 Fine schedule: 25 cents on first day overdue 
 
 50 cents on fourth day overdue 
 One dollar on seventh day overdue. 
 
 APH 20 1947 
 
 
 26Feb'57\V« 
 
 afltCBL OCT 1'1!> 
 
 LD 21-100m-12,'46(A2012sl6)4120 
 
ivl306053 
 
 OA 15^ 
 5 U "^ 
 
 THE UNIVERSITY OF CALIFORNIA LIBRARY