?3V ^J^A4-vt'cc^ C^^^t^^^^vx, Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/algebraforschoolOOevanrich ALGEBRA FOR SCHOOLS GEORGE W. EVANS Instructor in Mathematics in the English High School Boston, Mass. NEW YORK HENRY HOLT AND COMPANY 1899 Copyright, 1899, BV HENRY HOLT & CO. ROBERT DRUMMOND, PRINTER. NEW YORK A/^-3 t? PREFACE. In the arrangement of topics in this book an effort has been made to preserve the pupil from the besetting sin of conceiving algebraic operations as a species of legerdemain. This end could best be secured, it seemed to me, by making practical problems the point of departure, initially and at each new turn of the subject. With a concrete case in mind, the pupil can hardly fail to perceive not only the need for the process that he is set to study, but also its rational basis and its economy. In this larger appeal to the practical sense it will be found that there is no slighting of mental dexterity, no injurious deviation from accepted methods, and certainly no sacrifice of mathematical rigor. If the arrangement of chapters here adopted is not acceptable, it is entirely feasible to take the topics in the traditional order. The index will facilitate this rearrange- ment. Whether or not teachers agree with me in respect to the ordering of topics or the method of attack which I have found fruitful in class, they are sure to appreciate the very large collection of examples — some thirty-five hundred all told — which are not reprinted from other text-books. In the following features also I think the book possesses advantages : 1. The careful classification of problems, so that due em- phasis may be given to the several types of equations arising from them, iii ivi30B0»7 iv PREFACE, 2. The insistence upon a scheduled explanation of steps in the reduction of equations; and the clearness and brev- ity of reference and explanation obtained by denoting an equation by its serial number enclosed in a circle. 3. The introduction of supplementary sets of constants for some important problems, as on pages 28 and 39 and in Chapter V. 4. The thorough study of literal equations and general- ized problems; and especially of literal quadratics. 5. The prominence given to the solution of equations by factoring as a fundamental method; and the treatment of '' completing the square^'' as a method of factoring. 6. The treatment of Eules and Formulae in Chapter II, and of Theorems and Identities in Chapter IV. 7. The separation of Elimination into two parts suitable for linear systems and for linear-quadratic pairs respectively ; the discussion of simultaneous, independent, and consistent equations; the ^^ Equations of the New Set,'^ page 160. 8. The treatment of H. C. E. by elimination of highest and lowest terms. Much of the book has been used, in manuscript and in proof, for class- work, but I can hardly hope that all mis- takes have been corrected : notice of remaining errors will be gratefully received. Eor friendly aid and suggestion I am indebted to several of my colleagues; and especially to Mr. P. E. Gartland, who scrutinized for me every letter of the proofs. G. W. E. English High School, Boston, February 3, 1899. CONTENTS. PAGE I. The first use of algebra: abbreviation of expla- nations 1 II. The second use op algebra: abbreviation of rules 41 III. Transformations 56 IV. The third use of algebra: theorems and identi- ties . . 88 V. Factorable equations 104 VI. The first method of elimination: equations of the FIRST DEGREE 131 VII. The second method of elimination: linear-quad- ratic PAIRS. 163 VIII. Multiplication of fractions; highest common fac- tor 190 IX. Lowest common multiple; summation of fractions . 212 X. Indices; surds; roots 239 XI. Literal equations; generalization 274 XII. Proportion and variation 299 XIII. The progressions 310 XIV. Detached coefficients; the binomial theorem. . . . 328 XV. Logarithms 348 Table of Logarithms 363 XVI. Supplementary PROBLEMS FOR practice AND review. 366 ALGEBRA. CHAPTER I. THE FIRST USE OF ALGEBRA. 1. Algebra is a method of abbreviating the explana- tions of problems in arithmetic. It is also used to abbre- viate the statement of rules and the demonstration of theorems. 2. This chapter illustrates the first use of Algebra: for abbreviating the explanation of problems in arithmetic. Some problems are here given as examples; the answers to many of them are evident at a glance. The pupil must re- member that he is not expected to shorten the arithmetical work, for he cannot do that. He is to put down the ex- planation briefly and systematically. When that is learned he will find that he can do complicated problems with greater ease, because he can put down in black and white as he goes along the successive steps of his reasoning. 3. Model A. — A father is 6 times as old as his son, and their united ages are 42 years; find the age of each. Explanation, (1) The father's age -j- the son's age = 42 years. (2) The father's age = 6 X the son's age. (3) 6 X the son's age + ^^6 son's age = 42 years. (4) 7 X the son's age = 42 years. (5) The son's age = 6 years. (6) The father's age = 36 years. 2 ALGEBRA, Abbreviated Explanation, Let 5 stand for the number of years in the son's age ; then 6X5 will stand for the number of years in the father^s age. 6X5 + 5 = 42 © 7 X 5 = 42 ® 5 = 6 ® 6 X 5 = 36 EXERCISE I. 1. John is twice as old as Henry, and the sum of their » ages is 21 years. Find the age of each. 2. John and Will wish to divide 36 cents so that John shall have twice as much as Will. How much will each have? 3. Paid $220 for a horse and saddle, and the horse cost 10 times as much as the saddle. Cost of each ? 4. A sidewalk laid with a flagstone and a curbstone is 6 feet wide; the flagstone is 11 times as wide as the curb- stone. Width of the flag and of the curb ? 5. The electoral vote for the state of Pennsylvania is 5 times that for the state of West Virginia. Both count 36 electoral votes. Each counts how many ? 6. There are 7 times as many sheep as lambs in a pasture, and in all there are 96. Number of each ? 7. Pole 9 feet long partly peeled of bark; bark part twice as long as bare part. Find length of each part. 8. It takes a man 4 times as long to run the first three- quarters of a measured mile as to run the last quarter; he goes the whole mile in 6 minutes. How long for the last quarter ? 9. Mercury weighs 13 times as much as water; a quart of mercury and a quart of water weigh in all 28 pounds. Weight of each ? THE FIRST USE OF ALGEBRA. 3 10. A bag of oats, hung on one end of a steelyard beam, weighs 20 times as much as the iron weight attached to the other end of the beam ; the total downward pull on the steelyard is 52J pounds. Find the weight of the bag of oats. 4. An equation is a statement that two things are equal. 5. The algebraic abbreviation for any quantity, or for any combination of quantities, is called an algebraic ex- pression. 6. It takes two algebraic expressions to form an equa- tion. The two expressions said to be equal are called, re- spectively, the first, or left-hand member, and the second, or right-hand member, of the equation. 7. The parts of an algebraic expression separated by the signs + aiid "~ ^^'^ called the terms of the expression. 8. In order to fix the student's mind on the nature of the processes by which his equations are obtained, it is well for him to indicate beside each equation how it arose from the preceding equations. The additional trouble is slight and the scheduled '* explanation " is logically complete. 9. Model B. — In a certain collection of insects there are twice as many butterflies as moths; of both there are 39. How many of each ? Let X = the number of moths. Then 2x* = the number of butterflies. 2.T + X = 39 (D 3x = 39 same as @ x = U (D -^ 3 ® 2x = 2Q (D X 2 Ans. 13 moths; 26 butterflies. ^ When two quantities are written together without any sign be- tween them, multiplication is indicated. For representing in any prob- 4 ALGEBRA. EXERCISE II, 1. One man is twice as heavy as another, and both weigh 339 pounds. Weight of each ? 2. The number of cows and sheep in a certain farmyard is 75, and there are 4 times as many sheep as cows. Num- ber of each ? 3. The doctor has twice as many books as the minister, and they both have 2100. How many books has each ? 4. In the dog-show are 36 St. Bernards, twice as many long-haired as short-haired. Number of each kind ? 5. A locomotive weighs 6 times as much as a car, and both weigh 14 tons. Weight of each ? 6. A man is twice as heavy as a boy, and both weigh 200 pounds. Weight of each ? 7. A's house cost 2^ times as much as B^s house, and both cost $7000. Cost of each ? 8. Paid $100 for two lots of tiles; one lot cost 4 times as much as the other. Cost of each lot ? 9. Walked 4 times as far in the afternoon as I did in the forenoon ; all day together I walked 15 miles. How far in the forenoon ? 10. It costs 4 times as much to go from New York to Chicago as from Boston to New York; from Boston to Chicago it costs $25. How much from Boston to New York? 11. The president of a stock company owns twice as many shares in it as his brother; and both own 165. How many does each own ? 12. A man had a big bill in his pocket, and asked the price of a house and barn; he found that the house cost lem numbers of which the values are not given, the latter letters of the alphabet are generally used, most often x. These are matters of universal agreement among writers and students of Algebra (conven- tions). THE FIRST USJE OF ALOEBRA. 5 7 times the value of the bill, and the barn 3 times the value of the bill, both coming to $5000. Find cost of house and of barn. 13. Three times as many barrels in one cellar as in an- other; in the store above half as many as in both cellars. In the whole building 216 barrels. How many in each part ? 14. A fortress is garrisoned by 5200 men; and there are 9 times as many infantry and 3 times as many artillery as cavalry. How many are there of each ? Valuation Problems. 10. Model C— Settled an account of $48 with $2 and $5 bills, using twice as many 5's as 2's. Find the number of bills of each denomination. Let X = the number of $2 bills. Then 2x = the number of $5 bills. ® lOo; + 2x = 4:S^ (2) 12iz; = 48 same as (D a: = 4 © -^ 12 ® 2:?: = 8 (D X 2 A71S. Four $2 bills; eight $5 bills. EXERCISE in. How many bills of each kind must I use to settle the following accounts in the manner described for each ac- count ? 1. To pay $88 I use 5's and 2's only, 3 times as many 2's as 5's. 2. To pay $72 I use 5's and 2's only, 2 times as many 5's as 2's. * The statement is about tlie number of dollars in the values of the bills, not about the number of bills; the value of the $5 bills is 5 times their number, 5 X 2aj or lOiC. 6 ALGEBRA. 3. To pay $42 I use lO's and 2^s only, twice as many 2's as lO's. 4. To pay $80 I use lO's and 5's only, twice as many 5's as lO's. 5. To pay $26 I use lO's and l^s only, 3 times as many I's as lO's. 6. To pay $45 I use 2's and l^s only, 7 times as many 2's as I's. 7. To pay $78 I use 5's and I's only, 5 times as many 5's as I's. 8. To pay $84 I use lO's and 2's only, 4 times as many lO's as 2's. 9. To pay $75 I use lO's and 5's only, 7 times as many lO's as 5's. 10. To pay $183 I use lO's and I's only, 6 times as many lO's as I's. 11. Five horses and four donkeys weigh all together 9600 pounds. Each horse weighs 4 times as much as a donkey. Find the weight of each animal. 12. Bought apples for 39 cents; twice as many red ones for 5 cents apiece as green ones for 3 cents apiece. Num- ber of each ? 13. Engineers examined a bridge that broke down under the weight of a crowd of people, and decided that the breaking strain was 36 tons. Supposing that there were 4 times as many men as women, the average weight being 150 pounds for a man and 120 pounds for a woman, what was the number of each ? 14. Tea of first quality at 60 cents per pound, and 3 times as much tea of second quality at 45 cents per pound, came to $13.65. Number of pounds of each? 15. Mules at $40 apiece, and 7 times as many horses at $125 apiece, cost $2745. Number of animals in the drove ? 16. Drove westward in a buggy for 10 hours, then rode THE FIRST USE OF ALGEBRA, 7 in a train 5 times as fast for two days and nights ; the total journey was 1500 miles. How fast did the buggy go ? 17. Lead weighs four times as much as marble; three leaden globes and two marble globes, all of the same size, weigh 56 pounds. AVeight of each globe ? 18. A team is made up of oxen and mules, one of each in a pair; supposing that an ox can pull 4 times as much as a mule, and the whole team is used to pull 30 tons, how much of this load do the oxen pull ? 19. A team consisting of 2 oxen and 5 mules, in which each ox pulls twice as much as a mule, is used to raise a safe weighing 18 tons. What would be the strain on each trace of each mule ? What would be the strain on the ox- chain ? 20. A horizontal iron beam weighing one ton is sus- pended by 3 chains, 2 at one end and 1 at the other. What is the pull on each chain ? 21. Paid 91 cents with dimes and three-cent pieces, of each the same number. Number of each ? 22. Three times as many nickels as two-cent pieces in my purse; in all 51 cents. Number of each ? 23. Paid $1.80 with quarters, dimes, and nickels; twice as many dimes and 3 times as many nickels as quarters. Number of each ? 24. Bought twice as much coffee at 33 cents as tea at 54 cents, and paid in all $2.40. How many pounds of each were bought ? 25. Bought horses at $120, sheep at $14, and chickens at 50 cents; 3 times as many sheep as horses, 7 times as many chickens as sheep. All cost $517.50. How many of each ? 26. Two travellers started towards each other from oppo- site ends of a straight road 132 miles long; one went 4 miles an hour, the other 7. How long before they met ? 8 ALGEBRA, 27. The crew of a towboat consists of an engineer, 2 firemen, 3 deck-hands, and a cook, whose wages are re- spectively $2, $1.50, $1, and $1.20 per day. For a certain voyage the pay-roll was $73.60. How many days out ? 28. In a certain mill are employed men at an average wage of $1.17 per day, 5 times as many women at an average wage of 63 cents per day, and twice as many chil- dren (as men) at an average wage of 31 cents per day. Tlie weekly pay-roll of this mill is $296.40. How many men, women, and children are employed in it ? 29. Three families, with 8, 5, and 3 members respect- ively, divide among themselves a square mile of land, so that there are equal shares for all the individuals. What is the share of each family ? 30. A sidewalk is laid with 3 flags and a curb, the flags being each 7 times as wide as the curb. The width of the sidewalk is 11 feet. Find the width of flag and of curb. Difference of the Unknown Numbers Given. 11. Model D. — A father is 50 pounds heavier than his son, and both weigh 248 pounds. Weight of each ? Let X = the number of pounds the son weighs. Then x -\- 60 = the number of pounds the father weighs. © a: + a; + 50 = 248 @ 2:r + 50 = 248 same as ® (D 2:?: = 198 © - 50 ® a; = 99 (D -f- 2 (D a; + 50 == 149 ® + 50 Ans, Father 149 pounds; son 99 pounds. EXERCISE IV. 1. Two cannon-balls and a six-pound weight balanced on a scale 50 pounds. What was the weight of each canno.n-ball ? THE FIRST USE OF ALGEBRA. 9 2. A marketman sold 11 sheepskins, and then lost $2 of the purchase-money; he had $7.90 left. How much did a sheepskin sell for ? 3. On a bill of 1181 part of the account is paid with equal numbers of $10, $5, and $2 bills, and a check for $28 is given for the remainder. How many bills of each kind were used ? 4. Two mules and a horse were bought for $400, and a horse cost $40 more than a mule. Cost of each ani- mal ? 5. Two bucketfuls of water fill a 15 -gallon tub all but 5 quarts. What is the capacity of the bucket ? 6. A man starts on a journey of 163 miles; he walks 13 hours, rides twice as fast for 20 hours, and finally has to stop 4 miles short of his journey's end. How fast did he walk ? 7. A grocer mixes tea that costs him 20 cents a pound with 3 times as much tea that costs him 38 cents a pound, and sells the mixture for $10, clearing a profit of 62 cents. Number of pounds of each ? 8. My farm is three times as large as the one next to it, and both together, with a ten-acre lot across the road, just make a quarter-section — 160 acres. Size of each farm ? 9. Two trains start from opposite ends of the same 13- mile track, one going 20 miles an hour and the other 24. They stop when 2 miles apart. How long were they going ? 10. Quarters in one pile, twice as many dimes in another, and $1.13 besides — in all $6.08. How many quarters and dimes ? 11. Bought peas at 59 cents a peck, and 1 peck more of beans at 30 cents a peck; paid 25 cents for lettuce — in all $5. How many pecks of peas and of beans ? 12. Bought sugar at 5 cents, 3 times as much coffee at 10 ALGEBRA. 50 cents, and paid $1.83 for a ham. The whole bill was $12.68. How much coffee ? 13. Sold a house; then sold 5 more at double that price; then sold 35 tons of coal at $5 per ton. Eeceived all together $64,448. Price of first house sold ? 14. Three times as many nickels as half-dollars; in all $4.55. Number of each ? 15. Three times as many dimes as nickels, 5 times as many three-cent pieces as dimes; in all $5.60. Number of each ? 16. Three times as many dimes as dollars, 4 times as many cents as dimes, 5 times as many nickels as cents; and $17.68 all together. Number of nickels ? 17. I have 10 cents more than my uncle, and we both have $2.90. How much has each ? 18. Suppose each of the 36 boys in a class has the same sum, and the teacher has 15 cents more than all of them together. All the money counts up $79.35. How much has the teacher ? 19. Three times as many lO's as 2^s, and $7.35 besides, make $167.35. Number of 2'o ? 20. Seven times as many I's as 5's, and $1.45 besides, make $37.45. Number of I's ? 21. Three times as many 2's as 5's, 4 times as many lO's as 2's, and $11.11 besides, make $666.11. Number of 2's ? 22. Eode 6 miles an hour, then walked 7 times as long at 4 miles per hour, then went 23 miles by train; 91 miles in all. How long did I walk ? 23. Made a mixture of water, 3 times as much wine at $1.20 per quart, twice as much bark extract as wine at 75 cents per quart, and one-fourth of a pint of beef extract costing $3.25. Sold it all at a profit of $5 and received $24.45. Number of quarts of bark extract ? 24. Five times as many nickels as two-cent pieces, 5 THE FIRST USE OF ALGEBRA II' times as many dimes as nickels, 7 times as many quarters as nickels, twice as many dollars as dimes, and a check for $53.92, making $300. Number of dollars ? Reduction of Equations. 12. Consider the problem: A is twice as old as B; 22 years ago he was 3 times as old as B. What are their ages now ? Let X = B's age now. Twenty-two years ago A's age was 2x — 22 and B's age was x — 22. So the equation is 2a; - 22 = ^x - 22). Expressions like 3(^ — 22) are new to the pupil. Suppose a farmer contracts to deliver 3 bushels of oats and 5 bushels of barley every day. To get the number of bushels of grain delivered in several days he multiplies not only the quantity of oats but also the quantity of barley by the number of days. Suppose a farmer receives daily x bushels of grain, and delivers daily 3 bushels. The increase of grain in his store for, say, 7 days would be found thus: c?; bushels daily for 7 days, .... received 72; ; 3 bushels daily for 7 days, .... delivered 21. Subtracting the amount delivered from the amount re- ceived, Ix — 21 is what the daily increase of a; — 3 amounts to in 7 days. Consideration of similar cases will make the following principle evident. 13. Whenever an expression of two or more terms is multiplied, each term of that expression separately must be multiplied. 14. The equation 2x — 22 = 3(2; — 22) now becomes 22; — 22 = 3a; — 66, the only change being a change of form, not of value. This equation 22; — 22 = 32; — 66 differs from those we 12 ALGEBRA. have been used to in that it has negative terms on each side of the equation, that is, terms to be subtracted. The left side is, not equal to ^x, but just 22 short of it. In the same way the right side is 66 short of ^x. If 22 is added to each side of the equation, the shortage on the left side will disappear, but the right side will still be 44 short of 3x. That is, the equation will be 2x = 3^ — 44. This shortage also disappears when 44 is added to each side, giving 2^ + ^^ = 3a;, an equation like many we have been solving. But if 66 is added at first the entire shortage on the right is made up; and on the left it only takes 22 out of the 66 to make up that shortage, leaving 44 to be added to the complete value of "Hx, The shortest way, then, when there are two similar shortages, is to add the larger. Shortages and Multiplications. 15. Model E. — A is twice as old as B; 22 years ago he was three times as old as B. What are their ages now ? Let X = B's age now ; then A's age = 2a;. © 2a; - 22 = 3(a; - 22) © 2a; — 22 = 3a; — 66, same as ® (D 2a; + 44 = 3a; © + 66 ® 44 = a; © — 2a; ® 88 =. 2a; © X 2 Ans. A^s age 88; B's age 44. EXERCISE V. 1. A is 5 times as old as B, and 5 years hence will be only 3 times as old as B. Ages now ? 2. Eleven years ago A was 4 times as old as B and in 13 years from now he will be only twice as old. Ages now ? 3. Rice costs 2 cents a pound more than sugar; 3 pounds THE FIRST USE OF ALGEBRA, 13 of sugar and 10 pounds of rice come to $2.40. Cost of each ? 4. I can walk 3 miles more in a forenoon than in an afternoon; and between Monday noon and Friday night I can walk 75 miles. How far can I walk in one afternoon ? 5. A wall 13|: feet high is built with seven courses of foundation-stone and 41 courses of brick ; the courses of stone are 9 inches thicker than the brick. What is the thickness of the foundation-stones ? 6. A wall is laid with bricks of two thicknesses, 3 inches and 5 inches ; there are three more courses of the thicker kind than of the thinner, and the wall is 9 feet 3 inches high. Number of courses of each ? 7. Paid 12.20 in quarters and nickels ; two more nickels than quarters. Number of each kind of coin ? 8. A boy starts from his room in college at noon one day to walk to his home, 41 miles off ; at two o^clock of the same day the father starts on horseback for the college, riding 3 miles per hour faster than the son can walk ; they meet at 5 p.m. How fast can the boy walk ? 9. A pile of 27 cannon-balls, in three sizes, weighs 254 pounds ; 10 of them are twice as heavy as the 13 lightest ; and the others are each by 2 pounds the heaviest in the pile. Find the weight of each size. 10. A tank holding 5940 gallons is filled in 3 hours by three pipes, the first of which carries twice as much, and the other 3 gallons less, per minute, than the third. Number of gallons per minute through each pipe ? 11. A is three times as old as B ; 15 years ago he was five times as old as B. What are their ages now ? Find the number of coins of each denomination used in the folloiuing payments : 12. Paid 39 cents with three-cent pieces and five-cent pieces; 5 more 3's than 5*s, 14 ALGEBRA, 13. Paid 85 cents with nickels and dimes; 2 more nickels than dimes. 14. Paid $2.05 with quarters and nickels; 1 fewer nickels than quarters. 15. Paid $4.75 with quarters and halves; 5 fewer quarters than halves. 16. Paid $1.45 with quarters and dimes; 4 more dimes than quarters. 17. Paid 68 cents with dimes and two-cent pieces; 10 more 2^s than dimes. 18. Paid $2.30 with quarters and three-cent pieces; 2 more 3's than quarters. 19. Paid $1.09 with two-cent pieces and three-cent pieces; 3 more 3's than 2''s. 20. Paid $1.01 with dimes and three-cent pieces; 12 more 3's than dimes. 21. Paid $6.40 with nickels and halves; 4 more halves than nickels. REVIEW. I. What is the use of Algebra ? What is an equation ? What is a term ? II. State the first and second members, and the first, second, and third terms, in the following equations: i. 3:r - 5 + lOiT = 7 - X. 2. X — y + 5 — 3y + a, S. x-x + l + bx — y + 6=x+5x+l. 4. 4x=z3 + 5x- y + Q. III. Solve the following problems: 1, Thirty horses and 40 mules weigh 54 tons; on an average, each horse weighs 100 pounds more than a mule. Average weight of the two kinds of animals ? 2, There are in a purse three times as many nickels as dimes, and in all $1.50. How many nickels ? THE FIRST USE OF ALGEBRA. 15 S, Four cows, 3 calves, and 10 sheep cost $168; a cow costs five times as much as a (;alf, and a calf costs twice as much as a sheep. Cost of each ? ^.* A man wishes to pay $35 with dollars, halves, and quarters, of each an equal number. Number of each ? 5. A grocer mixes tea that cost him 25 cents a pound with four times as much tea that cost him 30 cents a pound, and sells the mixture for $16, clearing a profit of $1.50. Number of pounds of each ? Rule for Solving Simple Equations. 16. An Axiom is a general statement not requiring proof. Examples of axioms are : 17. If equal quantities are ijtcreased or dimin- ished BY THE SAME AMOUNT, THEY REMAIN EQUAL. 18. If equal quantities are multiplied or divided BY the same amount, THEY REMAIN EQUAL. 19. Upon these hvo axioms is based the following rule for solving simple equations : (1) Perform any indicated multiplications and unite similar terms. (2) If there are any terms with — before them, add to each member the quantities lacking. (3) Subtract from each member the smaller unknown term. (4) Subtract from each member the known term that stands beside an unknown term. (5) Divide each member by the coefficient of x, 20. Where a quantity may be separated into two factors, one of these is called the coefficient of the other; but by the coefficient of a term is generally meant its numerical factor. 21. Similar terms are those that have the same letters as factors. Similar terms may be united; e.g., * Reduce all sums to tlie denomination of quarters. 16 ALGEBRA. dx + 4.x = 7x 3x -]- 4:X — 2x — 6x 3x + 6x + 6 + 2x + 2 = 10x + 7 7X+6-2X+3-X-4r = 4:X+4: Terms not similar cannot be united; e.g., 3x -\- 4:y is not equal to 7y nor to 7i^; it cannot be simplified. EXERCISE VI. Perform indicated multiplications : 1. 3(x + 5). 4.13(1-^). 7. 0(3:?:- 17). 2. b{x + 4:). 6. 11(3 -4a:). 8. 8(19 + 5a:). 8. 7{2x - 3). 6. 8(4 + 3a:). 9. 201(10 - 71a:). 10. 3(a: - 5) + 5(3 — x). Unite similar terms : 11. a: + ^ + 7a; + 13y + 5 + a:. 12. 3a: - 5 + 4a: + 2 - 5a: + 7 - a:. 13. 7 - 7a: + 3 - a: - 11 + 9a: + a: - I. 14. 3 + 5a: - 8a: - 2 + 7a; - 10 + a: + 15. 15. x + y+l + x-y + l + x + y-l-x + y + 1, 16. 13a: - 2a: + 17 - 5a; - 11 - a: + 7.a: - 23. 17. 1 + a: + 6 - 3a: - 10 + 7a: — 5a: + 4. 18. 4 + a: + 4a: + 20 - 3a; - 8 + 2x. 19. 208 - 29a; - 191 + 17a: - 17 + 53a;. 20. 1089a; - 2001 + x+ 1987 - 787a: - 400 + 350a:. Perform indicated multiplications arid unite similar terms : 21. 3(4 -x)+ l{x — 3) + 5a: - 4. 22. 5(1 - x) + 13(a; - 3) + 11a: - 19. 23. a: + 5 + 3(2a: - 1) + 7 + 2(1 - 5a:) + 10a:. 24. 3 + 4(a; - 5) + 3a; + 7 + 13(4 - 3a:) + 11. 26. 4(a: - 5) + 9(3a: - 2) + a: + 4(1 - bx), 26. 5(7a: - 2) + 13 - 5a; + 12(3 - 2a:) + 5. 27. 17(a: + 5) + 50(3 - a:) + 40a; - 10 + 3(a: + 1). 28. 40(2a: - 1) + 45(2 - a:) + 17 + 1^(^ + l*^)- THE FIRST USE OF ALOEBUA. 17 29. 30(2:?; + 3) - 15^; + 5 + 5(4 - bx) + 23. 30. I00{x + 100) - 10:*; - 10 + 10(2 - x) - 90a:. Find the value of x in the following equations : 31. 3(a; -^l)=2x - 11. 32. ^{x + 5) = lOo; - 6. 33. %x -\- b = ^X. 34. bx-1 =^ 3(a; + 1) _ 4. 36. lOx + 13 = ^x + 9). 36. 7(0; - 2) = 4t{x + 4) + 3. 37. a; + 5 = l{x — 10) — x. 38. 8:r — 1 = 5^: + 41. 39. 1{x + 2) = ic + 32. 40. b{l — x) = 1 — 3^;. 41. 5(:i; + 4) = 14a; + 2. 42. 3(a; + 3) = 3(3a; + 1) - 12. 43. 4(a; + 5) = 6a; + 6. 44. 5(a; + 1) = 6(3 - a;) - 2. 45. 2(a; - 3) = a; + 7. 46. 2a; = 5(a; — 1) — 2a;. 47. 17(a; - 2) = 2(2a; + 1) + a;. 48. 13(5 - a;) = 3a; -f 1. 49. 2a; — 2 = a; + 4. 50. 5a; — 8 = 6(a; — 3). 51. 3(2a; - 6) = 4a; + 1. 52. 25 - 4a; = 2(a; - 4) - 3. 53. 7 - a; = 3(4 - a;) + 1. 54. 2(a; + 5) = 5(5 - a;) + a; - 3. 55. 4 + a; + 4(5 - a;) = 3a; + 2(4 - x), 56. 2a; + 5 = 5(a; — 2) + 2a; — 10. 57. \+x + 3(2 -x) = 7(3 - x). 58. 13(8 - a;) + 2(a; - 6) = 2a; + 1. 59. 23(10 -x) - 4(a; -7) = 3a; + 3(a; -13). 60. 9(a; + 1) + 5 = 5a; + 6(a; - 1). 18 ALGBBMA. 61. Hx - 4) = 4.{x - 3) - 17. 62. 11(2: + 3) - 100 = 9(2; + 5). 63. 15(7^ - 3) = 8(13^?; + 7) - 20. 64. 7(19:?: - 21) = 11(12.-?; - 3) - 30. 65. 9(252; + 11) = 19(12a; ~ 5) + 2. 66. 51a; - 1135 + 13(103 - x) =z 11(2.t + 20). 67. 203 - 17a; = d{x - 505) - 2. 68. 1170 + {x- 4:) = ll(a; - 3) + 22. 69. 9(103a; - 820) = 206(2 - x) + 139. 70. 800(:?; - 10) + 2(lla; - 32 ) - 1 - 35a;. 71. 3000(a; - 5) = 10(100 - 27a;) + 355 - x, 72. 10(2a; - 3) = 2(a; + 15) + 6. 73. 3(3a; - 11) + 2(5a; + 2) = 5 + 9(2a; - 5). 74. 5(7a; - 2) + 2(5a; - 13) = 3(lla; + 2) + 26. 75. 17(3a; - 1) + 3(5a; - 17) = ll(5a; - 3) + a; + 5. 76. 13(a: - 11) + 17(2a; - 27) - 1 = 3(a; - 29). 77. 101(5a; + 17) = 203(3a; + 23) + 3a; - 3059. 78. 4:{x - 13) = 13(a; - 4) - 81. 79. 117(a; - 25) + 25(a; + 117) = 3a; + 50 + 3(7 - x). 80. 8(a; - 1) + 4(9 - 4a;) = 4 + 17(3 - x). 81. 17(2a; + 4) + 13(2a; - 5) == 3(a; + 17). 82. 51a; - 1137 + 13(103 - x) = ll(2a; + 20) - 2. 83. f (18a; - 252) = 2(15 _ 3^) _|_ 1. 84. |(24a; - 800) = f (45 - 5a;) - 5. 85. |(60a; - 875) = ^(35 + 10a;) - ^x, 86. f (1185a; - 5925) = -^\{Ux - 55). 87. i(93a; - 138) = |(10a; - 25). 88. 4(56a; + 42) = 5(2a; + 16). 89. f (42a; _ 48) - a; = 3(a; + 17). 90. 1(60 - 22a;) + ^^(30 - 50a;) = 0. 91. i(5x + 25) + |(4a; - 28) = |(3a; + 9). 92. |(21a; -57) = 2(6a; + 17). 93. 13a; + 3 + f (18 - 12a;) = 3(7 - a;) + 11 - a;. 94. 31(a; - 22) + |(21a; + 18) = 2a; + 883. THE FIRST USE OF ALGEBRA. 19 Sum of the Unknown Numbers Given. 22. Model F. — A merchant has grain worth 9 cents per peck^ and other grain worth 13 cents per peck ; in what proportion must he mix 40 bushels so that the mixture may be worth 40 cents per bushel ? X = number of bushels at 36 cents. 40 — a; = number of bushels at 52 cents. © 3Qx + 52(40 -x) = 1600 ©36:^ + 2080 -522; =1600 same as ® (D 2080 —Ux =1600 same as (T) ® 2080 = 1600 + 16a: (3) + IQx ® 480 = lQx ® - 1600 (D 30 = :i; ^ ® -f- 16 Ans. 30 bushels of the cheaper and 10 of the dearer. EXERCISE VII. 1. Twenty coins, quarters and halves, came to $5.75. Number of each ? 2. $4.04 in dollars, dimes, and cents; 44 coins in all; 9 times as many dimes as dollars. Number of each ? 3. 108 coins, dimes and cents, amount to $4.32. Num- ber of each kind ? 4. Fourteen coins, dimes and three-cent pieces, came to 77 cents. Number of each kind ? 5. A bridge broke down under a strain of 28,500 pounds, caused by a crowd of 200 people. The average weight of a man being 150 pounds, and of a woman 120 pounds, how many men were there in this crowd ? 6. Bought 30 pounds of sugar of two sorts for $2.27; the better cost 10 cents per pound, and the poorer 7 cents„ Number of pounds of each sort ? 20 ALGEBRA, ,._ 7. Bought 15 apples for 59 cents; red ones at 5 cents, green ones at 3 cents. ISTumber of each ? 8. Find four consecutive numbers whose sum is 94. 9. A grocer is offered $15 for 50 pounds of tea, and mixes 25-cent tea with 35-cent tea so as to make $1 on the transaction. How many pounds of each grade of tea in, the mixture ? 10. A merchant has grain worth 11 cents per peck, and other grain worth 15 cents per peck; in what proportion must he mix 20 bushels so that the mixture may be worth 48 cents per bushel ? 11. $1.97 in two-cent pieces, three-cent pieces, and dimes ; 3 times as many three-cent pieces as two-cent pieces ; 40 coins in all. Number of each ? 12. A man starts on a wager to walk 2000 miles in 50 days; after travelling for 30 days he finds that he can go 5 miles slower per day and still come out 20 miles ahead. How fast did he go at the start ? 13. One hundred coins, quarters and dimes, amount to 120.20. Number of each ? REVIEW. I. Find the value of the unknown number represented by the letter x in each of the following equations : 1. b{x -l) = Zx — ^. 11. ll(a: + 3) = 10{x -f 1) + 1. 2. ^x - 3) =^ 5^ - 3. 12. S{x -f 10) = n\x + 7) - 8. 3. ^{x -f 5)= 6:?; - 20. IS. 100(a; - 5) = 60(a; + 3). ^. 3(a; + 2) = 4:^-5. U. 4.{x -f 5) = 2(3^ - 10). 5. 6(a;- 3) = 5(:?;-2). 15. 3(a;-10) + 2(:?;+4) = 6^-22. 6. %[x - ^) = bx - ^. 16. 8(10 - x) =z b{x + 3). 7. b(x + 4) = Ix - 4. 17. 3.T+4(:?;-2) = l+3(2:z;-3). 8. ll(a;+l)==13x-fl. 18. l^{x-2) = ^^x-b). 9. l^x - 10) =: ^x. 19. 3(3:^; - 5) = 4.[x -f 5). 10. 9(:^:+6) = 10(22:+l). 20. \b{bx - 3) = 12(4:?: + 3). THE FIRST USE OH ALGEBRA. 21 II. Solve the following problems: 1, Borrowed some coins, and paid back 4 fewer coins of another sort; the coins borrowed v,^ere quarters, those re- paid halves; paid 25 cents too much. 2, Borrowed some coins, and paid back 5 fewer coins of another sort; the coins borrowed were quarters, those re- paid were halves; still owed 50 cents. 3, Borrowed some coins, and paid back 33 more coins of another sort; the coins borrowed were halves, those repaid were nickels; still owed $1.50. ^. Borrowed some coins, and paid back 52 less coins of another sort; the coins borrowed were three-cent pieces, those repaid were quarters; paid $1.30 too much. 5, Borrowed some coins, and paid back 3 less coins of another sort; the coins borrowed were two-cent pieces, those repaid were dimes; paid 74 cents too much. 6. Lost a few quarters, then found 5 more coins than I lost, only these coins were the old-fashioned twenty-cent pieces; on the whole I was the gainer by 65 cents. How many quarters did I lose ? III. What is the difference in treatment between an equation and an expression ? IV. What important principle of multiplication is illus- trated in the solution of the equations in I ? V. Give the exact meaning of the word coefficient; also the sense in which it is generally used. EXERCISE VIII. Find the ages of the jjersons described in the following prollems : 1. A is 12 years older than B; in 2 years he will be 4 times as old as B. 2. A is 6 times as old as B; in 4 years he will be only 4 times as old as B. 22 ALGEBRA, 3. A is 18 years older than B; in 5 years he will be 3 times as old as B. 4.' A is 9 times as old as B; in 6 years he will be only 6 times as old as B. 5. A is 7 years younger than B; in 4 years he will be half as old as B. 6. In 3 years A will be \ as old as B; now he is 20 years younger than B. 7. A^s age at present is \ of B^s; 10 years ago it was \ of B's age. 8. A was 8 times as old as B one year ago; he is now only 5 times as old as B. 9. A is 30 years older than B ; 5 years hence his age will be 6 times B's. 10. A is 7 times as old as B; in 10 years he will be only twice as old as B. The Problem of the Digits. 23. Model G. — In a number of two digits, the first digit is double the second; and if 27 be subtracted from the number, the digits are reversed. Find the number. Let X — the units figure; then 2^: — the tens figure. The value of the number is 10 X 2^; + x. Q 20x-\-x — 21 = lOx + 2:^ © 21x — 27 = 12a; same as © (D 21:^ = 12:?; + 27 @ + 27 © 9a; = 27 (D — 12:?; (5) x = Z © -^9 Ans, 63. EXERCISE IX. 1. In a number of two digits the first digit is 3 times the second; and if 36 is subtracted from the number, the digits are reversed. Find the number. THE FIRST USE OF ALGEBRA, 23 2. In a number of two digits the second number is 3 times the first; adding 54 reverses the digits. Find the number. 3. In a number of two digits the first digit is 4 times the second; subtracting 54 reverses the digits. Find the number. 4. In a number of two digits the first digit is twice the second; subtracting 36 reverses the digits. Find the number. 5. The sum of the two digits of a number is 11; and if 27 be added to the number, the digits are reversed. Find the number. 6. The first digit of a number less than 100 is 1 less than double the second; and if 18 be taken from the num- ber, the digits are reversed. Find the number. 7. The sum of the two digits of a number is 9 ; and if 45 be added to the number, the digits are reversed. Find the number. 8. The difference of the two digits of a number is 1 less than the units figure; and if 18 be taken from the number, the digits are reversed.* Find the number. 9. The sum of the two digits of a number is 11; and subtracting 45 reverses the digits. Find the number. 10. The smaller digit of a number is 5 less than 3 times the larger; and if 9 be added to the number, the digits are reversed. Find the number. 11. The difference of the two digits of a number is 1 less than 3 times the smaller; and if 45 be added to the number, the digits are reversed. Find the number. 12. The sum of the digits of a number is 11 less than 3 times the larger; and if 27 be added to the number, the digits are reversed. Find the number. * The fact tliat subtraction reverses the digits shows which digit is the less. 24: ALGEBRA. Numbers in Other Scales. 24. Model H. — The sum of the two digits of a number is 7; in the scale of 6 the number would be 16 less. Find the number. Let X = the figure in the tens place; then "7 — x =^ the figure in the units place. 10x-]-7 — X = the value of the number in the ordinary or decimal scale; 6x-\-7 — X = the value of the number in the scale of 6. © 10a; + 7 - :z; == 16 + 6a; + 7 - a; @ 9x + 7 =^x + 23 same as © (D 4a; = 16 @ - 7 - 5a; x = 4: (3)-^4 (D 7-a; = 3 Ans, 43. EXERCISE X. 1. The first digit of a number exceeds the second by 5; and if the number were in the scale of 7 instead of the deci- mal scale of notation, its value would be 21 less. Find the number. 2. The first digit of a number is less than the second by 3; and if the number were in the scale of 12 instead of the decimal scale of notation, its value would be 4 more. Find the number. 3. The sum of the two digits of a number is 9; and if the number were in the scale of 11, its value would be 7 more. Find the number. 4. The sum of the two digits of a number is 13; and if the number were written in the scale of 8, its value would be 12 less. Find the number. 5. The sum of the two digits of a number is 10; if THE FIRST USE OF ALGEBRA. 25 written in the scale of 8, the number would be 6 less. Find the number. 6. The first digit of a number is twice the second; if written in the scale of 7, the number would be 18 less. Find the number. 7. The first digit of a number is 3 times the second; if written in the scale of 11, the number would be 9 more. Find the number. 8. The first digit of a number is 1 more than twice the second; if written in the scale of 8, the number would be 14 less. Find the number. 9. The second digit of a number is 1 less than 3 times the first; in the scale of 11 the number would be 3 more. Find the number. 10. The sum of the two digits of a number is 1 more than twice the second; in the scale of 13 its value would be 12 more. Find the number. 11. The sum of the two digits of a number is 17; in the scale of 7 its value would be 27 less. Find the number. 12. The first digit of a number exceeds the second by 2 ; if written in the scale of 8, the number would be 14 less. Find the number. 12. The sum of the two digits of a number is 7 ; in the scale of 6 the number would be 20 less. Find the number. 14. The two digits of a number are equal; in the scale of 11 the number would be 1 more. Find the number. Current Problems. 25. Model I. — A man who can row 5 miles per hour in still water finds that it takes him 5 hours to row up-stream to a point from which he can return in 4 hours. How fast does the current flow ? Let X = number of miles per hour the current flows; 26 ALGEBRA, Then 5 + a; = number of miles per hour the man can row down-stream, And b — X — number of miles per hour the man can row up-stream. ® 5(5 -x) = 4(5 + x) @ 25 — 5^; = 20 + 4^; same as © (D 25 = 20 + 9a; © + 5a; ® 5 = 9a; ® - 20 ® 1 = ^ ®-9 Ans, Kiver flows | of a mile per hour. EXERCISE XI. 1. A river flows 2 miles per hour, and a fisherman finds that he can row up-stream a few miles in 6 hours, but it takes him only 3 hours to come back. How fast can the fisherman row in still water ? 2. A man starts to row down the river to the steamboat landing and back in 4 hours; it takes him 2 hours to get down there, and at the end of his time coming back he is 4 miles short of his starting-place. The man can row 3 miles per hour. How fast does the river flow ? How far off is the steamboat landing ? 3. I lent my steam launch to a friend who did not know that it would run only 10 hours without refiring. He went down the river, which has a 2-mile current, for 6 hours, and on his return trip the launch stopped 30 miles away from the boat-house. What is the speed of the launch ? 4. A man who can row 5 miles per hour finds that it takes him 3 times as long to go up a river as to go down; find the speed of the current. 5. The two fire-cisterns in a certain town are of equal size, and are being filled from mains that pour into each THE FIRST USE OF ALGEBRA, 27 10,000 gallons per hour; but the one near the business part of the town, being considered the case of more urgent need, has a pump rigged from the other cistern so as to hasten its filling. Consequently it is half -full in 7 hours, while the other cistern does not get half -full till 6 hours later. What is the capacity of the pump ? Of the cisterns ? 6. I had 4 hours to spare, and started for a row down- stream ; but I forgot that the current flowed a mile an hour, so I took half my time before I turned to go back; the consequence was that I got home two hours late. How fast can I row in still water ? 7. A man can row 4 miles per hour in still water; going up-stream he goes only f as fast as he comes down. Find the rate of the stream. 8. A man who can row 6 miles per hour in still water goes 1 mile farther in 3 hours coming down-stream than he does in 4 hours going up. Find the speed of the current. 9. I had 6 hours to spare, and started for a row down- stream ; but I forgot that the current flowed 1 J miles per hour, so I took half my time before I turned to go back; the consequence was that I was only half-way home when my time was up. How fast can I row in still water ? 10. A centipede in a grain- elevator, crawling on a belt which moves 11 feet per second, takes 10 seconds to go the whole length of the building when he goes with the belt, and 2 minutes to return, against the motion of the belt. How fast could the centipede go if the belt were still ? 11. A flsh that can swim 15 miles per hour takes 12 minutes to pass up through a certain part of the river, but takes only 10 minutes to come down again. Speed of the current there ? ALGEBRA. New Work on Old Patterns. 26. In Problem I mark a small letter a over the number 5 in " row 5 miles/' a letter h over the number 5 in ^^ takes him 5 hours/^ and a letter c over the number 4 in *' return in 4 hours." Then solve the problem, putting instead of" the numbers marked a, b, and c, the values represented below, each set making a new problem. a b c 1. 4 3 1 2. 6 4 2 3. H 8 3 4, ^ 4 2 5. H 4 3 In example 1 of the preceding exercise, letter the numbers 2, 6, and 3 in the same way, and then use the following sets of values instead,* just as in the case of Problem I: a h c 1, li 5 2 2. 1 6 5 3. 3 8 5 4. 3 4 1 5. n 8 3 FRACTIONAL EQUATIONS. 27. When equations arise that contain fractions, it is necessary to multiply both sides of the equation by num- * For some of the problems suggested here it may prove more reasonable to describe the powers of a fish instead of a fisherman. THE FIRST USE OF ALGEBRA, 29 bers such as will change the fractions to whole numbers, as in the following examples : ® X 3 ® X 3 © X 5 (D-IO One line of the work in example K could have been saved if we had multiplied both sides of (\) by 15 at once, instead of first by 3 and then by 5. Model L.— (T) - _ - r= - - i Model J.- -®i ■ =% © X = 6 Model K. -® 3 3 '' 5 ® 2y = 9 5 ® y = : 9 9 10 ^-. 2x X . ® X 2 @ 3a; - 2a; = y - 3 © X 3 ® «=|-3 same as © (D 2a; = 3a; — 6 ® 2a; 4- 6 = 3a; ® 6 = a; ® X 2 ® + 6 ® - 2a; Here if we had multiplied Q by 12, the least common multiple of the denominators, we should have had; Model!.— (T) --- = --1 '^ 2 3 4 2 ® ^x-A:X^^x-Q ® X 12 (D 2x — '6x — ^ same as @ ® 2a; +6= 3a; (D + 6 ® 6 = ;2; ® - 2a; 30 ALGEBRA, EXERCISE XII. Solve the following equations {that is, find the value of x implied hy them) : 1. |+|+3=«:-4. 6. 1-^^+17 = 0. a;a;l_ 2 a;_5a; — 8 ^- 2 + 3+3 ~^~'-^- ''• ~3 + 6 = ~36~* X ^_^ii 2a; — 1 1_K I a; ^- 3 ~ 7 ~ 6 + ^- ■ '• ~% 3 ~ ^ +6- XX 1 _x -{- 2 7a;— 6 x 3a; — 5 *-4 + 6~3-~3~- ^- ~T~ + 3"=~~3~ + ^~^• a; a;_a; 3a; — 7 3a; x — b '*-2~5-2~^- ^*'- ^ n = ^~- 28. Model M. — Bought a certain number of picture- cards at 2 for a nickel, and the same number at 3 for a nickel. Sold them all at a uniform price of 2 cents apiece, and lost a quarter. How many did I buy ? Let X = the number of cards of each kind; then — 2 would be the cost of one lot and — of the other. o © | + | = 4a; + 25 (2) 6x + — = 8x + 60 © X 2 (D 5x + ^0x = 24:x+160 ©X3 (4) 25.'?; = 2ix + 150 same as @ ® a; = 150 ® - Ux THE FinST USE OF ALGEBRA, 81 EXERCISE XIII. 1. What number is that | of which exceeds 27 by as much as ^^ of it is less than 6 ? 2. A, B, and C shared a sum of money so that A had $30 more than half, B $33 more than a fourth, and the remainder, which was $16. Find the shares of A and B. 3. A has $107, B $45. B gives A a certain sum and then has only ^ as much as he. How much did B give A ? 4. If a certain number be successively subtracted from 53 and 62, and then ^ of the first remainder be taken from 4 of the second, the last remainder will be 10. Find the number first subtracted. 5. A and B started eastward from Palmer. At noon A had gone f of the distance to Boston, B had gone ^ of it, and they were just 2 miles apart. What was the whole distance to Boston ? 6. From a cask of wine -^^ had leaked out, f had been drawn out, and there were 11 gallons left. How much was there at first ? 7. Bought pineapples at the rate of $4 for 7 dozen, and sold them at the rate of $2 for 3 dozen, gaining $4. How many pineapples were there ? 8. A man can walk 4 miles an hour or ride 7 miles an hour. How long will it take him to go 66 miles, if he rides half the time and walks half the time ? 9. The fore wheel of a carriage has a circumference of 6 feet, and the hind wheel a circumference of 10 feet. When the fore wheel has made 25 turns more than the hind wheel, how far has the carriage gone ? 10. A woman had a basket of eggs, and sold -J of them, with i of an Qgg besides; what she had left were 3 more than half what she had at first. How many eggs had she at first ? 11. A boy has an hour for exercise; how far may he ride 32 ALGEBRA, with his father, at the rate of 10 miles per hour, before getting out to walk back ? He can walk 3-J miles per hour. REVIEW. 1. What is an axiom ? Upon what axiom does the re- duction of fractional equations depend ? II. What would express the value of a number whose three figures were represented by x^ y, and z ? What con- clusion could you draw as to the relative value of these letters, if you were told that subtracting 198 reverses the order of the figures ? III. Solve these problems: i. Paid $3.75 with halves and quarters; 3 times as many quarters as halves. Number of each ? 2. Paid $2.45 with dimes and nickels; 5 times as many nickels as dimes. Number of each ? 3. Paid $1.36 with three-cent pieces and two-cent pieces; 7 times as many 2's as 3's. Number of each ? Jf.. Paid $1 with cents and two-cent pieces; 18 times as many cents as two-cent pieces. Number of each ? 5, Paid $2.26 with dimes, two-cent pieces, and halves; 3 more two-cent pieces than dimes; 25 coins in all. Number of each ? 6. Paid $7.50 with nickels, quarters, and halves; 1 more quarter than halves; 55 coins in all. Number of each ? 7. Paid $3.23 with two-cent pieces, three-cent pieces, and dimes; 14 more dimes than two-cent pieces; 49 coins in all. Number of each ? 8, Paid $5.90 with three-cent pieces, dimes, and quar- ters; 6 fewer dimes than quarters; 85 coins in all. Number of each ? P. Paid $2.24 with halves, two-cent pieces, and nickels; 5 more nickels than halves; 28 coins in all. Number of each? THB FIB8T USE OF ALGEBRA, 33 10, Paid $2.66 with cents, quarters, and dimes; 2 more dimes than cents; 29 coins in all. Number of each ? EXERCISE XIV. Find the value of x in the following equations: X — 1 X _ ,. 2x+ 1 4:X ^ 1. —, = 2a; - 10. 4. — ^ =2 -X, 4 5 7 6 X 4- 1 , X 4- 3 ^ X 4- 5 , X — 5 dx a:-2 3-x ^ ^ 7. 3x -\ ^ = — ^ h ^^. x-l 32;- 1 1 _ ix- 3 g;- 7 8. r^+ 5 +7- 5 +3* 9. 32; -f 5 H ^r— = h 52;. 32; — 1 , 15^; — 3 , ^ 10. IO2; = ^ 1 ^^ 4-7^-1. The Minus Sign before a Fraction. 29. In example 10 of the preceding exercise the equation would have assumed quite a different form if, say, the frac- ]^5^ 3 tion had been subtracted from each side of the equation. It would then be ^ in 15^;- 3 32;-l , ^ Q 10^' ^— = — ^ h ^^ - 1, and in solving this especial pains must be taken on account of the fact that when we multiply by 14 the fraction 34 ALGEBRA. jr becomes a whole number^ and that whole number has to be subtracted from the preceding term of the equa- tion. (2) 140a; - (30a; - 6) = 21a; - 7 + 98a; - 14. Here we have 30a; — 6 to subtract from 140a;. If we had to take 30a; from 140a;, the remainder would be 110a;; now, however, since we are taking 6 less than 30a;, our remainder will be 6 more, or llOo; -|- 6. The equation is therefore solved as follows : Q 1 0:. _ 1-^^ ^ ^-i + 7 a; - 1 © 140a; - (30a; - 6) = 21a; - 7 + 98a; - 14 ® X 14 (D 110a; +6 = 119a; — 21 same as @ © 110a; + 27 = 119a; @ + 21 ® 27 = 9a; © - llOo; EXERCISE XV. 1. o; - 1 a; + 3 2 . 2 5 ~5' .. + ^ + '-3 '" + '-. 2. 5a; - 4 3a; + 2 a; - 4 8 7 ~" 4 • X X — b a; — 8 ''^ 4 3 - 4 • 3. 2a;-la;-5 a; +3 3 ' 6 ~ 2 * _a; — 4 X — b x-\- 5 '•2 8 ~ 7 ■ 4. 5^- ^^ 8^+ 5 . x + 5 x-\-l x-\-i *• 6 8 ~ 4 ■ 20 - 3a; 10 — 3a; X '' 6 - 3 ~2" ,n - + ^ ^- + 1 _ 7a; - 11 15 25 45 TBE FIBST USE OF ALGEBRA. 35 11. bX'\'20- 2{x + 2) = 40 - 4.{x — 15). 12. 10(a; + 1) - {Sx + 5) = 30 - lOo; + 2{x + 35). 13. 30(a; + 6) - 10(a; + 3) = 311 - 3x, 14. 2a; + 3{x + 9) = 200 - 4(90 + Sx), 15. 5(a; - 6) + 17 = 110 - 2{x - 5). 16. U{x - 13) + 10 = 200 - 2(75 - x). 17. 200 - 7(60 - 5.^) = 3{2x - 11) - 100. 18. ^^x — 10(a: + 10) = 40 + :c - 8(2 - x). 19. 274(2:?; - 10) - 163(8 + ^) - 48^; =^ 0. 20. ^x - 103(a; - 17) + 17(3:?; - 103) + 135 = 0. 21. -V - 100(13:^ - 74) = 7(7:c - 2). 22. I - \\{^x - 25) = \(hx - 61). 23. 2.^ — 7 — — ^r — = 17 — 24. 25 2 ~^' 4 • a; + 17 _ 7.-r — 2 8:r- 13 2 ~ 3 + 9 * 10:?; - 7 , , ^ 62; + 33 3:r - 11 25. ^— + 13 = ^- + --^— . 4a; - 7 ^ 3a; - 11 a; + 17 26. — ^ ^=—j ^. 5(0; +1) / 2a; --7\ ,^ lla; + 19 6a; — 5 _ 6a; + 1 ^®- 6 3""" 6~^' 29. 31. .,N / X — 1\ X 4-1 7a; - 11 6a; + 5 9a; - 5 4 5 22 2a; + 7 3a; + 4 a; + 3 2a; - 13 x ^ 33- -5 rr" = 3~^- 36 ALGEBRA. Day's Work and Cistern Problems. 30. Model N. — A can do work in 4 days which B would take 12 for. How long for both ? Let X = the number of days for both to do the work. Then in one day both can do - of the work. A can do J of the work. B can do -^^^ of the work. ® i + r2 = x ® i + f2^"^ ®^'' (D 3:^ + a; = 12 © X 12 4^ = 12 same as (3) ® x = d ® - 4 A?is. Three days for both. EXERCISE XVI. 1. A can do work in 15 days, B the same in 18 days. How long for both ? 2. A vessel can be emptied by three taps; by the first alone in 80 minutes, by the second alone in 200 minutes, and by the third alone in 5 hours. How long if all three are open ? 3. A can do a job in 2J^ hours which B can do in If hours and C in 3J hours. How long for all together ? 4. A does I of a piece of work in 10 days; then he calls in B to help him, and they finish the work in 3 days. How long would B take to do the work by himself ? 5. A bath-tub is filled in 40 minutes and emptied by the waste-pipe in 1 hour. How long will it take to fill it with the waste-pipe open ? THE FIRST USE OF ALGEBRA. 37 6. A cistern can be filled in 12 minutes by one pipe alone, or in 8 minutes if the second pipe is also turned on. How long to fill it with the second pipe alone ? 7. A cistern holding 2400 gallons can be filled in 15 minutes by three pipes, one of which lets in 10 gallons more, and the other 4 gallons less, per minute, than the third. How many gallons flow through each pipe per minute ? * 8. A can do a job by himself in 6 days, B can do it in 10 days, and both together with C to help them in 2f days. How long for C to do it alone ? 9. A cistern can be filled from the water-main in 12 hours ; if also an extra pump is rigged from a neighboring cistern, it can be filled in 8 hours; and with all the fire- engines in town to help by pumping water from a lake near by it could be filled in 6 hours. How long would it take to fill the cistern if the main were shut off, and the filling depended entirely on the extra pump ? How long if the filling depended entirely on the fire-engines ? 10. A, B, and C are working together on a job that any one of them could do alone in 36 days; but when the job is half done C goes on a strike, and, by annoying A and B, hinders the work as much as he could have helped it if he had continued at work. When the job is done how many days must A be paid for ? How long did work ? How many days did his annoyance delay the completion of the job? 11. Starting at noon, I can get to town on a wagon at 1 :48 o'clock; if I walked, I could get there at 2 :15 o'clock. If I agree to walk half the time, what time must I get off the wagon ? ♦ The method of solving this example is quite different from the preceding six. It is introduced here lest the pupil should conclude tbat all '* cistern" examples had to be done on the same plan as Model N, 38 ALGEBRA. The Clock Problem. 31. Model 0. — At what time between 4 and 5 o'clock are the hands of a clock 9 minutes apart ? Let X = the number of minutes past 4. Then — = the number of minute-spaces traversed by the hour-hand in x minutes. If the hour-hand is 9 minutes ahead of the minute- hand, the equation becomes ©30 + ^ = ^ + 9 © 240 + a; = 12a; + 108 ® X 12 (D 132 = 11^ (2)-x- 100 ® 12 = 2; (D -^ 11 Ans. 12 minutes past 4. If the minute-hand is ahead of the hour-hand, the equation becomes 9 - 108 ® X 12 © + 108 - a; (D-11 Ans. 28/y minutes of 5. Of these two answers the first only can be realized on an ordinary clock, because the mechanism of the clock is such that 28y*y minutes of 5 is never indicated by the position of the hands. They move by jerks, one for each tick of the pendulum, and pendulums that tick elevenths of a second would not have any reason for existence other than the need of illustrating this problem. © 20 + 12 "" - X ' © •340 + X = V%x (D 348 = \\x ® 31t\ = X TUB FIEST USE OF ALGEBRA, 39 EXERCISE XVII. Instead of the numbers 4, 5, mid 9, in the statement of Model 0, use the following sets of numbers, and solve the resulting proble7ns : 1. 4; 5; 2. 8. 8 2. 3; 4; 4. 9. 8 3. 6; 7; 3. 10. 5 9; 7. 15. 10; 11; 21. 9; 4. 16. 12; 1; 22. 6; 3. 17. 4; 5; 24. 4. 7; 8; 2. n. 12; 1; 27. 18. 7; 8; 24. 5. 3; 4; 7. 12. 9; 10; 26. 19. 9; 10; 23. 6 5; 6; 14. 13. 1; 2; 21. 20. 1; 2; 17. 7. 6; 7; 14. u. 2; 3; 26. 21. A student of music always practised at the piano between 5 p.m. and 7 p.m. One day he noticed as he began to practise that the hands of his watch were exactly 3 minutes apart; he practised until they were again 3 min- utes apart, and then stopped. How long was he practising ? 22. In a mile race on an oval track, 11 laps to the mile, a runner has 200 feet the start of a bicyclist and only moves ^ as fast. Where will the bicyclist pass the runner the first time ? 23. Where will the bicyclist pass the runner the second time ? 24. If they both start at the starting-post, and go oppo- site ways, where will they meet the second time ? 25. If the bicyclist starts from the starting-post and the runner starts from the half-way post, and they go opposite ways, where will they meet the second time ? 26. One boy can run 5 yards while another runs 7; if both start opposite ways, from the same corner, to run around a block 75 yards on one street and 42 yards on the other, how far from the furthest corner will they meet ? 27. The hands of a clock are together ten different times between midnight and noon. Find when. 40 ALGEBRA. 28. The hands of a clock are at right angles twenty-two different times between midnight and noon. Find when. 29. The hands of a clock point opposite ways eleven different times between midnight and noon. Find when. 30. The hands of a clock make with each other an angle of 30° twenty-two different times between midnight and noon. Find when. 31. Find when on an actual clock the hands are exactly one minute-space apart. At what times are the conditions of the last four problems realized on an actual clock ? CHAPTEK n. ABBREVIATION OF RULES. 32. The second important use of Algebra is for abbre- viating rules in Arithmetic. For example, the rule for finding the number of square inches in a circle, when its radius is known, is: *' Square the radius, and multiply by 3.1416.''^ By Algebra we abbreviate this rule to where r is understood to mean the radius of the circle and 71 stands for the number 3.1416 (or 34- as it is sometimes written).* Such an abbreviation of a rule is called a formula. Translating Formulae into Rules. 33. Model A. — Express as a rule the formula a — V{h -\- I)){h — h), which enables us to determine the length, «, of one leg of a right triangle when the hypote- nuse, h, and the other leg, Z>, are given. Here ^ + ^ is the sum of the hypotenuse and given leg, and A — ^ is their difference; so, to find the required leg, — Multiply the sum of the given sides by their difference and find the square root of the product. * The exact value of the famous number represented by 7t cannot be expressed in figures. To ten places of decimals its value is 3.1415926535. The value 3^ is near enough for ordinary use, and 3.1416 is more accurate still, being correct to within j^nrirnF- 41 42 ALGEBRA. EXERCISE XVIII. Translate into rules the following formiilce : 1. For the circumference of a circle, when the length of the radius is given : %nr, 2. For the area of an equilateral triangle, when the length of one side is given : — — — . 3. For the volume of a circular pillar, when the radius and lieight are given : nrVi. 4. For the volume of a square pyramid, when the height and one side of the base are given : — . o 5. For the volume of a sphere, when the radius is given : 4;rr^ ~~3~' 6. For the diagonal of a rectangle, when the length and breadth are given : ^V" -\- V^. 7. For the average diameter of a tree, when the girth is known: — . n 8. For the diameter of a ball, when the volume of it is known: \ — . ^ 7t 9. For the number of crossings made by a number of straight lines drawn at random: — '^ — -. (n stands for the number of lines.) 10. For the number of seconds required for a body to /25 fall: y — . {s stands for the distance in feet, and ^ stands for the number 32.2.) 34. In writing a formula, numbers that are given in the rule are written in figures, except when the numbers are of ABBREVIATION OF RULES. 43 very frequent use and very well known value. Such are the numbers n = 3.1416, g = 32.2, and a few others. Translating Rules into Formulae. 35. Model B. — Give the formula for the following rule: To find the number of gallons a pail will hold : Find the diameter of the bottom and of the top, and find the depth of the pail; then multiply the two diameters together, square each diameter, and add the three results; then multiply by 11 times the height and divide by 10,000. Let D = the diameter of the top of the pail ; d = the diameter of the bottom of the pail; h = the height of the pail. Multiply the two diameters, Dd ; square each, D^ and d^ ; add the three results, I)^ -{- d^ -\- Dd ; multiply by 11 times the height and divide by 10,000, ^^J.D'^ + ^^ + Dd). EXERCISE XIX. Oive the formulcB for the folloiving rules : 1. To find the area of an ellipse: Multiply half the greatest length by half the greatest breadth and multiply the product by 3.1416. 2. To find the tonnage of a vessel multiply the length of the keel by the breadth of the main beam, and by the depth of the hold in feet, and divide by 95. 3. To find the volume of a coifee-pot, multiply the greatest and least radii together; add to this product the square of the greatest radius and the square of the least radius; multiply the result by one- third the height; and finally multiply by 3.1416. 4. To gauge a cask, take the difference between the head and bung diameters, multiply by .62, and add the 44 ALGEBRA. head diameter; square this, multiply by the length of the cask and by \ of 3.1416. 5. To find the area of any triangle when the three sides are given: Add the three sides and divide by 2; subtract from this each side successively; multiply the four results together and find the square root of the product.* 6. To find the diameter of a dome when its surface is given, divide double the surface by 3.1416 and find the square root of the quotient. 7. To find the diameter of a cylinder when its height and volume are given, multiply the height by 3.1416 and divide four times the volume by the product; then find the square root of this result. 8. To find the depth of a conical hole when its cubical contents and its diameter are given, multiply the square of the diameter by 3.1416 and divide 12 times the volume by the product. 9. To find the velocity in feet per second of water flow- ing from a pipe, when the discharge is given in gallons per minute, and the diameter is given in inches, multiply the square of the diameter by 2.448 and divide the discharge by the product. 10. To find the velocity of water pouring from a hole in the side of a tank at a given depth (in feet) below the surface, multiply twice the depth by the number 32.2 and find the square root of the product. Application of Rules. 36. Model C. — How many square feet of flooring are re- quired to cover a circular platform 100 feet in diameter ? Here r = 50, r^ = 2500, and Ttr^ = 2500 X 3.1416 = 7854 square feet. Ans. * Let s stand for one-half of the sum of the three sides. ABBEEVIATION OF RULES, 45 EXERCISE XX. 1. A large leaden ball was dropped into a tank com- pletely full of water, and 33510.4 cubic centimetres of water overflowed. What was the diameter of the ball ? 2. A circular floor contains 1256.64 square feet. What is its diameter ? 3. It takes 1413.72 square yards of roofing- tin to cover a hemispherical dome. What is the diameter of the dome ? 4. A cylinder 9 feet high and containing 763.4088 cubic yards is how wide ? 5. A conical hole in the ground is intended to contain 261.8 cubic feet of water, and it is 10 feet across. How deep must it be in the centre ? 6. Out of a lot of logs 12 feet long I want to pick out those not having less than 37.6992 cubic feet of lumber. What must be the least possible distance through ? 7. I find that the average circumference of a lot of logs is 9.4248 feet. Suppose I cut the logs up into chopping- blocks, how wide would they be ? 8. A horse attached to the arm of a windlass walks 10.56 feet short of 5 miles in making 300 turns. How long is the arm of the windlass ? 9. A pipe 5 inches in diameter is to deliver 102 gallons of water per minute. What must the velocity of the water be, in feet per second ? 10. How fast will water flow through a hole in a dam 32f feet below the surface of the water in the reservoir ? 11. A milk-pail a foot deep is a foot across the top and 10 inches at the bottom. How many quarts will it hold ? 12. A cask 30 inches long is 18 inches across at the head and 20 inches through at the bung. How many gallons will it hold ? 13. How many gallons per 100 feet of length can be con- tained in a water-main 20 inches in diameter ? 46 ALGEBRA. Application of Formulae 37. Model D.- — State the operations involved in the fol- lowing formula and calculate its value for the given values of the letters : W 12 ' L = 150; s = 8; W=S80, Statement of Operation, — Multiply the number L by it- self; multiply the number s by itself; add the two results, divide by 12, and multiply by the number W. Evaluatio7i, L2 = 22500 s^= 64 12)22564 W 1880i 380 7145261 EXERCISE XXI. State the operations involved in the folloioing formulcB, and calculate their values for the give?i values of the letters : 1 W- ^' 12 2. Wr2 ,L2 + s2 3. W- 4. W 5. W 6. W 12 t 2 2 21-2 W^SOO W= 210 W = 200 W=S50; W=:JfOO; W=80; L = 20. r = 5. L^25] r — 2. r = 20; r= 15. s = 2. R=z25. ABBREVIATION OF MULES. 47 7. w(l' + I') W= 20; L = 10; r = 2. T — t 8. IQ— Tjr- T=Jf59) t=m] Q=161100; 1=^15. 9. ^ F= ^^^^; ' ^ = SOO; V = 400. 10. C-j- 0=100; B = S; D=10; L = 15. 4LVd 11. L= 1500; n=1700; d=7.6S; g=981. 12-^^ P = 10; P=12. mr 13. 7 m = i.^; r = ^^ m — 1 ' 14. ^ — V=- ^\ P = 12. d 16. T—^ ^ = -^^^ d = 17. 1 + n Rr ^e- (R+r)(m-l) '^=^^ ^^^-^^ ^ = ^^- 17. 2dnm c^ = i7^^^; n = lUO; m = 12.6. 18. n(4L + d) (1=5; L = 30; n = 255. ^^- ^ ~ 87* 30 -^ "" '^^^' ^ = ^-^--^^ ^ = ^^• ^^- ^ ~ ^-"^ -^ = •^^^' ^ = i5; A = ^4. 360L 21. L = 40; c = 900. c 7ra 180 180u 7t 180L n = S.I4I6; a =60. u = 2.2; 7t = S\. 24. — ;- L = 6; a = 10; 7t = S.I4I6. 48 ALGEBRA. 26. -^ Tt = S,U16; a = 12; r = 10. 27.^ a =.8; c=.100. q2 I rt2 1V,2 28. ^ ^ — ^ « = 4; Z' = ic5; ^ = i5. 29. ^^^-^ a=z 10; h = 17] c = 21. K2 30. -TT ^ = ^; ^^= ^. h ' 31. ;r(E + r)(K - r) R-25\ r = 20. 32. ih(L + l) i = 3.5\ 1 = 85.25] 1=2^.75. 33. 4:7ih{R + r) 2r = 8; 2E = 8. 34. 7rh(R + r)(E~-r) h = 10; E = 5; r = 8.5. qK i-r ,-r*i-L; \m=42; h = 9. (Eead ^^ B one/' ^' B two '^ ; or ^^ B sub one/' '^ B sub two/') 36. jTrr^h ;r = .^.i^i^; r =^ 10; li = 8. 37. 27rrh n = S.U16; r=z8; h = 7. {vZ. ^*- V-w 1 V=21. Q - 110; r= 5; n = S.U16. 39 -^ 4r^7r 40. Trfab + -y ] a=^ 18; h = 15. RA 41 a — A \ r = 15 {^: = i^6/; A =.125; a = .875; ^^- a - A ^^ ^ ^M ^ = 100; R' = 110. (Read " R prime " ; or '' R large prime.") * The same letter can be used to denote different numbers even in the same formula by using distinguishing marks, such as suffixes, as in 35, or accents, as in 42. ABBREVIATION OF RULES. 49 ^^' UA-W c = S; E = 100; R = 21. Rr + R'r + R'R ^^' ^^'^ R = 100; r=120; R'=1000. E 45. 5 E=1.8; x = 2; y=5; R= 10; r=100. 1 / a b \ 46- o -^ i; a = .002; b=.015; n = 60. 2 W — a n — b/ ' ' *'■ t fe ~ ^J T=8; E = .75; t=10. x(T - t) ( ^^2^s; T=16; t = 12; *"• W(Q - T) I a: = ;2^5; T= 16; i \Q = 96; W=1^5. D f c = .002; t = 20; H= 60; l_|_tc \h = S; d = 1.5; D = 13.5. (Q - t)(w + W) |y=9.S; ^ = iO; (2 = ^6; Tf=5W; r=i^(S. c, W(T - Q) V { V = 20; w = S00; W=10; "• » Ty(Q_t) w 1 8^19; t=10; Q=1S.8; T=20. W 62. ^(T - q) - s(r - t) - S(q - r) f s = 5; S=1;W=500; w = 2Jfi; t = 0; \T=U; q = 6; r = 5.5. ':i { .,?„_,_.(T-t±i W= 1000; w = 25.6; T= 100; q= 20; t = 4; s = 6. M J) = 10; d = 1.6; M = 2. 60 56. 67. 58. 59 60. W(t, - 2t, + t) d(W^ - w) W- w W + yJ abc 4K K s — a ALGEBRA. {P = 12U) W = 2; t^ = U^S; \t^ = 11.5; t = 10. (w = S2; W= 132; W = 1390; \ d = .995, V = 500; V = 2000; n = 5. a=15; 1=28; c=^l; K=126. a -\- h -\- c s = 2 a = 19: h = 20; c = 37; K= llJf. Formula for the Area of a Triangle. 38. An extremely important formula is that for finding the area of a triangle when the lengths of the three sides are given : the area of the triangle being represented by K and the lengths of the sides by a, h^ and c, we have the formula Kz=^ Vsi^s - a)(s — h){s — c), where s represents a -^-h -\- c EXERCISE XXII. By this formula find the area of the triangles whose sides are : 1. a = 12; l = H; c = 25. 2. a = 25; ^ = 51; c = 74. 8. a= 6 ; J = 25; c= 29. 4. a = 16; Z> = 25; c = 39. 6. a = 17; h = 25; c = 28. ABBBEYIATION OF RULES, 81 Formula for Square Root. 39. The rule for square root can be conveniently mem- orized by means of a formula: Ride, I. Separate the number into 5' 61' 69 2 periods of two figures each, beginning at __4 the decimal point. Begin with the left- 161 hand period, subtract its square, and bring down the next period. II. Let a stand for 10 times the part of the root already found. Write 2a for a new divisor, and call the quotient Z>; then subtract ^(2a + ^) from the new dividend. 5' 61' 69 4 00 5(2« + ^') = 1 61 29 2Z=^a + h 40 = 2a 43 = 2a + ^> 32 III. Bring down the next period for a new dividend. Let a stand for 10 times the part of the root already found, and write 2a for a new divisor; call the quotient b, and subtract l{2a + b) from the new dividend. ["40000 + 12900 =■] j I L52900 = 2302 J ( ! 5' 61' 69 4 00 00 61 29 00 h{%a + 1)) = 3269 3269 237 = « + 2> 40 43 460 = 2a 7 = b 467 = 2a + h 40. This process can be shortened by leaving off the cipher on 2a, or writing it lightly, so that when b is added 52 ALOBBRA. it can be put in place of the cipher; then the completed work would read : Model E. 5' 61' 69 4 237 1 61 1 29 43 = 2« + J 3 =1 3269 3269 467 = 2a + & 1 = h Of course the pupil must remember that at each new stage of the process a new meaning is given to the letters a and h. EXERCISE XXIII. Find the square roots of the following numbers : 1. 6889. 2. 44944. 3. 138384. 4. 245025. 5. 375769. 6. 546121. 7. 958441. 8. 982081. 9. 1948816. 10. 1292769. 11. 79.21 12. .049729 13. 11.9716 14. .00710649 15. 9545.29 16. 811801. 17. 3684.49 18. 1.062961 19. 418.6116 20. 19796.49 Approximate these square roots to three decimal places : 21. 2. 22. 3. 23. 5. 24. 6. 26. 10. 26. 9.87 27. 1.032 28. 22.5 29. 7.921 80. 688.9 EXERCISE XXIV. Evaluate the following formulce : 1. V's(s — a)(s — b)(s - c) a = 29; b = 60; c = 85. 2. ^ (3-a)(s-b)(s-^c) ^^^^. ^^^^. ^^^^^ ABBREVIATION OF RULES. 63 4^8(8 — a)(s — b)(s — c) 6. |/r2 _ ^^2 ^ _ ^^^. ^ ^ ^^5^ 7. |/li2+ ik2 ]c = ISO', }l^7. 2kr 8. , ^^^ r=^^; k = 9. 1 ./ "a^ ( 1=25', a = 10; h —.981-, ^^' "21^ ;rr2k ( n= 3,U16\ r = 5', h = 8. 11. d|/^ t? = i.5; ^ =r ^^6?; /= .i5. ^^^ ^(a + b)ab ^ ^ ^ ^^^ ^ ^ ^^ Va2 + b2 ^^- '^'^ ;rld I Z = 4; ^ = 13.6. 14. '/2r2 - r|/4r2 - k^ r = 100; h = 160. i/PvTTT^aT+t ( a = 273; t = 27; D = . 00188; 15. f ^(1.41)-^ \p^^^OJ^^,00. 41. Model F. — In the following formula what value must be given to n when the other letters have the values T= 210, iJ = 85, c = .0002 in order that T' may have the value 210.75 ? T' = {T- n)+n{l + c{T-t)}. 54 ALGEBRA, Let the required value of n be represented by x] then (210 -x) + x{l-\- .0002(210 - 85)} = 210.75 210 - a; + :?:{1.025} = 210.75 210 + .025^ = 210.75 .025a; = .75 25a; = 750 a;= 30 EXERCISE XXV. __ P(l + k) + mi~k) ^* 4h G = 86.3895] h = 2-, h = 20, Find I. 2. G=i;rh[3(r,2 + r,2) + h2] r,-= JfO\ r, = SO; h = SO; G is foimd to he 131,880. What value was used for n ? 3. |(B + 3T) = V When a = 3^0, B = 12,025, and V = 1,615,000, what must he the value ofT? *• ^ y When V is 3000 and v = 15, what value of n will mahe D ^ 10? 5. ;r(ab + -^) = S What value must he given to a in order that S may he 7854 when h is 50 f {n — 3.U16). The Interest Formula. 42. In the formula j9r^ = i, p stands for the number of dollars in the principal, t for the number of years, r for the rate per centum expressed as a decimal, i for the interest. ABBREVIATION OF RULES, 55 Model G. — Find the interest on $300 for 1 year and 3 months at 5 per cent. p = 300; r = .05; t = 1.25 i = 300(.05)(1.25) = 18.75 A?is. 118.75. 43. The formula for the amount is a =p -\-prt, which can also be written j9(l -\- rt) = a. For the amount at compound interest the formula is a=p{l + ry. EXERCISE XXVI. 1. What is the interest on $532 for 3 years 1 month and 15 days at 5 per cent ? 2. What is the amount of $298 at 3| per cent for 6 months ? 3. What sum of money will return $75 interest at 2 per cent in 2 years 6 months ? 4. If $1015 is put at interest at 7 per cent, how long before it earns $100 interest ? 5. If $784 earns $15.68 interest in 3 years, what is the rate exacted ? 6. What sum of money, if put at interest at 3 per cent for 5 years, will amount to $1000 ? 7. What rate of interest must be charged on $1500 for 5 years to make it amount to $2047.50 ? 8. How long before the principal, p, will double itself at 5 per cent simple interest ? 9. How long will it take $1500 to amount to $1591.35 at 3 per cent compound interest ? 10. Show that the formula for r in compound interest is CHAPTER III. TRANSFORMATIONS. 44. It is often necessary to change the form of a formula or other algebraic expression, or to perform some algebraic operation upon it. In order to be able to do such things intelligently we must investigate the laws and the methods of addition, subtraction, multiplication, and division, in order to see clearly how they differ from the similar laws and methods of ordinary arithmetic. Negative Quantities. 45. The expression a — b is the algebraic symbol for the result of subtracting b from a. But if b is greater than a the subtraction is impossible. The question is in that case what does a — b represent ? 46. Take a particular case. Suppose I send an order for 10 quarts of berries. If the grocer has a quarts in stock he will have after filling my order a — 10 quarts. Suppose his original stock was only 7 quarts. He would send me those, but his condition would not be as if he never had had any berries, nor any orders for berries; for he now has an unfilled order for 3 quarts. To put these circumstances into arithmetical shape : 47. It is required to subtract 10 from 7. The operation is impossible. But 7 can be taken from 7, leaving 0, 56 TRANSFORMATIONS, 57 and there are still 3 to be taken from any number that MAY HEREAFTER BE ADDED TO THE EXPRESSION; jUSt aS the grocer is expected to fill his order from any goods that may hereafter come in. 7 - 10 = - 3 48. The expression — 3 is new to the student. It means 3 to be subtracted from any number that may hereafter be added to the expression. If we add 5 to the expression in 0, we have ®7-10 + 5=-3 + 5 (D - 3 H- 5 = 2 If we add 3 to the expression in 0, we have 0-3+3=0 49. The expression — 3 differs from all expressions known to arithmetic in this particular, namely, that you can get zero by adding to it. 50. When two quantities added together give zero, one is called the negative of the other; but generally by a negative quantity we mean a quantity preceded by the minus sign. That is, 3 is the negative of — 3 just as much as — 3 is the negative of 3 ; but if asked which of the two was the negative we should say — 3. 51. Terms preceded by minus signs are also called minus or negative terms ; and terms not preceded by minus signs are called plus, or positive terms. 52. Thus it may be said that any positive term is the NEGATIVE of the Corresponding minus term ; the apparent contradiction being explained when we remember the defi- nition. 58 ALGEBRA. 53. Since it is immaterial in what order two numbers are added, we may as well add — 3 to 3 as 3 to — 3 ; this would lead to the expression 3 + (— 3)=::0, where the new kind of quantity is added. 64. Finally, we may translate the sign — by the words ^' the negative of." EXERCISE XXVII. 1. li a -\- h = what is the negative of a^ ? 2. Find the value oi c -\- x when c = 10; x = a — I'y fl^ = 3 ; J = 5. 3. What is the arithmetical meaning of — 11 ? 4. What is the negative of ~ 11 ? 5. Translate the equation — (— 4) = 4. 6. Simplify -(-«);_(- h). 7. Simplify - [- a - h). 8. — {x — a)', simplest form. 9. Find the value of x -\- y -\- a when x= — 5; y = 2; a ■= c. 10. It X -\- y — z = 0, what is the negative of a; ? Reciprocals. 65. When the product of two numbers is 1, either num- ber is called the reciprocal of the other. Thus ii ab = 1, a is the reciprocal of J. The reciprocal of 7 is |; that of 2^ is f; that of x is -; that of .25 is 4; that of 1.125 is f ; and so on. X 66. The reciprocal of the reciprocal of a number is the number itself. EXERCISE XXVIII. 1. It ah = 1, what is the reciprocal of a ? 2. If 5a; = 1, what is the reciprocal ot x? 3. What is the reciprocal of 2.5 ? TRANSFORMATIONS. 69 4. \ = a\ what is the reciprocal of « ? 5. What number added to its reciprocal gives 10.1 ? Rearrangements. 57. In the following ^^ chain of additions and subtrac- tions '^ there are ten terms, some positive and some negative : 5 + 17-3 + 2-30 + 15-3 + 5 + 2-1 In such expressions I. The terms may be rearranged in any order, without changing the value of the entire expression. II. The terms may be grouped in any way, the values of the groups found, and all the results added together, giving the same result as the original expression. That is, 5 - 3 + 5 - 30 - 1 + 2 - 3 + 15 + 2 +17 = 9 and (5 - 30) + (17 - 3 + 2 + 15) + {- 3 + 2 - 1) + 5 or - 25 + 31 + (- 2) + 5 = 9. EXERCISE XXIX. Arrange each of the following expressions in six different- ways without changing the value of the expression : 1. x+by -\- z. 4. lOA - 20^ + 21r. 2. X -\- y — z. d. z -{- 2a — 2b. S. p — q + s. Make a group of ttvo terms in each of the following expressions tuithout altering the value of the expression ; six answers for each : 6. a -\- b -{- c ~\- d. s. ^x -\- 2y — c — 5d, 7. h — k — j -\- X, 9. 13^ — 31^ + 14r — 4l5. 10. 12:?: + 23y - Slz + iw. 58. The same truths hold of a chain of multiplications and divisions, and may be restated as follows, understand- ing by the word ^' term " one of the numbers together with the X or -^ sign preceding it : 60 ALGEBRA. I. The terms in a chain of multiplication's and diyi- siOKS may be rearranged in any order, so long as each divisor remains a divisor and each multiplier a multiplier. II. The terms in a chain of multiplications and divisions may be grouped in any way and the groups multiplied, so long as each divisor remains a divisor and each multiplier a multiplier. l~3xl5x3-^5x7-^14x3-h7x21-^2 X 6-r-9 may be written 1 ^ 5 -f- 14-^7 X21x 7-^3xl5x2 x3-v-2x6-^9 or (1 -h 3 X 15 X 2) X (-^ 5 X 7 -4- 14 X 3 -^ 7) X (21 -^ 2 X 6 ~ 9) or 10 X ^h X "^ which equals 3; under- standing that -^ 5 X 7 is the same as 7 -^ 5. EXERCISE XXX. Arrange the folloiving products each in six different ways: 1. ahc, 5. ^{x — y){a — b), 2. 5ab. 6. x{a — i)(x — z), 3. V/xy, 7. {a — l)(l) — c){c — a). 4. ^x{a — h). The purely numerical multiplier., or the coefficient usually so called, is generally put first ; with this restriction rear- range iji six luays {including the order give7i) the follo7ving : 8. bxy{a — c). 9. 21st{u — v). lo. 301^(^ ~ j){j — ^)- Show why the last two multipliers in 10 are not alike, SUMMATION. 69. The introduction of negative quantities makes addi- tion and subtraction in algebra more difficult than in arithmetic ; and the necessity of having terms similar before they can be united introduces a further complication. TRANSFORMATIONS, 61 60. The actual operation of addition in arithmetic con- sists in taking two numbers that have been counted up separately, putting them together, and counting them up as one number. For example, two distances are measured from the beginning of the first to the end of the first, and from the beginning of the second to the end of the second; when they are added the beginning of the second is put on the end of the first and the united distances are measured from the beginning of the first to the end of the second. 61. When expressions are added in algebra they are written one after another, each term retaining its proper sign, and the similar terms are then united as if all the expressions were one. Model A. — Add the following expressions: 3a; + 5 - 2^ + 1; — 2 -\- 2x — 10 -\- bx] 2;+ll — ^x+1. Operatio7i, — ([)^x + b — 2x ^l — 'Z + ^x — 10-{-bx + x-\-ll - 3a; + 1 @ 3:?; - 2a; + 2:z; + 5a; + a; - 3^ + 5 + 1 — 2 — 10 + 11 + 1 (3) 6a; + 6 The form @ came by rearranging Q according to I, § 58, so that all the a;-terms would be together; then (3) came by uniting the two groups of terms in (2) according to § 21. The result cannot be further simplified till we know the value of the letter x, EXERCISE XXXI. Add the following expressions, and indicate the three steps of the process in each example : 1. 3a; + 2 + 4a; + 7; a; + 3; 3a; - 12; 1 - 10a;. 2. 5a; + 7a; + 13a; - 5a;; 3 - 17a; + 2 — 3a;; a; — 5; — a;. 62 ALGEBRA. 3. 8 + 2:r + 3 + 9ir — 7 — a; -- 4 — 3^; 2; 4 5 — 8a;; 2a; + 1; — x\ — X. 43^ 5x + Ux + 17 - Sx + 5 — lOx; 4a; + 2-3a; + 7; 34 -a;. 5. 171a; - 243 + 318.^' - 411a; - 111 + 150; 38a; + 117; 301 - 100a;; 102a.' - 299. 6. 3(a; — 5) ; 5(a; — 5) ; 8(5 — x). 7. 17(2a; — 3); 11(4 -3a;); (a; - 1)3. 8. 7a + 341 + 2a - 100 - 200« - 200; 10{a - 3); 3(3 - a), 9. 3./ + 7; (7 + f/)3; 7(3 + 7/); -(-3); -3-7--y. 10. -(-35); -(-8); -(-3^-8); d{s + 8). 62. Only similar terms can be united. EXERCISE XXXII. Add the following expressions: 1. 3a; + 8; 3^ + 8; 2a; + 4?/; a; -y - 15; -5a;~3y + 3. 2. 2x -\- y — z; x — 3y -\- z; 4y — 3a; + ^. 3. 13a; — 4:1/ -]- z; 4y — 10a; — bz; 4:Z — 4:X -{- 4:y — 4. 4. - 131a; - 100^ - z; - {lOOz - 100a;); 50a; + 150y. 5. x — 2y; 2y -\- x; —x; 2x — y. 6. 22) — 3g; 2^- q; ^p -2q; p + q +r + s. 7. dx + Uy -(- 17); 20 - 10a; - lOy; x-y-^l', -(-2). 8. 10a; + r+2;; a; + 10r + c; a; + r + 10;2;; x -{- r + z + 13. 9. dp + 33r + 3335; 2r + 22s + 222j9; s + ll^j^ + lllr; — lOOr — IOO5 - lOOp. 10. -1397^-128^; h - g; g + lOOh; 120k - h - 2g; 5^ + 5;^ + 5^ + 5. Subtraction. 63. In subtraction the minuend is the quantity from which the subtrahend is taken to leave the remainder. TBANSFORMATlom, 63 When any quantity is added to the subtrahend, the efect on the remainder is precisely the same as if the same quan- tity was taken from the minuend; and when any quantity is taken from the subtrahend, the effect on the remainder is precisely the same as if the quantity was added to the minuend. For instance, the following subtractions: Minuend. 50 50-2 50 50 + 2 Subtrahend. 30 + 2 30 30 - 2 30 Kemainder. 20-2 20-2 20 + 2 20 + 2 64. It is possible, therefore, without changing the value of the remainder, to take any term out of the subtrahend and write it as a part of the minuend; only we must remember to change its sign, from + to — , or from — to +. And if we can so transfer any term, we may transfer them all, and thus convert our example in subtraction into an example in addition. Model B. Minuend. 7:?:— 3^^+ 5 —ba—lOx^ Subtrahend. ^ — d>xy-{-2x-\-ba— 2x^ Remainder. (To be obtained.) may be converted into Minuend. '7x—3xy-{-5 — 6a-'10x'^—3+Sxy—2x—5a+2x^ Subtrahend. (All these terms have gone into the minuend.) Remainder. (New minuend simplified.) 6x-\-6xy-\-2 — 10a—8x^ 65. The rule for subtraction in Algebra is therefore given briefly as follows : Change signs in the subtrahend and unite similar terms. 66. To save time in calculation the signs of the subtra- hend should be changed IK youk mind, and the terms united without writing the expressions over again. 64: ALGEBRA, EXERCISE XXXin. Perform the following subtractions : 1. From x^ — 7x^ + 16x — 12 take x^ ~ dx^ + 2x — 48. 2. From a — b take a — b — c, 3. From x — y take y, 4. From h take — 5. 5. From x^ + ^x^ + 2:^; — 48 take x^ - 4:X^ — 8x + 8. 6. Take x^ — 5x^ — 2^; + 24 from x^ + 2x^ + Ax + 3. 7. Subtract 24 + Ux - 29x^ + 6x^ from 2x^ + dx^ - 13a: -12. 8. 2 - |/3 from 6 + 51^3. 9. Take a from a — b. 10. From 9^ — 5^ + 4r take S^^ — 2^ + 2r. Parentheses. 67. To save using words like subtract, tahe^ etc., the operation of subtraction is often indicated by putting the subtrahend in a parenthesis with the sign — before it. In such examples the parenthesis, with its sigk, serves only to mark the subtrahend and distinguish it from the minuend. EXERCISE XXXIV. 1. 2a-2b + 3c-d- {6a - db + 4c - 7^). 2. x^ + 4^3 _ 2^2 _|_ 7^ _ 1 _(^4 _[_ 2x^ - 2x^ + Qx- 1). 3. — {a^ — ax -\- x^) + 3a^ — 2ax + x^. 4. 4a;3-2a;^+:r+l-(32:3-a;2-2;-7)-0T3--4:i;2_[_2a;+ 8). 6. lOa^'b + SaP - 8a^^ - b^ - {ba% - Qab^ - la^% 6. ^x^y ~ Zxy'^ + 7^^ + 2;^ - (8x^y - 3xy^ + 9?/^ + lla;^). 7. -JajS — \xy - 1/ -(- l:?:^ + ^y - 2/^). 8. \d' - ^a-1 -{- 1^2 + t? - i). 9. f'^^^ - i^ + i - (i^ - 1 + ^). 10. I^'?^^ — -|^:f — (i — i^^ — f«^). TRANSFORMATIONS, 65 68. Expressions often occur in Algebra where it is desirable to take out parentheses from them and simplify by uniting terms. To do so it is necessary to remember first that a parenthesis with — before it indicates subteac- Tiajq", and the signs of all the terms within must be changed when the pakenthesis, with its sign^, is removed; and, secondly, that a parenthesis with + before it indicates addition, and no change of sign is necessary when THE PARENTHESIS, WITH ITS SIGN, is removed. 69. When parentheses occur within parentheses, it is best, to avoid confusion, to begin within the inner one, that is, not to remove a parenthesis which has another parenthesis within it. 70. The sign of the first term in a parenthesis is not the sign of the parenthesis; thus in the two expressions Zz — x — y] bz—{X'\-b\ the sign of a; is +. EXERCISE XXXV, Simplify the following expressions : 1. a; + 1 + (5 - a:) - (3 - 13^). 2. a; + 2/ - 5 - (:c - y + 5). 3. 2a - 3Z> + (3a - U) - (2a - U), 4. a — l -\- G,— {a -\- h — c). 6. 14a + 21h - 13 - (7a - llOh - 17). 6. 20;r + 30y - 16z - (I5y + 16z - 17x). 7. 1108 + 135.^ - 780^ - (39:?; + 42y - 109). 8. 3^2 _|_ 33^ _^ 333 _ (333^2 _ 33^ _|_ 3) 9. 2700^2 _ 908^2 _ 137^^ - (1900a; + 3000^;^ + lOOy + 1000^^)^ 10. x^ -\- y^ -\- 2x -\- 2y — {x^ — y^ — 2x -\- 2y). * 11. 2x + 3y - 5 - 9{5 - y + x). * See § 13. 66 ATMJSBnA, 12. a - bh -\-2>c + {a - h) - 2{6b - c - a). 13. X — y — 6{'dy — 172:). 14. 2x + 3.^2 + 1 + 2{x^ + 1) - 5(1 + x^). 15. 4:{x - 1) - 5(2 - 3x). le, a - b - 3{2a - b) + 7(a + b), 17. X -]- 2y + {x — a) — 2{a — 3y). .18. iz; + 5 - 3(1 - 2:r). 19. x^ + y'^ — 2{x + y). 20. « + 5^ + 3c — ir — 2(a; — a — 5J + 3c). Nests of Parentheses. 71. Model C. a-[b-{c+{d-e-f)}] ci-[^~{c+{d-e+f)}] a^[b- {c+ d-e+f}] a — [b — c — d -{- e —f] a— b -{- c-\- d — e -\- f EXERCISE XXXVI. Simplify the following expressions : 1. X - {y - z) + x{y - z) + y - {z -{- x). 2. 7a - [2b + \a-{b~\- a)}]. z. a - [3a - [bb - {4.c - Sa)}]. 4. [h- {s - t)\ -^ [s - {t - h)} - [t - {Ic - s)] - (h + s + t), 5. 2x — (6i/ + [4^ - 2:?:]) - {^x - [?/ + 2;?]). 6. - [ - { - (^ + c - 2«) } ] + [ - I - (c + a - 4^) } ]. 7. -(-(-2a))-(-(- (-3a;))). 8. 4.x - [2x + 2y - {x -\- y -{- z - {2x + 2y -\- 2z + 7c)]\ 9. - 3a - [4a; + {4c - {by + 4a; + 3a))]. 10. - l^x - (12y - 4a:)] - [6y - (4a; - 7^)]. Where the answers to the foUoiving equations are not tvhole nmnbers, express them in decimal fractions : 11. a; + 3 - (2a; - 17) = 4. 12. 2a; - 7 - (a; + 2) = 2a; - (2a; + 9). TllAN8FOltMATlONS. 6t 13. 3^ + 2 ~ (2:?; - 5) + (9 - 7:?:) = a; + (2 - 5^^;). 14. 13 - (5 - 2x) - (1 - Ix) = 25 - (2 + 7x). 15. 300 - 2x - (115 - 3^) = 1000 - (Sx - 61). 16. 100:r - (39 + Sx) - 10 = 12a; + 345. 17. X - {Sx - 2x + 1) r= 8 - (3a; + 7). 18. 12a; - 3(a; - 2) = 20 - (5 + 9 - 7a;). 19. 100 - (2 - 15a; - 23) = 98 - (20a; - 24 - 5a;). 20. 25a; - 10(1 + a;) = 3 {x - 5x ~ 7) + 8a;. 72. To put terms of any expression into brackets similar precautions are necessary. If a + sign is used before the bracket, no change is necessary; but if a — sign is used, that takes part of the minuend and makes it a subtra- hend ; consequently every + must be changed to — , every — to +. Of course there are various ways of bracketing the terms of any expression. Model D. ax^ — 2/^0;^ + ^y^ — ^^^ ~ ^fy + ^ ^^J be written {ax' - 2hxy) + {by' - 2gx) ^ {^ 2fy + c) or {ax' + hy' + c) - {2hxy + 2^a; + 2/^) or {ax' - hxy - gx) - {lixy - ly' -\- fy) - {yx +fy - c) or in many other ways. EXERCISE XXXVII. In the folloiinng examples get tiuo results, first with a + sign hefore each bracket, then with a — sign : 1. Bracket terms containing x: lex -f cibx + CLcy — cibc, 2. Bracket terms containing a in the same expression. 3. Bracket terms containing «J in the same expression. 68 ALGEBRA. 4. a^hc^ — ah^c^ + a/hh — aWc — a^hc^ + aWc^ + (^y^(?\ bracket terms containing a^. 5. Bracket last two terms : W'c^ — cPW' — aV. 6. Bracket last three terms: x^ — y'^ -\- %yz — z^, 7. Bracket separately terms containing a and terms containing b: ax -\- ly — ay — Ix -\- az — Iz, Bracket like powers of ^: 8. ax^ -\- hx^ -{- c -\- hx^ + cx^ -{- a -\- cx^ -\- ax^ + ^ + ax -\- hx -{- ex, 9. ax^ + 5a;3 - aV __ 2^^3 _ 3^2 _ j^4^ 10. ^2;^ + ^^^^ — ^^^ ~ ^^'^ — ^^^« 73. The letters a^ Z>, c, J9, and q in the following expres- sions are called constants, and the letters x, y, and z are called variables. EXERCISE XXXVIII. Group the variable terms in one parenthesis, obtaining t200 results as before^ one with a + bracket and the other with a — bracket, 1. a^ + b^ + x^ + y^- 3a:^bx - 2ab^x + ba% - '^x^y, 2. a^ + c2 + Z>2 _|_ x^ — '^ax - 2ab-2bc+2bc + 2bx+2cx. 3. aV + 2bx^-x^ + 2a^ + 4:P - 2bx - a^ - 2ab + 4. a{x + c) -p{a + y) - c{x - y+p - q). 6. d{x - a) + 5{a -2x- 3y) - 17{b + c- 2z). 6. 2a{x + b-2c)- 2c{y +q - 2b), 7. bx{x + y — a) + lb{x — a -\- 2p — 6q) — Spq. 8. 7a{2x -p + 2q) - 3p{2y - a + 2bc) - ba^, 9. 2(4a — 32: + 2p) - b{2b - by-\-'dq) - 7^1 10. I2x(a^ - b^) + 6a{a^ - P) - 2a{a + b- 2px). ax. TBANSFORMATIOm, 69 MULTIPLICATION. 74. The multiplier and the multiplicand are called the FACTORS of the product ; in continued multiplication there are more than two factors. E.g., 3 X 5 = 15; 15 X 7 = 105; 105 X 11 = 1155. 3 and 5 are the factors of 15; 15 and 7, or 3, 5, and 1, are the factors of 105; 3, 5, 7, and 11 are the factors of 1155. It appears from the principles stated in § 58 that the factors of any product may be written in any order, so that, e.g., bah = bha, 75. When more than one of the factors of a number are alike, it is sometimes convenient to indicate the fact as follows : 2X3X3X3 = 2X3^ = 54; ^aaaWbl = '^a%K 76. The small figure denoting the number of equal fac- tors is called an index. 77. The product of equal factors is called a power. 78. One of the equal factors of a power is called a root. For instance, 27 is the third power of 3, 2 is the fifth root of 32; 64 is a sixth power, 2 being the root and 6 the index; 36 is a second power, 6 being the root and 2 the index. 79. The second power is generally called the square, and the second root the square root; 8 is the square root of 64. The third power is often called the cube, and the third root the cube root; 5 is the cube root of 125. The expressions a^, b^, c^, cP . , , p^ are read respectively a square, h cube (or third), c fourth, d fifth , , . p kth.* 80. Model E.— Multiply 6x^^ by dxY^^- bxxy.dxxxxxyy^yzz is what we get for the factors of the * Note the difference iu meaning between a', a^, a^ and «i , aa, o^s . . . (Read a two, a three, a four, etc.) 70 ALGEBRA. product. These can be rearranged so as to bring the numerical factors together, and bring all like letters together : 5x3 xxxxxxx yyyyy zz which can be simplified thus : Ibx^y^z^, EXERCISE XXXIX. 1. Multiply ^x by 4^. 4. Multiply a^ by a?. 2. Multiply 'dxy by '7xy. 5. Multiply a^^ by a. 3. Multiply 3abc by ac, 6. Multiply 3a^b^ by 4a^Z>2. 7. Multiply Sa^c by 5a^c^. 8. Multiply 12a¥c^ by 16a^c\ 9. Multiply ^a^c' by 4:a^bc^. 12. Multiply 2ahx by 7aJ:r. 10. Multiply a^ by 3«^. 13. Multiply xyz by 15^^^;. 11. Multiply d^x by aa:''^. 14. Multiply pqrs by 7j9^ W. 15. Multiply 192;^y;2 by lllx^y^z. 16. Multiply ] 3ai3 by 7a'. 17. Multiply 301aV by 2a^i2;l 18. Multiply 117 aWk^ by ll^^^y. 19. Multiply 2naVz by 112a^xh^K 20. Multiply 71^iV^ by 17^V. Multiply together the folloiving expressions : 21. x^y] xy ; xy, 24. S^r.'r^; 6a^xy; 6aPx. 22. i^^^^;; a;^^^; 2;^^2;; i?:^2;^. 25. lOA^A;; 157^P; 2^^^^. 23. aWc; ^^Z>6'; aZ>V. 26. 2a^^(?; 3a^bc; 4:a^bc, 27. lla^^^c; 12aZ^i2^; 6ab(^. 28. lOl^i^^V; 111^2^1 V; 13ai^Z^23^29^ 29. pqrs; p^qrs^; 7pqhh\ 30. SOl^soi. 103/^103 ; lOUi^^i^. 81. When an expression is separated into two parts by a plus or a minus sign, the quantity is called a binomial. When an expression is separated into three parts by plus or minus signs, the quantity is called a trinomial. TRANSFORMATIONS. 71 When an expression is separated into two or more parts by plus or minus signs, the quantity is called a polynomial. When an expression is not separated into parts by plus or minus signs, the quantity is called a monomial. The expressions multiplied in the preceding examples of this chapter are all monomials; the multiplication of bino- mials and other polynomials depends upon the multiplica- tion of monomials. 82. Model F.— Multiply ba^ by (3a; - 2a), ba\^x - 2a) = 5d^ . 3x - 6a^ . 2a = 16a^x - lOaK EXERCISE XL. Perfor7n the following multiplications : 1. 7x{x - 1). 6. 2{x - 5). 2. M\a - x). 7. 3A(4FA^- - 2^2/2). 3. 10h^k{2h - 5F). 8. 10Wk'{lW^k^-21hW), 4. llp^qi^p^q - 3prf). 9. 38hij{23hji + 12ijh). 5. lxyH2xy'^ + 4a:'y). lo. 10ax{100bx + lOOOcrr) Multiply together the following quantities : 11. 3; a; - 5; 2. 13. \0x^] 2y^', ixy^ + ^x^y, 12. 3a(a — ^); 2x; 3ax. 14. 3a^; a^ ~ P; 2ab, 15. lopq^; 9qr^; i^f"^ — -hV^- 16. 7a^^; ^La^p\ lbhq^\ i-^^P ~ ii^M- 83. The pupil has noticed that where a polynomial is multiplied by any number, each term is multiplied; that is, 3{x - 5) = 3a; - 15. 84. This principle may be illustrated again as follows:* A shopkeeper sends every week x dollars to the bank; his messenger uses every week II for necessary expenses; his wife draws out every week y dollars for her personal use; his son remits every week z dollars, which is added to * See § 12. 72 ALGEBRA. the shopkeeper's deposit. The increase of the shopkeeper's account each week^ then, amounts to x — 1 — y -\- z. In 5 weeks the increase would be 5(^ — 1 — ?/ + 2;). But in 5 weeks the shopkeeper would send bx, tlie messenger would use up 5, the wife would draw out by, and the son would remit bz. Therefore b{x — 1 — y^z) = bx — b — by -\- bz, 85. This is the third of the important and fundamental principles of Algebra. It applies also to minus signs; e.g., — {x — z) =^ — X -\- z] — [x — y-\-z — a — h-^-c] = — x-^y — z-{-a-{-b — c. It may be stated as follows : The product of a polynomial by a single term is found by multiplying each term of the polynomial successively. The negative of a polynomial is found by taking the negative of each term, successively. 86. The three principles stated on pp. 59, 60, and 72 are called the thkee fu:n^damektal laws, and they are known as I. The Commutative Law. (§ 57, I; § 58, I.) II. The Associative Law. (§ 57, II; § 58, II.) III. The Distributive Law. (§ 85.) Upon these laws are based all the transformations of Algebra. EXERCISE XLI. Multiply the following expressions : 1. bx{x — 5 + y); ^V 2. ^x^y; 2x^y + 12x^y^ — Ibxy^ — y* — 4a;*. 8. — {x^ -\- y^ -\- 2^y) ; ^^y- 4. - 2{W^h - daP + a^ - b^); 4:a\ 6. WJc - 217^3/^ + 42^*F; b¥k\ TBAN8F0RMATI0NS. 73 6. lOpqr:, bp^ + 3g^ + 7r^ + '^p^q + ^q^ + 4^g^ 7. x^ + 4^^ + 5.T — 24; 2a:i/^. 8. x^ + r/:r^ + Z>:?:^ — ex — d\ abcdx^. 9. - ?>li\%'lc + Ft + ^2y + jH + Fy + ph). 10. - 452^5^(75^ + 205^ - 155^ + 5^2 _!_ 20i(3 _ 15^4)^ Distributive Factoring. 87. A glance will show when a monomial is a factor of every term of an expression, and such an expression can therefore readily be separated into factors. Thus in the expression W - 15aZ> + 20^2 - 5« ha is a factor of every term, and the whole expression is the product of ha(a - 3^ + 4^2 _ i). EXERCISE:XLII. Find the factors of the following expressions, or show that they are not factorable : 1. ax + ay. 4. 3.r — 15y. 7. ^ + x^. 2. ax — ay, 5. a^ — ah. 8. 6x + 25a;^ 3. 6x + lOi/. 6. x^ -\- x^y. 9. 13ic + 91«^^. 10. i^^^y — ^y^. 11. 7x + 49ic2 + 343^:^ 12. 3a; + 6^;?/ + 2yz. 13. 13^^ _|_ 55^2^ _^ 117^:^1 14. 343a;3y + 9SxY + ^Sxy"^ + Sy\ 15. 15a;2 + 9a;y + 25«/l le. 382;3 + 67x^y + 19xy\ 17. o;!^ + lOx^y + 45a;y + 120a;y + 210a;y. 18. 45a;y - 120a;y + 210a;y - 270^^^ 19. n^ — 71 + n^ — n^. 20. ^5 — 2rh -\- rst^ -\- rh^. 21. y^ — 2mxy -\- x -{- y. 22. x^ — 2a;y + y^* 23. ic^ — 3a;^y + ^^y^' 24. ic* — 4tX^y + 6a;2^^ — 4a;y^ + 2/** 25. 5a;y* - 10x2^3 _[_ 10^3^2 _ 5^4^, 74: ALGEBRA. 88. The process of ^^distributive factoring ^^ is often used in rearranging terms of a long polynomial. Model G. x^—ax^ — 'bx^—cx^-\-ahx-\-acx-{-'bcx—al)c becomes, when we bracket like powers of x, x^ — (ax^-^^ l)x^ + cx^) + {ahx + acx -\- hex) — abc, and then, factoring each parenthesis, x^ — x'^{a + ^ + c) + x{ah -{- ac -\- he) — ahc, EXERCISE XLIII. Rearrange hy poiuers of x the first four expressions : 1. x^ — d^'ji? + hx^ + cji? — ahx -\- aex + he. 2. x^ \- 5a;^ — 4a;2 — ^:ax -\- 4^ -f ax^ + ^^-'^^ — ^^^ — ^^^« 3. cx'^ + nx^ + ^•'^^ + ^"^^ + ^^' — ^^^ + ^^^ — c?/LT^ + cnx^ — ^ma;^ + ^^^* 4. 2a;* + 4:r^i/ + ^x^y^ + 4:^^/^ + ^* - ^xH + 24:z;V - 32^^3 + 16^^ 5. From «V + h^y''- + a^?/^ + h'^x^ + ^^^^ + ^^^^ make two groups, with x^ and y'^ for coefficients. 6. From the same expression make three groups, with (T, V, and e^ for coefficients. 7. In the sum of the expressions in examples 1 and 3 find the coefficient of x^, 8. In the expression ax^ — hx^ + ^^^ ~ c^^ + (^^ — cib^ find the coefficient of a^ and that of Z>^. 9. In ax^ + 26«ir^ + al? + 27^x?/ + llihy + 2Azfx + 2A^/^ + hf + 2^^^ + ^t^' + 2^^ + 2^^ + 2/y + 2/^ + ^ find the coefficient of x and of y, 10. In the expression a{bc — 2ca) + ^(^^ — 2) -{-c{ah—2,hc) find the coefficient of a^ and of a. 89. This sort of factoring is of especial use in the re- duction of literal equations. TRANSFORMATIONS. 75 Multiplication of Polynomials. 90. In multiplying polynomials, the multiplicand, consid- ered as one quantity, is multiplied by each term of the multiplier; then each of these partial multiplications is car- ried out. Model H. (5« + 3Z>) (2« - 7^>) = 5a(2a - 7Z>) + U{2a - U) = 10a2_35aZ>+6aZ'-21Z>2=10a2-29a^-21Jl Model I. {2a — hx){'^a — ^x) ='^a(^a — 4:x) — bx{2>a—4:x) = 6«2 — ^ax - {15ax - 20x^) = Ga^ - 4.ax — 16ax + 20^^ = Qa^ -Wax + 20:6-1 Model J. {a-\-d~c){a—h-\-c) = a{a—I)-i-c)-{-b{a—b-{-c) — c{a — h -\- c) —a^ — ah -\- ac-\- ab —d^-{-bc—{ac—bc+ c^) =rt^ — ah -{- ac -\- ah — W-\-hc — ac-\-hc—c^ ^a^-h^-^'lhc - c\ EXERCISE XLIV. Multiply : 1. 2x—y; 2y+x. 11. a^+ah-{-h^; a—h, 2 p^+3pq+2q^; Ip — bq. 12. p^—q^] p^-\-q^, 3. p^-pq-^q^\p^^n^(f' 13. li^-lc'^ ¥+¥k^+]c\ 4. Ji^~lik+21c^\¥+]i]c-2¥. 14. x-\-y-\-z] x-^y—z, 6. x^^%xy-\-y'^\x^~\-2xy—y'^. 15. a^-^-^th"^-, a-\-4th. 6. x^~4:] x^-\-b. 16. x^—2x^-\-S\ x-\-2, 7. x^—^x-^^] x^—Qx-\-b. 17. a —2h-\-c] a-{-2h—c, 8. 0:^+52:— 3; x^—bx—S, 18. a^—7a-\-5; a^~2a-{-3. 9. a;+15; x—S. 19. 3x—6y; dx^—iy^. 10. x^ — dx+2; x^-3x^+2. 20. 20+:x;; a;2-10. Simplify these expressions : 21. {dx'^ + 2x + l){3x - 5). 22. (x^ + 5x+ 25){x -5). 23. {x^ -7x- U){x + 11). 24. {dx^+4x+5){Sx-10). 76 ALGEBRA. 25. (6a;2 -^x-{- 2)(2.'?; + 1). 29. {U^ + lU-\- 49)(2if~7). 26. {^x^-%xy-y'){'ix~y), 30. {^lf-QU'H'd){^t-\-'i). 27. {x^+?^xy+4:y'^){^x-4.y), 31. (16-28i?+49jt?2)(4+7i?). 28. (:z;^+2:^''+4:^;+8)(^-2). 32. (38^2_i9^2)(2^2_|_y2)^ 33. {^x^ + 2x - 6){2x^ + 5x - 1). 34. {^x^ + 4ic2^ — 3xy'^){5x^ — 4:r^ + Sy^). 35. {x^ + ^x^ + 9x + 27) (^2 — 2x — 3). 36. C'^^ - 2x^ + 4:X - 8){x^ + X - 2). 37. {x^ + 32; + 9)(^ - 3) - {x^ - 2x + 4){x + 2). 38. (m^ + n^ - 1)2 - (m* - ^^ _^ 1)1 39. {x^ + 2x - Sy - {x^ ^ 2x + 3)2. 40. (17a2 ^ iQab + 15Py - {Ua^ + 16aZ> - 17Z>2)2. The Law of Signs. 91. It is convenient to notice a law in regard to the sign of the separate terms of the product. Where any term of the multipher is plus, the partial product obtained by it has for each of its terms the same sign as the corresponding term in the multiplicand; where any term of the multiplier is minus, the partial product ob- tained by it has for each term the sign opposite to that of the corresponding term in the multiplicand. To express the four possible cases in a table: Multiplicand. + + — --' Multiplier. + — + — Product. + — — + 92. From this table the following law is easily derived : In multiplication like signs give plus and unlike minus. 93. The same rule holds for division, as will hereafter be shown. TBAN8F0RMA TI0N8, 77 Cross-Multiplication. 94. Simple products like {x — b){x-\- 2) have important features. Model K. I 4 i -10 '^^ -j^ ^ '% X — ^ a; + 2 H^l ^^1 J>^ ic2 -3a; —10 The products of terms directly under each other are called STRAIGHT PRODUCTS ; of terms diagonally opposite each other cross products. The straight products are not similar, and so cannot be united ; the cross products are similar and can be united. In this example the straight products are x'^ and -— 10; the cross products are — ^x and + ^^^ and their sum — Zx\ the entire product is x^ — Zx — \^, EXERCISE XLV. In the foUoiving examples name the straight products^ the cross products, the sum of the cross products, and the en- tire product : 1. {x + 2){x-7). 16. L. (11-^70(2 + A). 2. (« + 5)(« + 9). 17. (4^ - \){U -3). 3. (a + 3)(a-10). 18. {%x + 3)(2:r + 3). 4. {b+7){b-6). 19. (3y - 7)(3^ + 7). 5. (/i + 3)(A + 8). 20. (65 + l)(55 + 10). 6. {h-ll){h+2). 21. (^ + 2.y)(a; - ^y). 7. [h - l){h ~ 4). 22. (a-\- hl)){a-\-U), 8. (^ + 10)(a; + 10). 23. {3a + 2Z>)(5a + 2J). 9. (y-3)(y + 3). 24. (c + i^)(3c + 2^). 10. {s + l)(s + 99). 25. (3/(;-4(/)(5/^ + ^). 11. (2^ + l)(7^-l). 26. (IO2; - 2/^)(32; + 570. 12. (5a + l)(9« + l). 27. (4^> - 3>10(3Z^ - 2^). 13. (3a + 2) (5a - 2). 28. (3:r + 5^)(3.^ + 5iy). 14. (c + l)(3c+3). 29. (3^ - ll.)(3^ + 11.), 15. (*-7)(5yfc + l). 30. (65 + 50(105 + 70- /^ mc^ of the folloiving examples ascertain tvhat vahie X must have in order that loth expressions may be equal ; where the answers are not whole numbers, express them as common fractions : 31. {x — 6)(x~-^); {x—2){x — 7). 32. {x - 2){x - 8) ; {x- 3){x - 4). 83. {x'-7){x+l); {x + 3){x-6). 34. {x+8){x-5); (:«;-10)(a;- 8). 35. {x-2){x + 4); (^^ + ll)(.r - 4). 36. (22; -5) (^-2); {2x+5){x-2). 37. l2x-3){x + 12); (22: + l)(.T-ll). 38. {2x + 7) {2x - 5) ; {ix - 6){x - 5). 39. (3cc- 5)(7:?; + 2); {3x - 5)(lx+ 3). 40. (17^ - 29) (^ - 3); (17:^; - 29)(:?; + 2). 6. Products like those here formed by cross-multiplica- tion are called quadratic expressions. TRANSFORMA TIONS, 79 Factoring by Cross-Multiplication. 96. Model L. — In the product x^ — ^x -\- 14 the straight products are x^ and 14 and the sum of the cross products is — ^x. The terms that give the straight products must have like signs, and therefore the cross products must have like signs; both cross products, then, are minus. To get the first straight product we have XX x\ to get the second 2x7; then the signs must be chosen so that both cross products are minus; that gives us {x — 2) (re — 7). EXERCISE XLVI. Name the straight ^products and the sum of the cross prod- ucts i?i the follotving expressions, and find their factors : 1. cc^— 5.T+6. 2. x^—2x-\-l, 3. :«;2_3^_}_2. 4. X^ — 4:X'{-3, 5. X^-\-4:X-{-4:. 6. x^-\-5x-^4t. 7. x^+7x-\-12. 8. x^—7x-^6. 9. x^+7x+10. 10. x'^—8x-{-l^. 11. x^Sx+7. 12. x^-\-Sx-]-15. 13. .^'2-20:^;+19. 14. x^-20x+Q4:, 15. x^+20x-{-'d6. 16. ir^+202;+51. 17. x'^ — 20x+'76. 18. cc2_^20x+84. 19. x^—14:X-\-4c9. 20. a;2— 14a;+13. 21. x^—14:X+33. 22. 2;2— 14^4-24. 23. X^-]-14:X+4:5. 24. 2;2-f 14^-f48. 25. x^-8x+16. 26. ^2-121a;+120. 27. 2;2-121a;+238. 28. x2-121^+1210. 29. 2;2-121^+3550. 30. a;2— 10002;+999. ^ 97. Model M. — In the product x^ —6x — 24: the straight products are x^ and — 24 ; — 5x is the sum of the cross products. The terms giving x^ for a product had like signs, and the terms giving — 24 for a product had unlike signs ; then the two cross products had unlike signs, and the minus cross product was the greater. To give x^ for a W'^ __- 80 ALGEBRA, straight product we should have to multiply x and x] to give 24 we could multiply 1 and 24, 2 and 12, 3 and 8, 4 and 6 ; and in each case the larger number would have to be minus. x + l i» + 2 a; + 3 X — '^^ x—1^ x— d> -\-x — '^4:X + 2^ — l%x -\-^x — ^x Trying each in succession, the first pair of factors to give the right cross products is {x -\- 'd){x —8). EXERCISE XLVII. Nayne the straight products and the sum of the cross products m the following expressions, and wherever pos- sible find their factors : 1. X^ — X~2, 11. ^2-26:^-407. 21. Sa:^-x-10. 2. x^-x-Q. 12. ^2_4^_437^ 22. 2x^+'dx-9. 3. X^-dX-4:, 13. x^+4:X-221. 23. bx^-9x—2. 4. x^+x-Q, 14. .'^2_10.^'-231. 24. 6x^+x-15. 6. x^+2x-l5. 15. ^2_ 14^-6840. 25. 10:^2-9:^—9. 6. x^+3x—2S. 16. 2:2-282:-1100. 26. 15.t2-72;-2. 7. X'^-4:X-21. 17. ^^2-1-91:?;- 900. 27. Ux^+llx-15. 8. x^'-4:X—32, 18. ^2_j_9Q^_1944^ 28. 2l2;2+:?;-10. 9. x^+3x-70. 19. :?:2+ir-10302. 29. 10:?;2_29^_21. 10. x^+4.x-96. 20. x'^-54.x-3(j27. 30. 6x'+17x-U. 98. In the equation 2:r — 7 = 0, 2; must have the value ■I; and if we substitute this value for x the expression 2x — 7 becomes zero. Model N. 6a;2 — 5a; — 6 factors into (3a; + 2) (2a; — 3). To make 3a; -f 2 = we must have a; = — f ; To make 2a; — 3 = we must have x = ^, TRANSFORMATIONS. 81 EXERCISE XLVin. What value 7nust we suhstitute for x in each factor of the folloioing expressions in order that the factors may severally hecome equal to zero ? 1. x^ - 3x + 2. 6. Ux^ + 5:?: - 1. 2. x^ - 6x+ 4. 7. 2lx^ - Ylx + 2. 3. ^x^ — X- 2. 8. 3.^2 - 14a; + 16. 4. "^x^ - 3ic + 1. 9. Vlx^ -7:^+1. 5. x^ — X — 2. 10. 15^:2 — ll:r — 14. 99. This sort of factoring is of especial use in the solution of quadratic equations; it is also used with other devices in the reduction of fractional expressions. Arrangement in Multiplication. 100. In multiplication it is often, if not always, conven- ient to arrange the work somewhat as in Arithmetic, with the partial products on separate lines, as follows* Model 0. Model P. x^ — xy -\- 3^2 X — h X — "ly X -\- "l x^ — x^y + 3^y^ ^^ — 5x - 2xhj + 2xy^ + 6y^ ^ + 2x - 10 x^ — 3x^y + 5xy^ — 6y'^ x^ — 3x — 10 101. The advantage of this arrangement is that similar terms in the different partial products may be more readily seen. A further help in this direction, for long examples, is the arrangement of multiplier and multiplicand ^^ac- cording to powers '^ ; that is, taking for the first term of each that with the highest power of some one letter, for the next term the next power below, and so on; e.g., — 4:5xY + 210a;y - lOxif + 210^4^^ + x^"^ - 120^^^^ + Uxy + y'^ - 1202;y - 10^% - 252^y 82 ALGEBRA, could be arranged according to powers of x thus : * + 2\0xHf — UOxy + 210:^;2?/«- 10a;/+ ^^^^ or according to powers of y thus : yio _ iQ^y9 _|_ 45^,2^8 _ i20xhf + 2102:4^^ _ 252a;y + 2102;y - 120a;y + 210xY-10xhj+x^^ EXERCISE XLIX. Rearrange the folloiving expressions hefore multiplying : 1. {a^x^ + x^-\- a^){a^ + x^ - aV). 2. {x^ + y^ + ^y){y^ + ^^)- 3. (a' + ^a'h'' + ^6 + Wa?){a' - Z^« + Sa^^^ _ 3^2^4)^ 4. (V - Z^5+ 5^^^- ba'l) + lOa^b'^-\Oa%^){d'-\-h^-'Uh), 6. (6a;V + 2;^4-a^ —4:X^a — 4:Xa^)(^a^x — ?>ax^~a^ + a:^). 6. (W + /^4+54)(A2_^2), 7. (5^2 _ 2.Ty + 32;2)(2:^?/ + S^;^ - 5/). 8. (3/iF + ¥ -'^Wh - ¥){¥+k^-QJi'k^-^¥lc-A:h¥). 9. (21/^2_|_35y^3_|_35y^4_|_7^6,^_^y_|_^7_|_21^/^5_j_ 7^^6) (y_|.^3_|_3^2^_j_3^^2), 10. (i^^^— 6^^y + 132;^— 12^^^+ 4?/^) (o;^ — ^'^ -]-xy^ — y^). 11. (^'+l^ + 4)(|-i^ + ^'). 12. (3a2 + ^-2«)(5^2__ 1 _ 1^), 13. (a;2 + a:y + iy^){y^ -^y + x% 14. (fr2-i:.+ 5)(^:,. + ^^2„^). 15. {j\a^ + 2^2^ + -%U^ + |a&2)(A^2 _^ 1^2 ^ 2aZ>). DIVISION. 102. In division we are given one facror of a number to find the other. The given factor is the divisor, the re- quired factor is the quotient, and their product is the divi- dend. With regard to the signs of the separate terms of the quotient, they must be such that when multiplied by TRAN8F0RMA TI0N8. 8 3 the terms of the divisor they produce the signs given in the dividend. To express the four possible cases in a table : Divisor. + + — — Dividend. + — + — Quotient. _{____[- 103. Hence the rule: In division like signs give plus and unlike signs minus. EXERCISE L. Divide : 1. 12x^ ly 2x. 5. ^Sx^a ly l^ax. 2. Mx^ ly 4:x\ 6. 400a^%^ by IQab. 3. 9x' hy ^x\ 7. na^hVi^h by ISaWhh. 4. 4^5 by 5x\ 8. d25a^'z^ by Ida^hK 9. {a + by by {a+bf, 10. '72a'{Il^ - P)10 Jy 8^3(^2 _ p)7^ 11. abc-^ {- c). - d4.'da^{a - by 12. l^ah ~ 3ah. ^^' - 49(a - by ' - 39axY - 76{¥ - Uy ^^- Uax^y^' ^^' 19[h^-U) ' — 348a;y 17. 333^}'^ -r- (— 9ik), ^^- - V2xY ' 18. - 469a{b - c) -^ 27a. 19. 1002^(5 - c)3 -r- { - 6ib - cy\. 20. (-400) -- (- 3.2). 21. 12a^ + Qab + 30a by 6a. 22. 2a + Uah + 6a^hJc - 26a^ by 2a. 23. 21xYz^ — 4:9xYz^ — dx^z^ by 3xyz. 24. 3ij + Qi^h — 9H + 12ijk by 3i. 25. - 3a¥h + 6aPF + 9ah¥ by - 3ahh, 26. l^m^n^ — 42m^^^ + 28mn'^ — 3bmhi^ by 7mn. 27. ISJc^i^ + 1) + 9h\]c^ + 1) ~ 12/^(7^ + h) by - 3k. 28. 2^(5 + ty - 6t{s + ty + sk{s + ty by {s + ty. 84 ALOEBRA, 29. r(r2 + s^) + 5(r2 + s^) - t{r^ + s^) by - {r^ + s^). Long Division. 104. Model CI. — Divide x^ — y^\)^ x — y. The correspondence of multiplication and division may be seen as follows. The arrangement given for division is recommended as the most compact. Divisor. ^ — y Quotient. x^ + ^y + y^ Multiplicand. ^ — y Multiplier. x'^-\-xy-\-y'^ Dividend. x^ — y^ 1st subtrahend, x^ — x^y x^y - y^ 2d subtrahend, x'^y — xy'^ 1st partial product, x^ — x^y 2d partial product, x'^y — xy'^ 3d partial product, xy'^ — y^ Product. x^ — y^ xy^ - y^ 3d subtrahend, xy'^ — y^ EXERCISE LI. Divide : 1. x^ — y^ by X -\- y, 2. x^ + ^^y^ + y^ iy x^ — xy -\- y^. 3. x^ -\- x^y 4" ^^y^ + cc'y^ -\- xy^ + y^ by x^ + y^- 4. Sc^ - 22c^x +43^2^:2 - 3Scx^ + llx'^ by 2c^-3cx+4:X^ 5. x^ - 3x^ + 9a; - 27 by x^ + 9. 6. a^ -]- b^ -\- c^ — 3abc by a -\- b + c. 7. s^ -25^ + 1 by s^-2s + 1. 8. x^ — xV — x^ ~{- a^ by x^ — xa — x -{- a, ^ ^6 — ^3^3 _ ^3 _|_ ^3 J^y 1^3 _^ ^2 _ ^2^ _|_ ^ _ ^5 — ^^ 10. h^"" + JiY + ^'^ by Iv" + hg + g^. 11. (K + ^) - (i^ + !)• 12. (f«^ - \ab' -n^)-^ {la + b). 13. (1^4 + |«&^ - m -^ (« + 1^). TRANSFORMA TI0N8, 8 5 14. (K + l«^^^ - ^^^'^ - l«'^') -^ (-2^ + «^)- 16. (K + i«'^' + m - (i^' + \^^ + i^'). TF^ere ^^ere is a remainder in the folloiving divisions, write the remainder as the numerator and the divisor as the denominator of a fraction which is one term of the complete quotient : 16. {x^ + y^) -^{x- y). 17. (x^^ + x^y^ + y^^) -^ {x^ + 2/5). 18. (a^ + a% + \ah^^^^)^{^---a\ 19. 1 {x^ - yy + xy{7? + y^ + dxy) \ ^ {x^ -\. xy -^^ y^)- 20. (3a* + 27a^3 - 10^*) .^ (^ _ 2^>). EXERCISE LII. Perform the operations indicated in the following expres- sions : 1. {a' + h^){a - h){a^ + ah + V'). 2. {a:' + ^ah + h^){a' - 2ab + P). 3. {a^ + a% + aV" + W){a -h). 5. (a3+3a2Z>+3aZ>2_^Z>3)(a+J)-(6-a)(a34-a2^4-^52_|_j3)^ 6. [a^ + 3a2(Z> + 6?) + 3a(^> + cf +{b + cf] ^{a-\-h + cf. x^(y + z) -{- y^{z -\- x) -\- z^{x -j- y) — x^ — y^ ~ z^ — 2xyz 2; - (^ - ^) • h\k + s) + h^s + h) + s\h + lc)-\- Uhs ks -\- sh -{- hk (g2 -{-st^ t^){s^ -st + t^){s^ - t^) - {3sH^ - 3g¥) {h^ ~U + 9){¥ + 6h + 26){¥ -2h- 1 5) {h^ - 8^2 + 24.h - 45) (^* + a;^^/ + xY + xf^ + ^4)(:r2 + i?;y4-y2)(^_^y {x^-y'^Y + ^xY ci^ - 3a^Z> + 3a^>^ - Z>^ + g^ a;2 — ic^ -4- ^2 • • ^ _ ^ _|_ ^ 7. 8. 9. 10. 11. 86 ALGEBRA. EXERCISE LIII. Substitute : 1. ^ = c?; + 2 in the expression dx -\- 4:y — 25. 2. ^ =. 2y — 3 in the expression 3a; -f- 3;2; — 21. 3. ;^ = 3 — ^ in the expression ^ + 10 — 5^. 4. y = 2x -{- 3 in the expression o;^ + 3xy + ^^« 5. X = y — 1 in the expression 3^ + 2xy + 2a;^. 6. ;2 = 1 — 2/ in the expression y^ -i- z^ -^ 3xy -^ y -{-z, 7. y=2 — 3^; in the expression ir(y— 3)— ^(2;— 3)+^'^^^+^^. 8. ^ = 2y + 7 in the expression x^ + xy -\- 2y^. 9. y = — x^ in the expression xy -{- 1, 10. y = a:^ + a; + 1 in the expression y + xy — x\ 11. a = :X^ -12a; + 16; Z> = = a;3-12a;-16: ; c = a;2-16 m — . c 12. ^ = :X^ — 2a; + l; k = a;3 _ 3^2 ^. 3^ . - 1 m -^^ — -3x k + 2) 13. h = a' + a;3; c ^ a^ + , 2- K«' (T^o; + a;^ m -7-5- c{a^ • -f ax — ax 14. S = X^ - 4a;3a + 6a;V - - Axa^ + a'; t ^^ ■ - 2x^a + 2xa^ - ■a' . s{x'^ + 2aa; + a^) 16. X = a -\- b -{- c; y = {ab -\- be -\- ca) in \-l * 16. s = a^\ t = be ]n -^ '—— . s -{- t 17. x-\-Z for 2; in the equation 4:Z^ + 9^^ = 24;^. 18. a; — 5 for z in the equation .2^ = lOz — ?/^. TRANSFORMATIONS, 87 19. ;2 + 5 for y in the equation IQx^ — 2by^ + 200xy = 0. 20. h — 1 for X in the equation 3^^ — ^xy = 1y — %hy, 21. a—1) for y in the equation %a^ — y'^ = y{a-]-b) + 2a^. //i each of tlie following examples, ivhat value must x have in order that the given expressions may he equal to zero ? x^ - 11:^: + 14 x^- 14^2 ^ 17^ _ (5 22. 24. 26. 26. 27. 28. 29. 30. 31. + 2a;- 7 * ic^ + Sa;^ - 5^; + 2 x^ - n * 3a;5 - 16:^;4 _ 35^3 ^ 2a; - 14 ^x^ + 5^3 + 2 '• 6a;^ - 15a;^ + 4:^;^ + 4^;^ + :^; - 2 62;^ - 3^=^ - 2a;2 + 1 6:z;5- 33:^;^ - 10^+55 3:^4-5 45a;5 - 261a;^ - 260 :^ ;^ - 33^ + 220 15^* +T3:? - 11 blx^ - ^bx^ + 63^2 - 171a; + 110 19a;^ + 21a; - 22 6a;^ — 4a;^ - 16a;^ + lOa;^ - 9a; + 15 • 2a;4 + 2a;3 - 2a;2 - 3 ° 81a;5 + 306a;^ + 33a;^ - 6a; - 22 27a;^ + 3a;3-2 CHAPTER IV. IDENTITIES AND THEOREMS. 105. In regard to the following equations: I. 3^ + 5 = 22; + 7 II. 3(.^; + 5) = 3^ + 15 the pupil will notice that while the first equation is true only for the particular value x = 2, the second is true for any value that may be chosen. Again, that the expression 3^; + 5 cannot, by any means we know of, be transformed into 2:?; + 7; while 3{x -\- 6) is in a very simple way trans- formed into 3x -\- 15. 106. These two kinds of equations have distinct names. An equation which is true only on condition that the let- ters in it have particular values is called an equation of condition; while an equation which is true for any values whatever of the letters in it is called an identical equation. Equations of condition are the equations ordinarily met with in solving problems. Identical equations (or equations of identity, or simply identities) may be recognized by the fact that one member can be transformed to the identical form of the other member. The two members are said to be identically equal, and sometimes the sign = is used instead of =. The Proof of Theorems. 107. A Theorem is a general statement requiring proof. Theorems in algebra are often proved by stating them as IDENTITIES AND THEOREMS, 89 equations, and showing that one member can be trans- formed so as to be exactly like the other; that is, showing the equation to be identical. 108. Model A. — Prove the following theorem: The product of the sum and difference of two numbers is equal to the difference of their squares.* Proof, — Let a and d represent any two numbers, then the theorem is expressed by the following identity : {a-^l)){a- h)^a^-l)'^. In transforming the first member we get for straight products a^ and — ^ ^ ; the cross products are ah and — ah, and their sum is zero ; so that the entire product is a^ — h'^, EXERCISE LIV. Prove the folloiving tlieorems : 1. The square of the sum of two numbers is equal to the square of the first number, plus twice the product of the two, plus the square of the second. (The identity is {a + hf^a"" + lah + V.) 2. The square of the difference of two numbers is equal to the square of the first number, minus twice the product of the two, plus the square of the second. 3. The square of any polynomial is equal to the sum of the squares of the separate terms, added to twice their products, taken two at a time. (The straight products are the squares; show that the cross products are double.) 4. The difference of the squares of two consecutive numbers is one more than double the less number. (Let a be the less number, ^ -j- 1 the greater. ) 5. The difference of the squares of two consecutive numbers is equal to their sum. * This tlieorem will be hereafter referred to as Theorem A. 90 ALGEBRA, 6. The sum of the squares of two consecutive numbers is one more than twice their product. 7. The difference of the cubes of two consecutive num- bers is one more than three times their product. 8. The sum of the cubes of two numbers, divided by the sum of the numbers, is equal to the sum of the squares of the two numbers minus the product of the numbers. 9. The difference of the cubes of two numbers, divided by the difference of the numbers, is equal to the sum of the squares of the two numbers plus the product of the numbers. 10. The product of three consecutive numbers is equal to the difference between the middle number and its cube. (Let a — 1, a, and a -\-lhQ the numbers.) 11. The product of two consecutive numbers is equal to the smaller number plus its square. 12. Prove Theorem A, letting a be the smaller of the two numbers, and h the difference between them. 13. Prove the same theorem, letting a be the larger of the two numbers, and h the difference between them. 14. The difference of the squares of two consecutive even numbers is twice their sum. 15. The difference of two numbers formed by the same two digits in opposite order is always divisible by 9. 16. The difference of two numbers formed by the same three digits in opposite order is always divisible by 99. 109. Identities may be translated into theorems. Model B.— In the identity {a-^h)'^ = a^+b^+3al){a + l)), a and h stand for any two numbers, because an identity is true for all numbers; a ^ h is the sum, a^ and b^ are the cubes, ab is the pkoduct of any two numbers. So the theorem may be stated : The cube of the sum of any two numbers is equal to the IDENTITIES AND THEOREMS, 91 sum of their cubes plus three times their product multi- plied by their sum. Similarly for the identity (a - hf = a^-¥- ^al){a - h). Many algebraic theorems are so complicated that they can be conveniently stated only as identities. Such are the following. EXERCISE LV. Show that these identities are true : 1. {a^ +al) + ¥) (^2 -ah + b') = a' + a'}? + IK 2. «5 + Z>^ = (a + l){a' - a% + a^W - ah^ + ¥). ^ a^-.h^=\a- h){a' + a^h + a^l)' + aW + b% 4. a'-b'= (V ^ ^2)(^ _^ ^)(^ _ j^Y 6. {a + h-^ cf = a^ + ^a^{h + c) + ^a {b + c)^ + {b+c)'. 6. la + by E= a^+ 4.a^b.+ QaW + 4.ah^ + b\ 7. {a + b-\- c) {be + m + ab) = d\b + c) + h\c + a) + c\a + b) -\- dabc. 8. ^« + y^ = {x^ + y^){x^ - xY + y^)* 9. a;« - / = {:^ + ^) (cc - y){:x^- xy + y^) (a;2_|_ ^.^ _|_ ^2)^ 10. 0>^ + ^>15 = (^3 _|_ ^3) (^12 _ ^9^3 _L ^6^6 _ ^3^9 _[_ J12) APPLICATIONS OF THEOREM A. 110. Theorem A is by far the most important of those here given. Its application leads to one or two convenient processes in Arithmetic. In Algebra its applications, to be hereafter shown, are of even greater importance. Finding Squares by Theorem A. 111. By Theorem A the square of any number can be readily calculated when the square of any number near it is known. 92 ALOEBBA, Model C— To find the square of 23 : 23 + 20 = 43 23 - 20 = 3 129 := 23^ - 202 232 =z 529 Model D.— To find 59^ : 60 + 59 = 119 60 - 59 = 1 119 = 602 - 592 .-. 3481 = 592 EXERCISE LVI. Find in this way the squares of the following numbers : 1. 45. 3. 78. 6. 29. 7. 249. 9. 988. 2. 37. 4. 117. 6. 198. 8. 1011. 10. 793. A somewhat different method is as follows : Model E.— To find the square of 23 : 23 + 3 = 26 23 - 3 = 20 520 = 232 - 32 529 = 232 Model F.— To find 592 : 59 + 9 = 68 59 - 9 :=3 50 3400 = 592 - 92 .-. 3481 = 592 In applying this theorem the pupil must try to find some number which, added to or subtracted from the given num- ber, will make an easy multiplier. IDENTITIES AND THEOREMS. 93 EXERCISE LVII, By the second mstJiod find the square of: 1. 111. 3. 98. 5. 85. 7. 1998. 9. 288, 2. 57. 4. 52. 6. 49. 8. 311. 10. 63. EXERCISE LVIII. Find by Theorem A the difference of the squares of each of the folloiving pairs of numbers : 1. 3 and 73. 6. 121 and 120. 2. 9 and 109. 7. 8133 and 8131. 3. 575 and 425. 8. 2731 and 269. 4. 339 and 319. 9. 101 and 99. 5. 1723 and 277. 10. 10001 and 1. The pupil may with profit invent arithmetical illustra- tions for the other theorems. Factoring by Theorem A. 112. Wherever, in the product of two binomial factors, the cross products disappear, the factors will be seen to be, respectively, the sum and the difference of the same two quantities, or multiples of them ; so that such a product is also an illustration of Theorem A. Model G. — For example, I'^x^ — 8 is the product of ^x — ^ and 62; + 4, which may also be written 3^ — 2 and 2(3:c + ^)? ^^^ i^ ^^^^ ^^ original expression might have been written 2(9:^;^ — 4) and the parenthesis factored by Theorem A: since it is the difference of the squares of ^x and 2, it is the product of their sum (3^; + ^) ^^^ their difference (3:^; ~" ^)' 94 ALOEBBA, EXERCISE LIX. Factor the foUoimig expressions : 1. 4.^2 - 9. 2. 8^:2 - 162. 3. 18^2 - 32. 4. 27x^ - 147. 5. 1728a;2 - 12. 6. 49a;2 - 16y\ 7. 4aW _ i21a;2. 8. dQxY - 25;^^ 9. 1007^^ - 36P. 10. 1210«3Z,3 _ iQab^ 11. 243a;^V — l^x^z. 12. fl^^^^ — 96'^. 13. aV - 49. 14. 1 — X^'^. «- 15. 4 — a^ 16. 9 — 16xY- 17. 1 - lOOA^F. 18. 75a;^o __ 48^8^ 19. 9a^'c^ - 9x^\ 20. 1 - 100a'h'c\ 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. z 4- y)'^ — ^^. z — yY — x^. X + y)2 - 4. X + 1)2 - 6«l x-\- 2^)2 - ^l 2a; + 3«)2 - 9/. :^ + 6a)2 - 4^4^ 2a; - 5.^)2 - ^c\ 9a; — d^y — a^. 31. x^ — {y -\- zy, 32. x^ — {y — zy. 33. 4 — (a;^+ ?/)l 34. ^2 — (x + 1)2. 35. ^2 - {x + 2yy, 36. 9/ - (2a; + 3a)2. 37. 4/ - (^ + 6^)2. 38. 49^2 _ (^2 _^ 7)2^ 39. 9^2 - {2a; - 5^)1 40. a^ — (9a; — a2)2^ What relation exists between the expressions iiumhered 21 and 31, 22 and 32, 23 and 33, e^c. .^ PTAa^ relation exists between their factors? 41. {a + J)'^ - (^ + ^)'. 42. {a — by — {x — yy, 43. {x — ^)2 — {a — by, 44. (a - hy - (Z> + ^)'. 45. (2a; - yy - {x^ + 2^)2. 46. {a — 3a^)2 — (4y — 2^)2. 47. {a + 2by - (4a; + 5^)1 48 («^ + by - {3a + 2>2)2. 49. {x^ — x^y — {x — 1)2. 60. (V - a3Z>)2 - (a2^2 _ ^^3)2^ 51. a;2 4- 2a;y + ^^ — («^ + ^)^- 52. a2 — 2a J -]-P—{x — yy. IDENTITIES AND T1IE0BEM8. 95 53. 9^2 - Qah + Z>2 _ {^x - 3^)2. 64. a^ - 2ah + h^ - {x^ + 2z)^ 65. «^ + 2a^ + r^ - {iy - zf. 56. €? - {V - Uc + ^2)^ 67. 4a^ — {y^ — 2yz -\- z^), 58. c^ - (25^2 - dOab + 9Z>2). 59. {x - yf - («2 + Qah + W). 60. x'^ + ioax + 25^2 _ [iQz^ + 8;2 + 1). 61. 9.^2 - 4^2 + 12«Z> - W. 62. 1 - «2 - 2a^' - Z>1 63. 16:?;^ — x^ — 14:xy — 49«/^. 64. a^ — 2ah -\-V^ — x^. 65. 9c^ - 4^2 + 4.ai - P. 66. x^ -\- 2xy + ^^ — «^ + 2(^^ — P, 67. tt^ - 2ab -^V - c^ - 2cd - d'\ 68. x^ — ^ax^-\- ia^ — 1)^ -\-2hy — y'^, 69. 4.^2 _ i^i^ ^ 9^2 _ 9.^,4 _!_ 30^2^2 _ 25^,4^ 70. x^ + 2x-^l- a' - 2d^h^ - ¥, 113. In the ten examples last preceding the groups of terms that must be bracketed together to make a perfect square are easily found; but even where carelessly arranged they may be picked out by selecting the cross products first, then the straight products ; and by remembering that in a perfect square the straight products are +, and conse- quently the negative straight products must appear in the negative brackets, i.e. the subtrahend, the second of the two groups into which the expression is to be divided. Model H. — In the expression x^ — x^ — ^— 2aV + a'^-\-6x 2a^x^ and 6x are the cross products, and the two groups are x^ — 2aV + a^ and — 2;^ + 6^ — 9, of which the second must go in the — bracket. x^ - 2aV + a^ - {x^ - 6x + 9) = {x^ - a^f - (x - 3)2. The factors of this expression are {x^ -a^ + x - 3)0^2 - a^ - x + ^). 96 ALGEBRA, EXERCISE LX. Factor the following : 1. x^ -{- y^ -^ 2xy — 4:X^y^, 2. ia'^b' — Sab — a^ — \m. 3. a^ - 9aV - iab + U^ 4. x'^ — a^ -{- 1 — 2x — 4:ab — ^b^. 5. ^a - Mx-\-l - x^ - 166^2 _^ 9^2^ 6. x^ — a^ ^ y'^ — ¥ — 2xy -\- lab. 7. ^x? - 12ax - c^ - ¥ - 2cJc + 9al 8. a^ + ^bx - Wx^ - \ + 4. /S^o?7i6 (?/* ^/^e folloioing examples have more than tivo fac- tors : 11. {a^ - 'dVf - 4^2^^ 16. (62;2*- 35^2)2 _ ^2^2^ 12. (2^2 - 3^2)2 _ ^2^2. ^^ (7^2 _ 11^)2 _ 36^6^ 13. {x^ + 12)'^ - 492^2, 18. {^x^ - 10^/2)2 _ ^2^2, 14. {x^ - 5x)2 - 16^>l 19. (3.^2 + xtj)'^ - lOOy. 15. {li^ - ^hf - 36. 20. (67^2 _ 177^^)2 _ 9^2^ Completing the Square. 114. If we know that an expression is the square of a binomial, we need know only two of its terms ; the missing term can be constructed from those two by the use of Theorems 1 and 2.* Model I. — Thus if we know that 9^^ ^nd 49 are the first and last terms of a perfect square, we know that the terms of the binomial are ^x and 7; and the binomial must be then either 3:?; + 7 or 3:?; — 7 ; in the first case the middle term is +422; (twice their product), and in the second case the middle term is — 42^;. * See p. 89. IDENTITIES AND THEOREMS. 97 The complete square, then, of which ^x^ + 49 are the two straight products, would be either ^x^ + 42a: + 49 = {^x + 7)^ or 2x^ - 42x + 49 = (3^ - 7)2. Model J. — Again, if we know that x^— 24:X are the first two terms of a perfect square, the last can be found by remembering, according to Theorem 2, that while x^ is the square of the first term of the binomial, 24:?; is twice the product of both terms; so the second term of the binomial is — 12; and the last term of the square, then, would be 144. The complete square, then, of which x^ — 24^ are the first two terms, is x^ — 24a; + 144 = (^ — 12)^. 115. The same facts may be more clearly seen by con- sidering the diagrams for cross-multiplication. In the first case the double cross product is missing; in the sec- ond case, since the middle term is known to be a double cross product, the single cross products are each half of it. EXERCISE LXI. Complete the square implied in each of the following expressions : 5. x^ — 18r?;^ 6. 49^2 + 625. 7. 9X^ — 24:X. 8. 64^2_|_4ooy. 9. 4:X^ + 729. 10. 16:^;^ — 200x^y, 1. x^ + 112a;. 2. x^ + 2dQ. 3. x^ — 144a;. 4. a;^ + 16. 116. We are sometimes enabled, by means of these facts, to transform an expression so that it can be treated 98 ALGEBRA. as the difference of two squares, and factored by Theorem A. Thus Q^x^ + 625 becomes a perfect square by addhig 400a:^; and if we also subtract 400^;^ we obtain the original expression unchanged in value, but transformed so as to be readily factored by Theorem A. Model K. Ux^ + 625 = Ux^ + 4:0Qx^ + 625 - 4002^2 ^(8:?;^ + 25)2- {20xy = (8:z;2-20:z;+25)(82;''^+20iC+25). EXERCISE LXII. In the same way factor : 1. x^ + 4. 6. 4x^ + 1296;2l 8. 324^^4 + 625. 2. 4:^4 + 81. 6. 642;* + 81. 9. 1024^* + 81^. 3. x^ + 324. 7. 2500:^4 + 1. 10. ^x^ + 6561. 4. 4.x^ + 625f/4. 117. Some trinomials can also be treated in this way. This is the case when the number to be added to the middle term to make it a double cross product is itself a perfect square. Model L. — Factor x^ + x^y"^ + yK x^ + x^ -\-y^ = x^-\- 2xY + y^ - ^y = {x^ + y'^f - {xyf = {x^ + y^ + xy){x^ + y'^- xy). EXERCISE LXIII. In the same way factor : 1/^4 _|_ 9^2 _^ 81. 3. x^ + xhf + y^. 6. 1 + ^x^ + ^\x\ 2 x^ + x^ + 1. 4. 16a;* + 4x^ + 1. 6. I62:* + 36x^ + 81. 7. x^+xY+y^ (3 factors). 13. x^ — "Ixhf + /. 8. 256c«;8 4- 16:^;* + 1. 14. x^ - 2x^ + 1. 9. ^5 _^ 9^3^2 _|_ 81^J4^ j5^ ^4 _ 27^2^,2 ^ J4^ 10. 48^;* + 12^2 _[_ 3^ jg^ 16^4 „ 24a2 + 1. 11. x' - 6x^ + 1. 17. 81^* + 8a;2 + 16. 12. x^ - WxY + 2/*- 18. 625a;4 + a;2 + 1. IDENTITIES AND THEOREMS, 99 19. x^ - 66:^2 _[_ 625. 25. 2500:?;4 _|_ 4^2 _|_ 4^ 20. a;«+14a;*+625 (4 factors). 26. 3aV - 51aV + 48^2. 21. x^ — ll2;3 + X. 27. 80^;^ + 115a:'' + 405a;^ 22. a'l)^ - 27^2^4 _|. j6^ 2g 7^^4^4 _49a2:y+567a^/^ 23. 80^5 - 120a3 + 5a. 29. 4^* + 162a;8 + 2. 24. 243:^;y + 24^;^ + 48^1 30. 4:X^ - "^b^x^ - 1, Factoring Quadratics by Completing the Square. 118. Model M.— Factor x^ + 56^ + 768. x^ + 56a; + 768 = a;^ + 56a; + 784 - 16 = (a; + 28)2 _ 42 = (a; + 28 + 4)(a; + 28-4) = (a; + 32)(a; + 24) EXERCISE LXIV. In the same 2vay factor : 1. a;2 — 62a; + 945. 11. x^ - 200a; — 281600. 2. a;2 + 68a; + 1155. 12. x^ + 400a; - 422400. 3. a;2 - 10a; - 704. 13. x^ - 700a; + 105600. 4. a;2 - 50a; + 616. 14. x^ + 800a; + 153600. 5. a;2 - 12a; - 1260. 15. x^ - 900a; + 201600. 6. a;2 - 14a; - 1176. 16. x^ - 800a; + 158400. . 7. a;2 - 82a; + 1512. 17. x^ - 800a; - 614400. . 8. a;2 - 10a; - 875. 18 x^ + 800a; + 134400. 9. a;2 - 8a; - 768. 19. x^ - 900a; + 170100. 10. a;2 + 74a; + 1344. 20. x^ - 1400a; - 633600. 119. This process becomes somewhat more difficult when the coefficient in the middle term is an odd number. Model N. a;2 + 11a; - 726 e: a;^ + 11a; + ( ) - 726 - ( ) = a;2 + 11a; 4- 30.25 - 756.25 = (a;+ 5.5)2 _ (27.5)2 = {x+ 5.5 + 27.5)(a; + 5.5 - 27.5) = (a; + 33)(a; - 22). 100 ALGEBRA, EXERCl (SE LXV. In the same tvay factor ' 1. x^ + 6x- 594. 16. x^ - lOlo; + 2520. 2. x^ - 9x - 792. 17. x^ — 57^ - 2268. 3. x^ - 152; - 1134. 18. x'^ + 135:c + 4536. 4. x^ - 51x + 648. 19. x' + 145a; + 5184. 5. x^ + 81:^ + 1458. 20. x^ - 21x - 4860. 6. x^ - 76x + 1386. 21 x^ - 2lx - 1960. 7. x^ — 75^ - 13356. 22. x^ + 117a; + 3240. 8. x^ + 87^ + 1782. 23. x^ - 119^; + 2940. 9. x^ + 67^ + 1120. 24. x^ - l^x - 2880. 10. ^2 _ 79^ _^ 1350. 26. x^ + 165:?; + 6804. 11. x^ + SSx + 1512. 26. x^ - Id^x + 4800. 12. ^2 _ 37^ _|_ 1620. 27. x^ + 31x - 2520. 13. ^2 _ 99^ _j_ 2240. 28. 2:2 _^ 139^ _^ 3730^ 14. x^ — 3x — 5400. 29. .2:^ - 99^; + 1944. 16. a;2 + 129:^; + 3780. 30. x^ + 9x - 6300. 120. A still further complication is introduced when the x^ term has a coefficient greater than 1. Model 0.— Factor %x^ - II2; - 72. The most convenient way to attack an example of this kind, if it is not practicable to factor it by inspection, is to divide the expression throughout by the coefficient of x^, so as to reduce the expression to the form of those we have just been considering. It must not be forgotten that this divisor must be restored to the other factors when we find them. 2 11^ >---^^+(gr---(ir llo; 121 _ 1849 6 "^ 144 Tsr 43\2 _/ IIV /43\ =l"-r2) -(12) IDENTITIES AND THEOREMS. 101 + isjl^ - 13 -i 11 These factors, multiplied by the coefficient we divided by at first, will be thq factors of the given expression. Qx^ - 11a; - 72 = ^{x + %){x - f) = ^{x+%),2{x-^) = (3a; + 8)(2ic-9). Model P. 400:?;^ + 678:?^ — 135 is somewhat harder to factor by inspection. Dividing by 400, 678 _135^ 2 I 6^8 /339y _135 /339\^ ^ + 400 ^ 400 "^ "^ + 400 ^ + UoO / 400 \ 400 ) _ ^ _^ t) /«^ + 160000 160000 - r + 400/ \400y = r + 400 + 4ooy r + 400 400/ - r + 4ooy r 400; Whence the factors of 400:?;2 _j_ 673^ _ ^35 ^^e = (8:?; + 15)(50:^-9). 121. Practice in the application of this method of fac- toring may be had by trying some of the harder examples of the preceding chapter. It is not well to spend much time on such examples as the following, because an easier 102 ALGEBRA. method will be given later, in which the factors can be found by a formula. EXERCISE LXVI. Factor the folloioing dy completing the square : 1. 30^2 _|_ 56^ _|_ 24. ^ 24:X^ + Q7x + 45. 2 21x'^ - 1042: + 60. 7. 362;^ + 81^ - 40. 3 36.^■2 + 231^ - 345. 8. 150^^ _ 175^ _ 294. 4. 200:?;2 _ 9^^ _ ^35^ g 43^2 _|_ 73^ _ 3^5^ 6. 210a;2 - 23x - 72. 10. 4:00x^ - 651^; + 108. 122. This method can be used to factor any * quadratic expression, even when the second number in the trans- formed expression is not a perfect square; because we can find the approximate square root of any number expressed in figures. Model a.— Factor 3^^ _|_ g^^: + 1. 2 , 8:r , 1 „ ,82: , 16 13 ^ +y + 3^^ +-3 +^--9 13.0000 9 3.605 400 396 6 40000 36025 7205 5 397500 X + 3 y\ ' 3 3 y = {x + 1.33 + l,2){x + 1.33 - 1.2) = {x + 2.53)(ic + .13) * An apparent exception occurs wlien tlie transformed expression becomes the sum of two squares, instead of the difference. Thus x^ — 10a; + 29 = {x - 5)2 -f (2)2, to which Theorem A does not apply. This difl&culty will be dealt with later. IDENTITIES AND THE0UEM8. 103 EXERCISE LXVII. In the following examples figure the coefficients of the factors to two places of decimals : 1. x^ -{- %x — 1. 16. x^ -{- ^x — 2. 2. x^ -^ ^x-\- 2. 17. x^ - Ux - 7. 3. x^ + Qx + 7. 18. x^ + I'lx - 10. 4. x;^ — Sx + 14. 19. x^ - 21a; - 20. 5. x^ - 16x + 61. 20. x^ - 332; — 2. 6. x^ — 10:^; + 20. 21. x^ + x — 1. 7. x^ — 20:^; — 20. 22. 2x^ + 3:^ — 4. 8. x^ + 12x + 30. 23. 3a;2 -\- "^x — 2. 9. x^ -^ Sx — 1. 24. 4a:^ + 9^ "~ ^• 10. x^ — ^x — 2. 25. 2:?;2 + a; — 1. 11. x^ + 3a; + 1. 26. a;2 + 3a; + 1. 12. x^ - \lx + 6. 27. 3a;2 + 10a; + 1. 13. a;2 - 7a; + 11. 28. bx^ + 8a; + 2. 14. x^ - 15a; + 42. 29. 2a;2 + 8a; + 5. 15. a;^ + 5a; + 3. 30. 4a;^ -\- bx — ^. CHAPTEK Y. FACTORABLE EQUATIONS. 123. A theorem has been defined as a general statement requiring demonstration; and an axiom as a general state- ment not requiring demonstration. One of the most im- portant axioms in elementary algebra is the following : 124. The product of two or more factors can never be zero unless at least one of those factors is itself equal to zero.* Two Answers to One Question. 125. Model A. — A square box 7 inches high has 160 square inches more in its lateral surface than on its bottom. What is the size of the bottom ? Let X = length (and breadth) of the bottom. Then Ix — area each side and x'^ = area bottom. Q 282; - x^= 160 © = 160 - 28^; + x^ (T) - 28^ + x^ (Z)0 = (^x-8){x- 20) © factored f ® = ^ - 8 from (3) by Ax. A ® S=^x ® + 8 (6) = :?; — 20 from (3) by Ax. A ® 20 = i?; © + 20 * This axiom will be hereafter referred to as Axiom A. f When an equation is so arranged that all its terms are on one side and zero on the other — in other words, so that we have an algebraic expression equated to zero — then the factors of that expres- sion are sometimes loosely called the ''factors of the equation." 104 FACTORABLE EQUATIONS. 105 126. This example has two answers, or, as we sometimes say, there are two values for x which satisfy the equation. This does not mean that the same quantity x can be equal to two different numerical expressions at the same time; the box canuot be 8 inches square and also 20 inches square. But a box 8 inches square would have the prop- erties described in the equation, and so would a box 20 inches square; there are two sizes of square box that could be made so as to be 7 inches high and at the same time to have 160 square inches more in the lateral surface than on the bottom. EXERCISE LXVIII. The following examples have each two answers : 1. Both ends and one side of a rectangular field require 41 rods of fencing, and the area of the field is 200 square rods. Find its dimensions. 2. A field 17 rods long is made square by cutting off 72 square rods from one end. Find the width of the field. 3. A farmer trades grain with a seedsman for a basket of new seed, and agrees to fill a basket for him for every quart the basket contains; he filled the basket 12 times, and then found that he had given just one bushel too much. How many quarts did the basket hold ? 4. A man contracted to pay 15 cents a foot for gilt picture-moulding and 20 cents per square yard for straw matting used in furnishing a square room in his house. He was astonished to find that the picture-moulding cost him $4 more than the matting. What was the size of his room ? 5. In another room, where there were 2 yards more in the length than in the breadth, the moulding at 15 cents per foot cost 14.20 more than the matting at 20 cents per square yard. What were the dimensions of the floor? 106 ALGEBRA, 6. There is a cylinder 8 inches high, whose lateral sur- face (2 TrrA) exceeds the area of its base {nr^) by 88 square inches. Find its radius. (Use n = d\.) 7. A square reception-room 11 feet high has walls and ceiling papered; 8 more rolls of paper (36 square feet to the roll) are required to paper the walls than are required for the ceiling ; what is the size of the room ? 8. A man sold a horse for $102 and found that his loss per cent was one-eighth of the number of dollars he had paid for the horse. How much had he paid ? 9. A stock-raiser bought sheep for 1210, but after he had lost 5 by sickness, he sold what he had left of the flock for 1150, a loss of $1 a head. What did they cost him apiece ? 10. A man bought 9 tons of coal in an inland town, and sold enough to get his money back at a profit of $4 per ton; afterwards the price rose $6 per ton, and he could have got his money back by selling 2 tons less. How much did he pay for his coal ? How many tons did he sell ? 127. If we represent by a the number which is given as 7 in the last illustrative problem, and by I? the number which is given as 160, we can state five new problems by putting the following values for a and b in the statement of that illustrative problem : a = 11; b = 340 a = 8; b = 60 a = 5; b = 91 a = 9; b = 160 a = 4:; b =z 62 In the same way successive sets of figiires are given for each of the ten problems in the last exercise : Example 1. {a) 27 42 34 39 148 (a = 41; Z> = 200) {b) 91 220 144 187 2400 FACTORABLE EQUATIONS, 107 Example 2. {a) 23 30 28 40 44 (a = 17; Z^ = 72) (Z>) 132 200 75 375 480 Example 3. {a) 13 22 11 16 15 (a = 12; Z> = 4 pecks) (Z*) 5 5 3 3-i 4i Example 4. {a) lOcts. llcts. 8cts. 13cts. 25cts. (a = 15; ^ = 20; (Z>) 15cts. 16cts. 18cts. 36cts. $1 c = $4) {c) $1.05 $2.30 56cts. $1.20 $2 Example 5. {a) 2 ft. 2 ft. 2 yds. 2 yds. 3 yds. (a = 2 yds.; Z^=15cts.; (b) lOcts. 7ets. lOcts. 15cts. llcts. c = 20cts. ; d= $4.20) (c) 21cts. 15cts. 18cts. 27cts. 36cts. (d) $1.05 $1.08 $2.00 $3.00 $2.78 Example 6. (a) 25 5 6 9 13 (a = 8";^ = 88sq.in.) {b) 154 66 110 176 418 Example 7. («) 12 10|^ 9 10 6 (a = 11; Z> = 8) {b) 7 10 5 11 3 Example 8. (a) $144 $147 $105 $192 $105.60 (a = $102; ^^ = $1) {b) i i i iV tV Example 9. (a) 120 215 256 210 280 (« = $210; b = 6; (b) d S 2 8 3 c = $150) (c) 85 140 210 135 225 Example 10. (a) 10 12 50 30 32 {a=:9; b = U; (b) 3 2 1 4 3 c;= $6; d = 2) \c) 4 3 5 2 2 (J) 2 2 15 3 4 Both Answers Alike. 128. Model B. — In example 1 in the preceding exercise suppose the room had been 6 feet high, and that 4 more rolls of paper were required for the walls than for the ceil- ing. Let X = the length (and breadth) of the room. 108 ALGEBRA, Then Qx = area of each side, and x^ = area of ceiling. © 24:x - x^ = 144 (2) x^ - 24x + lU = ® - 24:X + x^ ® {x - 12)(:z; - 12) :=: (D factored ® i?; — 12 = from ® by Ax. A ® 2; == 12 ® + 12 Here, on account of a peculiar selection of figures, the two answers to the problem are alike. Similar results are obtained if, instead of 6 feet high and 4 more rolls, we say 9 feet high and 9 more rolls, or 12 feet high and 16 more rolls, or 15 feet high and 25 more rolls. Try the effect, in problems 1, 2, 3, 4, and 6, of the numbers given below : Example 12 3 4 6 (a) 60 18 16 20 rv (b) 450 81 2 bush. 24 154 (c) 16 Answers Apparently Different. 129. If we find two different numbers as the values of x in solving a quadratic equation, we naturally conclude that there are two answers thus implied for the problem which gave rise to the equation. We sometimes find, however, that the two answers thus indicated are the same. Model C. — To divide 28 into two parts, whose product shall be 75. Let x = one part ; then 28 — a; = the other. Q x{2S — x) = 76 (product of both parts) © 282; — x^ =75 same as (T) (D :c2 - 28ir + 75 = ® + x^ - 28a; ® {x-26){x-d) = ©factored ® x - 25 = (6):z; = 25 (T)x - 3 = @x = 3 FACTORABLE EQUATIONS, 109 Here the first answer means that one part can be 25; then the other must be 3 ; or that one part can be 3 ; then the other must be 25. The two answers are therefore really the same. EXERCISE LXIX. 1. A rectangular field containing one acre* requires 924 feet of fencing. What are its dimensions ? 2. Two cubical bins, side by side, extend the whole length of a 16 -foot wall, and contain 9|^ cords * of kindling- wood. What is the size of each ? 3. The number 30551 has two factors whose sum is 360. What are they ? 4. A rectangular room requires 68 feet of picture-mould- ing, which runs above the top of the windows; and the same room requires 32 yards of carpet. What is its size ? 5. A similar room requires 20 yards of border f half a yard wide, and 18 yards of carpet one yard wide. What is the size of the room ? 6. A wharf which projects 27 feet into the water is made of two square platforms, and its total area is 377 square feet. What is its greatest width, and its least ? 7. Forty-six rods of fencing are required for a field whose diagonal is 17 rods. What is the size of the field ? 8. The government of Utopia once enacted that each liquor saloon should pay $10 for its own license, and the same amount for every other license granted in the same ward, and at the same time refused to grant licenses out- side of the two central wards. Thus 48 licenses were granted, yielding a license tax of $12,800. How many licenses in each ward ? * An acre is 10 square chains, and a chain is 66 feet long ; a cord of wood measures 4 X 4 X 8 ft. t Not including mitres. 110 ALGEBRA. 9. Two years later the number of licenses had increased to 208, and the total tax had become 1217,040. How many then in each ward ? 10. A rectangular corner lot and two square lots adjoin- ing contain in all 9972 square feet, and have altogether 114 feet frontage on each street. All the lots on a street have the same depth. What are the dimensions of the lots? 11. Two numbers are reciprocals and their sum is 2y\. What are they ? 12. Two men are separately hired to lay the curbstone on opposite sides of the same street, and receive for the whole job 8 days^ pay between them; one third of the job was accomplished the first day. How many days did it take each man to do his own work ? 13. By one pipe a tank is filled, and then by another immediately emptied, the whole operation requiring an hour all but 10 minutes; then both pipes are used to fill the tank again, which is done in 12 minutes. How long for each pipe ? Meaning of Negative Answers. 130. Sometimes one answer is negative and still can be interpreted as a reasonable answer. It is generally neces- sary to change the sense of some word, so as to have it mean just the opposite of its meaning in the statement of the problem. Model D. — Two laborers applied for work on a farm, and were each sent to build a rod of fence, working by the hour. After they were through, the overseer decided that one of the men could build 3 rods a day more than the other, and that between them they had earned half a day's pay. How many rods could each build per day ? PAGTOHABLE JSQUATlOm. Ill Let X = the number of rods the poorer laborer builds in a day; then a; + 3 = the number of rods the better laborer builds in a day. , ^ x'^ x + d 2 (2) 2x + 6 + 2x = x^ + '^x ® X 2x{x + 3) @x^ — X — 6 = ©— 4a; — 6 ® (^ - 3)(i^; + 2) = (D factored Whence we obtain x = 3 and :?;-j-3 = 6; i^==— 2 and r?;+3=:l. The answer a; = — 2 becomes intelligible if we assume that to build — 2 rods means to destroy 2 rods, and tliat the destructive laborer has to pay by the hour for his amuse- ment. While the better laborer is building his rod and earning thereby a day^s pay, the other fellow gets through with destroying his rod and has to pay for the half -day that he uses up in so doing. If we represent by a the number 3 in the above problem, and by b the number ^ , we can substitute the following numbers for a and &, and obtain in each case a new problem. («) 5 7 9 33 39 (*) i -h io i^ EXERCISE LXX. jV 1. A field 20 rods by 18 is made 352 square rods in area by cutting off a strip of uniform width on the side, and adding to it a strip of the same width on the end. What is the width of the strips ? 2. If a fraction whose numerator is 70 be inverted and multiplied by 6 its new value will exceed its original value by 1. Find the fraction. 112 ALGEBRA, 3. A boat which is able to go 5 miles an hour goes 21 miles southward in a certain river and then 24 miles north- ward, the whole time spent in travelling being 11 hours. How fast does the stream flow southward ? 4. Both pipes at a certain water-tank are opened, and the tank, being empty at noon, is filled at 2 o'clock ; but if the pipes were opened separately, one each day, the tank would be filled 3 hours later by one than by the other. At what time is the tank full in each case ? 5. A manufacturer employs his son as superintendent at $3 per day, with the understanding that the average amount saved by him in the daily expenses of the factory is to be added to his daily wages. When the balance due to his savings has reached $400 on the books, he finds that the concern owes him, including his wages, five times the average daily saving of the factory. What is the average daily saving ? For these examples the following numbers will serve to state oieiu problems : Example 1. {a) 20 16 25 32 35 (a = 20; ^ = 18; {b) 13 10 22 30 12 c = 352) {c) 230 105 540 880 342 Example 2. (a) 8 72 16 45 80 (a = 70; & = 6; {b) 4 36 144 900 16 c = 1) {c) 3 5 7 11 15 Example 3. (a) 3 7 6 13 8 {a = 5; & = 21; {b) 12 34 30 62 22 c = 24:; d=ll) (c) 10 33 32 63 30 (d) 8 10 19 10 8 Example 4. {a) 3 6 4 6 8 (a = 2; 6 = 3) (b) 8 5 15 35 63 FAGTORABLE EQUATIONS, 113 Example 5. [a) 4 5 2 6 10 {a = ^;b = ir, W T^ i i H 6^ c = 400; d = 6) (c) 80 900 1152 3840 2500 (d) 8 90 10 6 13 Answers Suggesting Related Problenns. 131. A negative answer to a quadratic equation may not be in terpre table as a reasonable answer to the problem which gave rise to the equation, but may suggest a new problem, closely related to the given one, and having the negative and the positive answers interchanged. Such is the following : Model E. — A farmer bought a herd of cows for 1400; if there had been 4 more in the herd, the price would have been 15 less apiece. How many were there ? Let X = the number of cows in the herd. ^400 400 _ © 400^; + IGOO = 400a; + hx^ + 20a; ®^x{x ^ 4) (3) hx^ + 20a; - 1600 = ©-1600-400a; ©a;2 + 4a;- 320 = ®-^5. (5) a; = 16; a; = — 20. Now we can change the algebraic sense of the problem by changing words like ^^more'^ to ''less " and vice versa. The new problem would read : A farmer bought a herd of cows for $400; if there had been 4 less in the herd they would have cost $5 more apiece. How many were there ? Let X = the number of cows in the herd. ^ 400 400 ^^ a; a; — 4 Whence a; = 20 or — 16. 114 ALOEBBA. EXERCISE LXXI. Of ihe following 'problems, the negative anstvers suggest related problems in the opposite algebraic sense : 1. There are 40 rooms in a house, and 3 more rooms on a floor than there are floors. How many floors ? 2. Comparing a pail with a 140-quart tub, a man con- cluded that the tub held as many pailfuls as the pail held quarts. He found, however, that the tub held 4 pailfuls more. What was the capacity of the pail ? 3. In order to save 4 hours on his regular trip of 126 miles, a stage-driver finds it necessary to go 2 miles an hour faster than his usual rate. What is his usual rate ? 4. What is the price of eggs when 10 more in a dollar^s worth lowers the price 4 cents a dozen ? 5. A rectangular field contains 1^ acres; if its length is increased by 20 feet and its breadth diminished by 18 feet its area will be diminished 2340 square feet. What are its dimensions ? For these examples the following numbers will serve to state new problems : Example 1. {a) 35 50 60 44 112 {a = 40; Z> = 3) (^) 2 5 4 7 6 Example 2. {a) 88 176 99 90 180 {a = 140; Z> = 4) (P) 3 5 2 H 3 Example 3. {a) 6 12 4 2 5 (a = 4; Z> = 126; (^) 176 135 120 180 150 0=2) {c) 3 4 li 1 n Example 4. {a) 5 2 25 50 20 {a= 10; J = 4) (^) 1 1 8 2 5 FACTORABLE EQUATIONS, 116 Example 5. (a) 3 3 1 4 n {a = 1|; 5 = 20 [V) 34 14 12 -60 -45 c = 18; cZ=3340)(c) 20 30 10 -16 -30 id) 3120 7680 780 6240 -3600 Meaningless Answers. 132. In each of the following problems an answer will be obtained which satisfies the quadratic equation, but has no reasonable interpretation in the problem which gave rise to that equation ; and the answer may not even suggest a related problem. EXERCISE LXXII. 1. A square box 7 inches high and without a cover has 288 square inches of inside surface. What is the size of the bottom ? 2. Of two large wheels, one makes 48 turns more than the other in rolling a mile ; their tires, straightened out and laid end to end, would reach 21 feet. What is the circum- ference of each ? 3. Two sprinters start opposite ways from the middle post of a straightaway 440-yard track; in one second they are 21 yards apart and one of them finishes his 220 yards 2 seconds ahead of the other. What is the speed of each ? 4. A cistern containing 4 gallons is filled by one pipe and then immediately emptied by another, both operations requiring just 5 hours. One pipe delivers 3 gallons per hour more than the other. How many gallons from each pipe alone ? 5. A man rows 5 miles down-stream and back in 4 hours. The stream flows 3 miles an hour. How fast can the man row in still water ? 116 ALGEBRA, For these examples the folloioing numbers will serve to state new problems : Problem 1. {a) 8 10 15 20 3 {a = '7; b = 288) (^) 185 176 544 249 36i Problem 2. {a) 40 10 88 96 88 {a = 48; J = 21) (^) 23 32J 22 24f 27' Problem 3. (a) 400 300 1000 900 500 (a = 440; h = 21; (^) 18 22 201 20i 241 = 2) (0) 5 H 22^ 10 t Problem 4. {a) 6 8 10 4 21 (« = 4; & = 5; W 5 7 3 3 13i c = 3) {0) 7 9 7 5 5 Problem 5. (a) 5 24 30 25^ lOJ (a = 5; & = 4; (^) 9 14 12 10 5 c = 3) (0) 4 5 6 7 2 133. The answers to a quadratic equation are not always commensurable; in that case they can be approximately stated as decimal fractions. Model F. (Tjx^ + dx = 11 © :^2 _|_ 3^ _ 11 ^ ® - 11 (D ^2 _|_ 3^ _j_ 9 _ 5_3 ~ same as © ® (^ + 1)^ - (3.64)2 ^ same as ® ® (^ + I + 3.64)(:c + I _ 3.64) = ® factored (D {x + 6,l^){x - 2.14) = same as ® X = 2.14 or X = - 5.14 The following solutions are worthy of attention : Model G. ® a;2 = 16 © a;2 - 16 = ® - 16 ® {x + 4:){x — 4) = ® factored ®a; + 4 = 0;^=-4) ® ^ - 4 = 0; ic = 4 [ ® Ax. A Ans, FACTORABLE EQUATIONS. 117 Hence there are two square roots to 16 (or to any arith- metical number); one is + and the other — . Model H. ®x^ - 'dx = © - 3a; ®x{x — ^) = (2) factored Notice that the value a; = satisfies the equation. EXERCISE LXXIII. 1. What number divided into 81 will give itself for a quotient ? 2. Three times the square of a number is equal to 10 times the number itself. What is the number ? 3. A room is twice as long as it is wide, and its floor con- tains 144^ square feet. How wide is it ? 4. A square room 9 feet high takes twice as much paper to cover the walls as to cover the ceiling. What is the size of the ceiling ? 5. In a current of 2 miles an hour a man takes 8 hours longer to row 24 miles up-stream than to row the same distance down. How fast can he row in still water ? 6. A field 20 rods by 18 is made 353 square rods in area by cutting off a strip of uniform width on the side and adding a strip of the same width on the end. What is the width of the strips ? 7. A square box 9 inches high has 300 square inches more on the sides than on the bottom. What are the di- mensions of the box ? 8. One man takes 4 hours longer than another to saw a cord of wood, and both, working together, can saw it in 3 hours. How long for each ? 9. Two wheels, the sum of whose circumferences is equal to 8 yards, roll the same distance; and the number of turns 118 ALGEBRA. made by one wheel, added to the number of turns made by the other, gives a sum numerically equal to the number of yards traversed.* Find the circumferences of the wheels. 10. A line 100 centimetres long is divided into two parts, such that the longer part is contained in the whole line as many times as the shorter part is contained in the longer part. What is the length of each part ? A New Kind of Number. 134. The solution of certain quadratics, of which the simplest \^ x^ = — 1 or x^ -\- \ — 0, leads to an expression which has no interpretation according to the rules of algebra, so far as we have studied them at present. This expression is V— 1, and it arises as follows: Model I. Q a;2 = - 1 ® ^-'^+1 = ® + 1 @ x'^ — {— 1) = @ in form of x^ — y^ ^2 _ ( i/ZTiy = (5) (^x + V~l){x- V~l)= ® factored ® ^ + y— 1 = 0; X = - ^^H ^ Ax A When we remember that two factors must be exactly alike in order that their product may be a perfect square, and also that two numbers which are alike in respect to sign can never have a negative product, we see that it is impossible for a negative quantity to have a square root like any other algebraic expression such as we have met hitherto. It is customary to represent this new and strange expression, V— 1, by the small letter i, so that P = ■— 1. * Have a letter for the number of yards traversed ; also one other letter. FACTORABLE EQUATIONS. 119 Then i^ is a part of our algebraic system, and while we can- not say that i is, we will assume that it can be dealt with by ordinary algebraic laws. Expressions in which i appears in the first degree are called imaginary. Model I. — The solution of the equation o;'^ = — 1 now becomes : ® a;^ = v' (D x" -{'=0 (i)-v> ® {x + i){x -i) = © factored (J) X + i = 0; x = — i^ ® x-i-0; x — i 1 (D Ax. A Model J. @ ^2 - 16|2 zz: Subst. i^ @ {x + 4:i){x — 4:i) = a; -f 4^ = ; .T = — 4i I @ X — 4:1 = 0; X = 4:1 ) -lin® (D factored (D Ax. A Model K. (i)x'^+6x + 13 = (2)x^-\-6x-\-9-{-4 = same as ® ®x^+6x + 9 — 4f'^ = subst. *^ = — 1 in @ (A) {x + d + 2i){x + 3~ 2^) = (3) factored ® x+3+2i=z0; x=:-3-2i ®x+d-2i=0; x=-d + 2i_ ® Ax. A 135. The great advantage of the substitution of i^ at this stage in our study is to enable us to change the sum of two squares to the form of the difference of two squares, and thus to obtain a complete rORM of solution to any quad- ratic equation. 120 ALGEBRA, EXERCISE LXXIV. Solve tlie following equations: 1. —x^=^ 9. 6. ^x^ -\-x^ — x^-^x. 2. ^x — x^ — 13. 7. X — \h — 5 — X 3. 20a; = x^+ 121. S. x + 90i = ^(6 - x). 3 , ^, 1 16 - :r 10 - ^ X ' Ax X 4 2x-7 18 6. = ri;. 10. - 2 ' i^i?i^ algebraic solutions to the following proUems : 11. Two numbers are reciprocals and their, sum is -J. What are the numbers ? 12. Of two equal fractions the first has a numerator 5 ; its denominator exceeds the numerator of the second frac- tion by 7, and exceeds the denominator by 9. Find the two fractions. 13. What is the length of a rectangular garden, 29 square rods in area, which is enclosed by 20 rods of fencing ? 14. In building a wall each laborer employed worked as many days as there were men on the job. If there had been 10 men less, the job would have taken 20 days. How many laborers were there ? 15. A rectangular board 12 inches long can be made square by cutting 45 square inches off the end. How wide is it? FACTOBABLE EQUATIONS, 121 The Quadratic Formula. 136. Any quadratic equation, if simplified as much as possible and so transposed as to give an expression equal to zero, will be in the form of ax"^ -\- hx -\- c = Q, which is called the standard form of the quadratic equa- tion. 137. The letters a, h, and c are the three coefficients of the quadratic, and of course for any given set of values for these there will be two and only two answers. Thus for the following ten equations the values of a, J, and c are as given, and the answers may be found to be as in the table. x^ — bx^^ = 2^2 -_ 3ic _ 2 = 2:?;^ — 5a; + 2 =: 6^2 — 5^; — 6 = x^ + 10:z; + 23 = ^2 _ 81 = x^ + 81 = 92;2 - 16 = %x^ + l^x = x^ + 2a; + 10 = a 1 I c - 5 6 2 -3 -2 2 - 5 2 6 - 5 - 6 1 10 23 1 - 81 1 81 9 - 16 9 16 1 2 10 Answers. 3; 3 2; i 3 . 2? 9i — 3.586; 9; 9i; 0; -V- l + 3i; -1 6.414 ■3i p 138. It will be shown now that these two answers to every quadratic may be obtained by the formulae X = — J + Vb^^ — 4:ac . 2a ' X = 2a 4:ac, 122 ALOEBRA. these are usually written, to save space, in one formula, with the so-called ambiguous sign, which can be read either plus or minus, each reading giving one of the two correct answers. _ -1)± VU^ - 4:ac ""-■ 2a 139. This is called the ftuadratic Formula, and is obtained as follows : Q ax^-{- bx -{- c = (2) x^ + -X + - =0 ®-^a ^ '^ a \2a) a ^d^ the square ® l^ + 2-J "l^^) = ^ sameas® ® factored ^^ ^ 2a 2a 2a ® ^+2^ 2^ = '-' "= 2« ® Ax. A Whence the formula x = 2a FACTORABLE EQUATIONS. 123 Solving Quadratics by the Formula. 140. Model L. 2iz:2 ^ 3 - 3^ © 2^;''^ + 3a; - 3 = Q + 3^ ~ 3 a = 2 Z>^ = 9 Z> = 3 4(^c = - 24 c = - 3 1)^ - 4:ac = 33 VU'- 4:ac = 5.745 X = - - 3 ± 5.745 4 _ 2 .745 - 4 '' 8.745 4 = .686 or - 2.186 Model M. ©I-"- -360:^ = 604:X + 959 © 360 - 360^;'^ = 504:?;''^ + 959:?; © X ^ ® 864^;^ + 959a; - 360 = © + 360a;'^ - 360:c a = 864 P = 919681 864 h = 959 4ac = - 1244160 360 c = - 360 P - 4.ac = 2163841 5184 2592 VP - 4:ac = 1471 311040 1244160 - 959 ± 1471 ^ - 1728 512 ^=1728 '' "=^ 2430 1728' which reduce to 8 x^^ ov x=- 45 32 124 ALOEBRA. EXERCISE LXXV. By means of the Quadratic Formula, find the Uvo roots * to each of the following equations : (Commensurable roots.) (Incommensurable roots.) 1. Sx^ - ^^x + 135 = 0. 11. "Ix^ - 10a; - 9 = 0. 2. 362;2 - 133:^: = 120. 3. 72^2 _|_ 161^ _ 200 == 0. 4. 54:^2 _ 345^ ^ 500 r= 0. 5. 180a;2 + 351a; = 50. 6. 300^'^ - 385a;+73i = 0. 7. 480:^'^ - 6822;+225 = 0. 8. 128:?;2 - 24^; = 495. 9. 320a;''^-1114a;+675=0. 10. 1056:^2 = 1610:?;+1125. 141. The two roots of a quadratic equation are some- times represented by a and (i, that is, 12. \Zx^ ^^x ^ 10. 13. "IXx' = 22x - 23. 14. lOOo;-^ = 25:c + 36. 15. x2 + 3 = 2:^; + 1. 16. x^ ^ x-\-\. 17. 10x2 - IOO2; + 203 - 0. 18. 64.^2 - 33x = 100. 19. 202:2 r= 30:^: + 41. 20. 100^2 _ looo; - 101 = 0. « = ta ' ^ = %a • EXERCISE LXXVI. 1. Prove that a -\- B — . a 2. Prove that a^ — -. a 3. By means of examples 1 and 2 show that a(x — a){x — §) — ax^ -\- Ix -\- c, 4. A quadratic equation whose first term is ^x^ has for answers f and — | ; what is the equation ? 5. In the equation '^x^ -|- 4x = 3, find a — p. * The numerical value of x which satisfies an algebraic equation is called a ROOT OF the equation. FACTORABLE EQUATIONS. 125 Factoring Equations of Higher Degrees. 142. Sometimes, but by no means always, equations of degree higher than the second can be factored ; and in that case the factors lead to solutions by the same theory as ordinary quadratics. Model N. © :c3 = 27 @ 2;3 - 27 = ® - ^'^ ® {x-d){x^ + ^x + ^) -0 (D factored ®x — ^^0]x = ?^ (D Ax. A (5) :^;2 _|_ 3^ _|_ 9 ^ (3) Ax. A (§) X = — ^ — ^- — ® Quad. Form. Ans, X = 3; x — 31 ^ 1; x — 3V ~ — These are the three cube roots of 27. Model 0. (Q x^ - Ux^ + 40 = @ {x' - 4:){x^ - 10) = ® {x + 2){x - 2){x + Vl6){x - |/10) = (i) X + 2 = 0; X = ~ 2 (5) X - 2 = 0; X = 2 ® X + VTO = 0; X = - |/10 Q) X - VIO = 0; X = l^io EXERCISE LXXVII. 1. Find the 3 cube roots of 64. 2. Find the 4 fourth roots of 81. 8. Find the 6 sixth roots of 64. 4. x^ - 34a;2 + 225 = 0. 6. 36a;4 = 17x^ + 144. 126 ALOEBRA. 6. x^ - Ix^ r= 18:^;. 7. {x^ - 9) (3:^-2 _|_ 8a; - 3) r= 0. 8. {x^ - 3^; + l)(.^'^ - 3^; - 2) = 0. 9. (2;2 - ^x)(x^ - bx - 10) = 4(2;2 - 2^;). 10. 3:^2 + 2ic - 5 = 3a;2 + 2a;^ - 5a;. Factoring by Parts. 143. This is an extension of distributive factoring, and sometimes serves as an elementary method for the solution of equations of higher degrees. The factors of Mx^ — 3Mxy + 6My^ are M{x^ — 3xy + 5y^), This conclusion would be true whatever the size and shape of the letter M, and even if the letter M repre- sented a different algebraic expression altogether. Thus the factors of {da - 7d)x^ - 3(3a - 7b)xy + 6{3a - 7b) f are {3a — 7b){x^ — 3xy + 5?/^). Here the letter Jfis replaced by the expression (3^^— 7^). Now if in the multiplication (3a — 7b)(x^ — 3xy + ^y^) the second factor is taken as the multiplier, the three ex- pressions (3a — lb)x^, — 3(3a — 7b)xy, and 5(3a — 7b)y^ are the partial products; distributive factors removed from them leave the multiplier, which is seen to be the same in all of the partial products. Model P. — In the expression 01^ — 5x^ — 9x -\- 4:5 the first two terms and the last two terms may be taken as partial products; a^-bx^ + {- dx+U)=x^{x- 5) + 9(- x + 5). FACTORABLE EQUATIONS, 12 Y If the sign before the second partial product is changed the multiplicand will appear the same in each partial product : x\x - 5) + 9(- :?; + 5) = x\x - 5) - 9(^ - 5) = (x- 5)(>'2- 9). The same expression may be factored by grouping the first term with the third, and the second with the fourth, to form the two partial products : x^ — bx^ — ^x -\- 4:^ ~ x^ — ^x — bx^ + 45 = x{x? - 9) - b(x^ - 9) ~(x^ -^)(x - 5). Of course the prime factors of this expression would be (x + Z){x - 3) (a; - 5). EXERCISE LXXVin. Factor ly parts : * 1. ax + by — ay — Ix. 6. ^'^ — 3a:^ — 2hx + 75. 2. 2x-]- xy — %y - x^. 7. x^ + 10^:'^ - 8I2; - 810. 3. x^ — \^ -\- ax — Aa. 8. 112 + 16a: — 7x^ — x^, 4. ax + 4:X^ + 4a + 16x. 9. 99 - 9^: — 11^'-^ + x\ 5. x^+6x + 6-x^- 2x\ 10. 121 + I2I2; - x^ - x\ 11. {2x - y){ix^ + 2xy + y^) - {2xy + y'){2x -y). 12. (1 - 2^)(1 + 2y + 4.y^) - (1 - 2^)1 13. y^l - 4^ 4- 4/) - (1 - 2yY. 14. a^(6 - a) - (36 ~ a'). 15. {lOy - 1)100/ + (lOOy^ - Wy) + {lOy - 1). 16. {a+h+cY + Sa + 3b + 3c. 17. {x - 17){x + 7) + x{x^ - 49). 18. a^ -\- ab -j- ac + be. 19. ace + % + ^^ + ^y- * It may be necessary in some cases to rearrange the terms. 128 ALGEBBA. 20. 6a + &2 _j_ GZ> + d^ + ^al). 21. ax ~\- J)z — (bx -\- az), 22. 6ax -\- hy — {hay + hx), 23. pr — qs -\- qr — ps. 24. px^ — ag^ — {ap — g'^x^), 26. ."^^^^ + x^ — {y^ — x^). 26. x^ + X — 2y{2y + 1). 27. a^x + «^y + ^^^ + abx. 28. 2«a; + 3^y — (2^y + 3Z>:^). 29. ic^ — 3?/^ + {3xY^ — a;''^^). 30. 3fl^iz;^ + (^ + .V)(^ ~~ y) ~ '^axy. 31. «^ + 32;''^^(a; — 5?/) — hay, 32. a;y — 9 + ^'^ + 3?/. 33. ab + a''^^'^ — - (x^ + ^0- 34. d^ + dx -{- a — 9x\ 35. /^y + (^ + '^)'^ + %• 36. a^'(^''' - a^) + 3aZ^(aZ> - A''^) - ¥{P - a^). Model ft. — Solve the equation: Qx' - 6x^ + S =^ x^ - 4:X @x' - 6x^ + 8 - x^ + ix := (T) - x^ + 4:X (3) {x^ - 2){x^ - 4) - x{x^ - 4) = ® (x^ - ^){x^ - 2 - ^) = ® {x + 2){x - 2){x + l){x - 2) = The four answers to this equation would then be X = 2; X = 2; X = — 2; X =^ — 1. EXERCISE LXXIX. Solve the equations: 1. x^ + 2x^ = 9x + 18. 6. c?:^ - 27 = 19(a: - 3). > 2. a;3 — 12 = 4:c — 3x\ 7. a;3 + 64 =r 5a;^ + 22rz: + 8. 3. 100 - 2)3 rir 42;2 - 25x. 8. 2^4 -13^2+36 = 3.^(0;'^- 9). 4. 16a; + 80 - cr^ = 6x^ 9. ^4-13o;2-48 = 4o;3-64o;. 5. 180 + x^ = 6x^ + 36a;. lo. x^ - 2bx^ = 9x^^- 225. I FACTORABLE EQUATIONS. 129 144. In some examples (for instauce, in the tenth of the preceding exercise) easier methods of factoring may be seen on simplifying the equation; in others, the simplified form obtained by uniting similar terms is harder to handle. EXERCISE LXXX. Solve the foUon^ing equations : ^' + ^^ o ^ c, 1,1 1 X — 2 X X + 3 5 x^+x-2 x^-x+2 _l 2 _ 3 ^' 3 I ~6* ^' ^^^^~^'^' x^ - 6x _2 x^ - 6 _ dx-l ^' 2x^ -1~3' ^' 2 ~ 4 • 1 + ^ + 2^:'^ 1 13 4. — -z = 2 + 3a:. 9. - = 7 + x. 1 ^ X XX 3a; — 1_2:?; — 1 x _ 3 ^' x-2 ~ X -3' ^^' 3 "" X -3' 11. In the equation x^ -\- xy -{- 2y^ = 74, if 3/ = 5 what is the value of a; ? 1 1 14 12. In the equation — I — =-— , if x — ^ what is the a; ' ^ 45 value oi y 1 13. In the equation 4z{x -\- y) = 3xy, \i x = 4z what is the value of y ? 3x 14. In the equation x + 2y -\ =16 substitute 5 for X and then find the value of y. 15. In the equation xy = 2400 substitute 100 — x for y and solve for x, 16. In the equation - -j- - = 2 substitute 2 — x tor y x y and solve for x. 130 ALomuA. 17. In the equation 35 = 3/i -| ^^ — - — -d substitute 2 for d and solve for n, 18. In the equation s = an -{- d ^ substitute s = 119, a = 2, d = 6, and solve for 7i, 71 19. In the equation 5 = — (a + /) substitute 5 = 105 , a=— b^l=— h-\-{n— l)d, d = ^, and solve for n , In the equation H= - = X — 2, and solve for x. 2Pp 20. In the equation H = let ZT — 12, F = x -\-d, CHAPTER VI. THE FIRST METHOD OF ELIMINATION. 145. The equations of condition * already studied have been true only on condition that the unknown quantity had a particular value, or one of two or three — one of a few — particular values. Thus the equation X -\-7 = 2a; — 3 is true only when x = 10. x^ -|- 12 —7x is true only when a; = 3 or 4. {x^ — Qy^ = x^ is true only when ^' = 3 or — 3 or 2 or — 2. But when an equation contains more than one unknown quantity, the condition implied by it becomes much more indefinite. The equation 2x — dy = 6 is true for any value of x whatever; but only on condition that when x has any value, y has a particular value which is said to correspond to the stated value of x. Suppose, for instance, x = 6. Then 2:^; = 10 and the equation 2a; - 3^ = 6 becomes 10 - 3y = 6. Then 10 = 6 + 3y ; 4 = 32/ ; H = y. If x = 5, then the equation is true only on condition that y = 1^, It X = 6, y must equal 2. Ux=7, y =21; iix =8, y = ^l; it x = 9, y = 4:. * See p. 88. 131 13^ ALGEBRA. 146. There is no end to the solution of this equation; we might continue to write corresponding values of x and y as long as we chose, and the solution would still be incomplete. The Graphical Method, 147. A method of completely exhibiting the solutions of oh is kind of equations will now be described. The same method is often used for other purposes in science and in practical affairs. 148. Different values of one letter, like x, may be repre- sented by equally distant points along a straight line; taking a particular point to represent zero, positive whole * * * * - * » * — 1 — I — I — ! — I — I — I — I — I — I — I — \ — I — \ — i — r — I — I — 1 — I — I — rn — .n_10-9-8' -7- 6-5-4-3-2-1 1 2 3 4 5 6 7 8 9 10 11 Ftg. 1. numbers are represented by equally distant points running out towards the right, while negative whole numbers are represented by equally distant points running out towards the left. Fractional numbers of all kinds are represented by points lying between the whole numbers; thus 5 is five spaces to the right of zero, ~ 6 is six spaces to the left of zero, 10^ is ten and one third spaces to the right of zero, — 4^ is four and one half spaces to the left of zero, etc. On the accompanying diagram the stars mark approx- imately the points which represent — 10, — 6, — 4^, 0, 5, 8^, 10^, which may be different values of the letter x. The only numbers that have no points on this scale to represent them are the imaginary numbers.* * See § 134. THE FIRST METHOD OF ELIMINATION, 133 149. Now we want a method of representing at the same time by one point a value of x and also a value of y. This may be managed by taking two scale-lines like the one just described, and placing them perpendicular to each other with the zero-points together. X-- -7- . G P A " 4: o ^ i> •^ ^ ^ -i > 1 ^a ns 0/ X I 1 t _ 3 -^ '{ ^-i I - 10 1 2 3 4 6 i -1 -2 o A -4 pr -0 -C -=-7 __ _ _ _ _ --X Fig. 2. Call the horizontal one the ^-scale, or axis of x^ and the vertical one the y-scale, or axis of y. Then imagine vertical lines drawn through each point of the axis of x, and horizontal lines through each point of the axis of y, A point which represents x= — 2 ; y = 4 must be the point which lies opposite — 2 in the axis of x, and opposite 4 in the axis of y. That would be the point marked P in Fig. 2. 134 ALOBBRA. 150. The points in Fig. 3 that represent the following values of x and y are marked with the corresponding Koman numerals: lx = 4:]y = 2 llx=. —2]y = 'd llVx = 3 ; «/ = 5 IV^=z 6;^ = - 3 ■a; = 21; y- VII U=-3 r^= - 5; U=-3f -VH II.. 4 -S -2 -:i VI -4- '5 III I-V- -X Fig. 3. 151. Eeturning now to our equation 2x — ^ — 6, we : 5 i 4 o o 1 -I t- 3-^ 2-1 I i 5 ^ ( » r 6 ^ 1 ) -5 j 5 10 -1 -2 -3 -4 ^5 , L -X Fig. 4.— Partial list of answers to the equation 2x — Sy = 6. find the following set of answers, represented by their respective points, running up from left to right on Fig. 4; THE FIRST METHOD OF ELIMINATION, 135 X - - 4.; «/ = - 4| x = 4; y = 1 a; = - 3; y = - 4 a; = 5; «/ = 1^ x= -H; y ^ - 3 a;= 6; y = 3 X = 0; y = - 2 a; = 7; y = 2| x=l; y= -1^ ^ = 8; y = 3i x = 2; y =^ ~ f a; = 9; «/ = 4 X = 3; y = An Infinite List of Answers. 152. It would be found on trial that all the other an- swers to this equation would lie on the same line, i.e., their corresponding points would fall into line with those already put down. In fact, the straight line drawn through two ^ — — - —^ — — — — ■~~ y ^ / /^ y / y y /^ y y y ^ y y y y' y y y y y ^ y ^ ^ Fig. 5.— Complete list of answers to the equation 2a; — Sy = 6. points that represent any two answers to the equation is a complete list of all possible answers to the equation. In order to be complete, of course, not only the two axes, but also the line AB, are supposed to be endless in extent. 136 ALGEBRA. Model A. — In a similar way construct a complete list of answers to the following three equations (see Fig. 6) : I 7a; + y =14: II 2:^; + 6^ = 3 III 2ic + 5y = 10 y 1 r \ 1 lU \ I TT ei \ ^ \ "-^ \ \ I I"" -^ ^ ^ I ■^ -^ ^-^ -J ^ "V ^ X X _ p > ^ t ^-v. ^ I ^ "^ \ \ \ -5 \ T \ 1 f\ \ -10 ■•' r .__. ... Fig. 6. [Only two points need be found; then a straight line can be drawn through them.] THE FIRST METHOD OF ELIMINATION. 137 EXERCISE LXXXI. For each of the following equations construct a complete list of answers^ and put each pair on a separate sheet : * 1. x + 2y =^ ^] dx + ly = 17. 2. 15a: + 3y = 39; llx — y = 31. 3. *Kx -\- 4:y — 41 ; 3a; — ^ = 4. 4. 'Zx -\- y = 11] ^x + ly = 22. 6. 5a; + 6^ = 51; 6a; - by = - 12. 6. 2a; - «/ = — 4; 7a; + 6?/ = 62. 7. 8a; — ^ = 27; a; + 8«/ = 44. 8. 15a; + 7^ = 162; 9a; + 2y = 84. 9. 14a; - 3^ = 44; 6a; + 17^ = 92. 10. 5a; - 7^ == 0; Ix + by = 74. 153. It will be found that each of these pairs of lines intersect in the points given below; the points are num- bered to correspond with the pairs of equations: 1. a; = 1; y = 2. 6. a; = 2; y = 8 2. a; = 2; y-3. 7. a; = 4; y = 5 3. a; = 3; y = 5. 8. a; = 8; y = 6 4. a; = 5; y = l. 9. X --= 4; y = 4: 5. a; = 3; y = 6. 10. a; =. 7; y = 6 154. There is only one point, then, that will satisfy two equations of condition of the kind used so far in this chap- ter. That is, two conditions serve to determine the values of two unknown quantities. 155. It must not be rashly assumed that every equa- tion with two unknown letters will have for its complete * Paper ruled in squares, called cross-section paper, can be bought of dealers in drawing materials. 138 ALGEBRA. list of answers a straight line ; for example, Fig. 7 gives the list of answers for (I) x^ — dy -\- 10^ and for y I \ \ y ^ ^^ v,_ L ~y \ / \ ( \ / \ \ -\ \ / 1 — / \ v \ / — ^ A \ ^^ k / ^ ^ ^<: z^ Fig. 7. (II) 9:^2 4- 25?/2 = 225. There are evidently four dif- ferent answers that would make both these equations true. To construct these curves, successive values of y are sub- stituted in each equation, and the values of x calculated by solving the equations so obtained. Thus, in (I), if y = 2, x — ^OYX — —^\ if^ = — 2, :^=2ora; = — 2; ify=:l, X— ± i^Ts = 3.606 or— 3.606; and so on. But if y is taken equal to — 4, x — \^ — "l — 1.414i or — 1.414^, neither of which can be represented by a point on the dia- gram. In (II), if x is taken greater than 5, or y greater than 3, the results are also imaginary. Such curves as these form the subject-matter of an ex- tensive and very interesting branch of mathematics, called Analytic Geometry. Not much time can be profitably spent on them here. THE FIB8T METHOD OF ELIMINATION. 139 SIMULTANEOUS EQUATIONS. 156. Model B. — Six horses and 11 cows sell for $810; 13 horses and 5 cows sell at the same rate for $1190. Price for each animal ? Let X = the number of dollars for one horse; y = the number of dollars for one cow. (i) 6x + lly = 810 (II) Idx + 5y = 1190 Model C. — Six pounds of tea and 11 pounds of coffee sell for $9.23; 13 pounds of tea and 5 pounds of coffee sell at the same rate for $11.90. Price of each ? Let X = the number of cents for one pound tea; y = the number of cents for one pound coffee. (III) 6x + Uy = 923 (iv) Ux + 6y = 1190 In the four equations that come from these two prob- lems X and y do not have the same meaning; in (i) and (ii) X and y stand for a number of dollars, in (iii) and (iv) for a number of cents. When the unknown letters in two equations stand for entirely different quantities, as in (ii) and (iii), they must be treated as if written with different letters; the terms which seem to be similar are really n^ot similar. In (i) and (ii) we should find that X = $80 and y = $30 ; while in (iii) and (iv) we should find that x = 76 cents and y = 4:3 cents. 157. When in two equations the two letters stand for the same unknown quantities, the equations are called simultaneous. Terms having the same letters are similar terms. The members of either equation can be added to or subtracted from the corresponding members of the other equation,* and if the x terms (or the y terms) are the same, * See § 17. 140 ALGEBRA. a resulting equation may be obtained free from x (or y). Thus, for Model B : (i) 6x+ 11^ = 810 (ii) 13:?; + 5y = 1190 @ 30x + bby = 4050 (i) X 5 143^; + bby = 13090 (ii) X 11 The y terms are now alike. Subtracting (3) from 0, — ® lUx = 9040 © - (D ® .T=: 80 ® -^ 113 To find the value of y we may return to the original equations, (i) and (ii), make the x terms alike by multiply- ing (i) by 13 and (ii) by 6, and then subtract the resulting equations. Or, since we know now that x — 80, Qx = 480, we may write, instead of (i), ® 480 + lly = 810 ® substituted in (i) ® lly = 330 ® - 480 (9) y = 30 ® ^ 11 The answer to Model B, then, is $80 for each horse, and $30 for each cow. In the same way we find, for Model C, 75 cents a pound for tea, and 43 cents a pound for cotfee. 158. This is the method by which the points of crossing were found for the ten pairs of lines that the pupil was asked to construct. One equation, in any pair, imposes upon X and y the condition that their point must lie in a certain line; the other equation imposes also the condition that the point must lie in another line; and since the point must lie in both lines, it must be the point where the lines cross. 169. That is, taking the first pair of equations in the set of ten mentioned, the two conditions contained in the equa- tions x-{-2y = 5; 3x -{-Hy = 17 are the same as these two conditions: ^ = 1 ; y = 2, The algebraic work consists in reducing the given conditions to the form of particular values for x and y. THE FIRST METHOD OF ELIMINATION, 141 Elimination by Combination. 160. The process of getting rid of an unknown letter is called '^eliminating^' that letter. The method used above is called Elimination by Combination. Other methods will be described later. EXERCISE LXXXII. Solve the follo2vi7ig equations:^ 1. 2:c + 7^ = 59 ; 3a; + % =: 43. 2. bx + ^y = 34; ^x + ^ = 24. 3. 13^ + 17^=107; '^x-\-y=10, 4. 14a; + 9t/ = 100; 7a; + 2^ = 30. 6. a;+15^=37; 3a; + 7^ = 35. 6. 15a; + 19?/ = 79; 35a; + lly = 157. 7. 6a; + 4?/ = 90; 3a; + Iby ^ 123. 8. 39a: + 21 y = 213; 52a; + 29y = 249. 9. 72a; + Uy = 170; 63a; + 7y = 112. 10. 101a; + 26y = 886 ; 103a; + 39y = 941. tu. 2a; + 7^ = 49; 6x ~ 6y = 17. 12. 7a; - 4y :=:= 2; 25a: - 13^ = 24. 13. X -{-y = 90; x — y = 13. 14. 4:X+9y = 60; 7x - lly = 22. 15. bx — 3y = 17; V2y — lx= 23. 16. 171a; - 213^ = 642; 114a; - 326?/ = 244. 17. 9^ - 7a; = 43; 15a; -ly = 43. 18. 12a; + 7y = 176; 3y - 19a; = 3. 19. 43a; + 2y = 266; 12a; - 17^ = 4. 20. 5a; + 9y 188; 13a; - 2y = 57. 21. 4a; + 3^ = 22; 3a; + 5^ == 11. 22. bx - 2y = bl; llx - 3y = 162. * That is, reduce the given conditions to the simplest form. f It may happen that the x terms (or the y terms) are of opposite signs in the two equations ; in that case the equations must be added to get rid of that letter. 14:2 ALGEBRA. 31. 32. 33. 34. 36. 23. ^x - by =^ 51; bx ~\-ly = 39. 24. '^y — 6x = 143; 3:^; + 5y =: 89. 25. 4:?; + lly = 144; 82: + 17y = 198. 26. 2a; — 7^ = 8; 3y — 9:^: = 21. 27. 17^; + l^y = 59; 19^; - 3^ = 148. 28. 8a; + 3^ = 3; 4:^ + 3?/ = 1. 29. 59y - 17a; == 123; 2a; - 13y = - 17. 30. 3a; + 2y = 42; 13a; + 23y = 225. 80x + 50y = 22190; 70a; =11(60^). C2x 3y 3 "^ 4 = 700; J/ +100 _ bx ~ 6" ra; + 2/ = 7; I5 + X 3 ^9 + y 4* x-\-y = 15 16* x-y = 13 16' C2x y 3 "^2 = 60; = 8. 36. 37. 38. 40. ^ + 60 = ^; x-\-y =: 120. a; + 2 _ 1 X _1 "^3- + 3 a;+3 _ 2/+3~ a; — 3 39. a; — y = 5 ; 12 - a; _ 2 17^" 7* |a; + y == 7; |l0a; + y + 9 = 10?/- EXERCISE LXXXIII. 1. For 8 cows I get $20 more than I pay for 40 sheep; and for 16 sheep I pay $21 more than I get for 3 cows. Price of each ? 2. Find two numbers whose difference is -^^ of their sum, and 3 less than \ of the larger number. 3. In 10 hours A walks 1 mile less than B does in 8 THE FIRST METHOD OF ELIMINATION, 143 hours; and in 6 hours B walks 1 mile less than A does in 8 hours. How many miles does each walk per hour ? 4. If A^s money were increased by 36 cents he would have 3 times as much as B; if B's money were increased by 5 cents he would have half as much as A. How much has each ? 5. A pound of tea and 6 pounds of sugar cost 72 cents; if sugar were to rise 50 per cent and tea 10 per cent, the same quantity would cost 84 cents. Price of tea and of sugar ? 6. Find two numbers such that 3 times the greater exceeds twice the less by 29, and twice the greater exceeds 3 times the less by 1. 7. Three men and 16 boys earn $107.25 in h\ days, and 4 men with 10 boys earn $192.50 in 11 days. What are the daily wages of men and of boys ? 8. A farmer bought land for $7500, part at $80 per acre and part at $50 per acre. The cheaper he sold at a loss of 10 per cent, and the dearer at a gain of 10 per cent. On the whole he profited $50 by the transaction. How much land did he buy ? 9. If I divide the smaller of two numbers by the greater, the quotient is .36 and the remainder .64; if the greater is divided by the smaller, the quotient is 2 and the remainder 27. Find the numbers. 10. The sum of the ages of a father and son will be doubled in 25 years; the difference of their ages is \ of what the sum will be in 20 years. How old are they ? 11. Find two numbers such that 3 times the greater ex- ceeds \ the less by 439, and 3 times the less exceeds \ the greater by 49. 12. Eight years ago A was 5 times as old as B; 7 years hence he will be only twice as old. How old are they now ? 13. A merchant offers me 8 pounds of black tea and 24 144 ALOBBIRA. pounds of green tea. for 110, or 14 pounds of black tea and 17 pounds of green tea for the same sum. If it makes no difference to him which offer I accept, what are his prices per pound ? 14. A and B buy a horse for $550; A can pay for it if B will advance J the money he has in his pocket; and B can pay for it if A will advance J the money in his pocket. How much has each ? 15. A sum of money was divided equally among a cer- tain number of persons; if there had been 10 more persons, each would have received $2 less; if there had been 10 less, each would have received $3 more. How many persons, and the share of each ? 16. If a certain lot of land were 8 feet wider and 2 feet longer, it would contain 960 square feet more; if it were 2 feet narrower and 8 feet shorter, it would contain 760 square feet less. What is its area ? 17. A farmer bought eggs at 2 for 5 cents, and others at 3 for 8 cents; wishing to sell them to a relative at cost, he named a price of 31 cents per dozen, but found he would lose 5 cents; his relative then suggested 5 for 13 cents, and the farmer accepted that, but made a profit of 10 cents on the transaction. How many dozen were there of each kind? 18. It takes me 16 times as long to walk around the edge of a long rectangular field as to walk directly across it; and the next field, which is 30 feet wider and 150 feet shorter, has the same area. What are the dimensions of each field ? 19. A cruiser 386.8 feet long passes a battleship going in the same direction in 1.12 minutes; returning, she passes the same battleship in 12 seconds. The length of the battleship is 352.4 feet. Supposing each ship main- tains the same speed throughout, how many miles per hour do they go (5280 feet to the mile) ? TEE FIB8T METHOD OF ELlMmATlON. 146 21. 22. I 20. Two men who lived in towns 70 miles apart started at the same time and rode towards each other; when they met, they exchanged horses and continued as before. One horse could go 10 miles per hour, the other only 8. The slower horse arrived while the faster horse was still 5 miles from the journey's end. Find the time each traveller took on the road. Find the fractions descriled as follows : If 4 be added to the numerator, the value of the fraction will become 1; If 3 be added to the denominator, the value of the fraction will become i. If 1 be subtracted from the numerator, the value of the fraction will become -|; If 11 be added to the denominator, the value of the fraction will become ^. If 8 be added to the numerator, the value of the fraction will become f ; If 8 be added to the denominator, the value of the fraction will become i. If 17 be added to numerator and to denominator, the value of the fraction will become |; If 1 be subtracted from numerator and from de- nominator, the value of the fraction will become i. If 3 be taken from the numerator and 1 be added to the denominator, the value of the fraction will become J; If 3 be added to the numerator and 5 to the denomi- nator, the value of the fraction will become -J. Find two fractions with numerators 4 and 11 re- spectively, such that their sum is 2i|, and if their denomi- nators are interchanged their sum is 2||. 27. A fraction is such that when it is multiplied by f the sum of its terms is 107; if 8 is added to each of its terms the value of the fraction becomes |. 23. ^ 24. 25. 26. 146 ALGEBRA. 28. A proper fraction is such that if the numerator is halved and the denominator increased by 3 its value be- comes \] if the fraction is multiplied by | the difference of its terms will be 53. 29. The sum of the terms of a fraction, divided by their difference, gives 2 for a quotient and 5 for a remainder. If the fraction were divided by 2 the sum of its terms would be 53. 30. Two fractions have denominators 20 and 10 respec- tively. The fraction formed from these two by taking for its respective terms the sums of the corresponding terms of these two fractions is f ; and the fraction similarly formed by taking the differences is -f. Find the numbers described as follows : 31. A number divided by the sum of its digits gives 7f for a quotient; if 54 be subtracted from the number, the digits are reversed. 32. If I divide a number by the sum of its digits, the quotient is 6 and the remainder 7; but if I invert the order of the digits and then divide by the sum of the digits, the quotient is 4 and the remainder 6. 33. A number exceeds 5 times the sum of its digits by 8; if the order of the digits were reversed, the number would be 15 less than 7 times the sum of the digits. 34. A number is 4 times the sum of its digits; and if 27 be added to the number, the order of the digits is reversed. 35. A number is 2 more than 5 times the difference of its digits; if its digits are reversed, the resulting number is 8 times the sum of its digits. 36. Two numbers have the same digits in opposite order; the difference of the numbers is 3 times the sum of the digits; and the sum of the digits is 2 more than 2| times the difference of the digits. 37. If 45 be subtracted from a number, the digits are THE FIRST METHOD OF ELIMINATION, 147 reversed; but the same result might have been obtained by subtracting 7 from tlie number and then dividing by 3. 38. Two numbers which have the same digits in opposite order differ by 18, and the smaller number is 4 times the sum of the digits. 39. Subtracting 27 from a number reverses the digits; in the scale of 8, instead of the decimal scale, the number would be 18 less. 40. If a certain number is divided by the sum of its digits, the quotient is 4 and the remainder 6 ; if the digits are reversed, the resulting number divided by 5 gives 4 more than the sum of the digits. 41. A number consists of three digits, the middle digit being zero. If the digits are reversed, the number is increased by 396 ; if only the second and third digits change places, the number is increased by 3 more than 6 times the sum of its digits. 42. A number of three digits, the middle digit being 6, has its digits reversed if one adds to it 22 times the sum of its digits ; and the number so formed is 72 less than double the original number. 43. A number has four digits, of which the second is 6 and the fourth is 3 ; reversing the digits increases the num- ber by 909 ; but if only the first and third digits change places, the number is increased by 2970. 148 ALQBBBA. Solving for Reciprocals. 161. In equations like the following it is better to make the X terms (or the y terms) alike without first clearing of fractions : Model D. ^32 3 ^5,4 11 ^6 4 4 ® X 3 ®l = k (D-® 0) 2l = x ® X 31a; similarly for y. EXERCISE LXXXIV. 1. A and B together can do a piece of work in 8| days; if A worked 3 days and B 5 days, only half of the work would be done. How long for A alone ? How long for B? 2. Five boys and 10 men could do a certain job in 3 days; one man and one boy would take 24 days to do it. How long for one man alone to do the work ? One boy alone ? 3. Twenty-four pails of water and 20 cans of milk will just fill a certain tank; 6 pails and 14 cans will half fill it. How many pails to fill it ? How many cans ? 4. Two pipes fill a cistern in 20 minutes; if one of the pipes were twice as large * and the other half as large, the cistern would be filled in 15 minutes. How long for each pipe ? * The word ** large " here refers not to the width but to the capac- ity of the pipe. THE FIRST METHOD OF ELIMINATION. 149 5. Two pipes can fill a cistern in 5 minutes; if one of the pipes is closed half the time it will take 7|^ minutes. How long for each pipe alone ? 6. A and B could dig a well in 12 days; but at the end of the third day B quits, so that the job lasts A 6 days longer. How long for each alone ? 7. After working 2 days on a certain job with B, A says to him, '^I can finish this job alone in 10 days.^^ B re- plies, *^If we work together one more day, I can finish it alone in 5 days/^ If what they say is true, how long will it take each alone to finish the job ? 8. A can row 11 miles down-stream for every 7 against the stream; he rows down-stream for 3 hours, then rows back, and at the end of the 6 hours he is 5 miles from his starting-place. How fast does he row, and how fast does the stream flow ? 9. B rows 9 miles down-stream in 45 minutes; he rows back, near the bank, where the current is only half- strength, in one hour and a half. Speed of boat and of stream ? 10. C rows 6 hours down-stream and 15 hours up- stream, covering in all 72 miles. His speed up-stream is f of his speed in still water. Speed of stream ? 11. D rows down-stream for an hour and a half, but it takes him 3 hours to row back. He rows in the first 3 hours 12 miles. Speed of boat and of stream ? 12. A and B run a mile race; A gives B 12 seconds start and beats him by 44 yards; then A gives B 165 yards start, and is beaten by 10 seconds. Speed of each ? 13. A traveller started on a journey of 330 miles, having 4 hours and 12 minutes to spare on a connection he ex- pected to make at the end of that journey ; but an accident occurred when he was 2 hours out, which not only held up the train for 2 hours, but diminished its speed for the rest of the run. Then he missed his connection by just 3 min- 160 ALGEBRA. utes. If the accident had happened 6 miles further on, he would have been barely in time. How fast did the train go before and after the accident ? 14. A was sent to a town 147 miles away, and 7 hours later B was sent after him. After travelling 71 miles B was handed a letter to deliver to a person living 17 miles out of the town to which A had been sent. He overtook A just as he was entering the town, and handed him the letter; the letter was delivered just 9 hours and 40 minutes after B received it. Speed of A and of B ? Where Elimination Fails. 162. In studying equations of one unknown letter, we found that the given equation could be reduced to a very simple equation giving a particular value for that letter. That simple equation was the condition that the given equation should be true. We afterwards found that if there were two unknown letters in an equation of condition, another equation had to be given before we could get particular values of x and y. When only one equation is given, either of the two letters can have any numerical value; there must be given two equations of condition before their values are limited to a particular set. 163. In this connection the following problem is in- structive : Model E. — A certain number of two digits is equal to 4 times the sum of its digits ; if the digits are reversed, it is equal to 21 times the difference of its digits. Let X = the tens digit; y = the units digit. ([) 10x + y = 4.{x + y) ®lOy + x = 21(^ ^ y) (D lOo; -^^ y =^ 4:X -{- 4:y same as Q THE FIRST METHOD OF ELIMINATION, 151 ® %x — ^ — (D - 4^ ~ 4y ® \0y + X = 21x — 21y same as © ® 31^ - 20^ = ® - 21x + 21y If we compare equations ® and ® we shall see that both cannot at the same time be true ; ® says that x — \y, while ® says that x = ^^y. 164. Such equations are called inconsistent, and of course cannot be used to determine the values of the unknown quantities. They point to some mistake in the statement of the problem, or in the pupil's understanding of it. 165. We tacitly took for granted, in forming equation @, that the tens digit was larger than the units. Let us see if that is the source of the inconsistency. ® 10a; -}- y = 4:X -}- 4:y © 10^ + ^ = 21y - 21:r ® 22a; - 11^ = @ + 2I2: - 21y (T) 2x - y = ®-^ll ]N"ow the equations ® and @ are consistent, — too much so in fact ; for the second equation reduces to one precisely like the first. In this case also, then, we cannot determine, from the two equations given, the particular values of x and y. 166. In all equations of this kind, then, we must require that they be consistent, that is, that both cak be true at the same time; and that they be independent, that is, that both must not reduce to the same equation. Non-Algebraic Conditions. 167. It happens sometimes that a problem is given, like the above, which implies only one algebkaic condition, but still the answer is restricted to a few or perhaps even 162 ALGEBRA. to only one particular value, by a condition, implied in the statement of the problem, such that it cannot be stated algebraically. In Model E our digits must be from the nine Arabic numerals; we found by trial that our equations were in- consistent and the problem unsolvable unless the second digit were the larger; and our algebraic conditions both reduced to the fact that the second digit was twice the first. That leaves, for the only possible solutions, — X = Ij ^ =1 2; the number 12 ^ r= 2; y = 4; the number 24 cc = 3; y = 6; the number 36 a; = 4; y = 8; the number 48 Four particular answers to the problem. 168. Equations which do not give a solution from the algebraic conditions alone are called indeterminate equa- tions; and the problems which give rise to such equations, although they may contain enough non-algebraic con- ditions to determine the answer, are often called indeter- minate problems. More Than Two Unknown Letters. 169. The equation 3x -\- 4=7/ -{- Qz = 41 has three un- known letters in it. If we take the value of x to be 3, the equation reduces to 2y + 3,^ = 16; and if we take the equation x -{- 2y -{- dz = 19 and eliminate x thus: (£} ^x + iy + 6z = 41 (2) X + 2t/ + dz =19 ® 3x + 61/ + 9z = 57 (2) X ^ ® 2y + ^z = U ® - ® we come to the same result. In this case we have imposed the same condition in two different ways; and from our experience with equations THE FIRST METHOD OF ELIMINATION. 153 like 2y + 82; = 16 we know that the values of y and z can- not be determined unless we have one more condition. Counting the given equation of condition, the second that we imposed at random (since a; = 3 and x -\-'ly -\-Zz = 19 are not independent conditions we have only im- posed one), and the third that we know we must have, there are three conditions necessary to determine three unknown values. If the values turn out to be :?; = 3; y = 5; = 2, or whatever else they are, these answers would themselves be THKEE independent equations; and to get them we must have three independent equations to start with. 170. Whatever the number of unknown letters, the same number of independent equations of condition is required to determine their values. Elimination with Three Letters- 171. Model F. © 32; + % + 5:2 = 39 ^ (D 2: 4- 2y -f- 3;2 = 19 V Three given conditions, (3) 4a; + 5y + 7^ = 51 j ® 32; + 6^ + 9^ = 57 © X 3 (5) 2^ + 4^ = 18 ® - ® d) ^ + 2^ =9 ®-f-2 We have now eliminated x from the first two equations. We might have eliminated y or ^ from the same two equa- tions, or we might have taken ® and (3) or @ and (3), instead of the first pair. Now that we have started with ^•, however, we must con- tinue to eliminate this letter; (6) is an equation with y and z for letters, and we need another equation to determine their values; so we must eliminate x again; and we must take a different pair of equations, either Q and (3) or (2) 154 ALGEBRA, and (3), or else our second equation in y and z will not be independent of (6). ® 4a; + 8^ + 12;^ = 76 © X 4 (D 3^ + 5^ = 25 ® ~ ® (6) and (D are two independent equations with two unknown letters; and the rest of the solution is familiar ground. ® 3?/ + 6;^ = 27 © X 3 @ ^ = 2 ® - © @ y _[_ 4 = 9 (g) substituted in ® ® y = 5 (0) - 4 @ 2; + 10 + 6 = 19 (10) and (iD subst. in © (U) a; =:: 3 @ - 16 ^^5. a; = 3;y — 5;;2; = 2. 172. When we have obtained two equations from which one of the unknown quantities is eliminated, these are called the equations of the new set. In the preceding solution equations © and © are the equations of the new set. EXERCISE LXXXV. i^x + ly-2z-l'i, iy - x-\- z = ~ 6, 1. \sx -\- ^ -\- z = 11, 6. \z-y-x=—^^, ( ^- - 4^ + 10^ = 23. ix -\' y + z = 35. i^x~2y-\-bz = 16. i 3.T - 2y + 5^ = 26. 2. •< 32; - 2y + 4^ = - 10. 7. \ x - 2y + 3^ = 6. ( 4:z; + 2?/ - 5^ = - 2. ( 2a; + 3^/ - 4^; = 20. r 5;^; _j- 3y _ 6;^; =: 4. r 4:?; - 3?/ + 2;^ r= 40. 3. X%x ~ y ~\-'%z = 8. 8. \hx-\-^-lz = 47. ( ^ - 2^ + 20 = 2. ( 92; + 8y - 3;^ = 97. f 2a; - 2?/ + ;2 = 1. ( 3a; + 2y + ;^ = 23. 4. -j 2a; + 3y — = 5. 9. -j 5a; + 2^ + 4;^ = 46. ( a; 4- y + ;^ = 6. ( 10a; + 5^ + 4^ = 75. ( 2a; + 3?/ + 42; = 20. pa; - 6^ + 4^ = 15. 5. -j 3a; + 4y + 5;^ nr 26. 10. K 7a; + 4y - 3^; = 19. ( 3a; + 5y + 60 = 31. ( 2a; + 1/ + 6^ = 46. THE FIRST METHOD OF ELIMINATION. 155 Solve for reciprocals first : * 11. 12. 11 1 ^ + ,- + 7^ = 36. ^2 2 7 "60* ■< ^ + ^-1 = 28. X ' y z 13. < 3 4 2 ^ '^y~"z' 7 "30* i + 3^ + 2l = ^«- 5 3 2 3 ~10* X ^ y z X ' y ' z = 36. , X y ' z 14. - 1 , ^^i = 2S. ^-^ + ^ = 23. 11 1 ^x +3^+2^ ^ 20, 15. -< ' 3 4 iz; ~5^~^ 1 1 1 38 ~z~ 5* 2 83 ^~12' * 4 ^5a; 1 "2^ + 4 121 ;2 ~ 20' 172 1. Model G.— I 'rom the fou r equal expressic >ns in continued equation ^{x -\- z - b) = y - z ^'^x -11 = ^ — {x + 2z) we could form six different equations, but only three of them would be independent ; from these three the values of x, y, and z can be determined. (^ :^(x-\- z - b) = y - z @ y — z = 2x — 11 ® ^x - 11 = ^ - {x + 2z) ®x + z — b=^^ — 'Zz ® X - ^ -\-^z = b (§) 2x — y + z = 11 ® 32; + 2^ = 20 * See § 161. ® X2 ® - 21/ + 2^ + 5 ® -y + Z+11 @ + :«; - 2;2 + 11 156 ALGEBRA, Here are three equations, ®, (6), and (7), for ttiree unknown letters; but one of the equations has only two unknown letters in it. The letter missing from that is y, so we combine ® and (6) to get rid of y, and thus get two equations with two unknown letters [® and (9)]. ® 4^ - 2y + 22; = 22 (6) X 2 ® 3a; - ;2 = 17 ® - ® (7) ^x + 22; = 20 ] ^^ ^^ ^ New set ]■ ® 'dx - z = 11 Whence we obtain x = Q\ y = 2] z = \, Ans. EXERCISE LXXXVI, 1. a; + 20 = ^ + 10 =r 2;^ + 5 = 110 - (3^ + ;^). y + z z -\- X X 4- y 2. ^^ = ~- = -^; x + y + z = 27, 3. ^--, — = ^-^ — = bz — ^x\ y + z = 2x + 1. ^, X + 2y = 6z - lOx = y + z =z 600. 5. x-^- = Q; y _ ^ ^ 8; ^ - | .:. 10. 2,1 33 2 ^1,1 4 6. - + - = 5-; =2; - + - = -. a; ^ 2z z y x z 3 7. 2a;-3y=8; ^~3;^=-ll; x - 2y -\- ^z = Yt. 1,2^34 .34^ 8. -+- = 5; -— _zzr_6; =5. X y y ^ z X 2x - y _Zy -^ 2z _x ^ y - z _ 9. 10. 3 4 5 ~ X ~ y __y — ^ __ X + z _x - \- 10 3 4~ 5 ^i2~"' THE FIRST METHOD OF ELIMINATION, 157 EXERCISE LXXXVII. 1. A storekeeper exchanged with a neighbor 2 bushels of oats and rye, mixed half and half, for If bushels of corn ; and he told the neighbor that of two bills that he sent out for corn, oats, and rye, the first was for 5, 6, and 8 bushels respectively, and amounted to $10.30; the second was for 3, 5, and 8 bushels respectively, and amounted to $8.75. Price of each kind of grain ? 2. Suppose you and I together had 55 cents; you and some friend of ours had 62 cents; while the three of us had 97 cents. How much would each of us have ? 3. A pays to B and C as much as each of them has; then B pays to A and C half as much as each of them has after the first division; finally C pays to A and C one-third as much as each of them has after the second division. Counting their shares then, A has $12, B $84, and C $138. How much had each at first ? 4. A and B together can do a job in 12 days. After they get it three-quarters done, however, they call on to help them, and thus save one day. C can do as much work in 5 days as A can in 6. How long for each alone to do the entire job ? 5. A and B together can do a certain job in 5 days; if A works 2 days, B 3 days, and C 5 days, y\ of the job will be done; but if A works 6 days and B 3 days, the job will be half done. How long will it take each alone to do the entire job ? 6. A number consists of six digits, of which the first is 1 ; if the first digit is erased and written down after the other five, the result is 3 times the original number. Find the number. 7. A number consists of three digits; their sum is 13, and the middle digit is | of the other two; if 297 be added to the number, the digits are reversed. 158 ALGEBRA. 8. If 5 kegs, 3 cans, and 2 jars of oil be drawn from a full cask, the cask will remain \^ full ; if 4 kegs, 5 cans, and 8 jars be drawn from the full cask, the cask will remain ^3_. f^ii J and if 2 jars be filled from a full keg, the keg will then contain -gV of the cask full. What frac- tion of the cask full will the keg, the can, and the jar respectively contain ? 9. A farmer received $5745 for horses, cows, and sheep, at prices of $110, $62.50, and $7.50 apiece respectively; 8 more sheep would have brought as much as 6 cows; and the total number of animals was 5 times the number of cows. Find the number of each. 10. A, B, and C subscribed $100; if C had put in $2 more, and B ^^ more than he did, A could have com- pleted the sum by subscribing -^^ less; if had put in $18.50 and B ^ more than he did, A could have put in -J less. Sum each subscribed ? More than Three Letters. 174. Equations of four or more unknown letters are treated in a similar way. From the original set of equa- tions one of the original set of unknown letters is elimi- nated, and a new set of equations is formed, with one equation less ; one unknown letter less. This is repeated until we have one equation and one unknown letter ; and then the work is done. 175. Care must be taken that each equation of a new set is derived from a different pair of the old equations, so that they may all be independent. The following example shows an error that is often made. THE FIRST METHOD OF ELIMINATION, 159 Model H. 2a; + 3y -- 2^; + bu = 39 ® Sic -f 5i/ + 4^ - 3i^ := 28 (D 5^ - 2^ - 3;2 + 2t^ = 32 ® 4:^; + 4y — 5:2 — 4w = 1 Original set. ® 4a; + 6y — 4z + 10m = 78 ® X 2 ® 2«/ + + 14w = 77 ® -® ® Qx + 9y- 6«+15m = 117 ® X 3 ® 6a; + lOy +82 - 6m = 56 ® X 2 '® - y - 142 + 21m = 61 ® -® @ 13a; + 20y + 16« - 12m = 112 ® X 4 @ 12a; + l%y - 15? - 12m = 3 ® X 3 @ 8y + 3l2 = 109 @ -o ® 2y4- 2 + 14m = 77 1 ® — y — 14« + 21m = 61 >• New set. ® 8.y + 3l2 = 109 ) @ 6?/ + 3^ + 42w = 231 ® X 3 @ - 2^/ - 28^+ 42?/ = 122 (?) X 2 * (15) 87/ + 31^ = 109 ® - ® Since (j|) and @ are identical, the three equations (§), (e), and @ are kot iN"DEPE]srDE]S'T. Scanning the work we find that (6) was obtained by combining ® and Q, (9) from and ©, and (jD from ® and 0; equation (12) there- fore contains no facts that are not already in ® and ®. And if these three equations were independent, we should have determined our set of four unknown quantities from three equations, — which is impossible. 176. In eliminating with several unknown letters, then, one must be careful that for any number of equations of the KEW SET at least one more of the equations of the original set has been utilized. In the work given above, any one of the pairs ® 0, 0, ©, may be replaced by any pair containing 0. 160 ALGEBRA, Equation (Tz), obtained from (4) @, has the advantage that it can be used without change as one of the next new set; it would be better, then, to put some other pair instead of ® ® or @. A correct solution of the problem would be as follows : Model H. Q 2a; + 3y - 2^; + 5t^ = 39 (D 3a; + by + 4.z - 3^^ = 28 ®bx - 2y - ^z -\- 2u = 32 (a) ix -\- 4ty ~ bz — 4:11—1 (5) 4a; + 6y - 4^ + 10u= 78 *® '^y + ^ + l^^(^= 77 ® lOo; + Iby - 10^ + 2bu = 195 (8) lOo; — Ay — Qz + 4:U = 64 * (?) 19y - 4;^ + 21u = 131 @ \2x + 20y + \^z - \2u = 112 (Q) 12a; + 12y - 15z - 12^^ = 3 * @ 8^ + 31^ = 109 > Original set. ® 2y + Z+UU ® 19y — iz + 2lu @Sy + 31z = 109 = 77 ) = 131 - First neii @ Qy + ®SSy ~ © 32y — 3z -}- 8z + Uz = 4.2u : 42W : = 31 = 231 = 262 @ 8^ + @ 32?/ - 31^ = Uz = = 109 = 31 t Second new set. % 32«/ + 124« = 436 @ 135^ = 405 @ 2=3 © X 2 (D -® ® X 5 (D X 2 ®-(D @ X 4 ® X 3 @- ® ® X 3 ® X 3 (8)-(fD (iD X 4 (!D -dD (17) -^ 135 N.B. It is convenient, as fast as tlie equations of the new set are obtained, to mark them with an asterislf and thus save perplexity in looking for them. THE FIRST METHOD OF ELIMINATION. 161 ® % + 93 = 109 subst. ®m® ®1/ = ^ [(19) - 93] -^ 8 @i + d T^Uu = 77 snbst. ©and @ in (6) @) u= 5 [(2j) - 7] ■-- 14 @ 2^ + 6 - 6 + 25 = 39 subst. @, @) and @ in (?) @^ = '7 [@ _ 25] --- 2 ic = 7; ^ =r 2; ;^ = 3; w = 5. ^/^s. EXERCISE LXXXVIII. 1. dx + 2i/ + z + 6w=30 ^. 3x - 2z = 2 2x-3y-5z + w = -l5 by-7io= 2 x+ hy - 3z - 2w ==-6 2^ + 3^ = 45 bx — y ~-2z-\- 'dio- 9 4^2 + 3z^ = 56 2. 7iz; - 5^ + IO2; - 2z^= 29 5. 2x + 11;^ = 61 ^x-\-7y - bz + 10w= 57 3w - Ux = - 12 2:^;^+ 3«/ + 72; - 5w = 16 10^ + 32: = 59 10:?; _ 2^ + 3^ + 7«^ := 61 7u + 13;2 = 83 3. 2x+ 5y = 11 + 3z + 2w 6. ^x - 3z + 4:U ~ 23 __ bz-2y - 3w 6y - 3z + 2w = 12 ^ ~ 8 6«/ — 4a; - 3i^ = 12 22; — 3.^ — 5«(; 5y — 4«^ + 3?^ = 11 ^ 8 4;? - 9t(; = 30 ^ = 1 + i{3y + 2w - 8^) 7. 2x-3y + 4:z-15=z2y -3z + 4u-'7=^-^{2z-3u + 4:x) = M^ + y + ^ + ^) = ^ 8. 3^-52;~2^ + 6r=10^ + 2;2-3?^H-4=:5^ + 3;2 — 8«(; + 4 = 13x - 25«/ - 11?^ + 5 = 2;2 9. There is a number of four digits such that if the digits are reversed, the number is diminished by 1089; if the second and third digits are interchanged, the number is diminished by 90; if the first two digits are removed and placed after the last two, the number is increased by 693 ; and if the number were in the scale of 9 instead of in the 16^ ALOEBRA, decimal scale, its value would be decreased 912. Eind the number. 10. A traveller has a large number of foreign bills, all of the same value, also gold, silver, and copper coins, all coins of the same metal having the same value, and is vainly attempting to pay his railroad fare with them ; he oifers 5 gold coins, 6 silver coins, and 2 bills, but is $1.10 short; then he tries 6 bills, 2 gold coins, 7 silver coins, and 6 copper coins, and finds that makes $4.03 too much; 6 gold coins, 8 silver, 10 copper, and one bill is again $1.10 short; 5 bills, one gold coin, and 24 silver coins make $1 too much, and so do 5 gold coins, 20 silver, 100 copper, and one bill. What were his different kinds of money worth, and what was his railroad fare ? CHAPTER VII. THE SECOND METHOD OF ELIMINATION. Linear and Quadratic Pairs. 177. Model A. — A certain rectangular field, containing 480 square rods, requires 104 rods of fence to enclose it. What are its dimensions ? Let X = length in rods; then = breadth in rods. X Q) 2x + 2— =: 104 © a;2 + 480 = 52:?; Q X 2; -^ 2 (D :z:2 __ 52^ _|. 480 = (2) - 52a; and so on. Or otherwise. Let X = length in rods; then 52 — a; = breadth in rods. © a:(52 - x) = 480 © 52a; — x^ = 480 same as © ® a;2 - 52a; + 480 = © - 52a; + x^ and so on. Still another way would be to use two unknown letters, as follows: 163 164 ALGEBRA, Let X — length in rods; y = breadth in rods. ® xy = 480 © 2x + 2y = 104 (D x + y =52 ® Equations Q and (3) we cannot solve by the method of elimination heretofore tried. We can find the value of x from the first equation, — not the numerical value that sat- isfies both and (3), but a sort of formula for x, — and then substitute in the other equation; that is, write instead of X the formula for it. Then, if the formula for x had no letter x in it, we shall have an equation from which x has been eliminated. Thus, — ® X 480 © 480 + / = 52y (T) y^ ~ 52y -f 480 = and so on. Or otherwise, (J) X = 62 - y (5) y{62 - y) = 480 (6) 52y - y^ = 480 (T) = y^ - 62y + 480 ®-y (4) substituted in (3) ® X ^ ®- 52y ®-y (T) substituted in same as ® ® - 52^ + y^ Or we could in the same way have eliminated y] and the rest of the solution would have been identical with the so- lutions with one unknown letter which were worked out first. The use of the second unknown letter may often, as in this case, be a way of explaining how to get with one letter abbreviations for the two unknown numbers. TEE SECOND METHOD OF ELIMINATION, 165 Model B. — A cistern can be filled by two pipes, running together, in 2 hours 55 minutes; the larger pipe by itself will fill it 2 hours sooner than the smaller pipe by itself. How long for each pipe separately ? X = number of hours for smaller pipe; y = number of hours for larger pipe. (J) X - y = 2 ®^ + y = 35 (D d6y + 352; = 12xy © x ^5xy ® x = y + 2 (J)+y ® 36y + 36{y + 2) = 12y{y + 2) ® subst. in ® (6) 12y^ + 24^ = 70y + 70 same as ® 12/ - 46y - 70 = (6) - 70y - 70 (D 6/ - 23«/ - 35 = Q) -^ 2 (9) (6^ + 7) (2^ - 5) = ® factored .-. y =: 5, or - i 5 hours for larger, 7 for smaller. Ans. Model C. — A number of two digits is 9 less than the square of the sum of its digits; and if 45 be subtracted from the number, the digits are reversed. Let X = the tens digit; y = the units digit. ® 10x + y + 9 = x^ + 2xy + if © 10:?; + y - 45 = 10^ + iC @ 9^; - 9y = 45 @ — lOy — a; ®^-^ = 5 ®-^9 ®a; = ^ + 5 ® + y ® 10(^ + 5)+y + 9 ^^+10^+25+21/(^+5)+/ ©subst. in® ® \\y + 59 = 4/ + 2^y + 25 same as ® ® 4/ + 9;z/- 34 = ®_lly-59 ® (4y + 17)(^ - 2) = ® factored .-. i/ = 2; ^ = -1^ iz; = ^ + 5 = 7 Ans, 72 166 ALGEBRA, In the above examples, the equations of the first degree,* since they may be represented by a straight line in a dia- gram, are called linear equations. The other equation is of the second degree,* and is called a quadratic. The rule for this method of elimination (called Elimination by Substitution) is : 178. Find the formula for x (or for y) from the linear equation, and substitute in the quadratic. EXERCISE LXXXIX. 2. Q xy :=^ 1\ 3x — 6y — 2. 4. 5. "^ 2 -"^^ ^ x + 2 -^' 7x^ - Sxy = 159; 6x + 2y = 7. ^2 _ 2xy -^ y^ = 31; X + y = 13, 6. 3x - lOy = I; x^ -- xy = by^ + 79. 7. X + y = 7; x^ + 2y^ = 34. S. X = y - 2; 3x^ = 4y^ + 48. 9. ^ - 2x = Q; 4:X^ = 9{y^ + 12). 10. x^ + 24.xy = {3y + 4.y + S; x = 3y + 2. 11. - + | = 1; - + -=4. 2 3 X ^ y 12. 3x^ - 2xy = 15; 2x + 3y = 12. 14. 11«/ + 6x =z 63; x^ - xy + f = 13. 16. ^-^-'^~f = l;x + y + 3xy^93. * Tlie degree of an equation is the degree of its highest term ; and the degree of a term is the number of unknown letters that are fac- tors of it ; thus dx^y^ would be of the fifth degree, because it con- tains two factors x and three factors y. See § 75. THE SECOND METHOD OF ELIMINATION. 167 16. ^^^-^^ = ^; a;2+6a;«/ + 2«/' + 4x + 3«/ = 29. n. ]0- ?^±^ = ^; 3x^ - 6=. + 52/ = 20. 18. X - y - 1 y 20. ?^±|-^ ^ ^+_^; .V^ + 4r.^ :^ ^f + 20:. + 6. 21. Find two numbers whose sum is 4 times their differ- ence, and the difference of whose squares is 196. 22. A path around a rectangular garden is 7 feet wide and 1806 feet in area, which is 994 feet less than the area of the garden itself. Find the size of the garden. 23. A number is 7 times the sum of its two digits; and if the number is multiplied by its first digit, the product is 672. Find the number. 24. The difference of two numbers is -^^ of the greater, and the difference of their squares is 380. Find the numbers. 25. If a certain rectangular field were 75 rods longer and 20 rods wider, its length would be double its breadth, and its area would be double what it is now. Find its dimensions. 26. There are in a certain block of exactly similar houses 300 rooms; 5 more houses in the block than there are rooms in one of the houses. How many houses in the block ? 27. The front wheel of a bicycle makes 16 turns less than the hind wheel in going a mile; if the front wheel were 6 inches more in circumference, it would turn 60 times less than the hind wheel in going a mile. Find the cir- cumference of each wheel. 168 ALGEBRA. 28. When a certain train has travelled 5 hours it is still 60 miles short of its terminus; and on the whole trip 1 hour can be saved by running 5 miles an hour faster. Find speed of train and length of trip. 29. A and B start on a road race together; A is a sure winner, and looking back once on the road he sees B 60 rods behind. A crosses the line 4 minutes after that, and B comes in 9 minutes behind. When A looked back he had as far to ride as B had already ridden. Find their speeds. 30. The fore wheel of a carriage turns 132 times more than the hind wheel in going a mile; and 6 turns of the fore wheel cover 2 feet less than 5 turns of the hind wheel. Circumferences ? 31. Kowed 24 miles down-stream and back again; took 8 hours longer on the return trip, the current reducing the speed to -J of what it was going down. Kate of the current ? 32. Two trains start from opposite ends of a double- track railroad 300 miles long; after they pass, one train takes 9 hours, the other 4, to complete the journey. Speed of each train ? 33. If 1 is added to the denominator of a fraction, the value of the fraction becomes J; and if 2.1 be added to the fraction, the fraction is inverted. Find the fraction. 34. If 2 be added to the denominator of a certain frac- tion, its value becomes J; if 4 be added to each term of the fraction, its value is increased by ^. Find the fraction. 35. A rectangular field 238 rods in area loses 58 square rods by taking off a strip 1 rod wide all around the edge for roads. Find the dimensions of the field that remains. 36. Five leaks and a drain-pipe empty a cistern in 4 hours ; the average time for one of the leaks to empty it would be 24 hours more than half the time the drain-pipe THE SECOND METHOD OF ELIMINATION, 169 requires. How long will it take the cistern to be emptied if the leaks are stopped ? 37. When a certain kind of cloth is wetted it shrinks up ^ in length and -j^ in width. A piece of this cloth used for an awning was found to have shrunk 8| square yards in area, while the edging bought for it before it shrank, intended to go on one long side and both ends, was found to be 2 yards too long. What was the length and the width of the cloth before shrinking ? 38. The area of a rectangular plot of land is 16 feet less than a square plot of equal perimeter; and its breadth is one foot more than | of the length. Find the dimensions of the rectangular plot. 39. Invested in 5,^ stocks; then invested the same sum in 6^ stocks, paying $6 higher premium and getting 2 shares less; the income from the second investment is $18 more than that from the first. What was the sum invested ? 40. Two boys run in opposite directions around a rect- angular field the area of which is one acre; they start at one corner and meet 13 yards from the opposite corner; and one of the boys could go around the field 6 times while the other is going 5 times. Find the dimensions of the field. PAIRS OF QUADRATICS. 179. If we were to make diagrams for the equations in this chapter as we did for some of the equations in the preceding chapter, we should find that the points for an equation of the second degree formed a curved line, — a circle or an ellipse, or perhaps one of the two kinds of curves shown in Fig. 8, — while the points for an equation of the first degree formed a straight line. 170 ALGEBRA. 180. A straight line can cut one of these curves twice; that is, there are two points which represent an answer to both equations, and consequently we always expect two answers when we have one equation of the first degree and one of the second. When both equations are of the second degree the case is different: then both of the lines which represent the equations are curved lines, and they can cross " Parabola." " Hyperbola." Fig. 8. each other four times, as in Fig. 7, Chapter VI. In other words, two equations of the second degree have four answers, — four sets of particular values for x and y that satisfy both of the given equations. 181. This very fact, — that there are four answers, and consequently that the equation we should get after sub- stituting would have four factors, — is enough to show us that we cannot generally solve a pair of quadratic equa- tions, because we cannot generally factor an equation of the fourth degree. 182. Special cases arise, however, which can be solved by some special method, — sometimes requiring a great deal of ingenuity. Only Two Kinds of Terms. 183. In the following set of examples, since there are only two kinds of terms that are not similar, the method of combination may be used : THE SECOND METHOD OF ELIMINATION, 171 EXERCISE XC. ^' ( :?;2 _|_ 2?/2 ^57 ^- ( 3^2 _ ^2 ^ ^^ ( 3:^:2 _ 5^^^ 7 ^4 (3:^^+57/'^ =155 •\4.xy- 22;'^ =14 ' ( V - ^^^y = 15 # 6. j 15:?;2 _|_ 4^^ ^ 40 ( "ixy - 12:^2 = 2 Finding a Linear Equation. 184. The first effort of a pupil, in trying to solve a pair of quadratics, should be to work out from them, if possible, some equation of the first degree, which, with either of the original quadratics, will form a Lii^EAR akd quadratic PAIR. Model D. ® 2^2 _ 5^ .^ 3^ ^ 171 ® 3:z;2 ^_ 3^ _ 13^ :^ 239 (D 6:^2 _ 15^ + 9y = 513 ® X 3 ® 62^2 + 6:^; - 26^ =:^ 478 @ X 2 (5) 21^ - 35^ = 35 ® "- (D (6)3:c-5^=-5 ®^7 ® 2/ = -^^ [(D + 5 + 52/] - 5 ® 2:z;2 _ 5:c + 3 — ^ = 171 ® substituted in ® ® 10.^2 - 25:^; + 9ic + 15 = 855 ® X 5 @ 10r7;2 _ 16:?; - 840 = ® - 855 (0) 5:^2 - 8a; - 420 = ® -r- 2 @ (5a; + 42)(a; - 10) = . •. ic = 10 or — 4^3- * See footnote, p. 173. 172 ALGEBRA, In this example we can get only one equation of the first degree to substitute from ; and consequently here (that is, where the terms of the second degree are alike in any pair of quadratics *) we can get only two sets of answers. The values of y are obtained by substituting the values of X successively in the equation of the first degree by which they were obtained (in this case (f)). 5y=-5 126 , 5 %= 5 5^=35 126 + 25?/ = 25 25«/ = - 101 101 ^= 25 185. In order to show what values of x and y belong to the same answer, it is best to arrange the results in a table, thus : X 10 -¥ y 7 -Y/ (D X 13 © X 3 @ + ® ®-4-7 - 420 186. We might also have solved by eliminating y: (D 26:^2 - Qbx + 39^ = 2223 ® 9a;2 + 9x - 39^ = 717 ® ^bx^ - b^x = 2940 (6) bx^ - 8^ - 420 = whence as before. This method can be used when in both equations all the terms containing one of the unknown letters are similar. * In fact, whenever tlie expressions made up of terms of tlie second degree in each equation have a common factor containing x or y. THE SECOND METHOD OF ELIMINATION, 173 One Quadratic Factorable. 187. Sometimes one of the given equations can be factored : Model E. f — ^xy + 8i/ == 16a; — 7 — ^x^ (D x^ -f 2xy = 24 (D 4:X^-4:xy+y^-16x-irSy + '7=0 (T) + 7+4:X^-16x ® {2x — yf — 8(2a: — y) -\- 7=0 same as (3) (5) (2:r - ^ - 7)(2a; - y - 1) =0 ® factored (^^x-y-l=:0 (5) Ax. A ®^z=2a;-7 ® + y ®2x-y — 1 = (5) Ax. A ®y = 2^-l ® + ^ (10) ^2 _|_ 2a;(2:r - 7) = 24 ® subst. in @ ^ 5^2 __ 14^ _ 24 =0 ® ~ 24 @ {bx + io){x - ^) = @ factored .*. .7; = — I; :r = 4 (g) 2:2 + 2^(22; - 1) = 24 ® subst. in @ @ 52;2 - 2a; - 24 = @ - 24 @ {bx - 12) (a; + 2) = @ factored a; = -igS; :^; = — 2 Substituting these four values of x, EACH IN THE LINEAE EQUATION BY WHICH IT WAS OBTAINED, we get the fol- lowing four answers : X - 1 4 V- - 2 ■ y -¥ 1 -V- - 5 Of these four answers the first two depend on ®, and the others on ®. 174 ALGEBRA. One Quadratic Homogeneous. 188. The method just described can be applied when- ever one equation of the pair of quadratics is homogeneous, that is, where every term is of the same degree. A homo- geneous equation with two unknown letters is also called a binary equation. Model F. © x^ -\- xy + 2«/2 = 74 (D 24«/2 - "Zbxy = 2bx^ (3) 25x^ + 25xy ~ 24^^ _ q ®{bx + Sy){bx-^) = ®bx-\-Syz=zO 6x ®y = 8 ^_ 6x ®^-¥+^ = « @7ix^ = 74 X 64 Qx^ - 64: = .\ X = S; X = — S ® ^' + T + ¥ == ^4 @ 74.x^ = 74 X 9 @x^ - 9 = X = d; X = — 3 © + 25xy — 24^2 @ factored ©Ax. A [® - 5:^] H- 8 ©Ax. A [® + 3^] ^ 3 © substituted in © ® X 64 @ -^ 74 - 64 ©Ax. A ® substituted in © (JD X 9 @ -^ 74 - 9 Eemembering that the first two values of x were found on the supposition that y = — -^-, and the others on ®, we get for our table of answers THE SECOND METHOD OF ELIMINATION, 176 X 8 - 8 3 - 3 y - 5 5 5 - 5 189. The general principle of finding a linear equation from the pair of quadratics, either by eliminating one kind of terms or by factoring one of the equations, is illustrated by the following examples. EXERCISE XCI. Find four answers wherever possible : ^ ^Zx^ -2x-\- bxy = 96 ^ {x^ - ^0 = 2y2 + 2\y ( a;2 - IO2; + 25 = 4^2 ( ^xy — 5^ = 15 3. [ ^^' Vlx' ^x^ - Sxy + dy^ -8y + 29=0 2x^ -{- y^ = 4:xy — 7 ^ ( 3x^ - 7xy + y^ + 9y + Q6 = 'i 3x^ + 28y^ + 4:8 = 7xy ( x^ -{- xy -\- y^ = 19 r i ^^ "^ ^^^ ~ ^^ '\6x^- Uxy +6y^=: ( lUx^+ 291xy = 216y^ + 5xy — 6y^ = (xy + M = I Qx^ + Idxy + 6y' M xy = y x^ -f ^^ 2 _ 4 •■i 2^2 + 27/ = 21a;^ x^ + Zxy = 27 3xy 11. 18a;2 - 9xy = Ux^ + 28/; x^ + 2/ = 18. 12. IQx^ + Sxy = 16x^ — 15/; 2x^ -\- xy =: 15. 13. 24:r'^ + Idxy = 2/; x^ -{- xy — y^ = — 11. 14. 4:x^ + 12/ = 7x' + Ixy + 14/; x^ + 3/ = 28. 15. 69a;2 + 23^;^ - 69/ = 660;'-^ - 33/; ^x'^ - / = 23. 16. ^x^^^Oxy-^y'^ =lx^+21xy - 14/; 2/-a;2- 3x^=4. 17. Iliz;2-ll:r2/-176/=2a;2+22:y-16/; a;2+^«/=22+8/. 18. 5^^ + 5/ = 29x^ — 58^^:1/ > ^^ ~ ^^^ ^ ^' 176 ALOEBEA, 19. — 3a;2 4- Qxy — d2xy + l^y^] x^ — Ixy = 16. 20. l^x^ + 45^^ = — *7xy — 28y^; xy + 4^/^ = 30. 190. The two equations obtained by factoring a binary equation are either inconsistent or identical; and further, two binary equations can never be simultaneous, for the same reason. Find out ivhy. Finding a Binary Equation. 191. When in each equation the terms containing the unknown letters are all of the second degree, it is possible to get a homogeneous equation by eliminating the numeri- cal terms : Model G. ® ^2 _|_ ^2 ^ 5 ® Sx^ + 8^2 ^ 40 ® X 8 ® IQx^ -{• bxy -\- ^y^ = ^^ @ X 5 ® 2x^ + 6xy - dy^ = ®-® ® {2x-y){x + 3y) =0 ® factored (T) 2x — y =z ® Ax. A ® y = 2x Q) + y ® X + 3y = ® Ax. A ®y=-l [® - ^] -^ 3 ([}) x^ + 4.x^ = 5 ® substituted in ® (ID :?;2 _ 1 ^ @- 5 -1 @ {x+ l){x - 1) = @ factored ®x + l = 0.'.x=-l ©aj — 1 = 0.-. ^ = 1 1 (13) Ax. A @ x^ + ^- = 5 @ substituted in ® @ 10^'-^ = 45 @ X 9 @ a;2 ~ 1 = dD-io-i THE SECOND METHOD OF ELIMINATION, 177 Since the square root of f = 2.121+^ we can continue the work as follows : @ (^ + 2.121) (a; — 2.121) = @ factored @ a; +2.121=0; a; = ~ 2.121 ® Ax. A ©a; -2.121 = 0; a; = 2.121 (I9) Ax. A 192. The values of x given in @ and @ were obtained on the supposition that ® was true, that is, that y = 2x; and the other values of x were obtained on the supposition X that @ was true, that is, that y = — - ; consequently, we o must be careful to substitute (u) and (15) in (8) [not in one of the original equations], and @ and @ in @. In this way we get X 1 - 1 2.121 - 2.121 y 2 - 2 - .707 .707 Instead of figuring out the square root of f we might have represented it by the symbol V|; our table of answers would then have been ; : \ X 1 - 1 VI r% y 2 - 2 -iV% \v% 193. Whenever, as in this case, the root of a number cannot be exactly expressed, the symbol that stands for its exact value is called a surd, or irrational quantity. Later we shall learn methods of reducing and combining such quantities. 178 ALGEBRA. EXERCISE XCII. 1 i x^ + xy '\' y'^=2S { 2x' J^ xy + x = 21 I 2:z;y + y2 _ ^2 ^ 28 ' ( 9^'^ + 2x^ == ^xy a;2 + a;^ = 27 ^^' ( ^^ 4, 3^2 ^ 28 :?;2 + Uy + 2^/2 = 40 3 I ^^ + V = 32 2a;2 ~ 2a;y + y2 ^ 5 ^ ' ( 3:r?/ + 6/ - 4 = 32 C a;2 + 9a:y = 340 C 15^^ _ ^xy = 111 I 7a;^ - y^ = 171 ' \ 12(xy- /) r= 3x^ - :r.v ^^ + ^y + 2y^ =74 j 3a;y + 6y2 = 4:y + 4cO 2x^ -\- 2xy + y^ =^ 73 ' \ x^ + 6y^ = 6xy (bx^ + 3xy + 2y^ = lS8 ( x^ + 3xy := 4.0 ' \ x^ - xy + y^ = 19 {x^ + 3/ = 28 ^ (Sx^ — 3xy - y^ = 4:0 ( 6x^ + 31xy — Iby'^ = QQ ' \9x^ + xy + 2t/ =60 '^' \ 5x^ - y^ = 5 ( x^ - xy + y^=7 (2x^ - xy - 6y^ = 23 *• I 3x^ + 13xy +8^2^:,: 162 ^®' \xy = 21 i2x^-\- 3xy = 26 r 2a;2 + l\xy = 120 ( 3«/2 + 2a:^ = 39 ^ • I a;2 ~ 30^-^ = 105 10. i^ - '? = !J 20. i ''"^/''^ loV+'^) THE SECOND METHOD OF ELIMINATION. l79 The "Symmetrical" Method. 194. Where both equations are symmetrical, — that is, where x and y enter into each in such a way that if they were interchanged the resulting equation would be identi- cal with the given one, — a method of combination is em- ployed to reduce the two equations to the form x-\-y = , , . and a; — 2/ = . . ., as illustrated by the following examples: Model H. © a; + y = 15 © xy = 36 ® x^ + %xy ■\- y^ = 225 ®' ® ^xy = 144 ® X4 (D a;2 - Zxy + j,^ = 81 ®-® @{x~ yY - 81 = ® - 81 (J)x-y = 9; ®x- y = - 9 ® Ax. (D 2a; = 24 ® + ® @ 2y = 6 ® - ® © 2a; = 6 ® + ® © 2y = 24 ® - ® X 12 3 y 3 12 Ans, Model I. ®x^ + y^ = 74 xy = 35 (D x^ + 2xy + y^ = 144 ®x^ - 2xy + y^ = 4: ® {x + yY - 144 = © + 2 X® © - 2 X© (D - 144 180 ALGEBRA. ® ® ® (Q) ® ® (1^ (a; _ y)» _ 4 = ® - 4 x + y = 12; X — y = %; 2x = 14 2y = 10 2a; = 2y = 2x = 10 14 10 14 ®2y @2x = - U @2y= -10 (S)x + y = @x — y = ® + ® ®-® (D + ® ®-® ® + ® ®- ® ® + ® ® - ® - 12 - 2 (D Ax. A (6) Ax. A X 7 - 5 5 -7 y 5 - 7 ^ 7 - 5 Ans, Model J. (T) a;' + y'' = 185 ® a; + y=17 (D a;' + 2a;y + ^^ = 389 ©^ 2xy = 104 ®-® ®x^ — 2xy + y» = 81 ®-® ® (a; - 2/)^ - 81 = ® -81 ®a; -2/ = 9; (f) « - ^ = - 9 ® Ax. A ® 2a; = 36 ® + ® ® 2y = 8 ®-® (Q)3a; = 8 © + ® @ 22/ = 36 © - ® , X 13 4 y 4 13 Ans, THE SECOND METHOD OF ELIMINATION, 181 195. Sometimes the equations are symmetrical except that the interchange of variables produces a change of sign. The general method illustrated above is still appli- cable. The same interchange can of course be made in the set of simple equations that constitute the answers; and, consequently, in these as in all examples that can be solved by the symmetrical method, it is only necessary to find half the answers, — the others can be written down by making the interchange of x and y. Model K. © a; - 2/ = 3 (2) a;2 + y'^ = 65 (^ x^ - 2xy -\- y^ = ^ & © %xy = 56 ®-(D (D a;« + 2xy + y^ = 121 ® + ® (D (« + yf - 121 = ® -121 Ox + y = 11; ®x^y = - 11 ® Ax. A CD 2a; = 14 ® + ® @ 2«/ = 8 ®-® (0) 3ar ^ - 8 ®+® © 2y = - 14 ®- ( x» X 7 - 4 A y 4 ■ - 7 xi A Short Cut. 196. When a quadratic equation is prepared for factor- ing, if the first straight product is oc^^ the second straight product is the product or the two roots; and the sum of the cross products will be — a; multiplied by the sum of THE ROOTS.* For instance, the equation iz;^ — 7a; + 12 = has answers x = 'd and x — 4=. * See footnote, p. 124. 182 ALGEBRA, Now this property of the quadratic equation can be made use of in solving symmetrical equations; the values of cry and X -\- y^ or of xy and x — y, are early obtained; and remembering in what cases the two values of x are opposite in sign, we shall know when to make the second straight product negative: Model L. (\)x-\-y = n @ xy z=z ^% Auxiliary quadratic, ©Ax. A ® o;^ - 13a; + 36 = ® a; = 9; a; = 4 X 9 4 y 4 9 y=^l (3) Ax. A Model M. 0a; — ^ = 7 or a; ®xy = 1S @x^^lx — lS = or x'^ +7a;-18=0 Aux. quad. ®a;=9 I a:=-9 a; = - 2 i ^^* x = 2 Ans. 197. The auxiliary quadratic constructed in this way is precisely the same as would be obtained by the method of substitution. Irregular Devices. 198. Equations of higher degrees than quadratics can sometimes be solved. Often considerable ingenuity is required to get a solution; and it must be remembered X 9 - 2 - 9 2 y 2 - 9 9 9 THE SECOND METHOD OF ELIMINATION. 183 that it is only an accident when a pair, even of quadratics, is solvable at all by elementary algebra. One device of frequent use is to divide one equation by the other. Another device that is often of great service may be applied to example 14, below. Let z stand for x^ -\- y^ and u for ^xy. The two equations then become 3;^ — w = 27 ; 4;2 — 3w = 16. When u and z are found, equate their values to '%xy and x^ + y^ respectively and the rest of the way is familiar. EXERCISE XCIII. In the folloiving examples occasional hints are given to enable the student to choose a good method of elimination : IX — y = 8; a:y = 33. 2. x^ -\-y^ = 68; xy — 16. 3 ^3 _ ^3 ^ 37. ^z ^xy -\-y^ = 37. 4. x^ -{- y^ = 152 ; ay^ — xy -}- y^ — 19. 5. a;3 + ^3^ 407; x-\-y^\\. 6. x^ -y^ = 2197; :r - y = 13. 7. x^ + xY + y^ = 2128; x^ + xy + y^ = 76. ,,^- + l = 2i;x + y = 6. 34 15 . „ 9. ^ , 2 = — ; X + y = S. x^ + .V ^y 10. x^ -{- xy = i2; X -{- y = 5, 11. 2x -{- y = x^; 2y -\- x — y^, [Subtract and factor.] 12. x^ + xy = 210; if + xy = 231. [Add.] 13. xy{x -\- y) =^0] x^ -{- y^ = 35. [Divide and cancel.] 14. 3(2;^ + y^) - 2xy = 27 ; M.x^ + f) - ^xy = 16. [Solve for x^ -}" ^^ ^^^ ^^'^O 15. :c^ + 4:xy -f ^3 _ 33. ^ _|_ ^ _ 2. [Divide.] X -X- y 8 16. -y-^ = x + y + V ^^ ^ ■'^^* tSolve first for x + y,] 184 ALGEBRA. 17. a^ + y^ = —^ ] ^ + y 9. [Divide.] 18. x^ + y^ = -^; ^ + y = ^ [® -^ ® — ©^; solve for a;?/ and substitute.] 19. x^ -\- y'^ = xy -\-l^', X -{- y = xy — 6. [® + ^^y 5 solve for xy and a; + 3/'] 20. ^'^ + ?y2 = 74; 10(:z; + 2/) = ^^^^ - ^• [® + f®; solve for a; + ^.] Symmetry not Obvious. 199. The following examples may be solved by the sym- metrical method ; for though they are KOT symmetrical in X and y, they are symmetrical in functions* of x and y ; just as the equations of Model M are symmetrical in x and — y. For convenience we may adopt the same device as in example 14 above, letting z and w stand for the functions of x and y, 27^3 I ^.6 Model N. 9x^ •-- dxy' + y' = ^ ^ Let z = dx; w 10 28. y'^\ then the equations become z^ — z\o 4- w^ r= 28 ^3 _j_ ^3 ^ 280 whence we obtain ;^ = 6 or 4 ; «^ = 4 or 6. \iZx^^,x = 2; If ?/2 = 4, ^ =r ± 2;__ if 3a; = 4, :z; = |. if / = 6, ?/ = ± 1^6. X 2 2 i 1 y 2 - 2 VI - |/6 * A FUNCTION of X is an expression containing ic; e.g., 2iC — 1, 3x2 _^ 7^ etc. THE SECOND METHOD OF ELIMINATION, 185 EXERCISE XCIV. ■•{ ^x^ + 9«/2 == 181 xy = 15 x^ + "^xy^ + %4 ^ 273 :?;3 _ %y^ = 819 x^ + 9y^ = 873 '- + '- = 7 X y xy = 1 [Let z = 3x; w = 4«^.] 6. ■x-\-y'^ = 60 ^2 + y = 2522 7. •< 3:?; oy (6x+ 5y = 4: / i^;2 _j_ 4^2 ^ 4^ _|. 99 8. •) 2a:^ — x = 4:S ( [Let;2 = 2?/ - 1.] r a;^ + 9^2 - i2y r= 9 9. -I ^xy — 2x--Q { [Let 2; = 3^ - 2.] 10. < Let 2; = - ; w = 3y^. 4a;H 9^/2+122^-6^=175 2x — 9y — 6xy = 85 [Let z = 2x + ^] w = 3y - 1] Elimination by Comparison. 200. Still a third method of elimination, called ^^com- parison," is recognized by students of algebra, and is some- times of great convenience, though it is never necessary. It consists in finding an expression (a ^' formula ") for y (or for x) from each equation and setting them equal. Model 0. Q 7y + 29 = 10:?; © ^2 _^ 26 = 3xy + 2y ® y = ® y = 10^ - 29 7 x^ + 26 3x+2 [0 ^ 29] - 7 (2)~-{3x + 2) 186 ALGEBRA, ® 10:^- 7 29 _ ^2 + 26 3:?; + 2 ® 23:^2 - Qlx - 240 :i^ ® X^b', , X = 48 23 y = 3; i y = 509 161 (D and ® compared ® X 7(3:?; -I- 2) . . . ® Ax. A This method is evidently only a slight variation of the method of substitution. Sometimes it is useful to ' * com- pare " expressions more complex than the simple x or y. Model P. © a;2 + ^xy = 144 © ^xy + 36^2 ^ 432 ®xy + Gtf = n (D-6 144 ® x + Qy=t^ X Q -^a; 72 ® ^2/ a; y (T) and ® com- (T) 2y:=x ® X a;y -T- 73 ® 4y^ + 12/ = 144 ® subst. in .-. y =:r ± 3; :?; = ± 6 201. This example could also have been done^, like num- ber 12 above, by adding the two equations : ® x"^ + 12x1/ + 36y2 = 576 ® + ® ® X + 6?/ = ± 24 (D Ax. A. (5) ± Mx = 144 ® subst. in ® (D ± 144^ = 432 ® subst. in © .\x= ± Q; y = ±3 THE SECOND METHOD OF ELIMINATION. 187 EXERCISE XCV. Sot 1. 2. Ive hy comparison : y — x^ = 2; y ~{- 1 = 4:X, 6y = I3x; y — 1 = xK 3. x' -Z^y-] y + Z = x. 4. y = T ; y + 3 = a;. 6. 8.T + - = y ~ 11 ; 7y = 21 + 65a;. 21 6. 23 ~ a; = ly\ y x-\-b 7. :c^ -|- 4 = %xy] o;^ 4- 9 = 3:?:^. 8. a;2 + 36 = Qxy\ b{x - 1) == 4i/. 9. x^ — bx =^ {x -^ 3)^; a;— 3 = y . 10. y{x — 1) = x\ 2xy = 5a; — 2. Quadratics with Three Letters. 202. A system of three equations with three unknown letters can always be solved if two of them are of the first degree and the other is quadratic. Two of the letters must severally have their values expressed in terms of the third, and these expressions must be substituted in the quadratic equation. To this end it is in general necessary first to eliminate one of the three letters from the two equations of the first degree, and thus find one expression for substitu- tion; and then to eliminate another letter from the same two equations, and get the other expression. But it sometimes happens, as in example 1 below, that one or both of the expressions for substitution can be found directly from the given equations. 188 ALGEBRA. Model Q. 2^ + 3?/ - ^ = 15 ® ^' + 2«/ + 3^ = 12 (D 2i?^''^ — ^;2 = 48 ® 62; + 9^ - 3;2 = 45 ® 72; + lly = 57 (6) 11^ = 57-7:?; ^ 57 - 7:r ®^ = -lI- (D 4:?; + 6?/ — 2;^ = 30 (?) 3.T + 6^ + 9;^ :=: 36 @ 11;^ — i?; = 6 6 + a; ® X 3 © + ® — Ix ® -T- 11 ® X 3 ® X 3 ® -® (Q) ^== @ 2x^ — 11 57 - 7:?; 6 + x ^--11 = 48 subst. (T) and © in (3) 11 11 @ 2^2x^+7x^- 15x- 342 =: 5808 @ x 121 @ 249:i;2_ I5a^_ 6150 = @ - 5808 @ 83:?;2 _ 5^. _ 205O = (U) ^ 3 ® {83x + 4:10){x - d) = @ factored @ X = 6; x = —-^^- from (16) by Ax. X 5 - w y 1 3 w « 1 /^ u4w5. EXERCISE XCVI. 3:?; + 2^ = 12; 3y + 2;^ = 11; x' + 2f + 3z^ = 25. 5a; - 7?/ = 31; 3:^+7;^ = 27; 6xy - 7yz + 3:r;2 = 51. x^ + 3xy — Zxz = 39; 2:?; — 3^ = — 9; 3x + 4z — 13. 3^ + 2^ + ^ = 23; :?; - 3y + 5^ = 6; 72;2 _ 5^;?/ = 100. 5. x-y-\-z = 4:', 3x- 5y + 2z = 20; 3x^ - 5yz = 77. THE SECOND METHOD OF ELIMINATION. 189 6. ^ + y + ^-9; 22;+3y = 13; ^ + ?- = 5. 7. x^ -\-y^-\-xz= 24; ^ + 3y + 5^ = 22; 2y - Zz = 0. 8. X -{- ij + z = 10', x^ -\- y^ + z^ = 38; 3:?; + 2^ - 5^ = 4. 9. ^ 4- 3?/ = 11; 2x — z — 0; 3^/''^ — y^ = 15. 10. a: + ^ + 5=2/ + ;^ + 2= -^±^ = 15. CHAPTER YIII. MULTIPLICATION OF FRACTIONS; HIGHEST COMMON FACTOR. Fractions and Ratios. 203. Whenever the operation of division in Algebra can- not be carried out, or whenever for any other reason it is desirable to indicate division, the dividend may be written as a numerator, and the divisor as a denominator, and the whole result is called a fraction, — or, sometimes, a ratio. 204. It must be remembered that there are two kinds of division. For example : The expression -\^- may mean, — *^If I divide 30 separate objects into 5 equal parts, how many separate objects will be in each part ? ^' Or, otherwise, the expression -\Q- may mean, — '*How many equal portions, consisting each of five sepa- rate objects, may be gotten out of a collection of 30 sepa- rate objects ? ^' 205. In the first case the divisor is an abstract number and the quotient is of the same denomination as the divi- dend. • The indication of such division is in Algebra prop- erly called a fraction. 206. In the second case the divisor is of the same denom- ination as the dividend, and the quotient is an abstract 190 MULTIPLICATION OF FRACTIONS. 191 number. The indication of such division is in Algebra properly called a ratio. 207. In multiplication the multiplier is always an ab- stract number, and the product is always of the same denomination as the multiplicand. In reversing the pro- cess of multiplying (that is to say, in dividing), when we get the multiplicand for our quotient we have a fraction of the product, and when we get the multiplier, we have the ratio of the product to the multiplicand. 208. Again, whenever we ask how long, or how large, or how heavy anything is, the numerical answer to our ques- tion is always a ratio, — the ratio of the quantity we are asking about to the yard or the mile, to the acre or the square inch, to the ounce, the pound, or the ton. 209. The student has probably seen by this time that Algebra does not concern itself with denominations, so that we cannot tell in any example of division whether that particular division results in a fraction or in a ratio — unless we happen to know in advance just what our letters repre- sent. But it is not necessary to know. All the laws of transformation are the same for fractions as for ratios. Where it is very necessary to distinguish a ratio it may be written a : b instead of - ; but generally in Algebra such a distinction is not worth while. 210. The expression a : Z> (or -J is read ^^the ratio of a to Z>^'; ^ is the antecedent, the dividend, or the numerator, and b is the consequent, the divisor, or the denominator, of the ratio. The form - is generally read ' ^ a over b '^ or, more fully, ^' a divided by Z>." The term FRACTioisr, in" THIS CHAPTER Al^D THE SUCCEEDING OKE, WILL BE UNDERSTOOD TO INCLUDE RATIOS. 192 ALGEBRA. 211. The ratio of any number to one is the number itself ; consequently any number can be regarded as having a denominator one. EXERCISE XCVII. 1. The ratio of two numbers is |, and the smaller is 3; what is the larger ? 2. The ratio of two numbers is 9, and the smaller is 3; what is the larger ? 3. The sum of two numbers is 20, and their ratio is 3; what are they ? 4. The sum of two numbers is 100, and their ratio is the same as the ratio of one of them to 100. Find the value of the numbers within one- tenth. 5. One number is 12 greater than the square of another, and their ratio is 7. Find the numbers. THE LAWS OF FRACTIONS. 212. Theorems which are of use in this chapter and the succeeding one are proved by the use of the three funda- mental laws of Algebra.* 213. Just as the sign — means the reverse of addition, addition undone, so the sign -^ means the reverse of multiplication, multiplication undone. The sign — may be read ^^ the negative of,^' and the sign ~ may be read '' the reciprocal of." 214. The law of association for + and — signs implies such identities as these: a+l-c-\-d-e^{a+V)-{c-d^e)=a+{l-c)-\-{d-'e) * See § 86. MXILTIPLICATION OF FHAGTIONS. 193 In the same way the law of association for -~ and X im- plies : axi-^cxd-^e={axi)-i-{c-^dxe) = ax{l>-^c)x{d-^e) In fact, THE ASSOCIATIYE AND DISTRIBUTIVE LAWS ARE, SO FAR AS FORM GOES, THE SAME FOR X AND -f- AS THEY ARE FOR + ^^^ — • 215. Theorem I. Any fraction may have its numerator and denominator multiplied or divided hy the same numher without altering its value. Let — be any fraction and let m be any number. Then am ^ ,, . .: — = am ~- am = a X m -^ (o X m) om ^ ^ = a X m-^Z>-^m[the distributive law] = a -^ b X m -^ m [the commutative law] . 7. -^ = a -^ — Y hn h 216. Theorem II. Fractions are multiplied by multi- plying their numerators together for a new numerator and their denominators for a new denominator. a c Let 7- and -^ be any two fractions. a c -rX-r = «-^^ X G -^ d^ a X c -^ h -^ d d = aXc-^{l)Xd)~ U a c _ ac I ^ d^M 194 ALGEBRA, 217. Theorem III. The rule for dividing by a fraction is : Invert the divisor and then multiply. Let X be any number and — any fraction. h =. X X - a a i .*. X -^ T- - X X - a Another way of stating the rule is this: To divide by any number, multiply by its reciprocal. Reduction of Multiplications. 218. The first step in every example like the following is to factor numerator and denominator in every fraction : Model A. f x'' - '\x^- 5a; - 66 x'' - 2x f x'' - 121 x"^ -\- bx - 66 \ a.2 _ 5a. -j_ 6 '^ ic^ + 6a; - 72 ■ V a;^ - 9' '^ (c' + 9a; + isj ^ {X -^l)(aH-6) ^^ x{x - 2) ^ (a; + ll)(a;-ll ) (a;+ ll)(a;-6) . X / ^ a!2 + 6a; - 72 • x(x — 2) (a; + 12)0 X — 6)- x(x — ^ ^ X - {x -2)(a; - 3)^(a;-f 12)(a;-6) * (a; -f 3)(a; - 3) ^ (a; + 3)(a; + 6) ^ (a;-ll)(a;+6) a;(a; - 2) (a; + 3)(a;-3) ^ (a; + ll)(a;-6) - (x -2)(a; - 3)^(a;+12)(a;-6)^ (a;4- ll)(a; - 11) '^ (a; + 3Xa;-f6) According to Theorem II all the factors in the numera- tors, as the several fractions now stand, are factors of the numerator of the product which is the answer; and the same for the denominators. According to Theorem I we may divide the numerator and denominator of the product by the same number without altering its value; that is, we may strike out MULTIPLICATION OF FRACTIONS, 195 any factor that is the same in both. The expression in Model A then becomes (a;-ll)(a;+6) x{x - 2) {x + Z) { x - 3 ) (x-\-n ){x-Q) _ x {x-2){x-d) ' {x+12){x-Q) ' (a;+ll)(lc-ll) ' (ic+3)(a;+6) "" ic+12 Ans. EXERCISE XCVIII. Perform the operations indicated : 3x 7y . 4:X^ a — b n^ — P ^' 4^ ^12x'^ 9f' ^' a^ + ad ^ a^ - ab' x^ + X — 2 ^ x^ -{- 2x ""^^ - 7x ^ x^ - Idx -f 43* , a;^ + 3a: + 2 x^ - 7^: + 12 ' x^ — 6x -{- Q x^ -\- X ^ 4:X + 3 x^ — 9x + 20 x'^ — "Tx 3. x^ — bx -{- 4: x^ ~ 10:r + 21 x^ — 5x ' 1 1 a' - Via + 30 * a- 15* tt'^ — 2ay -\- y^ — W- ^ a — y -\-'b a^ + 2ay -\- y^ — V ' a + y — b' a' + a^ - SaW + 19^^?^ - lb¥ d^ + 'dab - bl)^ X X 9. :j — . [Multiply numerator and denominator by x^, ] X 3ax a^ — x^ f a^ -^ ax a —■ x iby c^ — x^ ' \bc -^ bx c — x. ^^- l,a2 + a^ + ^^V * a + Z^^ a^ + F 1 + 2/ ^ + ^^ ^^- 4a2_i -2^+1^ a* 196 ALOEBBA, 14. 15. 16. X^ — X — W X^ — X X 17. 19. 20. ■ 2 x + 1 x'^ + 'Zx-^'^ x^-\- bx X ^01? X X'' — 25 4:X^ -j- :r — 14 6x1/ — 14?/ x^ -]- X — 2 x^ ^x- 20 "^ \^a;^ - 2^ - 15 2:2 _ isx + 80 2;2 ~ 15^; + 56 X — 2 3x^ — X— 14: X^ — 4: ^^ 4:X 'x^ + 3^: + 2 X 7^ 2x^ + 4.x X -{- d x^ + 6x+4:' x^ •)■ X 1 18. 6x^ - ^bx-bO ^. - 2a^ x^-6x-7 * .T + 5* X — a 3ax + 2d^ ax — a'^ 2:^ — 64 9x^ — 4:d^ 2x -\- a ,2\ 'x x^ — dxy + 9y^\ {x + 3yy x^j^l2x-Q4: x^ _[_ 24tx + 128 x^ 64 {x - sy _X^-\- 4:X + 16 x^- U J * One Term Not Easily Factorable. 219. Sometimes in reducing fractions to lowest terms only one of the terms of the fraction can be factored by inspection. Then it is best to try the separate factors as divisors of the unfactored term. x^ + 2^: — 3 Model B. — Keduce to lowest terms 3^3 _|. 7^ __ 10' x^+2x - d _{x + 3){x - 1) 3^3 _^ 7^ _ 10 - 3x^ + 7a: - 10 • Evidently x + 3 cannot divide the denominator exactly because 3 is not an exact divisor of the straight product 10. Dividing by ic — 1 we get for the answer X + 3 3x^ + 3x + 10' X - 1 3^2 _^ ^^ _^ 10 3^3 _!_ 7^ _ 3x^ - 3x^ 10 3x^ + Ix 3x'^ - 3x lOx - 10 MULTIPLICATION OF FBAGTIONS, 197 EXERCISE XCIX. Reduce to loivest terms : Sd^ - 6ab x^ - 6x + 4. 2d^b - 4:ab^' ^' 4:X^ + 9x — 13* a{2b'^ - 2ba) 2x^ - 1032; + 35 b{ib^a - 9a*)* *'' x^ - Ux + 42 * 3^4 _|_ Q^^y _^ Q^2y 2 3a;3 - 78^; + 1 5 ^4 _j_ ^3^^ _ 2x'^i/'^ ' ®* 2a:^ — 13a; + 15* x^ + xy — 2y^ x^ — 3x — 10 x^ - f ' Ix^ - 232; + 10* 27a + «' 62;2 + c?; - 12 ! in 1 . 18a - 6a^ + 2a^ 12^:3 __ 23^; + 6 Test for Simple Factors. 220. It is sometimes easy to decide whether a very simple factor is contained without remainder in an algebraic ex- pression without performing the operation of division. The test depends upon the axiom that if one factor of an ex- pression is equal to zero, then the whole expression must be. Model C. — To decide whether a; — 1 is a factor of ^3 _|_ 5^2 _|_ ^ _ 10 it is only necessary to suppose that a; = 1 ; in that case a; — 1 = and x^ -\- bx^ -\- x — 10 = 7 — 10=— 3; hence c^:; — 1 is not a factor, because when X — 1 = the expression x^ + bx^ -\- x — \0 \^ not. In the same way x -\- A is not a factor, because, if a:= —1, x^-\-bx^-\-x~lQ becomes —1 + 5—1 — 10=: —7; and a; — 2 is not a factor, because, if a; = 2, x^-\-bx^-^x—10 becomes 8 + 20 + 2 — 10 = 20 ; but a; + 2 is a factor, because, if x = — 2, x^ -\- bx'^ -\- x -- 10 becomes _ 8 + 20 - 2 - 10 = 0. 198 ALGEBRA, EXERCISE C. In the same way decide luhether either or both of the binomials given loith each of the folloioing expressions are factors of that expression or not : Possible Factors. 1. ^3 _|.. 2x^ - ^x - 6 :^ + 1 X -2 2. x^ - 2x^ - 5x + Q x + 3 x + 2 3. x' - Ux - 12 x + 1 X - 1 4. x^ - 7x + 6 X — 2 X - I 5. ^3 _ 3^2 ^ 4 X — 2 x + 2 6. x^ - 19:?; + 30 X — 2 X - 3 7. ^3 _ 5^2 _ 2x -f 24 X — 2 X + 3 8. x^ - Ux^ -f Ux ~ 24 x + 1 X - 2 9. x'+ x^ - 10303:?;^ + ^ - 10302 x -j- 1 X - 1 10. 3^-4 _ 7^2 _ 20 x + 2 X -2 11. X'' — a" x + « X — a 12. x"" -\- a" x + a X — a 221. Theoretically the same test would serve to decide whether 3a; — 2 was a factor of 3x* + 13:?:'^ — 11:?; — 18; but practically it is easier, in this case, to perform the operation of long division than to substitute in the given expression x = ^. The pupil must use his judgment as to which test he shall apply in each example. EXERCISE CI. Fi7id the IT, C. F, in each of the following examples : 1. 2x^-3x^-5x-i-6; x^-x-2, 3. :?;2-3a; + 2; 7:?;^4-13a;2-20. 2. 5:c3-21a;2+16; a;''^-:2:-12. 4. 3x^-{-2x' -^7x-M]x:^-4, 6. 3x' + 7x-Q2; x^-\-29x-62. HIGHEST COMMON FACTOR. 199 Neither Term Easily Factorable. 222.* In the following example it is extremely difficult to factor either the numerator or the denominator by in- spection. ,, . , Tx T. 1 X . XX 182;3 + IZx - 14 Model D. — Reduce to lowest terms ^, » , ^ >. —, 24:X^ + 2:z;^ ~ 8 We must try, then, in some other way to find a common divisor for these two expressions. The way we shall adopt is to change these expressions into others which have the same common divisors and at the same time are easier to factor. Let us take as an abbreviation for the numerator the letter P, and for the denominator Q. Q has a factor 2, which is isroT a factor of P and is there- fore not a common factor. Divide Q by 2, and we get 12a;3 + x^ - 4:; multiply this by 3, and we get 36a;3 + 3a;2 - 12. Since we have not introduced or taken out common factors, any common divisor of F and Q is also a common divisor of F and 36x^ + dx^ - 12. In the same way 2 X P = 36a;3 -f 26a; — 28. Any number that will divide each of two quantities will also divide their sum or their difference. [This is merely one form of the distkibutive law. We may illustrate it, but we cannot prove it, because it is one of those funda- mental principles that have to be taken for a basis of all proofs.] * See also §§ 229, 230. 200 ALOEBRA. So any number that is a divisor of P and Q is also a divisor of 362;^ + 32;^ — 12 and of 362;^ + 262; - 28; and is therefore a divisor of their differekce, that is, of 32;2 _ 262; + 16. The advantage of this result lies in the fact that we have ELiMi^STATED, SO to speak, the x^ terms from our two ex- pressions, and so obtained a quadratic expression, which contains among its factors every common factor of P and Q. Factoring, 32;2 - 262; + 16 :E (32; - 2)(2; - 8). We see that 2; — 8 cannot be a common factor, because 8 will not divide 14, one of the straight products. Divid- ing numerator and de- nominator by 32; — 2, we get for the answer to the example : 62;'-^ + 42; + y 82;'-^ + 62; + 4 32; -2 32;- 2 62;2 + 42; + 7 82;^ + 62; + 4 182;3 + 132; - 182;^ - 122;2 14 242;^ + 22;2 - 8 242;3 - 162;'^ 122;2 + 132; - 122;2 - 82; 14 182;^ - 8 182;'^ - 122; 2I2; - 14 2I2; - 14 122; - 8 122;- 8 EXERCISE CII. Reduce to lowest terms : 32;^ - 112;^ + 18 ^* 22;3 - 232; + 15 ' 32;^ + 232;^ - 50 ^' 92;3 - 192; + 10 • 122;^ - 3I2; + 6 ^- 82;^ - 282; + 15* 212;^ - 9 2 :y^ + 8 492;^ - I82; - 4 ' 252;^ - 602;^ + 49 252;^ - 642;2 + 21* HIGHEST COMMON FACTOR. 201 More Difficult Examples. 223. When we have for terms of our fraction expres- sions of degree higher than 3, this process will generally have to be carried a little further. Model E.-Reduce ""^T^l^^l - Here we destroy the highest terms as before. For convenience we may number the expressions and make a memorandum of how we get them, just as we do with equa- tions : ®2x^ - Qx-20 © X 2 ® 5a;3 - 6z - 28 (D - © This (4) is an expression of the third degree, which we cannot easily factor. We know, however, that it contains all the factors common to and @. We can obtain an- other such expression by destroying (^^eliminating") the lowest terms. d) 4^;* — 12^ - 40 ® X 4 (D lOx"^ - 26x^ + 40 © X 5 © 14:x^ - 25:^3 - 12^ ® + ® This contains a factor x, and as no factor x occurs in either of the expressions we are investigating, we shall not affect our result if we cast it out. ® Ux^ - 25r?;2 -12 (2)^x. By the same reasoning as before, ® contains all the fac- tors that are common to ® and ©. Therefore the H.C.F. of ® and ®, if we could find it, would contain the H.C.F. 202 ALGEBRA. of and ®.* We now apply ourselves to a new problem, —to find the H. C. F. of © and ®. ® bx^ - 6:r - 28 ® I4.x^ - 25:^2 __ 12 ® 70a;3 - 842; - 392 ® X 14 @ lOx^ ~ 1252;2 - 60 ® X 5 (0) 1252;2 - 84^: - 332 ® - (TO) This expression (Tj), though only a quadratic, is still diffi- cult to factor. We are aided by the fact that and @, the given numbers, have for straight products x"^ and 2ic^ — 10 and 8, and consequently if there is a common factor which is also a factor of (?]), that factor may be a; + ^ or ic — 2. By trial we find the factors of @ to be (125^: + 166) (a; — 2). If there is a common factor, then, of and ©, it must be a; — 2. Dividing : ^j-2 a; -2 x^ -\- "^x^ -\- ^x -\- b 2x^—x^ — 2x~4i x*-3x — 10 2x^ - 5^;^ + 8 ^'r^^' ^^' 7 ^"^^ , ^3_|_2^2_|_4^_^5 22;3 _ 32; - 10 - 2;^ + 8 ^?^s. ,-f — rS"^- 22;^ - 42;'^ - x^ + 2x^ ix^x-Ax-^ 42;2 - 32; - 10 - 22;^ + 8 42;2 -82; - 22;2 + 42; 52; ~ 10 - 42; + 8 EXERCISE cm. Reduce to loiuest terms : x^ -f 32;^ - 272; + 14 1 + 2 2; ^ + x? + 2x ^ 1- ^4_i5^_^x4 • ^' 1 + 3^2_|.2^3^3^4- 42;* + ll2;2 + 25 62;^ 4- 2;^ — 52; — 2 ^' 42;* _ 92:2 _j_ 30^ _ 25* ^' 62;^ + 6x^ - 32; - 2' 9;^4 _ 6^3 _ 343 5. 92;* - 492;2 + 62; + 14 * So far as we have yet proved, it miglit contain other factors, further. HIGHEST COMMON FACTOR. 203 Theory of H. C. F. by Elimination. 224. If P and Q be the numerator and denominator, respectively, of a fraction which we are to reduce to lowest terms, the process used above may be carried out in sym- bols, and some general conclusions can be drawn from the formulae thus obtained. To destroy (or eliminate) the terms of highest degree we multiply P and Q^ respectively, by some suitable multipliers, say a and d, and subtract. The result, which we may call X, is an expression of lower degree than P and Q, (\)X=aP-bQ. Then destroying the terms of lowest degree, and letting h and Jc stand for our two multipliers : Now it is evidelit from and (2) that any common factor of P and ^ is a factor of X, and also a factor of F, that is, a common factor of X and Y, Let us take these two equations and find an expression for P and Q, to see if some other conclusion can be drawn. (D xY = hP - JcQ ® hX = ahP - bJiQ (b) axY = aliP — akQ (6) hX - axY = akQ - hhQ ^hX-axY ^ ® ah-hh -^ Here we have found an expression for Q by eliminating P, and from that value of Q it is evident that any common factor of X and Y is also a factor of Q. In the same way it would be found that ^^ - ak- bh ' ® X a; (J)Xh ® X a ®-® (D H- (a/fc - - bh) 204 ALGEBRA. and it is established that any common factor of X and Y is also a factor of P. Therefore, since X and Y have for common factors all the common factors of P and Q, and no others, the H. C. F. of X and Y is the same as that of P and Q\ and the elimination may be continued, taking each pair of results as a new problem, until the H. C. F. appears in two identical expressions. Model F. © Let P = 4^;* + 26a;3 + IW - 2x - U @ Q = 3x^ + 20^-3 + 32x^ -Sx- 32 (D 12.^4 + 78i?;^ + 123^;'^ - 6a; - 72 ® X 3 ® 12a;4 + 80a;3 + 128:?;'^ - 32x - 128 © X 4 ® 2:^3 + 5^"^ - 26a; - 56 X=:4(>-3P ® 16a;4 + 104a;3 + 164^;^ - 8a; - 96 © X 4 © 9x^ + 60x^ + 96x^ _ 24a; - 96 @ X 3 ® 7x^ + Ux^ + 68a;2 + 16a; ® - © ® 7a;3 + 44a;^ + 68a; + 16 ®~ x Y= ^ ^ a; Now since the H. 0. F. of Xand F is the same as the H. C. F. of P and Q, we may start on a new problem, namely, to find the H. C. F. of 20;^ + bx^ - 26a; - 56 and 7a;^ + 44a;^ + ^^^ + 1^^ which we may call P and Q respectively. @ Let P=2x^ + 5x^ - 26a; - 56 (0 Q = 7x^ + Ux^ + 68a; + 16 (jD i4a;3 _(. 35^2 _ 182^ _ 392 @ x 7 (@) 14a;3 + 88a:2 _f_ 2.36a; + 32 (ij) X 2 @ 53a;2 _|_ 313^ ^ 424 x= 2§ - 7P @ 4a;3 + 10a;^- 52a;- 112 @X2 @ i^x^ + 308a;'^ + 476a; +112 ([j) X 7 @ 53a;3 + 318a;'^ + 424a; ® + ® ® 53a;2 + 318a; + 424 -p^ 2P+7Q HIGHEST COMMON FACTOR, 205 Now since X and Y are the same expression, their H. C. F. is that expression, and that is the H. 0. F. of P and Q, — and that is the H. G. F. sought. EXERCISE CIV. Reduce ivherever possible : 2x^ - lla;''^ - 9 x^ — lOo;^ + 9 ^' 4:X^ + lla;^ + 81* ^' x^ + Wx^ + 20:?;2 _ iq^ _ 21* 9^4 __ 2x^ A-x^ ^^x — 6 x^+ 5x^ — x^ — 5x 8 4^4 _ 2x^ -f 3^ — 9 ' ^' x^ + ^x^ — x—'d ;3 _|_ 2 ^3 + a; + 3 ^5 _ 16^ _ 32 ■ x X *• 4^3 - 18:?;2 _|_ 19^ _ 3' 9. 32^3 _ 24^2 _ g^ + 3* 32;^ -3x3 -"Ix^ -x-\ x^ -x^-2x + 'il ^' Qx^-3x^-x'-x-l' ^^' U^-x-l * Another Application of H. C. F. 225. We have seen that an algebraic equation with one unknown letter can have no answers but those which make the several factors equal to zero. If there are two such equations, then, which can be satisfied by the same value of X, it follows that the two equations must have a common factor. Model G. — There is a number which will satisfy each of the two equations 16x* - 8x + 3 = 64^3 = 8 To find what that answer is, we first find the H. C. F. of the two expressions 16x^ — 8a; + 3 and 64^3 — 8. © 16x^ - 8x + 3 = © 64^3 -8 = (D 8x3 - 1 (D -^ 8 206 ALGEBRA, ® 24:X^ - 3 (D X 3 ® 16x^ + 242)3 _ g^ ® + © '^®2x^ + 3x^ - 1 ® -^8x (T) 16x^ - 2x ® X2x ® 6^ - 3 ® - ® ° * ® 2:?; - 1 ® -^ 3 If there is an H. C. F. it must be 2r?; — 1 ; we find on trial that this is contained in 2x^ -\- Sx^ — 1 exactly x^ -\- 2x -\- 1 times; hence 2:^ — 1 is a factor of both equations ® and ©. That is, each equation will be satisfied if 2:x; — 1 = 0; hence x = ^is a, root of each equation, EXERCISE CV. In each of the folloiviiig examples there is one numher that will satisfy both equations. Find that numher, 1. x^ - nx" + \^x - 12 = 0; 3^:2 - 14a; + 16 = 0. 2. x^ -M^ - 9;r + 27 = ; ^2 _ 2:^; _ 3 := 0. 3. 2x?-\-^x^ -^x- 9 = 0; 32;3 + "^x^ - \\x - 15 = 0. 4. x^ + 19a;2-22^ - 40 = 0; x^ ^ IW - 6^ - 40 = 0. 5. 4^3 - 2\x + 10 = 0; 2x^ + 9a;2 _ 25 = 0. 6. x^ - \W + 18:r - 8 = 0; ^x^ - 22:r + 18 = 0. 7. a;* - 2x^ - x^ - ix -^\2 = ^', 2x^ -Zx^ -x-2^ 0. 8. x^'-lx^^XW^Zx-X^^^) 42;3-21.t2+26:?;+3 = 0. 9. x^ ~ ^x^ - ^x^ + 36:?; - 27 = 0; x^-Zx^-'^x^^' = 0. 10. a;^+13a;3+332;2+31a;+10=0; ^x^-\-'^^x^-\-^^x^Z\=^^. HIGHE8T COMMON FACTOR. 207 H. C. F. of Three Expressions. 226. Model H.— Find the H. C. F. of Sx^ + dx^ - 64:X^ + 9; 9x^ - 64:X^ + dx + 8; and x^ J^ x^ - 8x^ - 5x + 3. QSx^ + dx^- Ux^ + 9 (2)x^ + x^ - 8x^ - bx + 3 (D 8x^ + 8x^ - 64:X^ -40^ + 24 ® X 8 ® 5x^ - 40a; + 15 ® - (D *®x^-8x + 3 ® -^ 5 ® 3x^ + dx^ - 24:X^ - 15:^; 4- 9 ® X 3 ® 5x^ - 4:0x^ + 15:?; ® - ® *®x^ - 8x + 3 (T) -^ 6x It is now clear that x^ — Sx -\- S is the H. 0. F. of two of the given expressions; it contains, then, all factors com- mon to those two. The H. C. F. of this result and the remaining one of the three given expressions will contain all the factors common to the three; so we seek the H. C. F. of x^ - 8x + 3 and 9a;* - 64:X^ + dx + 8. Qx^ - 8x + 3 ® 9x^ - 64:X^ + 3x + 8 @ 9x^ - 72.^'^ + 27x ® X 9a; ® 8x^ - 24a; + 8 ® - ® *®a;2-3a; + l ®-^8 The quadratic expression a;^ — 3a; + 1 is prime; hence if there is an H. C. F., this must be it. Dividing, we find that Hence a;^ — 3a; + 1 is the H. 0. F. required. 208 ALGEBRA. EXERCISE CVI. Find the H, C. F, of: 1. a^-x-%) ^x^ -x^ — lSx + ^] 2^* + 3ic3 + .r2-9:z;-9. 2. 2x^ - ^x^ + 13a; - 6; 4.x^ - 20^^'-^ + 31:^; - 15; 2a;3 - ll2;2 4- 19.-?; ~ 10. 3. 9^* - lOx^ + 1; 21^;* + 10:^:3 _ <^q^2 _ iq^^ - 1; 21^4 _ 4^3 _ 222;2 J[. 4:X + \. 4. 6a; - ll^''^ + 6a;3 - 1; 19:?;2 + l _ 4,x{^x^ + 2); 3:^;(3 + ^x^) - (1 + 262^^). 6. a;3 - 7a; -- 6; 2x^ - Ix^ + 9; "^x^ - Sic'-^ - 9a; + 18. H. C. F. by Long Division. 227. In Arithmetic the H. C. F. of any two numbers is always found by dividing the larger by the smaller, the divisor by the remainder, and so on, continuing the opera- tion until there is no remainder. Model L— To find the H. 0. F. of 62651 and 18377 18377)62651(3 = q^ 55131 r, = 7520)18377(2 = q, 15040 r^ = 3337)7520(2 = q^ 6674 7-3 = 846)3337(3 = q^ 2538 r^ =: 799)846(1 = q^ 799 r, = 47)799(17 = q, 47 329 329 HI0HE8T COMMON FACTOR, 209 A much more compact arrangement of the work is here shown. [The successive quotients are lettered q^^q^, ^39 • • • and the successive remainders r^, r^, r^, • • •] 1 = ?. ?. = 2 3 17 b = 18377 15040 62651 = a 55131 ?. = r. = 3337 2538 7520 = r, 6674 ?. = r. = 799 47 846 = r, 799 r. = 329 329 47 = r. 228. The theory of this method is substantially the same as that of the method by elimination of highest and lowest terms. If a and h are the dividend and divisor, and q and r the quotient and remainder, respectively, in any example in division, then (\) a = Iq -\- r. From this we see that a common factor of h and r will be a distributive factor of ^^ + r and therefore a factor of a ; that is, A common factor of b and r is also a common factor of a and h. Again, by subtracting bq from each member of 0, (f) a — bq = r. Hence a common factor of a and ^ is a distributive factor oi a — bq, and is therefore a factor of r; that is, A common factor of a and b is also a common factor of b and r. These two conclusions establish the fact that b and r have the same common factors as a and b^ and no others; in other words, instead of finding the H. C. F. of a and b, we may find the H. C. F. of b and r. 210 ALGEBRA. In Model I, therefore, the H. C. F. of a and h is the same as the H. C. F. of r, and h\ the H. C. F. of r^ and b is the same as the H. C. F. of r^ and r^'y the H. C. F. of r^ and r^ is the same as the H. C. F. of r^ and r,; the H. C. F. of r, and r, is the same as the H. 0. F. of r^ and Tg ; the H. C. F. of r^ and r, is the same as the H. C. F. of r^ and r^. Since r^ is contained in r^ without remainder, the H. C. F. of r^ and r^ is rj and r^ is therefore the H. C. F. of a and l. 229. This method, if applied to algebraic expressions, very often requires important modifications. Model J.— Find the H. C. F. of Ix^ - Sa;^ - Ho; - 66 and 3a;* + ^x^ - llx - 22 : q^= 2 a = 7a!'^ - 3a;» - Wx - 66 6a;* + 4.'c'-* - 34a; - 44 ri=a;* - 3a;» - 4a;2+ 28a; - 22 9a;34-14a;«-86a;+ 44 = ra 3a;H 2x^- llx - 22 = 5 3a;4 - 9a;=* - 12a;*-f 69a; - 66 3 = g. Here we find that while r^ and r^ have the same H. C. F. as a and b, their quotient will contain fractions whichever is used as divisor. Going back to first principles, we may say that since r^ does not contain 9 as a distributive factor, we may introduce it as a distributive factor in r^ ; then 9^^ and r, will have the same H. 0. F. as r^ and r^. Some such device as this may have to be used several times in one example, or in rare cases may not be needed at all. HIGHEST COMMON FACTOR. 211 gi=2 ^3= X ?4^41 6ar* + 4ic2 - 34aj - 44 7\ = ^- 3aj3 - Ax^ -f- 23aj - 22 9 9aj4-27a53-36a;«+207aj-198 Qaj-* + 14a;3 - 86a;« + 44a; r3=-41a;3+50x2+163aj - 198 -9 369ic=^-450aj«-1467a; +1782 369a!3+574a;«-3526a; +1804 3ic44_2a?«-17a;-22 = 6 3aj4_9a;3_i2a;5+69a;-66 9a;«+14a;«-8Caj+44 = r2 3 = ga r4=-1024a;2 + 2059a; - 22 1024a;5 - 2059a; + 22 Here again it would be profitable to depart from routine : rather than multiply r, by 1024, it would be easier to factor 1024^2 _ 2059:c +- 22. We find, in fact, that it is the product of a; — 2 and 1024;:^ — 11; and only the first of these will divide r^\ hence a; — 2 is the H. C. F. 230. This method is somewhat more perplexing at times than the method of elimination of highest and low- est terms; and the following cautions must be carefully observed : I. Remove distributive factors before beginning operations; if they are common to a and h, restore them at the end of your work. II. When you find that any remainder will not serve as divisor, carefully look for distributive factors, and if there are any, remove them; THEif introduce whatever factors are necessary into the last divisor. III. Continue to use the same divisor as long as possible. IV. When you get a quadratic remainder, factor it. All the examples given under the other method will serve for practice in this. CHAPTER IX. LEAST COMMON MULTIPLE ; SUMMATION OF FRACTIONS. 231. If one expression is a factor of another, the second may be called a multiple of the first. Thus x^ — 16 is a multiple of x^ -{- 4:, X -}- 2, X — 2, x^ — 4, x^ — 2x^ -\- 4:X — S, and x^ + 2x^ -\- 4.x -\- S, Again, a;* — 16 is a common multiple of x^— 2x^ -{- 4:X -- 8 and x^ + 2x^ -}- 4:X -\- 8, just as 192 is a common multiple of 48 and 32. 232. The lowest common multiple of two or more expressions is the expression of lowest degree in which those expressions may be found as factors. Thus x^ — 4: and x^ -\- 4: have ^* — 16 for their lowest common multiple; just as 9 and 16 have 144 for their least common multiple in arithmetic ; x^ -\- 4: and x -{- 2 have for their lowest common multiple x^ -\- 2x'^ + 4:^ + ^5 although x^ — 16 is a common multiple; and x^ — 6x-\- 6 and x^ — 4: have x^ — 3x^ — 4:X + 12 for their lowest common multiple. EXERCISE CVII. In the following table the expression in the third column is the L. C, M. of the corresponding expressions in the first two ; find out hy what each of those first ttvo expressions must be multiplied to give the L. 0, M, 212 LOWEST COMMON MULTIPLE. 213 1. x^ — dx -^2 x'^'-bX + 4: 2:^ - 72:2 + 14^ _ 3 2. x^-^x-lb 2:2-25 2:3 + 32:2 - 252: - 75 3. x^+x-^O x^ + lOx + 25 ^3 _!_ 6^2 _ 15^ __ iQo 4. x^ - 27 2;3+22;2+6a;-9 2:* - 2:3 - 27ic + 27 5. a;2 + 8a;+12 a;2 + 52: - 6 2:3 ^ 7^2 _|. 4^ _ 12 6. 2a;2 + cc - 6 22:2 _ 5^ _^ 3 22:3 -x^-lx + Q 7. x'^y — xy^ 2:*^2 _ x^y^ x^y^ — 2:2y^ 8. 3a;2-a;-10 22;2- 8 62:3 ^ 222:2 _ 282: -^ 80 9. ^x^ - 9:^: - 2 252:2 _ 1 252:3 _ 50^2 _ ^ _^ 2 10. lQx^-^x-% 502:3 __ 18^ 1002:*-1202:3-362:2+542: Very Simple Denominators. 233. In adding fractions it is most convenient to use the Lowest Common Multiple of the denominators. Model A. — Eeduce 2: — 5 _ 22;- 13 , 2^ _ 5 + 22; "T~ 6 ^ 8 12 The product of all the denominators would be a common multiple, but it would be a very large number; the least common multiple is 24. We can reduce each of these fractions then to 24ths. The first fraction has to have its denominator multiplied by 8 to give 24 ; then the numerator must be multiplied also by 8, in order not to change the value of the fraction ; similarly for the other fractions ; so we get 82: — 40 82: — 52 , 62; 10 + 42; 24 24 24 24 which reduces to 22:- 10 24 214 ALGEBRA. and this again reduces to V X — b '•f "^88— %.:;'■ ■ • , EXERCISE CVIII. hi the same way reduce : ^' 3 "'"■"5 30~' ^' 6 To ' l5~' a? - ^ _ 2:r^ - a; - 5 _ 5 - 2a: - ^x^ ^•4 6 12 3a - ^ _ 2^ - 7a _ 3a_ *• 8 ~~l0 20* x^ - 27 ^x^ - 5 + ^' - x^ - ^ ^'5 3 "*" 21 • {x - 2)(a; - 3) (2 - a;)(3 - x) _ ( :^; - 2)(3 - x) ^' 7 "*" 21 " 14 • (3a;-5)(2a;+3) (72;-l)(2:^;-l) _ (a;- 3) (3 -a;) ''• 2 "*" 6 18 ,(?£±^)(!£:^)_(.._3,)(?£^L^. ^«- I 18 )[-^) + l^ - ^Jl 36 )• • x-\-l %x—l a;^— 4 Sa:^ a; 1— a; ^^- "T" 6 "^ '^:^' ^^- ^^4 ^ i" ~ "To" • 3a:^+7 2a;^-3 __ ^+1 x^-y'^ x^ y__ ^^' a;+5 "^ 3a:+15 9 ' ^^' x^y +4a;+4^ + *2' _ a;+l _ 5 2a;+3 a;^ + 7 x -3a;-6 ^^' a;-3 9 2a:-6' ^^' 2a;-l "" 4^^ "*" 6a:-3 * 9. 17. LOWEST COMMON MULTIPLE. 215 3^:2 , a;2-3 , x-^ 2 5.4 20. x~-7 ' 49-7a; ' 14 ' 3:?;-2 32^-2 ' 12-18^ 5x Qx 7x dx^ X — y ~^ Ix — 7y ^x — Sy x^ — y'^* Factoring for the L. C. M. 234. The L.C.M. of several algebraic expressions may often be found by factoring. Model B.— Find the L. C. M. of ,^2 _ 9. ^3 _ 27; ^2 _ 4^ _|_ 3. ^4 _j_ 9^2 _|_ 81- ^3 ^ 27. Here the L.C.M. must contain all the factors of any one of the expressions; we find the factors to be: ©2;2- 9 = {x-{-Z){x- 3) (D ^3 - 27 ^{x- Z)(f + 32; + 9) (D ^2 _ 4^ _j_ 3 ^ (^. _ 3)(^ __ X) ^^4 _|_ 9.^2 _j_ 81 = (:^2 ^ 3^ _|_ 9)(^2 _ 3^ _|. 9) ® ^^ + 27 = (:?; -f i){x? - 32; + 9) The L.C.M. contains the factors of 0; then to make it contain the factors of (2), 2:^ + 3^; + 9 must be included, and so on; including in the L.C.M. successively the factors of each number that are not already in it. Starting with the factors of 0, the L.C.M. includes as factors (x^Z)(x.^Z) Then to contain © the L. CM. must also include (x?-{-Zx-^^) Then to contain (3) the L.C.M. must also include (2;— 1) Then to contain ® the L.C.M. must also include (x^—'6x-\-^^ Then to contain (5) the L.C.M. needs no more factors. 216 ALOEBBA, EXERCISE CIX. Find the L. CM, of: 1. x^ - 121; x' - 2x - 99; 6a^ - 55x\ 2. 27x^-8; 36a;2-16; 6:^2 + 11:^-10; 9a;2 + 9a; - 10. 3. x^ — S; x^ + 4:X^ + 16 ; x'^ + 4.x^ - Sx - 8. 4. a^ +P; a- b; a' + a^' + h\ 5. x^ — x{a -\-b) -\- ab] x^ — a^\ {x + a){x — by\ 6. x^ + Sx^; x^ - 9x; x^ + 27 ; {x + 3)^ - 9x, 7. x^ — y^] x^ — y^; ^ + ^• 8. x^^ — y^; x^ — y^\ x^ — y^] ^ — y\ x'^ -\- xy -\- y^, 9. 6x^ - I82; + 12; 4^3 _ 16^ _|. 12 ; 2x^ - 10.^ + 12. 10. x^'-{y + zY; y^-{x + zY; z^-{x + yf. More Complicated Denominators- 235. Model C. — Simplify the expression: 3a; - 1 3 ?>x-^l 3 ^3 _ 27 x^-]- 9x' + 81 ' x^+ 27 x;^ - 9' The L. C. M. is {x - 3)(cc2 + 3a; + 9){x + 3)(a;2 - 3a; + 9) = a;« - 729. The first denominator must be multiplied by (a; + 3){a;2- 3a; + 9) to give the L. CM.; then the numerator must be multi- plied by the same expression : 3a; - 1 _ 3a;^ - x^ ^ 8lx + 27 a;3 - 27 "" x^ - 729 * Similarly for the other fractions. The entire expression thus becomes 3a^-a;>-81a; + 27 _ 3a;^-27 Zx* -\- x^ -S\x - 21 _ 3a;< - 27a;'^ - 243 a;«-729 a;-^-729"^ a:« - 729 a;« - 729 _ 3aj* + 24a;2 - 162a; + 270 x"" — 729 LOWEST COMMON MULTIPLE. 217 EXERCISE ex. Simplify the expressions : 3a; 7 2_ ^- 4ic2-9 + 2a; + 3 2x-d' 2_+_^ ^_-3 1 -a; 2- ^'3 _ 4 + 3^"^r6 8:c + 16* X 3.4 ^3 _ 27 a;2 _ 5^ _|_ 6 ' 3^.2 _^ g^ _^. 27' a; + 1 1 — x a: — 2 2 — a: 2a;2 - 18 " x^-4:X^Z "^ a;^ _ 1 ~ x^j^'ix- 3* 2 a;2-l _a;2 + a; + l 2a; -3 3 ' :?;'^ _ 7a; + 10 x^ — 3a; - 10 3a; - 15' d^ + ^^ , ci h__ a + h ^' a^-h^'^ d' + ah + l?' a'-ai'^ ah-V' {x^ -\- y^Y x^ + y^ x^ x^ + x^y^ -\- y^ x^ — xy -\- y^ x^ ~\- xy -\- y^' 3a; + 2 __ 2a; — 3 a; + 2 ^^^"3a; - 28 a;^ - 16 "^ x^ - 11a; + 28' 2a;2 + 3 , 7a; . 2a; _ ~5a; + 15 ~ ' a; + 3' . Fractional Equations. 236. An expression which forms one entire member of an equation may have its value changed, without destroy- ing the equation, provided the other member of the equa- tion also has its value changed in the same way and to the same extent; but we must be careful not to change the value of an expression which is not a member of an equa- tion. 218 ALGEBBA, Thus in the equation "^ ' 7 ~ 14 ~4 ^28 we may ^^ clear of fractions^' by multiplying by 28; but in the expression x — b 3a; — 2 ~ir ^ 7 we can not clear of fractions ; we have no right to multiply by 14 or by anything else; we may reduce to one fraction 7a;- 5 6a; - 4 ^ 13a;- 9 14 "* l4~ "" 14 but we cannot lose sight of the denominator. EXERCISE CXI. Simplify the following equations : X — 1 3a; + 2 6 _ x' — x ^' "~2 ^ 6 '~ a; - 3 ~ x^^' 4a; + 7 , 3a; - 4 20a; - 3 5 ^ 15 15 4a; -3^ 3x+7y 6x-2y 9a; + 2i/_3?/ 2(a;+l) 3. i 14 21 ' 42 14 1,1 8 - a; 4. Z a + 6. x — 6 a: + 5 (a; — 6) (a; — 5) ' 1 2 _ 1 1 — a; 1 — a;^ ~ 2a; — 5' 16a; - x^ _3 + 2a;_2-3a;_l a;2-4 a;-2 ~ Y+x' ~ 8 ' 1 1 a;2+9a; + 20 ' a;2+12a;+35 a;2+lla; + 28 LOWEST COMMON MULTIPLE. 219 8. 1 1 x' — 13a; + 42 ' 9. 1 1 x^ + Ix -44 ' 10. ~^ __ 1 -T-1T + x^ — Ibx + 54 x^ — l^x + 63* 1 _ 1_ x^ — '^x — 143 ~ x^ — llx + 68* 2^ 1 a;-l :?;2 + 4rr + 3 ^ iz;2 _j_ 5^ _|_ g - ^_|. 3' The Three Principal Signs. 237. The entire numerator of an algebraic fraction rep- resents some number which may be + or — according as one set of values or some other set is assigned for the letters appearing in it; so also the denominator of an algebraic fraction may be + or — ; and the quotient represented by the fraction would have its sign + or — according as the entire numerator and the entire denominator had signs alike or unlike. In the expression x — the values a= 2by b = 8, q ~ 5 give to the fraction the value -\<^- =: 10, so that the value of the expression is a; — 10. The fraction is unchanged if it be written -, be- q — b cause in this case we have changed the signs of numerator and denominator both. If, however, we write the fraction ~ , its value becomes b-q 3Q — ^ — = — 10; and if we then wish the value of the ex- pression to be unaltered, we must change the sign of the term in which the fraction appears. Thus x — % b — q would be ic — (— 10) = a; -f 10; to keep the entire ex- pression unchanged in value we must write \i x -\- 1 . 220 ALGEBRA, So also if we write the fraction ■ ^, then we must q -I write the expression x + q-V 238. Whenever an algebraic fraction appears as a term in an expression, it is convenient to recognize as the THREE PRINCIPAL SIGNS of the fraction : I. The Sign of the Whole Numerator ; II. The Sign of the Whole Denominator ; III. The Sign hefore the Fraction ; and any two of the three principal signs of a fraction may be reversed without altering the value of the expression in which the fraction appears. Model D. — In the expression X — \ \ — x^ ^ \ -\- X it is desirable that the second denominator should be written o;^ — 1 so as to bring out more clearly the fact that the first and third denominators are factors of it. The expression may be written x^\^ x^ — \^ x-\-\ without altering its value. Model E x-\-l "Ix-l , 2 - 3:z: (x - 3) (:^; - 5) (x - 2)(3 - 2:) ^ (2 ^ x) (5 - x) may be written xJ^"^ 2a; -1 32^-2 (x - Vi(x - 5) ^ (ir - %){x - 3) (2: - 2)(a; - 5) without changing its value; as follows: LOWEST COMMON MULTIPLE, 221 in the first fraction there is no change; in the second fraction the sign of one factor of the denominator is changed; this changes the sign of the whole denominator, which is one of the three principal signs; so the sign of the whole fraction is also changed; in the third fraction we change the sign of two factors, which leaves the sign of the whole denominator un- changed; the sign of the numerator is changed, so the sign of tlie whole fraction must also be changed. Rearrangements. 239. The altered form of the above expressions will be found much more convenient in the process of uniting the three fractions into one. The principle of arrangement in this case is to have the ic-term in each binomial come first, so that one may apply mentally the process of cross- multiplication. 240. There is another principle of arrangement which it is sometimes expedient to follow. It is known as cyclic order, and is of importance to all students of Algebra. According to this principle, if we had four letters a, h, c, d, we should make h follow a, c follow h, d follow c, a follow ^ ^ dy h follow a, and so on; the several ^ v^^^^^ letters taking turns and beginning again at the first, as if they were arranged in a circle. Thus the expression (a - 1){1) - c) - (Z> ~ c){a -c)-\-(h-a){c- l) would be written, according to this principle, thus : {a - l)){h -c) + {'b- c){c --a) + {a- h){b - c) without change of value. 222 ALGEBRA, EXERCISE CXII. Simplify the following expressions : 1,2,3 • X - 2^ X + 2^2 - X ^* dx+l'^ 9x^- l~^l- dx 3 iz: . 1 — a; » I i 1 - 4:?;2 ' 2:^-1 '^ 22; + 1* 1 +,^,^^_^+ (ii; - 2){x - 3) ^ (:^; + 2)(x - 3) ^ 4 - :z;2- 2 3 5 i^;2 _ 7^ + 12 12 - a; - a;^ + 48 - 32;2' c , a c "• (« - ^)(«^ - c)^ {c - h){c - a)^ {h - a){h - c)' 7. 7:7— w^_-^ - /TT—Twr— :;^ + 10. 11. {a - h){h - c) {h - c){a - 6^) ^ (« ~ c){b - a)' a -^ h b + c {b - c){a - c) + {b - aj{c - ay {a - b){c -a)^ {c - b){c - a)^ {b - a){b - c)' c? b ^ {b - a){b - c) + (a - b){a - c)' a 2b 2b {b — c){c — a) {a — b){a — c) {b — c){b — a)' a; , 5 , 5 12. JZ K^l7. ^ + 77. ^u7. :7^ + 13. (^ - b){y -x)^{y- b){y ~ x)^ {b - x){b - y)' X 10 10 (5 - y){y -x)" (x- b){x - y). ~ (5 - y){b - x)' * Find the shortest way of doing this example. 14. LOWEST COMMON MULTIPLE. 223 (5 - y){x - y) (5 - x){y - x)' x^ y^ ^^' {y - ^){y ~ 7) "^ (^ -y){x- ly If y ^'i in the following expressions, what value must X have m each case to make the whole expression equal to f f ^1 1 , 2x X -\- y X — y x^ — y^' *i7 ^ ~- y I y"^ + ^^y . ^ + y ^ ' X -\- y y'^ — x^ X — y * a; + ^ y^ — x^^ X — \ x^ — y'^ xy — y'^ 19. *on x^ + y' _j^ +-^_. Solve the following equations : x^ — bx 2 21. 22. n;^ — 4rc — 5 3 3a;^ + 6a; _ 7 ii;2 + 4:^; + 4 "■ 3 * 2;2 -f 5:^; _|. 6 i?;2 _ 2a; — 3 = 5. 2a;2 + 13^ + 15 2x^ + 11a; + 5 ^ 04. 1 ! , -i- ■ — ' ! z=. 5 2a;2 - 2; - 1 4a;2 + a; - 14 _ 1^ ^^' 2x^ + bx + 2 ^ l(Jx^ - 49 ""5"' * First combine tLe fractions whose denominators are of the first degree. 224 ALGEBRA, 1 . 1 1 26. 27. cc+l ' {x-\-l){x-^2){x-\-^)~ {x+l){x-\-'^)~ ^-x 3 + 2a; 2 — 3a; 16a; -a;^ 1 "2 -a; "" 2 + a; + a;^ - 4 ~ II' a; 3 1 _2^ ^®* (a;-3)(a;+l) (:i;-|-2)(a;-3) + (a;+2)(a;+l)~"3* ^^- (3-a;)3(a;-2) + (2-a;)3(3-a;)2+(5a;-6-a;2)(a;-2)'^^' * ^ '^ 4-20a;_a:-2 a;+2 2 31. l-2x l+2a; 4a;2-l '~a;+2'^2-a; 4-a;2* 1 + L__- ^ a;2 - 9a; + 20 ^ a;2 - 11a; + 30 ~ ll(a; - 4)' 1.1 4 : + • a;2 - 7a; + 12 ^ 5a; - a;^ - 6 ~ 5(3 - a:)(4 - x)' 33 ^ , 1 . ^ -0 • 2a;2 - a; - 1 "^ 3 - a; - 2a;2 "^ 2(2a; + 1)(1 - x) 1 3 1 35 1 + a; - 2a;2 "T~ 6a;2 - a; - 2 (2a; + 1)(1 - a;)* 2 . 3 3 - 2a; 1 36. 37. X ' 1 - 2a; ^ 4a;2 - 1 a; — 2a;2' 113 a;-l a; - 1 a; + 2 (x-\-2f~{x- x^){x'^ + ^ - 2) 5 _ 1 24 5 2(a;+l) 10(a;-l) 5(2a; + 3) ~ 2a;2 + 5a;+ 3* 5 2 3a; 39. 5 + a; - 18a;''^ 2a;2 + 5a; + 2 ~ (2 + a;)(5 - 9a;)* 11,1 12 x+l (a; + l)(:r+2)^(a;+l)(a;+2)(a;+3) (a;+l)(a;+3)- * Simplify eacli member separately before clearing of fractions. LOWEST COMMON MULTIPLE, ^25 Modified Methods of Reduction. 241. There are certain types of fractional expressions and equations, of frequent occurrence, which can be much more easily reduced by modifying somewhat the ordinary straightforward method. For instance, examples 12, 13, 14, 16, and 26 in the preceding exercise. Model F. — Reduce 1.1. 6 54 3-^'3 + a;' 9 + ir2 '81 + a;^* In this example, if we combine the first two fractions, then that result with the third, and finally that result with the fourth, the work can all be done mentally. Sometimes the denominators are seen to belong together naturally in pairs: Model a. 3 2 3 2 2cc— 3 x + l "^x + ^ ^ X- 1 18 4 ■ {2x - 3) {^x + 3) ^ (x+l){x- 1) 18rr2 - 18 + 16a;2 - 36 Zix^ - 54 - . {4:3? - •d){x' - 1) - 4a;* - 13a;^ + 9" Model H. — Reduce x-%^ X- 3 x-&^ x-1 1 1 1 1 1 ^ x-2 x-l~'x-& x-Z ^ x-3 1 x-7 r q ® ^ _ 9^ + 14 - ^2 _ 93; ^ 18 °^°'° ^" ® 226 ALGEBRA. ® - 5{x^ - 9x+18) = d{x^ -9x + 14) ®X{x^-9x + U){x^ -9x + 18) (§) - 5x^ + 4:5x - 90 = 3x^ - 27^; + 42 same as ® ® 8a;2 - 72:?; + 132 = (5) + 5x^ - 45^; + 90 ® 2:?;2 _ x3^ ^ 33 ^ ® -^ 4 Whence by the Quadratic Formula x= 3.218; x = 1.718. The advantage of rearranging the equation as in @ is that the a;-terms destroy each other, and we save such multiplications as {2x - 6){x^ - 13:?; + 42) and {2x - ld){x^ - 6x + 6). Model I. — Keduce X —2 X — 1 __x — 6 X — 5 x-3 "^ x-2 ~ a;— 7 "^ x~^^' K we divide the numerator of the first fraction by its denominator, the quotient is 1 and the remainder 1; so that the fraction reduces to the mixed number 1 -\ -. X — 3 The rest of the reduction proceeds as in the preceding example. Model J. — Keduce x^ + 7x + 15 __ 2x^ + 3a; + 3 2x^ + 13x^ + 24tx - 10 4a;3 + 42;^ + 5a; - 4 LOWEST COMMON MXILTIPLB, 227 The reciprocals of these two equal fractions are equal, — ^ x^+lx+lb ~ 2a;2+32;+3 ' ^ Eeducing to mixed numbers, by carrying out the division indicated by each fraction, — (3)2^-1+ .^j; ,^^ =2^-1 + x^ + lx + lb ^ 2ii;2 + 3a; + 3 same as © ^ x^-^lx + lb _ ^x^ + Zx + ^ ^ ^^75 - 2^-1 ^-^ 5 5 (D a; + 2 -J -— = a; + 2 + r same as (5) a; + 5 2a; — 1 Whence a; + 5 = 2a; — 1 and a; = 6, Care must be taken not to apply the method of the example just preceding to equations where there are more than one term in either member; that is, if 3 , a;-3 _ a;3 + 3 a; — 2 a;^ — 5a? + 1 a;^ + 5a; + 2 IT IS NOT TRUE THAT a; - 2 a;^ - 5a; + 1 _ a;^ + 5a; + 2 3"^ a;-3 ~ a;2 + 3' In examples like number 1 of the next exercise clear of the numerical denominators first. 22S ALGMBMA, EXERCISE CXIII. Reduce to the simplest form : 11a; - 7 5x - 7 _ 7a; - 3 dx - 5 ^' 12 + 3 '- X - 1 ^ 5 1,1 4a; . 17 2a; - 3 ' 2a; + 3 4a;2 + 9 ^ 16a;* -f 81' 3a; _ 3a;^ + 2 4(3a; + 2) 1 Q^2 I Q I oi ^4 I 1 /? ' ^- 6a; - 4 3a; + 2 ISa;^ + 8 ^ 81a;* + 16 3 7 2 4 6. 2a; — 3'3 + a;'2a;4-3 3 — a;* 3 1 x + 4: x + 1 x + 1 2(a; + 2) (^ + l)(a;+ 2) "^ (a; + 2)(a; + 3)' 1 4,5 4,1 X — 3 X — 1 X X + 1 X + d' 2a; + 1 4a; — 7 x — 6 _:?:-f-53a; — 2 ''* 5 ^ a; - 4 + 3 ~ 2a; - 8 "^ 3 * a; - 1 , 20a; + 11 _ a: - 11 41a; ^ 4 9. a; — 3 ' 3 3a; — 9 ' 6 1.2 2.1 4 — a;' 2a; — 3 7 — 2a;'a; — 1* 1,1 1 1 10. - + a;5— -a; x + 1 4— a;' 2 2 2 2 11. a;— 2 a; — 7 x — 1 6 — x 1,1 1,1 13. 14. a; + 9'a; + 5 a; + 8a; + 6* 3a;2 — a; — 25 _ x^ + 7a; — 1 9a;3 _ 6^2 _ >^i^ j^ 18 3^3 _^ 20a;2 - 10a; + 6' 4a;^ - 4a; - 3 _ 2x^ -f 3a; + 4 4a;3 + 8a;2 - 10a; - 3 "" 2^ + 9a;2 + 13a; + 15* 15. 16. LOWEST COMMON MULTIPLE. S2& Ix^ + 10:^-1 _ ^x^ + ^^^ + 32 llx^ + 103^2 -Ux-S ~ 33x^+ 239x^+ 410:?: - 23* Qx^ - X + 13 _ 4:x^ + 12a; + 20 6^3 + lla;2 + 14:?; + 24 " ^x^ + 20^^ _|. 45^ _|_ 45- 3a; + 13 x -1 _ 4:X + 17 Sx + d ^'^' x + 3^x + l~~2x+Q'^4:X + 4t' 3a; - 8 , 4a; - 17 2a; - 13 , 5a; ~ 41 18. TT H 7- = ^ + 19. 21. a; — 3 a; — 4 x — 7 x — 8 8x - 27 2a; - 9 _ 6a; - 8 _ 3a; - 17 2a; - 7 "^ a; - 5 2a; - 3 " 'a; - 6 * 15a; - 2 12a; -- 41 ___ 14a; - 19 4a; — 8 3a; — 1 "^ 3a; — 11 "" 2a; - 3 "^ 2a; - 5* 1 1_ _ 2a ^__ X — a X -\- a x^ -\- a^ x^ -\- cC 23. 27. ,4' a;+4 "^ ^-^[6 ~ a; — 5 a; — 7' 4a; — 3 _ 3a; + 4 a; — 5 __ 3a; + 4 __ 10a; — 4 ^ 66a; - 49 4a; - 5 _ 14a; - 19 24a; - 43 6a; - 5 + a; - 1 ~ 2a; - 3 "^ 3a; - 5 ' x^ -f 28a; - 55 _ 3a;^ + 20a; - 47 7a;3+ 201a;2- 244a;— 245 ~ 21a;3+155a;2- 226a; -209* ^■2 -t- 2a; - 3 _ 67a; + 133 ^3 _[_ 3^2 _ 3^ _ 3 - 67^2 ^ 200a; - 1' a;^ + a; - 6 + 1 8a;^ - 18a; - 9 x-2 + 4a; + 1 _ 24a;^ - 55a; + 26 1 ~ 8a; - 13 * 2a; - 2* 2a;^ + 21a;^ - 2a; + 120 __ 4a;^ + 6a;^ - 2a;+ 105 x^ + lla;2 + 64 "" 2x^ + 3x^+56 ' 230 ALOEBRA, The Principal Dividing Line. 242. A fraction in which the numerator or the denomi- nator or both contain a fractional term is called a complex fraction. For example, the ratio f : 5 may be written as a com- plex fraction; so also 2 : |. When so written these ex- pressions cannot be distinguished except by the Principal Dividing Line of the fraction. ^,3 2 , ., 2 10 Thus r- = — ; while ^ = — . 5 15 d 3 5 243. In a complex fraction everything above the princi- pal dividing line constitutes the dividend, everything below constitutes the divisor. Plus and minus and equal signs which precede or follow a complex fraction must be on the same level as the principal dividing line of the fraction. 244. In most cases the readiest way to simplify a com- plex fraction is to take advantage of the principle that multiplying the numerator and the denominator by the same number does not change the value of a fraction ; and to choose such a multiplier as will render both num'erator and denominator integral. Model K. .^•^ _ x^ — if' _x^ -\- xy -\- y^^ x^ — y^ x^{x^ — y^) x^{x -f- y) 245. Sometimes one prefers to simplify the numerator and the denominator separately, reducing them each to a single fractional term, and then dividing the numerator by the denominator. LOWEST COMMON MULTIPLE, 231 Model L. a l aq — Ip V ^ — M _aq — Ip Ip — aq _ ah P ^ ~ pLzL^ ~ P^ ' ah ~~ pq ah dh EXERCISE CXIV. Simplify : HA 5. '' " 3 27 "^ 8 9. x^ — y^' ' 9a^ — 46'^' . b. • 10. x___p_ \_ 1 13 a h X y X a y ~~ h x — 2 x^-9 11. M X - - 5 1 4- 20 - - 9a;' ^ n^ a;^ ax - ■0? ^« a" - ■x' x^- 8 4 x'- X — 8 4 1 X ■ X 3 5 8- ^2 5^ _,_ g. ^2 ^ L_^ a;2-9 ^3__1 ic 6 y 13. ^ ~ ^ . * , 2z a 1 + 111 - — ^—-—^ — a c ' b ^ c ^ a 232 ALOEBBA. 17. ^ + ^ + 3- 1 5^ 3^ "T" Q^ + /^ 19. 18. a; + 3 2x+l x-d ' 2x-l x + 3 2a; +1 x + y^ X — 3 2a;- 1 a ~ h x^y a + h a ^ J) 5 ' 3a;' a— b X -- y a+ h a — h x+y x—y X — y~x + y Continued Fractions. 246. Complex fractions of the following peculiar type are called continued fractions : Model M. 3 + 2 + 1 4 - 4-i To simplify this expression, rewrite it as far as the last dividing line which indicates a complex fraction ; substitute for that its simplest form, and repeat the same process as often as necessary. 3 H 3 H ^ — 4-i 398 199 LOWEST COMMON MULTIPLE, 233 EXERCISE CXV. Simplify : 1 '1 . 1 1. X 4. ? 5.. 1 + ^ 1+ T- ^ + '^-— 1 1 + 2-3-^ a 1 2a 25 j^ _ 1 25 2a ,1 *■ 5 '-^ ^^- ,^2a-3 '• T- + 2a , , 1 1-^ 1 + 1 + -^ X 8. In the expression substitute for x the expres- 2 4-- sion — , and reduce to simplest form. ' + 4 + 1 X X 9. Find the value of the expression — : — when ic = 1 H -— and «/ = 1 — ; ' + 3+1 ^ + 3^i a;' is obtained from x by omitting the fraction indicated by 234 ALGEBRA. the lowest dividing line, and y' is obtained from y in the same way. 10. Find the value of the ratio — : — -, when y % , 1 y is the result obtained by substituting c -\- — ior c in the expression for a;; ^' is the result obtained by interchanging h and c in the expression for x, and ^' is the result of the same change in y. Another Application of L. C. M. 247. If we have an equation in the form of an integral expression equal to zero, the answers can be found if the equation can be factored. The answers of two or more such equations may all be found in the equation formed by setting the L. C. M. of the expressions equal to zero. Model N. — The equations following have the answers set down beside them : x^—^x = X = x^ — bx -\- ^ = X = 2 a;^ — 9 = X = 3 X = 3 X = d X = — 3 The L. 0. M. of the expressions x^ — 3x, x^ — 5x -{- 6, and x^ — 9, is x{x — S){x + 3){x — 2) ; and the equation x{x - 3){x + 3){x - 2) = has answers 0, 2, 3, — 3. LOWEST COMMON MULTIPLE, 235 EXERCISE CXVI, For each of the following sets of equations construct a neio equation, among whose answers can le found all the answers of each given equation; and let the resulting equa- tion he the one of lowest degree which can satisfy this condition : 1. 2a;2 = 3:?; + 2; 4^^2 = 1; x^ -\- 2 = ^x. 2. 8:?;3_27 = 0; 4cX = - X 3. 4:x^ = ± I; 2a; = 1. 4. x^ =z X + 20; x^ = 12 — x; x^ = Sx — 15. 5. a;2 = l(x + 1); x'^ = Z{x + 1) + J; x^ = ^x - - 12, 6. x^ = 8^3; x + 2a + ^^^ = 0; x = ^^\ 7. (3a: - 2) : (:r - 1) = 5 : 2x; Qx^ + 5 = Ux; ^x^ = 1. 8. 6x^=x+2; lSx{l-x) = ^x^+x+2; 5a;(3^+l)=:a;+l. 9. x^ = 27; x^ = 15a; — 36; x^ - ^x^ - 2x + Q = 0. 10. 5a;2 + 19a; = 4; lOa;^ + 19a; = 3 + 6a;; 15a;2 + 19a; = 13a;2 + 4(2a; - 3). Tests for Simple Factors. 248. Sometimes it is not possible to factor all the ex- pressions by inspection; and in such cases it often serves to get the factors of one expression and to try them as divisors of the other expressions. Model 0. — In the expressions 2a;2 + 7a; + 5; 2a;3 + a; + 3; a;^ - 6a; - 5 the factors of the first expression are (2a; + 5)(a; -f- 1); of 236 ALGEBRA, these 2iz; + 5 is evidently not contained in either of the other expressions; by substituting 2; = — 1 in the other expressions, we reduce each to zero, and hence conclude that :?; + 1 is a factor of each. Dividing we find the factors as follows : 2^3 _|. ^ ^ 3 = (^ _|_ i)(22;2 - 2a; + 3); x^ - Qx - h = \x -{- l)\x^ - X - b). Hence the L. C. M. would be {x + l){2x + 5)(2^2 _ 2a; + 3){2;2 _ a; - 5). EXERCISE CXVII. Rearrange the equations in each example so that each shall le an integral algebraic expression equated to zero. Then find the L, G, M. of their first members : 1. 2;2 - 4 = 0; x^ + X = 10. 2. x^ — 7x - 30 = 0; 2x^ + 15 = 13a;; 3x^ = 29a; + 5. 3. 2a;2=9a;+5; a;(a;"2-9)=:20(a:-l); a;2(2a;-l) = 5(2a;2-5). 4. a;2 = ^(a; + 10); a;(a:2 - 2) = |(a; - 4) ; Sa;^ + 4 = 7a:'^. 5. a;-2 = fa;2; a;^ - 12 = 15a;2 + 0;; a;^ = 16 + 2a;*'^ + So^^^ 6. a;2=|(9a:+4); o;^- 6a;2+ 32 = 0; ISa;^ - 19a;2 + 4 = 0. 7. a;3 = 8; x^ + 4a;2 + 16 = 0; x^ = x^ + 4. 8. lOOOy^ = 1; lOy^ = 99 f - 1. 9. 2a;3 - 6a;2 + 3a; = 9; 20:^ z= 19a; - 3. 10. 2a;3-5a;2=16a;-40; a;4-6a;2=:a;3-8a;+ 16; 2a;3=:a;2-j-25. Relation between L. C. M. and H, C. F. 249. The L. C. M. of any two expressions may be obtained by dividing one of them by their H. C. F. and multiplying the quotient by the other. For since the factors of the H. C. F. are contained in each expression, the first expression contains all the factors LOWEST COMMON MULTIPLE. 237 of the second except those that are not included in the H. C. F. Model P. — Of the two expressions 2x^ — ll:?;^ — 9 and 4^5 _|_ 11^4 _[_ gx the H. C. F. is iz:2 _^ 2^ + 3; the quotient obtained by dividing the first expression, ^x^ — llcc^ — 9, by the H. C. F., ^r^ + 2;r + 3, is 2^^ _ 4^'^ _|_ 2a; - 3; so the L. C. M. is (22;3 - 4:x'^ -f 2a; - 3) {^x^ + ll:r* + 81) ; if we divided the second expression by the H. C. F. we should obtain for the L. C. M. {4tX^-^dx^-l%x-\-'il){2x^-llx^-^)] in either case the product would be 8a;8+6:z;^-36x6+10^5-33:r4+162a;3-324:c2+162:c-243. EXERCISE CXVIII, Fiyid the L. C, M. of the following expressions : 1. x^^ + x^y^ + y^^] x^^ 4- x^y^ + x^y^ + x^y^ -\- y^^. 2. x^+x + Q; x^ + 2x^ + 9. 3. 2x^ + 66xy^ — 20y^; 9x^ + SOxy^ - Zy\ 4. x^+UxY-llxy^+l5y^^; 5a;5+lla;y+20a:y+9yio. 5. ^2a^x + 13a V - x^^; 36a^ - x^^ + 8a V - aV. L. C. M. of Three Expressions. 250. In finding the L. C. M. of three or more expressions, one finds first the L. 0. M. of the first two; then it is necessary to multiply by all the factors of the third ex- pression that are not contained in the first two. If the three expressions are represented by X, Y, and Z, and the L. C. M. of X and Y by m, we must multiply m by all the factors of Z that are not to be found in m already. To find these, we divide Z by the H. C. F. of m and Z, Model ft. — In the three expressions x^ — y^i x^ — y^; x^ — y^ 238 ALGEBRA. the L. C. M. of the first two is {x^-y^){x+y) ; the H. C. F. of this and the third expression is 2;^ — ?/^; hence the L. C. M. of all is {x^ - y'){x + y){x' + y^), EXERCISE CXIX. Find the L. C, M. of the five sets of expressions in Exercise CVI; also of the followirig five : 1. 4.x^-lx-^\ 2x^ - Ix^ -\- 9; 4.x^ — l^x'^ + 9:?; + 9. 2. x^-Qx^+llx-^)x^-'^x^-bx+^', 32;*— 13a;3+52a;-48. 3. a;3 + 2:^; - 12 ; x^ - x^ - l^] x^ - ^x - 3. 4. 4a;3 - 72; - 3; 1-^x^- lx^\ bx - Ax^ - 6. 5 ^4 _ 3^ _ 20; x^ - Ax^ + 15^; 25 - 4:X^ + 3x\ RearraJige the equations in each of the following examples so that each shall consist of an integral algebraic expression equated to zero ; then find the L, C, M, of those ex- 6. ^'+^1^ ^^2_|_i. x^-x-^^^^^x^-nx', ?^^^=8-^ b X X — ^: ^ x^-Y^ X^-\-6V 46a; + 7 = 9:?;^ 5 _ 2a; a;2 = — ^ 3a; 10. a;^ + 8a;2 + 282; = 7a;3 + 48; x^ - 14 = %x^ - 19a;; hx^ + 28 = 17a;2 CHAPTER X. INDICES; SURDS; ROOTS. 251. A Power is a number whose factors are equal. One of the equal factors is called a Root. The number of equal factors is called the Index. a^a^ = aaa, aaaaa = a^ Law I. a^a^ = a^ + ^ - o aaaaa „ a^ -T- «^ = = aa^ a^ aaa Law II. a"" -V- a^ = -„ = oj^*^. - a^ When y = X the quotient in (ii) becomes a^; When y ^ X, the quotient in (ii) being a^ " ^, the index a; — ^ is a minus number. Thus ^3 _^ ^3 __ ^3 - 3 _. ^0 252. For these new values of the index the old definitions will not apply; but if we consider them subject to laws (i) and (ii) we can interpret them in terms of the more familiar symbols. 239 240 ALGEBRA, ^O^m^^O + m^^m Law III. a-^a^ = a-^^'' = a^ = 1 a~V= 1; a-^~ -g; d-m^m ^ ^-m+m ^ ^0 ; a^ = Law IV. a-'^a'^ = 1: a-^ = — ; ^^^ = — - EXERCISE CXX. State in the simplest form the numerical value of the following expressions : 1.3-3. ,. (ip 3. (.01)-, ,._L. «.__!_. 6. 356(4)-3. 7. (16)^(6)-. 8. ^"(i). 9. 9^3- What is the reciprocal of each of the following expres- sions ? 11. 2. 12. 3. 13. 10. 14. 1. 16. 100. 16. \. 17. f. 18. f 19. .3. 20. .001. 21. 2.5. 22. ^. 23. 100.1. 24. 1.1. 25. 33.33. 26. a. a 27. -. X a" 28. -3^. a — h , „ 29. . 30. X + y^, c 253. From (iv) it follows that the reciprocal of any power is the same power with the sign of its index changed. INDICES; 8UBD8; BOOTS, 241 EXERCISE CXXI. State the reciprocals of the follotuing expressions with- out using the fractional form : 1. ^^; ^-^• \,\ ^3; ^'"^ 2. \' 3. {a^+x^)\ 7. 16-2. g 2«-^ 9. «-^>. 10. ic-2 + 3^3^ Reversal of Signs. 254. The law of reversal of signs in subtraction has a close analogy in multiplication : In subtraction : To use an expression as a term of the SUBTRAHEND is the same as to use its negative as a term of the MINUEND. In division : To use an expression as a factor of the DIVISOR is the same as to use its reciprocal as a factor of the DIVIDEND. Or, conversely: Any TERMS of an expression may be replaced by a sub- trahend containing the negatives of those terms. Any FACTORS of an expression may be replaced by a DIVISOR containing the reciprocals of the factors. 255. Dividend, numerator, and antecedent are in Algebra interchangeable names; so also are divisor, denominator, and consequent; every example of division being a fraction or a ratio, according to circumstances which are not always apparent. 242 ALGEBRA. EXERCISE CXXn. Transfer all x's and y's to the divisor, and all a^s, Vs, c's, and d^s to the dividend, wherever they occur in the following fractions {or ratios), changing the index lohen necessary : - x''^ x^ x^y axy ' „-2 • 2. —5. 3. ~~ro*7* 4. ttttq a * a —2\' 4. ~^. 6. a-^x^:h^cy-\ 0? .V + ^ ^"^ 6- ^- 7. 7 TTw.. 8. ^^TT-- 9. (a + ^,)-i- °- a-3 • ^- l + a= 2' ^"- a-^ + 52' ^^- Z>y2(^2 _ ^3)- 256. Model A. 3a^^>3,o^-^ ^ 3jb^^ ^ 24g^ ^~^g~^x^yz~''^ ¥dx^y l)^dx?y EXERCISE CXXIII. Change to expressions that have no zero or negative indices : 1. (j£^x-^ 2. — s. 3. -^ — y— 5. 4. a^^cd^e-^-^, x~^ x^y~^z~^ '' x?y~^ hx"'^ al)~^ (2a:^)~3 ^- ~ar^' ^ 3f^'' ^' 3^^' ®- Sx-'y^'' 2-'^x-Y (Qay)-' 9. TTT-T^- 10. — ^ ^^ Simplify the following expressions : y.2 ^j2 11. a^"^ + 3/^. 12. a'^ — I>-^. 13. ;::z2 — ^« X ^ — y x^ — 5^ + 6 a~^ A- h~ 14. —, ^T-V. 15. ^x-^ ' ■'*'• a-b + h^a'^' IJSrmCBS; sunns ; MOOTS. 243 ab-^ + b{a 4- b)'^ a(a — b) — a'^b' 17. (a-^ - b-^){a{a + b)'^ + Z>(a - b)-^]. 18. (a;"^ — y'^)(j^~^ — 2/"^)^- 19. (^ — y^x~'^){x^ + y^Y^i^y'^ '- ^V^ + ^~V) 20. (^y"^ — yx~^){y^^ — a;"^)"^. When a Root is itself a Power. 257. In 4 X 4 X 4 = 64; 2^ x 2^ x 2^ =: 64 there are two indices to consider . . . and factors within factors. (a^Y =z aVa^aV = aaaaaaaaaaaaaaa = a}^. Here for each of the large equal factors there are three of the smaller ones; in all five times three smaller factors. Here for each of the large equal factors there are x of the smaller ones; in all Qx of the smaller factors. Law V. {a^y = a^^. 268. Kef erring to (y), we have a power a^^ of which a^ is a root the index being q. This relation may be written as in (y), but also Similarly a^ = Va^, since a^ may be arranged in three equal factors thus : aa.aa.aa; and again a^ = Va^"^, since a'^^ may be arranged in seven equal factors, each of which 244 ALGEBBA. contains x factors a. In general, if we have Vc^, and if p is a multiple of q^ we may arrange the p factors of aP in q equal groups; each group will contain — factors a\ and in that case, if ^ is a multiple of q. EXERCISE CXXIV. Express as powers of prime mirribers: 1. (2i3)(84)(4326)(162o«). 6. (8)23 ^(16)1^ 2. (92ii)(27i^s)(6ioo)(833). 7. {(24)12(54)6} 11. 3. (12)3(36)^(192)43. 8. [(1000)« -- (25)io]i8. 4. (200)4(135)38(225)51. 9. V332 . 9^^ . 81. 6. (45)11(675)31(375)28. lo. V{VZy\lS') - 2^. Fractional Indices. 269. If ^ is not an exact multiple of g, then the expres- p sion «« becomes meaningless, as we cannot have a fractional number of factors. We are at liberty to assume that laws (i), (ii), and (v) apply to these new indices, and under that assumption to seek an interpretation for them. a^a^a^ = «3 + ^ + ^ = a* = a^^ Here c? appears as one of the three equal factors of a^\ that is, a^ = VdK \a^/ =a^, ax — ya^ (a^J ^aP; aa ~Va^ mDICES; SURDS; BOOTS. 245 In general any power whose index is a fraction may be taken as a root, and as many of them multiplied together as the denominator indicates; then the product so obtained will be a power whose index is the numerator. (J)' = a^ or, otherwise, Qr Law VI. UQ = Va^ 260. By (vi) any root may be exhibited as a power with a fractional index. 261. Again J = \Jf = ^J^P as well as a^ = {a^f = V~^. A fractional index, then, indicates both a root and a power ; the denominator indicates a root, and the numer- ator indicates a power ; and either the root or the power may be found first. EXERCISE CXXV. Express with radical signs : 1. «'. 2. x^. 3. {a — Z>)i 4. a^b^. 6. a^M. 6. x^2/'t 7. x^y^ (compare Ex. 6). 8. p^q^ (compare Ex. 5). 9. a^xh 10. (ax^y^)^, Express tuith fractional exponents : 11. Va^. 12. ^^a^. 13. i^x^a^. 14. Vax^y^. 15. \^{a-hf. 16. V^-b^ 17. t^Sy. 18. 'f^. 19. ^\/a^xh^y^, 20. ^ . yxy^ 246 ALOEBBA. Give the numerical value of each : 21. 8?. 22. 165. 23. 32S. 24. (i)?. 25. (|J)*. 26. (.36)5. 27. (.008)i 28. (1.728)1. 29. -(15625)i 30. (256) -8. 31.(^^7)-?. 32. (-8)-?. 33. (64)-Ki)- '. 34. (.0001)-J^ 35. (81)-5(.3)l 36. (i)-'(243)-S. 37. {i^)-^{Vll)-K 38. (216)i(T-V)-^(.l)-^. 89. (.0625)-3(i)-=. 40. l(128)5(V^)-?32-«6-2}-i. 41. 4-^ 42. 92». 43. 16". 44, 32«. 45. 256>2'. 46. iiY'. 4 (H)'-^. 48. (W)"'- 49. (2.25)^^ Simplify : 61. (aZ^^)i 52. (a^Z^-^)^ 53. (tt^Z^-^)^. 54. {a^^)h 56. (Vj-^)-i 56. (8a;y;2«)*. 67. {xVyf. 58. (6?2|/^6^ 59. ( VZQx^y. ^O.Va^Va^. 61. (V(«+a;)^^. 62. \^x\x-\-y)^, RADICALS. 262. A radical expression is one that contains a radical sign; a rational expression is one that can be written without a radical sign or a fractional exponent. 263. A surd is a radical expression that cannot be shown to be rational. 264. A surd whose index is 2 is called a quadratic surd; if the index is 3, a cubic surd; and if necessary similar names could be found for radicals of index greater than 3. Equiradical surds are those having the same index. 266. A mixed surd is a surd with a coefficient outside the radical sign ; an entire surd is one without such a coefficient. INDICES; SUBD8; ROOTS. 247 EXERCISE CXXVI. Express as entire surds : 1. 2 4/53. 2. 3 Vl. 3. 18 V2. 4. .01 4/1000. 5. 29 V%. 6. 2 V¥^. 7. Ill V'lT. 8. 3 i/87. 9. 87 fS. 10. 100 1^.00089. ii. 3 ^9. 12. Gf'S. IS. 11 V'lll. 14. 2 ^. 16. 4 f 2, 16. 2 f 3. 17. 3 ^2. 18. 17 f3. 19. 17 4/3. 20. 2f3. 21. 3 f 2. 22. 2 f 27. 23. 3 f 16. 24. 13 VTi. 26. VI H. 26. |/27f3. 27. V'^ V^. 28. ^2 Vs. 29. fS 1/3. 30. VI VI. 266. A surd is sometimes called an irrational expression. Similar surds are those whose irrational factors are the same. 267. A surd term is said to be in its simplest form when it is written with only one radical sign ; no rational factors in the radical ; no fractions in the radical ; no radical sign in the denominator. And in general any algebraic expression, to be in its simplest form, must have no fractional or negative ex- ponents. EXERCISE CXXVII. Express in the simplest form : 1. /i2. 2. v'Is. 8. V50. 4. V108'. 6. VaooT 6. 1^448^ 7. V'm. 8. 1^1584. 9. 1^1728. lo. ^"845^ 11. .0024/500000. 12. WVTQ, 13. ^^^^^ 58 • 248 ALGEBRA. 14. .004^6250000. i8. ( ~^^ 16. 13-^(l''3187)-*. 17. V^ -^ (V«'-9a2+27a-37)"'. 18. 4«'i5- ^I^S^sFl 19. 275:r^: Wlb^^f. 20. 15a2(^2_^2)-3. 3^-2|/(^_^)-3^ Z77iiYe similar surds in the following expressions : 21. VS- 2VE0+ V98? 22. 4/343"- ^27"- 4/12". 23. 1/507+34/12-8 1/75+ 1^507 24. 4/1000 + 4/360'- 4/207 25. v'800 - 4/8OO + 4/270 - V72". 26. 4/32 + I/32 + 4/108 - |/ia: 27. 4/48 - 4/8I - 4/675 + V'648T 28. >^320 + I/5OO - VT372 + 1/2560. 29. 1^6400+4/1445-4/405". 30. 4/29I6+ 1/432-1/250^ Distributive Law for Indices. 268. The expression a'^V contains 7 factors each of a, h, and c\ with one of each in a set, there would be 7 sets of factors, — a'Z>V = {ahc)\ In general powers of different roots, having the same index, follow this law : The product of the powers is equal to the power of the product. Law VII. a'^h^'x'' = {abx)\ 269, This law holds also for quotients : a'^P db ah f abV _ db ab __ lab y — c ' c ~ \ c I INDICES; SURDS; ROOTS. 249 270. The same law holds for roots: Va VI =Vah; because Va VI V~a Vb = Va Va'Vb Vb = ab. 271. Similarly for all fractional exponents: a'^w) ~ (aby p p p .-. aW={aby Hence the law (vii) holds for all exponents, positive or negative, integral or fractional. 272. Kadicals with different indices can be reduced to radicals with the same index by expressing them with fractional exponents and reducing the fractional exponents to a common denominator. Thus any two radicals can have their product or quotient written as one radical. Model B. EXERCISE CXXVIII. Reduce : 1. ^2^2. 2. VSf'IO. 3. 1^6^12. 4. V6V2. 6. V2V6. 6. V5V3, 7. V3V5. 8. VbVI. 9. VibVI. 10. VqVI, 11. V^V^V^, 12. VsVdVd. 13. V2V3VI. 14. V2Vf, 16. V2VJ. 16. VavW, 17. VaVb. 18. VaV^. 19. VaV¥. 20. V^ V^b. 250 ALGEBRA, Rationalizing a Term. 273. Any radical term may be converted into a rational term by multiplying it by a suitable factor. This operation, however, changes the value or the term. V^V2 =2 f 2 f 4 = f 8 = 2 If the radical term is in the denominator or in the nu- merator of a fraction, we may take advantage of this opera- tion WITHOUT CHANGING THE YALUE OF THE FRACTION. Usually it is desirable to have the denominator rational rather than the numerator. Thus —-= may be written V^V^ 2 V^VY Vi4 or — r=^ but the second form is easier to calculate, 7 being a simpler divisor than Vl4:, EXERCISE CXXIX. Rationalize divisors : _i_ _1_ J^ |i i- 2. -^. 3. —' 4. JJ. 6. -3^. i^I V3 3 3 1135 11. -^:^. 12. — —. 13. — —* 14. — -7=. 15. " 3 11^ nJ 32 3f3 ' 2V2 ' 4V2 * 2^2 ' N 32 d a^ a Vxy 16. — ;=!. 17. -7=. 18. ^ h-. 19. 7_. I^^» VaJ N^ yVic yVx ' V^3a INDICES; SURDS; BOOTS. 251 af8? ZaV^ a^hcVoFc \^32a^ |/i25 f 9^ 6. --=:^. 26. , . . 27. Vl5w : f225a;l 1/5;^; 4/3^^2 28. 5i/50a2& : 2VlOab, 29. 'f/l25«-«^>- 30. VlOOOa^-^ : 200a&. C7>2iYe similar surds in the following expressions : 31. V^-^l + ^l-^l' '^- \|i8 + \|50~\l50 + \27' '*• nJ32 + \|18 ~ sjas + Nir '®- \|i6 ~ \|¥ + sJm ~ "^s/so" \4« \|3a;^\|3" \J108a; ^ \|l00a;2 Sj: 39. V.03. + ^->^-^. ^^' \| 5 +nJ5Z>2^~\20c3* 5x 32* 262 ALGEBRA. Rationalizing Quadratic Surds. 274. Two binomial quadratic surds which consist of the same two terms with opposite signs between them are sometimes called conjugate surds. Their product is, by Theorem A, a rational expression. Where one such factor occurs as the denominator of a fraction we may utilize this fact to rationalize the denominator. Model C. Model D. 5 + 1/5 25-5 20 One important advantage of such reductions is evident when we seek the value of the expressions, having given the numerical value of the square roots. Thus ^=2.236; ^^=^,; ^)5- ^5)=?-(2.764); the reduced expression is obviously easier to calculate. EXERCISE CXXX. Rationalize the denomiiiators : 2+V2 2 - V^' "' 3 - W " 2- V2 3 + 21/5' 5. -T= V2- 1 v% + 1 4/2- 1 3 + V2" 3-51^5 9 8 + 4 V3" 3. 7 2 3-4/3" ■ -1-4^5" 10. 1-4/2" ' 10 V2 + 2 4/5" g ^ ' _. 9. ■ _. 10. ' _ • - 1 + |/2 6 - 2 V3 5 - 2 i^lO INDIGES; SUBDS; ROOTS, 253 Find the values of the expressions in the preceding exer- cise, assuming the folloiving approximate values for the roots : 1/2"= 1.414; V3 = 1.732; 4/5"= 2.236; |/lO = 3.1623. 275. Two sncli multiplications are necessary to rational- ize denominators which are quadratic surds of more than two terms. EXERCISE CXXXI. Rationalize denominators : 3 + 1/6^ + 2^2 |/6-_ |/io + 4/15 - 3 + V^ 4. 3 - l/6a + 2^2 i^6~+ VlO - VTb 1+ l/2 + Vs" 2-1/5" 1 Other Rationalizing Factors. 276. Rationalizing factors can also be obtained for bi- nomial surds that are not quadratic ; but their complexity often makes it undesirable to seek for them. Model E. V'^ — Vo suggests a^ - b^^(a - b)(a^ + ah + P) ; so that (|/2 - H)(H + ^10 + |/25) := 2 - 5 = ^ 3. Model F. 2 + i^3y suggests {a + b)(a^ - a^ - aP + b^) = a' - b'; so that (2 + f^3y)(8- 4 t'S^ - 2 ^9f + ^27p) = 16 + 3y. 264 ALGEBRA, EXERCISE CXXXII. Find rationalizing factors for : 1. 1-^2. 2, 2 + H 3. 1^2""- H 4. t^2 - 1. 6. V2 - V'S'. IMAGINARIES. 277. Imaginary quantities present an important excep- tion to the algebra of ordinary radicals, in this respect : V — 2 must by definition give — 2 when multiplied by itself; law (vii) can therefore not apply to imaginaries, because by its application we should have (- 2)^(~ 2y = [(- 2)(- 2)]* = (4)* = VI. This would allow V— 2 to be equal to 4^2. 278. If we allow i'^ co stand for — 1 and i for V— 1, as in a previous chapter, we shall have V — a =^ Vi^a = i Va\ then ( V— a){ V— a) = {i Va){i Va) = i^a — — a. The symbol i can be handled according to the ordinary laws of algebra and subject to the definition %' =: - 1. It follows from this that i^ = — t; t^ = 1 ; and in the expression i'^ we may subtract from n any multiple of 4 without altering its value. 279. An expression involving the square root of a nega- tive number is called an imaginary (or a complex) number; and i is called the imaginary unit. All imaginary expres- sions may be reduced to the algebra of ordinary radicals by substituting i ■— V — 1 as a coefficient of the radical; and of course no term containing i can be regarded as similar to a term not containing i. INDICES; SURDS; BOOTS. 255 EXERCISE CXXXIII. Unite similar surds : 1. V^^ + V^^^^^ - i/^=~T8". 3. V- 72 - V- 243 + f'- 338 - V- 1200. 4. 3 V^^ + 8 /^^ - l/6"( |/- 50 - V- 12). 5. 3 l/- 32 - 2 f/^^^^"^ - V^^nJ{ V^+ 2). /ory 6. V- 200 - 3Jy + v^Kt'"^^^^^ - '^135). - lb = 2xVZ. 4 The following equations yield surd answers, which must he reduced to their lowest terms 11. a;2 - 30a; + 9 = 0. X 21. —= Vx — 4:, 6 22. |- + 10 = 6(a; - 6). 23. |- - 10 = 6(a; + 6). 24. (3a; ~ 8)2 = 48a;. 25. a; = 8 + i|/44a; + 3. 26. 7a;2 : (35a; - 2) = 8 : 7. 27. 16(a;2 + 4) = 120a; + 1. 28. a;2 + 4.8a; + 1.84 = 0. _ 29. a;2 + 11 = 17.5a;. 11 — - ,,1 .0625 30. .16 =: 2— • X x^ Express the answers to the last 20 examples decimally, assuming V2 = 1.414; V3'= 1.732; V6'= 2.236; VW = 2.4495. 12. a;2 + 10 = 16a;. 13. a;2 + 20a; = 50. 14. x^ = 17 - 32a;. 15. x^ + 28a; + 71 = 0. 16. 61 = 26a; - x\ X^ — Q? 17. **" "^ — \r \ ^ 5 — 5a; 18. a;2 + 32a; + 106 = 0. « . ^^ 19. -^ = ^ + 18- 20. "'+'=2. ^. INDICES; 8UBD8; BOOTS. 265 ROOTS. Square Roots of Numbers. 285. In finding the square root of 383161, we first see that the result is larger than 600 (since 383161 > 360000) and subtract the square of that ; then, concluding that the remainder of the square root is not less than 10, we sub- 383161 619 ^^^^^ ^^^ ^^^^ ^^ '^^^ SQUAKE of 600 + 10. 36 Eepresenting 600 by a and 10 by h, we see that we have subtracted first a^, then {2a + l)h, in all {a + If or 6101 Next we decide that the remainder of the result is not 11061 1229 ^^^^ than 9, and so subtract the rest of the 11061 9 SQUARE of 610 + 9. If this process is not concluded by a zero remainder, it can be continued indefinitely. Square Roots of Algebraic Expressions. 286. The square roots of algebraic polynomials are found in the same way. Model J. 231 121 121 1 + Qa^b - a«62 - 30a6» + 25M a* + Zdb - 56« + ^a^b - a'b'' - 30a6» + 255-1 + 6a»5 + 9fl^'62 An%, 2a^ + Sab -f Sab 2a2 4- Qab - -562 - 562 - 10a''b'-S0ab^-j-26b^ -10a'b^-S0ab^-\-2^b^ '^ A large part of the work of this example consists in re- writing the same terms, as remainders in successive sub- tractions. The work may be contracted as follows : Model J. ^+J^^^- aW -^Qtt^ -\-^M^\d^ + ^ah - 5b^ -IQt^ 2^2 -f Sah 2^2 ^ Qah - 5&2 266 ALGEBRA, EXERCISE CXLI. Find the square roots of: 1. 9x^ + A2x^ + 37^2 _ 28:^ + 4. 2. 4:X^ — 4:X^ + 4:X^ + x^ -2x + 1. 3. 20x^ - 26a:2 - 122; + 9 + 25a;^ 4. 6x^ + 26^3 - 11a;* + 9x^ + Ux^ - 20a; + 25. 6. 49a;* + 56x^ - 2ix^ - ^2x^ + 9 + lQx\ Series for Square Root. 287. With algebraic expressions, as well as with num- bers, if there is no exact square root the process may be continued indefinitely. This leads to a succession of algebraic terms which will always be incomplete, however far extended. Model K. 1 -a; 1 X x^ x^ 5a;* 2 4 16 128 X ~2 x^ 4 X^ X^ X* "" 4 + 8" + 64 . 8 x^ x' 8 ~64 ^3 x^ X^ X^ 8 + 16 + 128 + 256 x' X' a? 16 5a;* x^ x^ 64 128 256 5a;* 5a;5 6x' 6x' 25x^ 64 + 128 + 512 ' 1024 ' 16384 x^ afl 5x* ^x* 138 INDICES; 8UMD8; BOOTS. 267 EXERCISE CXLII. Find series for : 1. VT+Jx. 4. V26 + X, 2. Vr+5x. 5. V9 -f Sx. 3. Vl + ^x- 3a;2. 6. |/a;2 + 1. Cube Roots of Numbers. 288. The process for cube root is closely analogous to that for square root, and is suggested by the identity {a + hf = a^ + Za?h + ^ah^ + l\ Model L. 1/584277056. First we decide that the cube root is not less than 800; and subtract the cube of that : 584 277 056 512 000 000 WO = a 72 277 056 Call 800 a and the next part of the root that we expect to find I, Since we have subtracted a^, the rest of the cube of (a + l) is 'da?!) + ^ah^ + Z>3 = j(3a2 4- ^al + V^)\ and the remainder, 72,277,056, will be larger than that. Now if we knew the value of ^a^ + 3a^ + V^ we could find h at once by dividing. Although we do not know it, we are helped in guessing at h by the fact that Za? is much the larger part of the divisor. In this case da? = 1920000, and we conclude that h is 268 ALGEBRA, not less than 30; on the supposition that a = 800 and J = 30 we construct 5(3a^ + ^^^ + ^^) ^'^^ subtract. 584 277 056 612 000 000 72 277 056 59 787 000 SOO = a 30= b 19200^6> = 3^2 nOOO = 3aZ> dOO = W' 199296>(9 = 3a2 + 3a^> + 12 490 056 We have now subtracted the cube of 830; we may call 830 a, construct 3a^ and proceed as before, until there is no remainder, or until enough figures of the root are obtained. Model L. 584 277 056 512 000 000 72 277 056 59 787 000 12 490 056 12 490 056 WO 30 6 = I 19200^0 = 3a2 720^^ = Zab WO = V^ Ans. 836. a = 80^ b = 30 19929(9^ = 3a^ + 3ah + 2066700 = 3a^ 14940 := 3ab 36 = P a = 830 b = 6 2081676 = 3a2 + 3ab + In practice the figures in italics in the preceding problem are omitted, so that at each successive step we seem to be dealing with tens and units. And it often happens that INDICES; SURDS; MOOTS, 269 the first guess for J is a wild one, — particularly if the first figure of the root is small. Model M. |/56623104. 56 623 104 27 29 623 27 872 1 751 104 1 751 104 384 2700 = 3a2 720 = 3ab 64 = b^ 3484 = 3^2 _|. 3^^ _|_ 433200 = 3a^ 4560 = dab 16 = P a= 30 b= S a ■- 380 b = 4 437776 = da" + Sab + ¥ Model N. i/29.23. 29.230 000 000 27 3.0804 2 230 2700 = 3a2 2 230 000 270000 = 3^2 7200 = Zah 64 = b^ 2 218 112 277264 = 3^2 + 3«Z> + Z*^ 11 888 000 28459200 = Za?' 11 888 000 000 2845920000 = 3a2 3696000 = 3a& 16 = &2 11 398 464 064 2849616016 = 3^^ + 3«^ + V 489 535 996 a = 30 a = 300 5 = 8 a = 3080 5 = a = 30800 5 = 4 270 ALGEBRA. EXERCISE CXLIII. Find the cube roots of: 1. 51895117. 2. 63044792. 3. 9181846584. . 4. 7587307125. 5. 6946005312. 6. 2700. 7. 54. 8. 100. 9. 2. 10. 20000. Cube Roots of Algebraic Expressions. 289. For cube roots of algebraic expressions the work is just the same in principle, but the result can generally be found by inspection — unless the polynomial is very long indeed — if one is sure that the given expression is a perfect cube. For the first term and the last term of the cube root are respectively the cube roots of first and last straight products; and the terms next to each of those in the root can be seen by dividing the next term in the given expres- sion by the ^' trial divisor'^ (3^^^). Model 0. — The first and last terms of 8«^+12a8_i86i'?+13a6+54a5-51a^-10«3+63a2_54^_|_27 are respectively 2a^ and +3; starting from the first term, 3^2 would be 3(2^3)2= 12^6; so h would be 12^^ -^ 12^^ = a^. rearranging the terms in reverse order, so as to start with the last term, Sa^ = 3(9) = 27 ; then h would be - 54a -^ 27 = — 2a. The root would be ^a? + a^ _ 2a + 3. To verify this result it is easier to go through the motions of finding a cube root than to multiply out the third power. INDICES; SURDS; BOOTS. 271 Model P. S^+12a8-18a^+13a«-f54a« 2a» + a« - 2a 4- 3 -51a4+10a3+63a'-54a+27 12a« = 3a' a == 2a8 M^ 6a5 = 3a5 6 = a« 12^8 -f 6a' + a« a^=:6« -24a'4-12a«4-54a5-51a4 + lO^^ 12a« + 12a^ + 3a^ := 3a2 + 63a'-54<* + 27 - 12a* - 6a3 == 3a& a = 2a^ + a' 4a'=.b'^ b=- 2a -24^''- 24a« + 18«5 -f 12tt4_8^3 36a« + 36a5-63<*4 _|_ ig^^ _^ ^3^2 12a«+12a5-21a4-12a2+12a2=3a2 -54a + 27 18a3 + 9a2-18a=3a6 9 = 6^ 86a« + 36a5 _ 63a4 + 18^' + 63a5 [a = 2a» + a' - 2a] - 54a + 27 [6=3] EXERCISE CXLIV. Find the cube roots of: 1. x^ + 96x - 4:0x^ + 6x^ - 64. 2. x^ + Ux^y + QOxY + 160:^)y + 2402:y + 192a:^5 + 64^«. 3. 27a:9 _ 54^8 _p 53,^7 _ 71^6 _^ 57^5 _ 35^4 ^ 22x^ - 9x^ + 3x - 1. 4. Sa^ - 12a^ + lSa' - Ua^ + 9a^ -3^+1. 5. x^^ - dx^^y + 3x^Y + '^^Y - ^^Y + ^^Y - ^^Y + ^x'^f' - "IxY - ^ccY"^ + ^xf'' - y'^. 6. 3285092;6 - 480861:?;^ + 283383^V^2 _ ^06b^h^ 7. 8x^2 _ i22;ii + 42a;io - Qlx^ + 99^:8 - 117^'^ + 126:?;« - 108^5 + 81:^4 - 4:7x^ + "IXx^ - ^x + 1. *8. la^ — ia¥ + |«%^ + Qak^ + Sb^ — |a%^ — I2ab^ - Qah^ + lah^ + lah^. 9. 16^V2 - 72:^«^i/6 + 324a;y4/2 - 1623/3|/6. 10. x^ — dxY^^y^ + ^xY^^^y — y^' * Rearrange. 272 ALGEBRA. Series for Cube Root. 290. As in square root, if the given expression is not a perfect power, the result will be an endless series of terms. Model a. frr X, 1 - X x x^ 5x^ 1 I-3--9- 81 - . . . — X 3 = 3^2 a = 1 — a; = 3a^ b=- X ~ 3 x^ 27 -; = .^ x^ x^ ~ 3 + 27 3-2. + ^^: = 3a=' « = !-! x^ 2^3 x^ x' x^ x^ ■~ 3" + 9" = 3a5 --1^ 3 "^ 9 " 81 729 x' SI' = 52 bx^ x^ 27 "^ 81 x^ ^ 729 x^ 3 - 2^ - -3 2a;8 + 9 + - EXERCISE CXLV. Fi7id series for : H + X. 2. yl + 2^ + :z;2. 3. VS V2>x. 4. 1^1 - x^. 5. VI — X + X^, Fourth and Higher Roots. 291 . By utilizing in the same way the identities (a + by ~a^+ ia^ + Qa^^ + 4:ab^ + b^ (a+by = a^ + 6a^b + lOa^^ + lOd^b^ + ba¥ + h^ and so on. INDICES; SURDS; MOOTS. 273 it would be theoretically possible to construct a rule, analogous to those for square and cube root, which would serve to find any desired root. Practically, such rules would be difficult of application, and of no possible utility either in numerical work or in algebra; because the re- quired roots can be obtained more conveniently by other means, as will be explained in Chapters XIV and XV. CHAPTER XI. LITERAL EQUATIONS; GENERALIZATION. 292. Referring to Model I in Chapter I, we see that the problem may be stated as follows; A man who can row a miles per hour in still water finds that it takes him h hours to row up-stream to a point from which he can return in c hours. How fast does the current flow ? As stated in Chapter I this problem had 5 instead of a, 5 instead of h, and 4 instead of c] and further on we find five other sets of figures, each of which could be substituted for a, l, and c in the statement of the problem here given. Since we have now learned to perform all sorts of alge- braic operations upon letters as well as upon figures, we can take the statement of this problem as given just above, construct our equation, and solve it, thus : Model A. Let X = the number of miles per hour current flows ; a -{- X = the number of miles per hour man rows down- stream ; a — X = the number of miles per hour man rows up- stream. ® h{a — x) = c{a + x) (2) ab — bx = ac -\- ex same as Q @ ah — ac = bx -{- ex (z) -\- bx — ac ® ab — ac = {b -\- c)x same as (3) ^^ ab — ac aib — ^ b -]r c b + c) c 274 ITEBAL EQUATIONS; GENERALIZATION, 275 293. An equation, like 0, which contains letters (in- stead of numbers) to represent the numbers that would, ordinarily, be given in the statement of the problem, is called a literal equation ; the answer, an algebraic expres- sion, is in fact a formula, and indicates all the arithmetical operations that must be performed upon the given numbers to get the numerical answer to the problem for any par- ticular numerical case. Thus if the problem were stated : A man who can row 5| miles an hour in still water finds that it takes him 3|^ hours to row up-stream to a point from which he can return in 2 hours. How fast does the current flow ? — we should substitute for a, I, and c in the formula above, 51(31 _ 2) and get x = ^^ ^ = 3|^ — 2 = 1^ miles per hour. EXERCISE CXLVI. Similarly, hy substituting in the formula, get the values of the speed of the current corresponding to the other sets of values of a, h, and c given for this problem in § 26. Obtain a formula and substitute as before for example 1 in Exercise XL 294. The degree of a term in a literal equation is the number of unknown letters that appear in that term as factors; the other letters representing such numbers as would be numerically given in the statement of the problem to be solved by this equation. 296. In literal equations of the first degree with one unknown letter, the known terms are arranged on one side of the equation, and on the other side the unknown letter, say X, is then recognized and separated as a distributive factor. The othee factor is the coefficient of x. 276 ALGJEBUA. Model B. Ix + Wx -\- a^ = ZIH + Z'^ + ax. a? — 1)^ = ax — hx — 3a^x + 3^^^ x{a - b - 3a2 + 3Z^2) ^ ^s _ ^3 a^- ¥ ci^ j^ ah J^y^ ^ ~ {a - h){l - 3a - 3Z^) "" 1 - 3a - 3^> EXERCISE CXLVII. X , X a-\-l) / , zx , 1 , z 1. — n+^i J.=T~^ — Tz.- 2. (a+c>)a;H -=:?;-f«— J. a; — Z>a;~-6;:?; — a_2; — a 1 1 c a ~^ h ~~ abc ac aV 4. ac~ — ^ -X — bx = ap — 36x, q a ^ i-L I 7x 3(y^ ^Tc) -hx 1 6. {a ^ x){p — x) = 3{b — xf. x^ — Jix -\- Jc x"^ ~ px -\- q 7. X — h X — p (x - af , {x - by , {x - cy ,^ , , ^, a:2 k — x * X + a -^ X — a,x — b,x — c xia -\- b -\- cY a-\-b^ b -{• c^ c -^ a {a-\-b){b + c){c + ay LITERAL EQUATIONS; OENERALIZATION. 277 Model C. — To pay a sum of n dollars, p coins, some of value a cents each and the rest of value l cents each, are found to be sufficient. How many coins of each kind ? Let X = the number of a-cent coins; p — X = the number of Z>-cent coins. (\) ax -{- hp — hx = lOOn (z)ax — bx = lOOn — hp (D x{a — b) = lOOn — bp ^^ lOOn — bp, » . ,, , . , (A) X =^ 7-^ (no. of coms worth a cts. apiece) ^5. 100^^ — bp ap — bp — lOOn + bp ®p — x —p =-^ = -^ ^ ^ '—^ ^^ ^ ^ a — b a — b _ap— lOOn, a — b -(no. of coins worth b cts. apiece) EXERCISE CXLVni. 1. A father is a times as old as his son ; p years ago he was b times as old. What are their ages now ? 2. There is a difference of q cents a pound in the price of two commodities; a pounds of one and b pounds of the other amount to h dollars. Price of each commodity per pound ? 3. Paid s dollars with coins worth a cents and coins worth b cents ; d more coins of the first kind than of the second. Number of coins of each kind ? 4. I have m dollars, in nickels, dimes, and 3-cent pieces ; a times as many dimes as nickels, b times as many 3-cent pieces as dimes. Number of each ? 6. A merchant has grain worth a cents per peck and other grain worth b cents per peck; in what proportion must he mix m bushels so that the mixture may be worth q dollars in all ? 278 ALGEBRA, 6. A man who can row 5 miles per hour in still water finds that he can row down-stream in a hours to a point from which it takes him h hours to return. How fast does the current flow ? 7. A man who can row a miles per hour finds that it takes him r times as long to go up-stream as to go down. Find the speed of the current. 8. Bought cards a for a nickel, and the same number of other cards J for a nickel; then sold them {a-\-l)) for a dime, and lost p cents. How many cards were bought ? 9. Walking, a certain distance can be accomplished in a hours; riding, in h hours; walking half the time and riding the other half, how long does it take for that dis- tance ? 10. The fore wheel of a carriage has a circumference of a feet, the hind wheel of l feet; when the fore wheel has made c turns more than the hind wheel, how far has the carriage gone ? 11. A can do a certain job in a days, B in Z> days; how long for both ? Factoring Literal Quadratics. 296. In solving literal quadratics the equation should first be arranged in the standard form ax^ ~\- Ix -\- c = and then factored by the method of cross-multiplication.* * In discussing literal equations it is important not to confuse the letters a, b, and c which represent the three coefficients of the quadratic with similar letters which may occur in the statement of the equation itself. LITERAL EQUATIONS; GENERALIZATION, 279 Model D. Solve x - o? = '^0' -i- V' - -^^,. x-\- a^ ® x^-'^o?x-Wx^Za^-\-Za''¥ © -Id'x-Wx-'itd'-^Za^^ ® a; - 3^2 = I from ® (5) ^ + (^2 _ ^2)*^ ) by Ax. A. (See Chapter Y.) @x — 36^2. ^. _ ^2 __ ^2^ ^^^^^ EXERCISE CXLIX. 1. x^ - (3^ - 4)i?; + 2^2 _ 5a + 3 = 0. 2. :^2 + (2a + V)x -]- a^ ^ ah - W = 0. 3. x^ -^ {a - h)x - (6^2 4- I3a^> + 6^>2) ^ 0. 4. a;2 4- 7:^; 4- 10 =: 4.ax + 17a - Sa^. x^ - {a? + 13Z^2) 6. ^2 - 60(^2 - J2) ^ ^(196 _ 11^). a^ -\- a — 1 + X a{a^ — x) 2 a^ + a — 1 7. a; + a2 ^ ±- 1 + ^ %, X -\- 1 — a -{- X 2 ^ or 12a ~^ 9 ~ ~T6 X + a^ I , 137a2 10. -J- = J^ + -IT 297. The operation of factoring quadratics becomes more difficult when neither straight product is a monomial. Model E. (9^»2 - 4a2)a;2 + {4:ah - 4.a^)x - {a^ - 2ah + J)") = 0. {Sh + 2a)x + (a - h) (di — 2a)x — {a — h) h — a h — a . X = ^ Ans, 2a + W 2a - U 280 ALGEBRA. EXERCISE CL. 1. (a2 ~ V^)x' - 2{a^ + i^)x + ^2 _ 52 = 0. 2. {a^ + ^a + 2)x^ 4- 5a; + 2 = 2a^x + 3a - a^. 8. a\2x^ 4- 3a; + 1) - Z>''^(a;2 + 4:X + 4:) = ab{x^ + 2x). 4. a\x^ - 1) = «^(2a;2 _ 7a; - 1) + ^>2(3a;2 _|_ 5^ _ 2). ^2(2a; + 3) __ d\2 + 3x) __ ah{x^ - 1) ^' 1 + 2a; ^ a; + 2 ~ (a; + 2)(1 + 2a;)* a2(5a; + 3) h\3x + 5) _ ah{x'' - 1) ^- 3 + 2a; "" 2 + 3a; ~ (3 -r 2a:)(2 + 3a;)" 7. x{iox+ll){a'+h'')-3lal{x^-\-l)z=:3ba'+h\(6-2^x'). {x^ - l){a^ + ah + P) _ Sab alx (a + ^)^ ~ a — 1)~ a -\- V 9. 2a(l - 10a;2) + 2 + ha^x? =: (8 - 5a - 76^^)^, «J + 6 2«(a;2 +1) x^ -\- a -I -\ 10. a + 2> ~ (a + l){\ - a;2) a;2 - 1 Solving Literal Quadratics by Formula. 298. Equations which are very difficult to factor by inspection may be solved by the quadratic formula. Model F. 2«*-5a3-27+15aa;(a2+3)::z2«a;2(a34_^_|_10)4-{17a2_i8):z; Here a = 2a^ + 2a^ + 20a J = _ 15^3 + Yia" - 45a - 18 c = _ 2a^ + ha^ + 27 J2 - 4ac = 16a8 - 40a^ + 241a«- 390aS+ 1023a* -QOOa^ + 197a2 - 540a + 324 4/^2 - 4ac = 4:a^ - ba^ + 27a2 - 15a + 18 LITERAL EQUATIONS; GENERALIZATION. 281 __ —h±V¥—4:ac ^" ¥a 2^3 + a2 _|_ 3^ I 9 3 _ ^ ^ = -irV — A \ . -.A — 5 or a; = — — ^. ^7i5. EXERCISE CLI. 1. («» - U^)x^ + 10«a; - (4a3 - 7a + 3) = 0. 2. aV + {d^ + 2a):^; + a^ + 6 = Saa;^ + (2^2 + 9):z; + 7a. 3. (2a3+ 5a2)a;2+ (Sa^- 2)a; = 2a3- 5a2+ 4: + ax - 2axK 4. (a^ - 1)^2 + (4a3 + 4)ic + 3a^ + lla^ 3= 4 + (a2 - a)ir2 + (2a - 4a2):r. 5. (3a2 - 4) + (2a2 + 3a + 2)x = {a^ + a^)x^ + a^+'^a^x. 6. a\x^ 4- 4a; + 3) - (9^; + 5) i?; = x{x - 4a2) + 25a + 6. 7. 2(2a;-l)(:r+2)4-a2(2iz;+a)2=-2a2^-a(4a;2+2ic+3). 8. a\2x^ + 5^ + 3) - (9a - %)x^ - 143a + 28 = X + (67a - ba^)x. 9. a\%ax^ - 3) + (5a2 + 7a-2)iz; = 2(a4+a2;2)-5-7a3i«;. 10. 2a3(a + x^) + %x{ax ~ 1) = 11a + 10 - a;(4a^ + ^a^ + 14a2 + 14a). 299. With practice even these may be factored by in- spection. In Model F, a and l may be presumed to have integral factors which can be found by trial; the factors of the straight products having been found, it should then be possible to arrange them so as to give the correct cross products. For the expert student it may safely be said that the most economical way to factor a literal quadratic is by inspection. 282 ALGEBRA. Literal Simultaneous Equations. 300. Model a ® (^x^ + 'b.y = c^ ® ci,^ + Ky = 0^ (D a^a.x + aj)^y = a^c^ ® X ^, ® a^a^x + afi^y = a^c^ ® X a, ® {(^A - (^A)y = «i^a - ^.Oi ® -® Similarly we obtain <^A - <^A It is generally cheaper to eliminate for each letter inde- pendently than to substitute a complicated literal expres- sion for one, to find the other. EXERCISE CLII. 1. c,x + 8^y = p,. 6. X + ay = h. c,x + s,y = p,. hx -{- y = a. 2. x-^ a,y ^ 1. 7. ax -\- h = y. a^x^y ^ 1. X -^ y = a. 8. y = ^:« + ^1. 8. ^ + y = (^,' y = ^,^ + ^a- hx -^ y = a^. 4. a b 9. a, 5, x + y = a. - + f- = l- a, I, 5. X + y = a. 10. a{l^x^)=:x^'\a + \). X - y = h. ax — by- LITERAL EQUATIONS; GENERALIZATION, 283 11. a^x + h.y + c,z = h, \ a^x + ^^y + o^z — Ic^K Find x only. 12. a^ + Z>2/ + ;2 = 6'. 13. a; + ^^ = ^. ^ + ^y + ^^ = ^- y -\- hz = c, bx -{- y -\- az = c. z -\- ex — a, l^. X — ay — h, i^, X -\- y -{- z = a,. y — Iz — c, X — y -{- z ^ a^, z — ex — a, X — y — z =z a^. 16. 2. + ^-.v-^. + |. 17. 1 + ^ + ^=3. y — X + 2z = a^ — ~, =5. « , y c a ^-^,+ y = a,-- _^_. 18. i^ + .V + ^ = ^' 19. i^^ + 2y = « + 45 + 3. i<; — y= ^j — 2a^ + ^3- y 4" ^^ == ^^ + ^^ + 2- ^+^!/+^^==<^3~"^^i"~^2- z -\- '2x — 2a + 3c + 3c 20. a; + 2/ + ;<; + ^^ = 2« + ^5- X — 2y -\- z =^ b — a ^^ w -\- 'dy — 2a. X -{- y — z — w ~ 2a — 2b — 2c. 2x{3y -j- x) _a 21. 2x + 3y = a; 9 ^+3^ 4 a+b X o , A ''■~^ = ^P^' 2^ + 4^ = «- -, X? — ab y + h 23. a; + a — 5 = ;?/; = "^ ' . ^ y — a 2 a + b 2a + b 2y + 3a 24. X — 3a = 2y\ 5—^7 ^^— - = -^—^ . -^^ 2y+-a^ a a 25. ^ = n^ + Z> + ^: -^^ + ^- = 2. ^ ' ' 2/ y — ^ 284 ALGEBRA, a -\- ^h y — X _x -\- a X -^ a — 4:b; X -{- )lb X — 2b a b X b 2%. X -{- y ^^ ba] xy \b ^=^ ha —b, 29, x?" -\- axy + ay'^ = xy ; x Ar y ^=^ Sa, 30. x^ -{- y^ =^ a^xy ; x -{- y = bxy. Constants and Variables. 301. In the Chapter on Elimination we saw that in the equation 2x — 3y = 6, while X and y are restricted, still the value of x may vary from any negative number to any positive number; and the value of y changes with every value of x. Similarly in the equation ax — by =^ c the values of x and y could vary, subject only to the restrictioii that, whatever the value of x, y must be given by the formula y = — 7— ; and a, b, and c, like the 2, 3, and 6 in the numerical equation, remain unchanged throughout. 302. In any problem the numbers whose values are given are called constants of the problem; and those whose values are required are called variables. Thus in Model A of this chapter the numbers «, b, and c are the constants; the (unknown) speed of the current is the YARIABLE. As the speed of the current is supposed to have a certain fixed value, which we are trying to find when we start to solve the problem, it seems strange to call that speed a VARIABLE ; but wc may state for this problem three LITERAL EQUATIONS; GENERALIZATION, 285 equations of condition, in any one of which, taken separately, x^ or y, or z, is a variable, subject to a restriction. If we let y = the speed of the boat down-stream, and z =z the speed of the boat up-stream ; then y = a -{- x ; z = a — x; bz = cy. So far as the first of these equations is concerned, x may have any value, so long as y is greater than it by a. But no algebraic restatement can make a, or b, or c appear to be vaeiable ; they are the constants of the problem. Discussion of Problems. 303. A problem is said to be generalized when one or more letters are used instead of numerical constants. 304. The investigation of a generalized answer, with a view to determining the effect upon it produced by different values or relations imposed upon the constants, is called Discussion of the problem. LIMITING VALUES. 305. In discussing a generalized problem we often meet values which by the ordinary rules of algebra have no mean- ing whatever. The two most important of such results are, first, that obtained when the denominator of a formula becomes for a particular set of values for the constants; and second, that obtained when the numerator and de- nominator both become 0. It is usual to consider the following three cases together : (i) Where the numerator is 0; (ii) Where the denominator is 0; (hi) Where both numerator and denominator are 0. These are called limiting values of the formula under discussion. 286 ALGEBRA. Model H.-^If we set _ x^ — Sx -{- 15 y ~ x^ - Ix + 12 we obtain the following limiting values : 25 - 40 + 15 _ (i) X = b] y = (ii) X = ^] y :=^ (ill) a; = 3 ; 2/ = 25-35 + 12 2 16 - 32 + 15 _ _ 1 16 - 28 + 12 ~ 9-24 + 15 _ 9-21 + 12 "" In considering (i) we may conclude at once that since if is divided into 2 parts, the result will be zero; and since if 2 is divided into zero,* the quotient and the remainder will both be zero ; then the value of - must be 0. When we come to (ii) we have no such easy task, since to divide 1 by is meaningless; and still more meaningless, in (III), IS -. We have to investigate then the three expressions (i) — ; (")o; (™)o- Let us consider at first the three expressions (i) — ; (ii) — ; (ill) — ; where x and y are variables, and a and h are positive constants. X (i) So long as x is positive, — will be positive; as x decreases, — decreases also; when x passes over from a * As, for example, wlien one is asked how many times lie can fill a two quart measure from an empty bin. LITERAL EQUATIONS; GENERALIZATION. 287 X small + value to a small — value, — does likewise. At the instant when x, in its progress from + values to — values, X assumes the value x = 0, — 2X the same instant assumes a the value 0. We may conclude, then, that a and also we may say that X (i) — is the limit which — approaches as the numerical ^ ^ a a ^ value of X is indefinitely diminished. X If a had been a negative constant, then — and x would have opposite signs just so long as x was -\- ; but at the X instant when x passed from -f to — , — would pass from — to -f , and, as before, we should have — = 0. (11) If now we suppose y to begin with a considerable + value and then to diminish, — would not diminish, but y increase, on the principle that, the dividend remaining the same, a smaller divisor gives a larger quotient. If y becomes — , — also becomes — . y As y passes through zero, from the region of + numbers to the region of — numbers, - passes also from the region of + numbers to the region of — numbers; but it does KOT pass through 0. We conclude then that (11) ^, like 0, is a boundary between -}- and — numbers. 288 ALGEBRA. Again, by causing y to assume a very small value, posi- tive or negative, and to become successively smaller and smaller, we cause - to assume a very lakge value positive or negative, and to become successively larger and larger. By choosing y suitably we can cause — to assume a value greater th^n any assignable value, however large; and still, if y is then taken weaker to in value, — becomes even ^ ^ y larger. Hence we also conclude: (ii) ?r is a limit towards which large positive or large negative numbers may approach if we suppose them to increase indefinitely. The symbol for this limit is oo and its name is infinity. (n)^-co. (hi) If we suppose the expression — equal to any con- stant h, we could still allow x to vary, and to assume any value whatever, positive or negative; y would always be determined, for any particular value of x, by the formula X y = --. AYe may cause both x and y to assume values less than any assignable quantity, however small, without X destroying the equation — = k. In other words, we may X . cause — to approach as near as we please to — without X destroying the equation — = h. These considerations will all hold good whatever the value of k ; hence we may LITEBAL EQUATIONS; GENEBALIZATION, 289 conclude that the symbol — does not of itself determine any value. (hi) jt is indeterminate. This conclusion does not mean that — = aky num- ber ; only that when substitution in a formula leads to this result, some special reduction must be resorted to for determining its actual value, if the value can be determined at all. Model H. — In the expression ^ ~ x^ -lx+ 12 .- _ ^ _ 9-24 + 15 _ '^ ^ ~ "^^ ^ ~ 9 - 21 + 12 ~ ' _ x? — ^x-^1^ __ {x — Z){x - 5) _ a; - 5 _ , ^ "" x^ -7^ + 12 ~{x- 3)(^ - 4) '^^^^~ ' Problems for Discussion. 306. Model I. — A cistern can be filled by one pipe in 12 minutes, and emptied by another in 36 minutes. How long will it take to fill it with both pipes open ? Generalized statement : A cistern can be filled by one pipe in a minutes, and emptied by another in b minutes. How long will it take to fill it with both pipes open ? X = number of minutes to fill, both pipes open. — = fraction of cistern filled by first pipe in one minute. 290 ALGEBUA, — = fraction of cistern emptied by second pipe in one minute. ^1 1 1 ^•^ a X @ bx — ax = ab © X abx ® x = ^^ ®^{b-a) Now if b is greater than a, this answer is positive, and its meaning is natural and obvious. Thus in the statement of the problem first given, « = 12, J = 36; then X = :^ ^ ^ — 18 minutes to fill. 36 — 12 If b is less than a, the answer is negative; on reflection, we see that if the outlet emptied in less time than the supply- pipe filled (the emptying would then be more rapid than the filling), it would be vain to expect the cistern to fill with both open. Assuming a = 24, b = W, we obtain 24 X 16 16 - 24 = - 48. We miglit say that wliile in the first case the instant of being full is 18 minutes after the instant of being empty, in the second case the instant of being full is 48 minutes before the instant of being empty. If b were negative, the numerator, ba^ product of unlike factors, would be — ; the denominator, b — a, sum of two negative terms, would also be — ; so x would be plus. Now for J to be a negative number is the same as to say fill in- stead of empty; and with both pipes filling, the answer would evi- dently be -j- and reasonable. Thus, if a = 28, 6 = — 21, - ^8 X (- 21) _ 28_X 21 _ ^~ - 21 - 28 ~ "*■ 49" ~ LITERAL EQXIATIONB; OBNEBALTZATION. 291 If a is negative, the numerator ah will be negative and the denominator l — a will be positive; so x will be — . But for a to be — is to have the supply-pipe also an outlet ; and in that case (as before when J < a) a; is — and the cistern is being emptied instead of filled. If a = Z>, X assumes a limiting value; t — a — and ah a^ h — a In this case one pipe empties as fast as the other pipe fills, and the cistern will never be any fuller than it is now. EXERCISE CLIII. In a similar way generalize and discuss the following prohlems : 1. A boat travelling 10 miles per hour passed south by- Highland light at noon; a steamer pursuing at 13 miles per hour passed on the same track 3 hours later. When will the steamer overtake the boat ? 2. Of two pipes running into a cistern each could fill it in 21 minutes; the waste-pipe could empty it in 35 minutes. How long would it take to fill the cistern with all pipes open ? 3. A person who can row 5 miles an hour in still water rows 21 miles down-stream and back in 10 hours, the stream flowing uniformly all the time. How many miles an hour does the stream flow ? 4. How much rye at 50 cents a bushel must be mixed with 50 bushels of wheat at 80 cents a bushel to make a mixture worth 70 cents a bushel ? 5. A bicyclist starts a ride, going 22 feet per second; half a minute later a second rider starts 60 yards behind the first man's mark, riding 25 feet per second. How 292 ALGEBRA. far from the first man^s mark will the riders be side by side? Discuss the solutions of the examples in Exercise GXL VI I L DISCUSSION OF EQUATIONS. 307. In the pair of linear equations (\) ax -^ ly = c @ px -{- qy =^ r we obtain by elimination cq — hr ar — pc aq — hp^ «^ — Ip ' Of these two answers, x — ii cq — hr = and aq — hp 4^ * ; ^ = if ar — ^c = and aq — hp ^ 0. If aq — dp — and cq — hr ^ 0, x = /0 15 18 18* I.e., not imaginary. LITERAL EQUATIONS; GENERALIZATION. 295 If W' — 4tac is — , Vh^ — 4:ac will be imaginary, and the values of a and /3 will be conjugate imaginaries.* Thus in 2a;2 + 3^ + 5 = - 3 ± |/- 31 X = . EXERCISE CLIV. State the character of the roots {whether real or imagi- nary, whether equal or different) in the following equations, without solving them ; and in each case give reasons : 1. 100a;2 -^Sx+ .001 = 0. e.f ^^ + ^p'^ = ^P^- 2. x^ + 2x+S=zO. t,\ p=:X - x\ 3. x^ + 2x — 3 = 0. 8.t x^ +p = 0. 4.t px^ + qx - r = 0. 9. 1 - VSx^ _ 2^ + 3 = 0. 9 ^ 16 -X 28 = ^' '''-^+T 5. 7^2 ^ 3^ ^ ^ ^ 0. 10. -^^ , f - lOo; = 15. Zero Coefficients in the QuadratiCn 313. If c becomes 0, while a and b do not, -b+Vb^- ■ o_ _ 2a^ h a 2a ~0 -b-Vb'- 2a — which are obviously the answers to ax^ -\- hx ^ 0. * See §274. \ In this example suppose p, q, and r positive integers. 296 ALGEBRA. 314. If ^ becomes 0, while a and c do not, a = — '—- and p = 2a 2a which reduce to Such would obviously be the answers to ax^ -\- c — 0. 315. If Z> and c both become 0, __ ^2a -h+ Vb^' -4.ac + VO = 2a 2a ^ - b ~ i^b^ - 4ac ^ ^= 2a = 2-«^'^ which are obviously the answers to ax^ = 0. 316. It a becomes 0, the equation reduces at once to an equation of the first degree, I)x-\- c = 0^ and x — — —, But it is interesting to note what becomes of the two answers, a and /?, as the coefficient a assumes values that are very small as compared with b and c. When a becomes YERY small indeed as compared with b and c, the quad- ratic becomes very kearly the same as the linear equa- tion bx -\- c =^ 0^ but still has its two answers, a and ^. Now in ax^ -\~ bx -\- c = ii a = 0, - b + Vb^ a= ^— ^- - b - V^ --2b P = 7^ = -7r- = «^ LITERAL EQUATIONS; GENERALIZATION. 297 The value of a cannot be determined from the formula, in this case; but if we rationalize the numerator of the formula by multiplying it by its conjugate surd,* we get ^l\ \/b''-4cac i-b- V¥-4:ac) _ h^-jh'^-^ac) la ' [-b^ Vb^-4:ac) '^a{-b- Vb^-4.ac) 4:ac 2c ~2a{-b- Vb^-iac) ~~ — b — \/b^ - ^ac If now in this formula we let a — 0, we get 2c c a -b-b- b' The meaning of these conclusions is this : in any quadratic if the coefficient of x^ becomes very small, as compared with the other two coefficients, one answer becomes very large, and the other answer becomes nearly equal to the value x would have if the first term were removed altogether. Model L. — In the equation \in z=zl, we have n' = — .502 and ^ — — 200 (nearly); if 7^ = 2, a: = .5 (very nearly) and § — — 20000 (nearly); and the larger we take n the more nearly a becomes = .5 and the larger ft becomes. 317. \i a and b both become 0, while c does not, b^ \fV — 4:ac 2a 0' but by the other formula a = , = I^_? = oo b + Vb^'-^ac - * See § 274. 298 ALGEBRA, and by a second formula for [3, similarly obtained, — %G —2c /? = -, =^=^ — — - — = 00 . b-Vb''-4.ac In this case, if a and b were really zero, we sliould Lave c (a con- stant) equal to zero ; that would be, of course, a meaningless state- ment. Thus- O.x'^ -\- 0.x -{- 5 = 0, being the same as 5 = 0, could not be true. But if a and b both became veky small as compared WITH c, it would take larger and larger values of x to satisfy the equation. 318. If a and c both become 0, while b does not, 2a 0-0 2o Q by the other formula, a = -— =:^ = -- = b+Vb^'-4:ac ^b- ^.'^ o -b-Vb^-4:ac -2b while p = , = —7—- = 00 . If a and c both become very small as compared with &, then a will become very small and /5 will become very large. EXERCISE CLV. Determine the limiting values of x as n increases in- definitely : 2^ 1 1. 10"^2 4- -— + 10" = 0. 2. ^''x'+ 2{3)-x+ - =0, J_U o 3 3n^2 _ 2» + »a; + ^„ = 0. 4. 10 V +^^ + ±=0. CHAPTER XII. PROPORTION AND VARIATION. 319. An equation between two ratios is called a propor- tion ; the theory of such equations can be investigated by means of a generalized type, and the laws so obtained ex- pressed as theorems. The general type of a proportion may be represented by Of these four quantities x, y, a, h, x and h are called the extremes, y and a the means; x and a are the anteced- ents, y and l the consequents; x and y form the first ratio, a and 1) the second ratio. The Three Simplest Theorems. 320. If we assume to be true, equations @, (3), and ® are at once obtained, and the theorems they express are consequently true of any proportion. I. The product of the means is (^hx — ay Q) X hy equal to the product of the extremes. aha II. The ratio of the antecedents is equal to tl consequents. XV u ®-=: - (T) X — is equal to the ratio of the 399 300 ALGEBRA, ^ y h , ^ III. The reciprocals of the ratios ^^ X a ^^ are equal. 321. Of these three theorems the first enables us to find any term of the proportion when the other three are given, — a method known to our grandfathers as the ^^Rule of Three. '^ The second theorem is known as the ^^Law of Alternation," and the third as the ^^ Law of Inversion/^ These three are the most elementary transformations of a proportion. EXERCISE CLVI. 2 7 1. Find X in the proportion — = — . 7 2. Find x in the proportion {x — 10) : 2 = 3. Find x in the proportion X -\-^ 2^; + 3 _ a; + 2 6 ^dx — 1' 4. By what successive transformations may we obtain from X : y ^=^ a : h the following forms ? X \ a = y : t h : y = a : X y '.h •=^ X \ a 5. Prove that ii x : y ^ a : d, then x : my =. a : mb. The Other Theorems of Proportion. 322. Keturning to the generalized proportion ^ X a PROPORTION AND VARIATION. 301 we may represent the value of each of these two equal ratios by some letter, say r. ^ y i Hence x — ry and a = rb. Then ®''-+^.^'i±y^':+l=.'Ltl . y y 1 ^ X -\- y r -{-1 a -\- h and (6) IV. In any proportion equal ratios are obtained by dividing the sum of the terms of each ratio by its ante- cedent or by its consequent. This is called the Law of Composition. 323. Again (7) = — ^ = -^i — = —r- y y 1 ^ ^^x — y r — 1 a — h and (D ^ = — ^- = V. In any proportion equal ratios are obtained by dividing the difference of the terms of each ratio by its antecedent or by its consequent. This is called the Law of Division. 324. Again ® ?i±^ - ':^ + ^ _ M^l _ a±. X — y ry — y r — 1 a ~ h that is — —^ = — ^■ X — y a — h VI. In any proportion equal ratios are obtained by dividing the sum of the terms of each ratio by their differ- - ence. This is called the Law of Composition and Division. 302 ALGEBRA. 325. Of the four terms of any proportion, two are said to be homologous if they are both antecedents, both conse- quents, or both terms of the same ratio. 326. Homologous terms, if not terms of the same ratio, may always be made such by Alternation; and since the value of a ratio is not altered by multiplying antecedent and consequent * by the same number, we have this trans- formation : yil. In any proportion, if any pair of homologous terms be multiplied each by the same number, the resulting proportion is true. 327. Keturning to our first proportion, — VIII. In any proportion, if like powers be taken of all four terms, the four results will be in proportion. 328. If we have two proportions x^'.y^ — a^ : h^ ; and x^\y^ = a^:h^\ then IX. The products of corresponding terms will be in proportion ; that is, multiplying the two equations, ^A'^i^. = «A:^^^.• 329. As in VI we may prove that if X a p h d _ y~b~q~k~s~''' then x^_ p^ + n + cl+ ._^ ^ x_ ^^^^ .^^ * Numerator and denominator. PROPORTION AND VARIATION. 303 X. In any series of equal ratios, the sum of all the ante- cedents divided by the sum of all the consequents is equal to the ratio of any antecedent to its consequent. The equation — = — is sometimes read '* As aj is to y, so is a to 5 " ; ^ y b or, '*x is to y, as a to 6'* ; and tLis sentence is the expression of tlie statement that the four quantities, x, y, a, and h, are in propor- tion; or that X and y are proportional to a and b. Instead of the equal sign, the sign : : was generally used; and the proportion was some- times called an " analogy." Dropping this cumbersome phraseology, much of the difficulty of the subject of proportion goes with it, and the different theorems are seen to be so many short cuts in the handling of a very simple type of fractional equations. A very complicated and difficult definition, known as Euclid's defi- nition of proportion, is gradually passing out of use. EXERCISE CLVII. Assuming that x :y = a -.ly establish the foJloiving equa- tions : 1. inx : mxy z= a : hx, 2. xy : ah = y^ : h^, 3. Icmx : myz — ok : hz, 4. px^-{-qy'^:y^—pd^-\-ql)^ : V^, 6. mx -\- y : ma -\- h =^ px -^ y : pa -\- 1). 6. px -{- qy : px — qy = pa -\- qb \ pa — qh, 7. x^ + 2/ : ^2 _^ 2h^ = {x -\- yY : {a + b)\ 8. 2a;2 + 3^2 . ^^z _|_ 3^2 ^ ^y . ^2>. 9. "^x^ - 3^2 . 2^2 - W = 2x^ - ly^ : 2a^ - W, 10. 2:?;2 _ 3^.y _ 2^2 . 2^2 _ 3^j _ 2^2 =2^^^+qy^'^pct^+qV^' Assuming that a \x ^=^ b : y = c : z, prove : 11. a -\- 3b + be : a — 2c = X + 3y -{- 6z : X — 2z. 12. px -\- qy -{- z : px + qz = pa -{- qb + c : pa -j- qc, 13. x^-\-y^—xyz:xy'^-\-yh-\-z^x=a^-\-b^—abc: aW-\-b^c-\-c^a, c? -^ W — be ac — ba 14. x^ + y^ — yz xz — xy' 304 ALOBBRA. X -\- 2y -\- z _ Vx^ + y'^ -\-~ z^ — 3xy Use the theorems to reduce the folloiving equations : 16. Apply to the proportion 2a+ bh + c + U _ 2a + bh - c - M '^a — bl) + c — M~ "^a - bt) — c -\- M the theorems VI, II, YI successively. 17. If 2; -h 2 : 7 = y + 3 : 9, find the ratio oix-bioy -0^. 18. Apply the law of Composition and Division to the ^X^+2X+1 '^X^ — X+1 r^, equation |^rz:^^-^ = 2^M^^-^- ^^^^ ^^^^^ for X. 19. Given x^ + 1 : 2x' =: 97 : 72; find x. 20. Solve 2a;2+10a;+53 : 18a:+ 45 = 101 : 99. (Apply VI.) 330. A proportion may be formed out of three quanti- ties if the first divided by the second gives the same ratio . as the second divided by the third. In that case the sec- ond quantity is called a mean proportional (or geometric mean) between the other two; and the third quantity is called a third proportional to the first two. In any proportion the second consequent is called a fourth proportional to the other three quantities. For a mean proportional the order of the extremes is immaterial; for a third or fourth proportional the other terms must be given AS they are arrakged ijs" the PROPORTION". 331. When terms in a series are so chosen that the ratio of any consecutive pair is the same as the ratio of any PBOPORTION AND VARIATION. 805 other, the quantities are said to be in continued proportion (or in geometric progression). Thns 16;8;4;2;l;|;i; etc., are in continued propor- tion ; so are a, J, c, d, e, /, . . . if a \ h =^ h : c =^ c \ d ^^ d '. e ^=^ e : f =^ , . . Of a series of terms in continued proportion any term is a mean proportional of the two terms adjacent to it. Thus in the two examples cited, 4 is a mean proportional between 2 and 8; c is a mean proportional between b and d. (See further in the chapter on progressions.) EXERCISE CLVIII. 1. Find a fourth proportional to 2, 3, and 4. 2. Find a fourth proportional to 3, 4, and 2. 3. Find a mean proportional to 2;^ and 9^. 4. Find a third proportional to 2^ and OJ. 5. 18, X, and 50 are in continued proportion; find a fourth proportional to them, eliminating x. 6. Find a third proportional to a^ and Zd^h] to ^aW and a?\ and of these two results find the geometric mean. 7. Of several numbers in continued proportion, if the second and third are 4 and 6, what is the fifth ? 8. Find a number to which if 2, 12, and 17 be sepa- rately added, the three results are in continued proportion. 9. Of two numbers one is 7 greater and the other 3 less than their mean proportional. What are the numbers ? 10. Of four numbers in continued proportion, 8 is the least and 27 the greatest. Find the other two. VARIATION. 332. In any algebraic investigation there may be two kinds of quantities: variables, whose values are subject to 306 ALGEBRA, change, either continuously or at intervals ; and constants, whose values are understood to be fixed and unchangeable throughout the investigation. Thus in the expression x^ -Sx-\- 15 x' - 7^; + 12 X may assume any value we choose to impose upon it; while 8, 15, 7, and 12 are constants. 333. The relation between the variables x and y in the equation X = ley {when the constant h is unknown) is sometimes expressed x cc y; to be read ^^ x varies as ^." A value of x and a value of y which when substituted together in this equation make it a true statement are called simultaneous values, or corresponding values, of x and y. They may be represented with equal suffixes, thus: x^ and y^, x.^ and y^^ x^ and y^ are simultaneous pairs, and we have the equations ^, = ^y.\ J = >^ Hence — = — ; or by alternation - = — . y. y. ^, ^. 334. If one quantity varies as another, a proportion may be formed by using, for homologous terms of the pro- portion, corresponding values of the variables. Model A. — The weight of wire of a certain kind varies as its length; if 1 metre of it weighs .8 grams, how much will 2.35 metres weigh ? PROPORTION AND VARIATION 807 Here/, = 1; w, = .8; /, = 2.35. 1 : .8 = 2.35 : w^ ^^ = .8 X 2.35 ~ 1.88 grams. Ans, EXERCISE CLIX. 1. li X varies as y, and x = ^ when y = 10, find the value of X when ^ = 11. 2. U X oc a, and x = 1.09 when y = .7, find y when a: = 23. 3. If X ex z, and z = p when x= q, find :?; when z = q, 4. If a a Z>, and «^ = 5'''^ — j.;^ when I) = §''^, find Z> when a=pq{p-q), _ 5. If h a ^, and ^ = 1^3 when h =■ Vb, find h when 335. One variable may vary as the reciprocal of the other, or as its square or cube, 'or as any algebraic expres- sion involving the other variable. The proportion will then be constructed by taking for homologous terms the same algebraic expression with the different values of the variable. Thus \i y i? sV? • • • 10. Find the last term and the sum of the terms in an A. P. of 11 terms of which the first two are a — x and a-y. Eliminating with the Formulae. 343. If any three of the five constants of an A. P. be given, substituting in the formulae (I) and (11) gives two equations with two unknown letters, which can be deter- mined by elimination. Model B.— Given ^ = 16; ^ = 4; 5 = 88; find n. © Z =: 16 + (w - 1)4 = 12 + 4/1 © 5 = 88 = |(16 + 1) @ 88 = -(28 + 4^) subst. ©in © ® 4^^2 + 287i = 176 ® X 2 ® 7i2 + 77i - 44 = ® -4- 4 - 44 ® (7i + ll)(7i-4) = 7^ = 4; n = —11 Of these answers ^ = 4 is the only one that satisfies the conditions of the problem. THE PROGRESSIONS. 313 EXERCISE CLXII. In the following examples find the constants that are not given : a d I n s 1. 5 ^ 10 2 3 25 2. 38 3 8 -3 5 1572 4. 787 80 5. 6 -34 -294 6. 11 .... .... 36 -2754 7. 4 57 16 8. -7 -36 -92 9. 397 200 19700 10. 11 13 182 11 11 22 231 12. 13. 14. 15. I 7 9 1- 100 "'9771* * 24* 37 - 6 11 16 if 5 17 3 17. 18. 19. 20. tV 25^ f -9f 21 25 - 993 -1 - 181i^ 21. i n 33| EXERCISE CLXIII. Determine formulm for each constant in terms of three others, as follotvs : 1. Find 5; given a, d^ and L given a, d, and n, [given a, I, and n ; formula 11]. given d, I, and n. 2. FindTi; given a, d, and I. given a, d, and s. given a, I, and s. given d, I, and s. 314 ALOEBBA. 3. Find Z; [given a, rZ, and 7i] formula i]. given a, d, and s. given a, n, and s, given 1. Again, suppose r = -J. n=^L^Tj~^"^'3+"9 "^27 + 8l"^243"^729"'' * * * The successive values of 5, for different values of n, are shown in the following table : Value of n 1 2 3 4 5 6 Value of 5 a 4a 3 13a 9 40a 27 121a 81 364« 243 Difference between s and — . a a "6 a 18 a 54 a 162 a 486 tb:b} progressions. 321 It is evident, then, that in this case, where r = i, the more terms we take, the closer their sum comes to the , 2a value —. 352. An infinite series in which the sum of the first n terms approaches a definite limit as n is indefinitely in- creased is called a convergent series; if the sum of the first n terms does not approach a definite limit as n is indefinitely increased, the series is said to be divergent. This distinction is of the utmost importance in the further study of algebra. Model F. — Find the sum of the infinite series 8, 4, 2, . . . n « = 16 l-r l~i Model G. — Find the value of the circulating decimal 3.7182* 3.7i82 = 3.7 + .Oi82 .0i82 =: .0182 + .0000182 + .0000000182 + etc. _ 182 182 182 ~ 10* "^ 10^ "^ 10^0 "*" • • Here a 182 _ ~10*' r = 103- Hence «' 182 ~ 10* ' ^(- 1 \ _ .0182 _ 10«y ~ .999 182 9990 Then 3.7 = nm- 3.7 + , .0182 ■ = mu + ifWo = 3i * This is an abbreviation for 8.7182182182182, where the repeti- tions of 182 are continued indefinitely. 322 ALQJEIBBA. EXERCISE CLXVIII. Find, wherever possible, the sum- of the infinite series: 1. 54; 18; 6; . . . 6. 1; .2; .04; . . . 2. .7; .07; .007; ... 7. 2.7; .9; .3; . . . 3. I; i; ^V; • • . 8. 1.728; 1.44; 1.2; . . . 4. I; I; 1; . . . 9. 7; 4.9; 3.43; . . . 6. f; 1; I; . . . lo. 130; 26; 5.2; . . . Find the values of the circulating decimals : 11. 5.7. 16. 3.736. 12. 5.73. 17. 8.18. 13. 5.735. 18. 10.201. 14. 5.73. 19. 702. 15. 1.01101101101101 ... 20. 100.8. 21. The sum of an infinite series in G. P. is 33^ per cent greater than the sum of the first two terms. What must be the value of r ? 22. The sum of the fifth and sixth terms of a series in G. P. is 12 times the sum of the second and third terms. What is the ratio of the seventh term to the first ? 23. The geometric mean of the first and fifth terms of a geometric series is 100; if the second term is 20, what is the fifth term ? 24. The geometric mean of the second and sixth terms of a G. P. is 500; that of the fifth and seventh terms is 20. Find the series. 25. In a G. P. the arithmetic mean of the first two terms is 5 times the sum of the second and third terms. Find r. 26. Three numbers whose sum is 38 are in G. P. ; if 1, 2, and 1 are added to them, in order, the results will be in A. P. Find the numbers. 27. In a certain G. P. the ratio of the third term to the THE PR00RBS8I0N8. 323 fifth is 7; what is the ratio of the first term to the seventh term ? 28. How many terms of the series 37, 33, 29, . . . amount to 187 ? Explain the two answers. 29. By how much is the sum of the infinite series 256, 64, 16 . . . diminished if all terms after the fifth are cut off? 30. In finding the sum of the infinite series 9 ; — ■ 3 ; 1 ; . . . what is the error if we stop with the 10th term ? 31. If in a given G. P. we find r = .1, find r for the series obtained by taking every fourth term of the given series. 32. Find an infinite series in G. P. such that each term is 4 times the sum of all the terms that follow it. HARMONIC PROGRESSION. 353. A series in which the reciprocals of the terms are in arithmetic progression is called a Harmonic Progres- sion.* Thus the numbers 60; 30; 20; 15; 12; 10 are in H. P. because their reciprocals •bV) -to? "O"^ "/o"? to 5 /o" are in A. P. 354. Problems in H. P. are solved by converting the given constants into the corresponding constants of arith- metic progression. * Usually abbreviated H. P. 324 ALGEBRA, Model H.— Insert 3 harmonic means between 6 and 30. This corresponds to an arithmetic progression in which i + 4^ = ^V 5 + 1206? =1 120^ ==44 ^ = - sV In A. P.: i; ,-\; -^-V; ^V; sV InH. P.; 6; 7^; 10; 15; 30. EXERCISE CLXIX. 1. What is the fifth term of a H. P. of which the second term is 5 and the fourth term is 10 ? 2. Find the sum of the first five terms of a H. P. of which the second and third terms are ^ and \, 3. Write down the fifth and sixth terms of a H. P. of which the second term is 1 and the third term 2. 4. The sum of three terms of a H. P. is If and the last term is |. What are the other terms ? 5. Insert four harmonic means between 12^ and 75. 6. The first term of a harmonic series is 15015 and the fourth term is 5005. What is the seventh term ? 7. Insert four harmonic means between 220 and 1540. 8. The first two terms of a harmonic series are — 3 and + 3; find the third term. 9. Insert four harmonic means between — 3640 and 455. 10. Insert five harmonic means between 1 and 8; five geometric means; five arithmetic means. The Three Means. 355. The harmonic mean of any two quantities is repre- sented by H\ the geometric mean by G', and the arithmetic THE PROGRESSIONS. 325 mean by A. They are determined by formulae derived as follows, in each case from the definition of the series. Let X and y represent the extremes. A] by definition oi A.,V., A — x = y — A %A=:x + y (T) j^x + A G'y by definition of G. P., G :x = y : xy = G^ (j) X xG G = Vxy H\ by definition of H. P., -— = —r ^ ^ H X y H H-x'^y- xy ^ ^ x '^ H H = -^ 1:®X2 356. For distinguishing among the three types of series investigated in this chapter, the preceding formulae furnish tests. Thus the three terms 60; 70; 84 60 + 84 are not in A. P., — because 70 :^ 2 and not in G. P., — because 70 ^ V60 x 84 1. ^ 4-1. • XT r» 1. ry^ 2 X 60 X 84 but they are m H. P., — because 70 = — ^^ , ^, — •^ 60 + 84 326 ALGEBRA. EXERCISE CLXX. 1. Prove by this test that the three means A, 0^ H, for AKY two numbers always form a G. P. 2. Prove that in the proportion x : y =p — x : y — p, p is Si harmonic mean bafcween x and «/.* Identify the following series as A, P., G.P., or H. P.: 3. i; 4; i> • • • 4. J; i; i; . . . 6. i; i; f ; • • • 6. 3; 6; 9; . . . 7. 4; 2; 0; . . . 8. 7^; 10; 15; . . . 9. 34; 5; 10;... lo. 2^; 5; 10; . . . il. 65;105; 273; .. . V2 ,— ,- V6 — 2 1 VE+2 12. 3—; iVdO; V2; ... 13. -^— ; 3-; —3^ 5 ... 14.1(^8 + 5); ^'<''\+ % U^S-S); ... 16. 275; 385; 539; ... I6. 924; 1309; 2244; ... 17. 4788; 3800; 3150; . . . 18. Insert 7i arithmetic means between a and b and find the sum of the series thus formed. 19. If a -\- b, b -}- c, and c -\- a are in G. P., prove that a -}- b _c ~ a b -{- c~ a — b' 20. If y — x-\-z — x; x — y; 1/ are in G. P., prove that y; x; z also form a G. P. 21. There are m arithmetic means between 2 and 47, and the ratio of the 4th to the {m — 3)**^ is |. Find m, 22. Find a number to which if 2, 8, and 17 be added the results are in G. P. 23. The arithmetic mean of two numbers exceeds the geometric by 24, and the geometric exceeds the harmonic by 30. Find the numbers. * This property is sometimes used to define H. P. THE PROGRESSIONS, 327 24. The arithmetic mean of two numbers exceeds the geometric by a, and the geometric exceeds the harmonic by h. Find the arithmetic mean of the numbers. 25. Given A and G for any two numbers, find a formula for the numbers. 26. Given A and iZ"for any two numbers, find a formula for the numbers. 27. Given H and G for any two numbers, find a formula for the numbers. 28. Find an A. P. whose first term is 7 and whose sec- ond, fifth, and tenth terms form a G. P. CHAPTEK XIV. DETACHED COEFFICIENTS ; BINOMIAL THEOREM. Detached Coefficients in Multiplication. 357. In multiplying expressions which contain only one letter, after the expressions are arranged by descending powers of that letter, each term in the product, as in the multiplicand and multiplier, is determined as to its degree by its position in the expression. This fact leads to a very convenient method in Algebra, called the method of de- tached coefficients, which has several very important appli- cations. Model A. — Arranging similar terms in columns : (3a;3 - 2a;2 + 5a; - 3)(2a;2 + 3a; - 1). 3^3 _ 2a;2 + bx - 3 2a;2+ 3a; - 1 6a;5 - 4a;4 + lOa;^ - 6a;2 + 9a;4- 6a;3+15a;2- 9a; - 3a;^+ 2a;2- 5a; + 3 6a;5 + bx^ + x^ + lla;^ - 14a; + 3 The entire significance of this operation is preserved if we write only the coefiicients, thus : 3__2+ 5- 3 2+ 3- 1 6-4 + 10- 6 9- 6 + 15- 9 - 3+ 2- 5 + 3 6 + 5+ 1 + 11-14 + 3 DETACHED COEFFICIENTS, 329 In the same way the multiplication (5^^ + ^x^ — 2ic + 3) {^x^ — 2a; + 3) may be written: Model B. 5+ 3- 2+ 2 3-2+3 15+9-6+6 _ 10- 6+ 4- 4 15+ 9- 6 + 6 15- 1+ 3 + 19-10 + 6 The answer being Ibx^ — x^ -\- Zx^-\- 19^^ _ ^q^ _|_ g^ In the following multiplication notice the significance of the zero coefficient in the result. Model C. {^x^ + bx^ - 2x^ + 7x- 2){5x^ + 2x^ - 7a; - 2). 3+5-2+7-2 5 + 2-7-2 15 + 25-10 + 35-10 6 + 10-4 + 14-4 -21-35 + 14-49 + 14 - 6-10+ 4-14 + 4 15 + 31-21-10+ 8-49+ + 4 The answer being written : 152:'^ + dlx^ - 21a;5 - lOo;* + Sx^ - 4t9x^ + 4 Model D. {3x^ + 5)(2:^;3 _|_ ^ _ 7). 3 + + 5 2+0+1-7 3 + + 5 3+ + 5 -21 + 0-35 3 + + 8-21+5-35 3x^ + Sx^ - 21^2 +5^-35 330 ALGEBRA, The ordinary notation of numbers is by detached coefficients ; thus 327 is really the same as W^ + 2^ + 7 where ^ = 10; and 5002546 is the same as 5^* + 2^* + 5^^ + 4^ + 6. It is complicated, however, by the fact that whenever the coefficient rises to the value of i the term becomes of higher degree. EXERCISE CLXXI. By detached coefficients multiply : 1. "^x^ +5:^2 + 7:^-2 by bx^ -Ix-^^. 2. 2,x^ - 5a; + 1 by 2x^ + ^x^ - bx — I, 3. {x^ + 3a; + 2) {^x^ -^^ ^x -\- ^){x^ - x - 1). 4. (30;'-^ + 5^; - 3)1 6. (5:?;* - 2a;2 + 1)2. 6. (a;« - 5a;2 + 3)2. 7. l^x^ + bx^y + Ixy^ - 2y^){5x^ - 7xy + 3^2). 8. {3x^ - bxy + 2/2) (2x4 _^ 3^y _ ^^yZ _ y^^ 9. (22;2 — xy -\- 2^2)3. 10. (5a;* - 32;2^2 _^ 2y4)2. Detached Coefficients in Division. 358. Model E. Divisor. 3-2 + 5-3 Quotient. 2 + 3-1 Dividend. 6 + 5 + 1 + 11 - 6-4 + 10-6 -14 + 3 9 - 9 + 17 • 9 - 6 + 15 - -14 + 3 -9 -3 + 2- -3 + 2- -5 + 3 -5 + 3 The answer being 2a;2 + 3^; — 1. DETACHED COEFFICIENTS, 331 EXERCISE CLXXII. Use detached coefficients in dividing the following : 1. {ISx^ - 45a;3 + 82a;2 - 67a; + 40) -^ {Zx^ - 4x + 5). 2. (:?;* - 6:^3 + 9a;3 - 4) -^ (a;2 - 3a; + 2). 3. (V + 0)3^ - ^xY + 19^:^^ ~ 15y^) -^ (a;^ + Sa;?/ - 5y2). 4. (32a;4 _|. 5^^3 _ gi^4) ^ (2a; + 3^). 6. (V - a;4 + «^ - ^^ + i«J - 1) -^ (^^ - 1). 6. (12~38a;+82a;2-112a;3+106a;4-70a;5)-f-(7a;2-5a; + 3). 7. (9a; + 3a;* + 14a;3 + 2) -^ (1 + 5a; + x^). 8. (8a;8 - 16a;6 - 34a;* + 32a;''^ - 6) ~ (2a;* - 7a;2 + 3). 9. (a;* + 4a;3y + ^x'^y'^ + Sa;?/^ + 2?/*) -h (a;^ + 3a;y + 2^^), 10. (5a;*+ 2a;3^ - 20a:y- 230;^^- 6/)-r-(5a;2+ 7a;2/ + 2y^). Detached Coefficients in H. C. F. 359. The method of detached coefficients often saves time in the process of H. C. F. Model F. H. 0. F. of 2a;* - 7a;3 + 12a;2 - Ho; + 4 Q2-7 + 12-11 + 4 ©3-8+5+2-2 (D 6 - 21 + 36 - 33 + 12 ® X 3 ®6- 16 + 10 + 4-4 (2)X2 *® 5 - 26 + 37 - 16 ® - ® *® 8 - 23 + 22 - 7 ® + ® ©10-52 + 74-32 ® X 2 ® 2 - 29 + 52 - 25 ® - ® ® 8 - 116 + 208 - 100 ® X 4 ©93-186 + 93 ®-® *([!) 1-2 + 1 (f6)H-93 (12) 16 - 46 + 44 - 14 ® X 2 (13)11- 20 + 7 + 2 @-® ©88-160 + 56 + 16 @ X 8 *@ 93-186 + 93 (14) + ® 3a;* - Sx^ + bx? + 2a; - 2. 3-8+5 + 2-2 1-2+1 H.C.F. Ans, \f - 2a; + 1] 332 ALGEBEA, Detached Coefficients in Elimination. 360. Space and perplexity may be saved in solving equa- tions of more than two unknown letters by using a method of detached coefficients; that is, by omitting the letters and arranging the coefficients of the same letter always in th^ same column. Model G. ^x -]- ^ - bz + w = 2^0 2x - 'dy -\- z - 4:tu = - 3 7x + 7y - 3z + 2w = 93 © ® ® ® @276 @393 X - 2 3 7 1 y + 4:z + 3io = 27 - 5 1 - 3 4 30 - 3 93 27 ® 6 - 9 3 - 12 - 9 ® X 3 ® 13 - 2 * - iO 84 @ + ® ® 8 - 12 4 - 16 -12 ® X 4 ® 7 - 11 * - 19 -39 ®-® ® 10 - 15 5 - 20 - 15 ® X 5 @ 13 - 13 * - 19 15 ® f ® ® 13 - 2 - 10 84) First ® 7 - 11 - 19 -39 [ New ® 13 - 13 - 19 15 j Set (Q) 6 - 2 * 54 ®- ® ® 26 - 4. - 20 168 ® X 2 @ 13 9 - 1 153 @ - ® @ 130 90 - 10 1530 @ X 10 ® 117 92 * 1446 ©- ® @ 6 - 2 54) 1446 j Second ©117 92 New Set ~ 92 2484 3930 (Q) X 46 @ + (!D DETACHED COEFFICIENTS. 333 From @,x=ilO Subst. in @, 60 - 2y = 54; 2/ = 3 Subst. in ®, 130 — 6 — IQw = 84:; w = 4: Subst. in ®, 10 - 3 + 43/ + 12 = 27; y = 2 ic = 10 2^ = 3 z = 2 W = 4r Coefficients of Powers. 361. The successive powers of {a + ^) maybe calculated by detached coefficients. Thus we obtain : 1 + 1 1 + 1 1 + 1 1 + 1 1 + 2 + 1 (second power) 1 + 1 1 + 3+1 1 + a + i 1 + 3 + 3 + 1 (third power) ' 1 + 1 1+3+3+1 1+3+3+1 1 + 4 + 6 + 4 + 1 (fourth power) Since the multiplier is always 1 + 1, the set of coefficients for the next higher power can always be obtained from those of a given power by using them twice as a partial product, displaced as usual one term to the right. Thus 334 ALOEBRA. the calculation for successive powers would appear as follows : 3d 4tli 5tli 6tli 7tli 8tli 9th lOth llth 12tli 1+ 3+ 3+ 1 1+ 3+ 3+ 1 1+ 4+ 6-h 4-f 1 1+ 4+ 6+ 4+ 1 1+ 5+10+ 10+ 5+ 1 1+5+ 10+ 10+ 5+ 1 1+ 6+15+ 20+ 15+ 6+ 1 1+ 6+ 15+ 20+ 15+ 6+ 1 1+ 7+21+ 35+ 35+ 21+ 7+ 1 1+ 7+ 21+ 35+ 35+ 21+ 7+ 1 1+ 8+28+ 56+ 70+ 56+ 28+ 8+ 1 1+ 8+ 28+ 56+ 70+ 56+ 28+ 8+ 1 1+ 9+36+ 84+126+126+ 84+ 36+ 9+ 1 1+ 9+ 36+ 84+126+126+ 84+ 36+ 9+ 1 1+10+45+120+210+252+210+120+ 45+ 10+ 1 1+10+ 45+120+210+252+210+120+ 45+10+ 1 1+11+55+165+330+462+462+330+165+ 55+11+ 1 1+11+ 55+165+330+462+462+330+165+55+11+1 1+12+66+220+495+792+924+792+495+220+66+12+1 THE BINOMIAL THEOREM. 362. The coefficients of any power of the binomial a -\- h are determinable by a famous rule, known as the Binomial Theorem. In the first place these coefficients are the same whatever letters we use instead of a and Z>, and, in the enunciation of the rule here adopted, it will help us at first to get the coefficients of powers of a; + 1. 363. In {x + 1)^ the first term is cc^. 364. For the other coefficients : Multiply the coefficient of any term by the index and divide by the number of the term, to get the coefficient of the next term. TSE BINOMIAL THEOREM. 335 Model H.— In writing (x + ly^: the first term is x^^. The coefficient of this term is 1; the index is 13; it is the first term. IX 13 -7-1 = 13 (coefficient of the 2d term). the second term is 13^;^^. The coefficient of this term is 13; the index is 12; it is term number 2. 13 X 12 -^ 2 = 78 (coefficient of the 3d term). the third term is HSx^^, The coefficient of this term is 78; the index is 11; it is term number 3. 78 X 11 -^ 3 = 286 (coefficient of the 4th term). and so on. EXERCISE CLXXIII. Write the first four terms of 1. {x + ly, 2. {x + 1)17. 3. {x + 1)26. 4. {x + iy^ 5. (:^ + l)^* 6. Write all the terms of {x + iy\ 7. A^erify the coefficients obtained by the rule for {x + iy and {x + ly. 8. Find the value of {x -\- ly when x=l; also the value of {x + 1)\ 9. What is the sum of the coefficients of any power of the binomial x -{-1? 10. How many terms in the expansion * of {x-\-iy? 11. The 10th term of (a + by^ is 92378aioj9; find the coefficient of the next term. * Where an algebraic expression of one term can be written out as a series of terms, that series is called the expansion of the expres- sion. 336 ALGEBRA. 12. The 8th term of (a + ^)i6 is llMOa^Z*^; find the coefficient of the next term. 13. The 11th term of {a + hf' is 184756«io^iO; find the coefficient of the next term. 14. The 6th term of (« + Z^)^^ is 142506^25^^; find tlie coefficient of the next term. 15. The 4th term of {a + h)^ is 6545a32^,3. ^^^ the coefficient of the next term. Symmetry of the Coefficients. 365. The coefficients obtained by this rule increase from the first term towards the middle, and decrease in the same way from the middle to the end, with exact sym- metry; so that if the set of coefficients were turned end for end, the entire expression would be unchanged. If we write then the expansion of (1 + xY beginning at the EKD, with x^y and understand by ^'the number of the term " the number counting fkom the end, the rule will give the same set of coefficients. N^ow the expression {a + hy has of course the same coefficients as {x + 1)" and (1 + xy. In applying the rule, if we count from the beginning, we must understand, by the words '^the index," the index of a\ and if from the other end, we must understand the index of Z>. EXERCISE CLXXIV. Find the first three and the last three terms of : 1. {x + ^)^^ 2. {a + l)"^. 3. {a + x)"^. 4. {x + aY\ 5. (^+2)1^. Powers of Differences. 366. In the expansion of (a — hy we must remember that — • 6 is a negative factor, and the terms will conse- quently be + or -— according as an even or an odd num- ber of factors I are contained in it; in other words, according as the index of l is even or odd. THE BINOMIAL THEOREM. SSiT EXERCISE CLXXV. First five terms of: 1. (a + hf\ 2. {a + ly^ 3. {a - iy\ 4. (« - ^)2i. 5. {a + Z>)23. JFVr^^ three and last three terms of: 6. (x + ^)i<^. 7. (^ - 2/)^^. 8. (a - xy^ 9. (a; - Z>)80. 10. (« + «/)iio. Irregular Coefficients. 367. It is evident that this rule will not apply to the numerical factors of the successive terms in expressions like (2a; - 3)^ Model I. {2x - 3)^ = 32^5 _ 24:0x^ + 720^^ - lOSOa;^ + 810a: - 243. Here 32 X 5 -^ 1 ^^ 240; 240 X 4 -f- 2 ^ 720; and so on. Such expressions are expanded by first writing the same power of a simple binomial as a pattern, and then substi- tuting. As in this case: (a - by = a^ - ba'h + IQaW - 10a?b^ + ^ah^ - W. Now in the required expansion we must have 2a; instead of a, and 3 instead of Z>; and making those substitutions we obtain the expansion as above. EXERCISE CLXXVI. In the same way expand : 1. (3a: - by. 4. (2a; - \y.^ 7. (Sa^ - 2a;)' 2. (2« - h'^y. 6. (3. + 1)^ ».(^+2-r »■ {- - ii (a 2\« 9.(2+^3)'- 10. {p - 5g)5. 11. {Zh - Uf. 338 ALGEBRA. The Binomial Theorem Generalized. 368. In generalizing the binomial theorem, we obtain by the rule: n{n - l){n - 2){n - 3) + 1.2.3.4 ^ w(w - l)(w - 2){n - 3)(n - 4) ■•" 1.2.3.4.5 w(n-l)(w-2)(w-3)(w-4)(w-5) ^~ 1.2.3.4.5.6 + - • • or, with two letters : 7K^-l)(n-2)(7i~3) + 1.2.3.4 "^ ^ . ^(->^ ~ 1)(^ - 2)(.^ - 3)(^ - 4) ' + 1.2.3.4.5 "^ ^+... The coefficient for a term further on in the expansion becomes very unwieldy, and a generalized formula, for any term of any power, would be very hard to remember. To simplify the expression a new symbol is introduced. Factorial n. 369. The factorial of any given number is the product of all the positive integers, beginning with 1 and ending with the given number. The factorial of 5 is written 5! or |5, and read ^'factorial five/' 6! = 1.2.3.4.5 = 120 6! = 1.2.3 .4.5.6 = 720 THE BINOMIAL THEOREM, 339 Higher factorials, like 20 !, give numbers that are incon- veniently large in the ordinary notation. In figuring the values of such coefficients it is best to express them with prime factors, in the order of magnitude. Thus : 20 ! = 1.2.3.2^5.2.3.7.2^3'.2.5.11.2«.3.13.2.7.3.5.2*.17.2.3M9.2«.5 = 2i8.38.54.7'».ll. 13.17.19 370. In finding the values of expressions involving fac- torials, the work of calculation can often be shortened by factorization. 8! Model J. — In the expression ^ the factors 5.4.3.2.1 are common to numerator and denominator; hence 8! 1; = 8 . 7 . 6 = 336 5! Model K. 10! - 5! = 51(10.9.8.7. 6- 1) = 120 X 30239 EXERCISE CLXX^MI. Find the values of the following expressions : 11! 11! - 7! 3!(8!- 6!) ^' ^' *• 6! • ^- 5! 10! ill__iiL ?'_!: ^- 5T5T* ^' 3T8! 2! 9!' ^' 5! 5!' 20! - 18! 100! - 99! 8. '• 187(10! 8! 5!)* **' 3(99!) ' 87! _ 13! - 11! ^' 29(86!) ^^' 55(9!) * The symdol 8 ! is in some modern loohs largely replaced by 18; the symbols have the same meaning^ and loth are in good use. They are used indifferently in the following : 340 ALGEBRA, 11. What factors must be multiplied into 8.7.6 to give 18 ? [Express in briefest form.] 12. What factors must be multiplied into 10 . 9 . 8 to give 111 ? 13. What factors must be multiplied into 20 . 19 . 17 to give 1 20^? 14. What factors must be multiplied into \n to give | (^ + 2 )? 15. What factors must be multiplied into n{n—l)(n—%) to give \n ? Find the value of n in the following equations : {n — 1)1 {n — b)\ n\ ^^ 105(^-4)! ^ , ^ 17. 7 ^TT-i = ^2. 20. -y-^^ wrv- =2^ + 5. {n — 2)\ {n — d)\ ' (^ + 1)! ..A f:\7^^(n-l) 1 18. / ^^rff = 110. 21. '-^ — - -4^ '• (w - 1) ! ~ '" \| |w ~ 3 ■ i«+_2 _ | w + l _ 9 Uw - 2 ■ I5|w - 4 ~ 4 ■ 4!(w-2)! . (w-3)! _ 48! 24. (^ - 6) ! • 46 ! (^ - 5) ! 47 ' {n +4)(^ ~ 5)! _ {n - 4)(y^ + 3) ! _ 5 (^ - 4) ! (^^ + 4) ! "6"* [(^ - 1) H' (^ + 1) ! (^ - 1) ! ~ For Any Term of Any Power. 371. To generalize the binomial theorem, so as to be able to write any term of any power directly from a for- mula, let us first write THE BmOMTAL THBOBEM. 341 where «,, a^^ a^, a^, etc., represent the successive coeffi- cients^ as obtained by the rule. 372. In each term there is a number (in the third term, for example, it is 2) which appears in three places: it is the index of y ; it is subtracted from n to give the index of x\ it is the suffix of a. This number is called the modulus of the term and is represented by h', in the first term h = 0, in the second term k = 1, in the third term ^ = 2, in the fourth term h = Z, and so on; in general, — 373. The modulus of any term is one less than the number of the term. 374. We may now state, and will hereafter prove. The Binomial Formula: CCi- = ^^^ M{n - k)\' Model L.— The 7th term of {x + yY^ is '-, ^^^ term. / X 1 y 16. — = 7^ ; 13th term. V V2y Vx / 17. (-^-J^)''; 7th term. \ V2x 6a I 18. TT-) ; 11th term. \ a dxj I 1 y^ 10. \2x^y —] ; 15th term. V ^yVxl 20. ^ \/x- |/^]''; 14th term. TEE BINOMIAL THEOREM, 343 375. According to the Binomial Formula, ^ ^ ^ ' ^ k\{n — ]c)\ Now if we could prove, - First: that the formula gives the correct coeffi- cients for the first few powers of (« + h)', and, Secondly : that if the formula gives the correct coefficients for any one power of {a + Z>) it must also give the correct coefficients for the next higher power; then we should have to admit that the formula was valid for all powers oi {a -{- h). Proof of the Binomial Formula. 376. Eef erring to the calculation of successive powers of a binomial, earlier in this chapter, where the method of detached coefficients was used, it is easy to see that in passing from one power of [a -\- 1) to the next higher power, the following rule holds : In any power of {a + i), starting with a term of given modulus, if we add to its coefficient the preceding coeffi- cient, we obtain the coefficient of the term having the same modulus in the next higher power of (a -j- h), 377. In {a -f 5)^, therefore, if the formula holds good, n\ the coefficient of the term of modulus h is and the coe is ^ — 1, is k\(:n-ky: and the coefficient of the preceding term, whose modulus ijc - l)\{n - k-\- 1)!* 344 ALGEBRA. The coefficient of the term of modulus Jc, in (a + hY'^^, would consequently be n\ I ^' {k - l)\{n-h+ 1)!+ k\{n - k)\ This reduces: _ n\ n+\ _ (^+1)! - (y^- 1) l{n-k) ! ' k{n+l-k) -kl{7i + l-k)l which is the coefficient that the formula would give. 378. Thus we have shown that, if the binomial formula holds for {a + Z>)", it holds also for {a + by + \ Now the formula does hold for (a + Z>)^, {a + by, and so on, as far as we choose to test it among the low powers of a -{- b. According to our proof, if it holds for {a + by, it holds for {a + by-y and if for {a + by, then also for {a + by; then also for {a + ^)^ and so on, step by step, through all powers of (a + ^) f^^ which the indices are positive in- tegers.* From the Formula to the Rule. 379. In the expansion of (x + yY the term of modulus 71 ! k is, by the formula, r—~ — '—m^" ~ V] and the next term, -^ kl{n — k)l ^ ' ' which has for a modulus k -{- 1, is y,n ~ k - l^.fc + 1 , y y (k + l)l{n-k-l)r ^ Tliis method of proof is called Mathematical Induction. THE BINOMIAL THEOREM, 345 the numerical coefficients of these terms, being represented by a^ and ^^^.^i respectively, are in the ratio %4.i n\ ^ n\ aj, ~ (^ + l)l{n -h - 1)! * 'k\(n-lc)\ _ Ic'.in — lc)\ _^n — h - (^ + 1) ! (^ - .^ - 1) ! "" f+t: Whence we obtain aje + i = -, a^c] and this agrees with the form of the theorem first enunciated, because, in the term of modulus h, 7i — h is the index of Xy and ^ + 1 is the number of the term. Negative and Fractional Indices. 380. Thus far we have considered the binomial theorem only for cases where 7i was a whole number, and a positive number. It holds also, with certain exceptions, for nega- tive and fractional values of u] but in such cases it always gives an infinite series. In such cases also we mast alter the binomial formula; for if n is fractional, or negative, n\ and {n — k)\ become meaningless symbols. 381. The coefficient Uj^ may always be written with k\ for its denominator, and with the product of h factors for its numerator, — these h factors being n{n — l){n — 2){n — 3) . . . {n - h + 1) That is: ,^ = -i--^)in-2) jn-k+l) 382. We may verify the formula in some cases as fol- lows: Model 0. Vl — X = {1 — x)i] expand in a series. By the formula : {1 — xY = 1 — Ix — \x^ — ^^x^ — . 346 Otherwise : ALGEBRA. |/1_ - a; a; x^ 1 1 "" 2 " 8 — - ^ X - - ^ + 4 "■ 2 x^ a;2 4 2 — a; - 8 a-2 a;3 a;* 4 8 64 ._3 8 ^ 64 2 — X - x^ 4 16 Model P.-Expand (1 + o;)"^ - ^ ^ /^ ^ ^2 ' By the formula: {l+x)-^^l-2x-\-?^x^-^x^-^bx^- Otherwise : 1 -\- "Zx -{- x^ 1 — 2a; + 3a:^ — ^x^ + bx^ — i \J^^X + X^ — 2x — x^ — 2x — 4:X^ —2x^ + 3x^ + 2a;3 + 3a;^ + 6a;^ + 3a;^ - 4:X^ - 'dx^ — 4:X^ — 8a;^ — 4a;5 6x^ + 4a;^ Model ft. — Expand {x + y) (_2)(_3)(-4) ^ic-2- 2aj-3y + 3aj- V - 4.'r- V+ • • • THE BINOMIAL TUEOEEM, 347 Model R.— Find the 11th term of {x — y)-^. (-a(-i)(-s(-a(-a(-f)(-f)(-f)(-f)(-iL .-,.( ,,,. 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 \ U) _J_ 4 . 7 . 10 . 13 . 16 . 19 . 22 . 25 . 28 ■ 31 gj ip — 310*1.2. 3. 4. 5. 6.7. 8. 9.10*^ ^^ _ 215.7.11. 13.19.31 ^^Q _ 2^5.7.11.13.19.31 ^lo f ^2 3^* -^uf-- - 31^ '—^0^ EXERCISE CLXXIX. Expand hy the Mnomial theorem, to four terms: 1- i^+y)'^' 2. (^+l)i 3. (rt+Z>)-ii 4. {x—y)-^\ 6. (l-c^)i 6. (:?^+2)"'. 7. (3:^+l)i 8. (2^ + 3^^)"^. Expand and verify, to four terms : 11. (^+y)"'- 12. (1+^)^. 13. (2-2:)-^ 14. (l+a;)i 15. (1 — ^)-^. 16. (\—x)-^, 17. In the series obtained by the binomial theorem for (1 — xY^ substitute x ^- \\ substitute also in (\—x)~^ = and compare the results; verify by the theory of geometrical progression. 18. In the same series substitute x = 2; is this series as clearly equal to the expression from which it arose ? Is the series obtained by division any more accurate ? Why does the theory of geometrical progression fail us here ? 19. Find the 8th term in {x + y)^. 20. Find the 10th term in (1 - x)^. CHAPTER XV. LOGARITHMS. 383. The work of calculating numerical products, quotients, powers, and roots is much simplified by express- ing all numbers as powers of some single number; all these operations then become addition, subtraction, multiplication, and division of the indices. In this chapter it is shown that such a simplification is practicable. Table of Powers of lO. 384. By successive applications of square and the following fractional powers of 10 can be calculated to three places of decimals: IOtb; iQi; 10^; 10^; 10^; lOi^; 10^; 10§; lO^'s 10^; 10^; lOi; 10^; lOH; 102; lO^^i; 10^'; 101; lOxi. Thus we get the accompanying table, in which, for example, IOtb (or 10-'^^) is shown to be = 1.154; IQi^ (or 10«^2^) = 6.494; etc. Indices. tV .0625 1 .1250 i .1667 t\ .1875 i .2500 A .3125 i .3333 1 .3750 tV .4375 i .5000 t\ .5625 1 .6250 f .6667 H .6875 i .7500 If .8125 f .8333 i .8750 n .9375 cube root Powers. 1.154 1.334 1.468 1.540 1.778 2.048 2.154 2.371 2.738 3.162 3.652 4.217 4.641 4.870 5.623 6.494 6.813 7.499 8.660 848 LOGARITHMS. 349 385. This table can be extended (by other means) till every number between 1 and 10, to two or three or even more decimal places, has been exhibited as a power of 10. Such a table is called a table of logarithms, or, more exactly, a table of common logarithms; for other tables might be constructed, in which numbers would be exhib- ited as powers of some root other than 10. In every such system, — 386. The common root of all the numbers is called the 387. The indices are called logarithms. Using the Tables. 388. In four-place tables, the first two figures of each number are given in the column marked N; and the third figure is given at the head of one of the ten columns of logarithms. Thus for the number 4.27 we should find the first two figures 42 in the N column, and on the same line, in the column marked 7, we should find 6304. [Decimal points are omitted.] We conclude, then, that 4.27 = 10-^^^*; or in other words, the logarithm of 4.27 is .6304. Similarly, if we had the logarithm .5092, to find the number which corresponds to it, we first look up 5092 among the ten columns of logarithms, notice that it is on a line with 32 in the column N and has 3 at the head of its own column, and so conclude that 5092 is the logarithm of 3.23. EXERCISE CLXXX. Find : 1. log 3.73; log 4.32 3. log 2.97; log 7.29 2. log 5.85; log 3.54 4. log 1.73; log 6.4 5. log 8.9; log 9 350 ALGEBRA, Find numbers whose logarithns are : 6. .4771; .8597 7. .9814; .7980 8. .1987; .4518 9. .5024; .4886 lo. .8645; .9562 Interpolation. 389. Model A.— Find log 2.573. This number is ^^ of the way from 2.57 to 2.58; and as we find from the table that the log 2.57 is .4099 and log 2.58 is .4116, we assume that log 2.573 is -^-^ of the way from .4099 to .4116. The difference between two successive logarithms in the table is called the tabular difference and is represented by D\ and D has different values in different parts of the table. The difference between the required logarithm and the WEAKEST TABULAR LOGAEITHM is represented by d. The assumption made above is that the dift'erence of two numbers, in the same part of the table, is proportional to the difference of their logarithms. This assumption is not absolutely correct, but leads to no serious error. 2.573-2 .57 _ ^ 2.58-2T57~5 3-^ d = (.3)i) = (.3)(17) = 5; log 2.573 = .4104 Model B.— Find log 3.736. - log 3.74 = .5729 log 3.73 = .5717 n^ 12 log 3.7'4 = .5729 d =z (.4) X 12 = 4.8 d= 5 log 3.736 = .5724 In most tables of logarithms the values of yVA toA YoD, etc., are given in the margin opposite the portion of the table for which they are correct, thus making it possi- LOGARITHMS. 351 ble to determine the logarithm mentally. These multiples of — are called Proportional Parts. Model C. — Find the number whose logarithm is .7470. log 5.58 = .7466 log 5.59 = .7474 D = S d= 4 d 4 Ans. log 5.585 = .7470. The table of proportional parts can also be used in this process. EXERCISE CLXXXI. Find : 1. log 1.832 2. log 2.471 3. log 5.234 4. log 4.788 5. log 7.323 Find 7iumhers whose logaritlwis are : 6. .2835 7. .3048 8. .9873 9. .5747 lo. .4891 Characteristics. 390. Although the tables give only the logarithms of numbers between 1 and 10, the logarithms of numbers be- low 1 and above 10 can readily be obtained also^ remember- ing that the logarithm of a number is its index considered as a power of 10. Model D.— Thus 395.2 = 100 X 3.952 = 10^ x lO'^ses^ since log 3.952 is, by the table, .5968. 10^ X 10*^^^^ = 10^*^^^^ Hence log 395.2 = 2. 5968 352 ALGEBRA. Every number may in the same way be separated into two factors, of wliich one is a number between 1 and 10, and the other is a power of 10, positive or negative. Thus 385400 = 3.854 X 10^ .0853 = 8.53 -^ 100 = 8.53 X lO'^. Model E.— Find log .003987 .003987 = 3.987 -^ 1000 = 3.987 X lO'^ = 10-3 X 10-6006 In practice this subtraction is not carried out, but in- dicated in a contracted form by writing the negative sign over the characteristic, thus: log .003987 — 3.6006. 391. The integral part of a logarithm is called the CHARACTERISTIC, and the rest of the logarithm, comprising all the figures on the right of the decimal point and con- stituting a proper decimal fraction, is called the MANTISSA. 392. The characteristic may be found by counting off the places from the first significant figure of the number to the units place. l^umbers between and 1 will have negative character- istics ; negative numbers have no logarithms. EXERCISE CLXXXII. Find : 1. log 283.5 2. log 30.13 3. log 78.45 4. log .07832 5. log 11.037 Find numbers whose logarithms are : 6. 5.8372 7. 1.1982 8. 2.6570 9. 3.6670 10. 3.4825 L00ABITEM8. 353 Augmented Logarithms. 393. The confusion which may easily be expected to arise with the use of negative characteristics is avoided by adding 10 to every logarithm complicated by such a characteristic. Thus log .003987 would appear in calcula- tion as 7.6006X; the X indicating an excess of 10 to be subtracted from the result. Calculation by LogaritFims. 394. Since powers are multiplied or divided by adding or subtracting their indices, so any numbers may be mul- tiplied or divided by adding or subtracting their logarithms. 395. In the same way roots and powers are obtained by dividing or multiplying logarithms by the required index. -- , , _ n^ ^ ^ (398.7)(.0983)(9.837) Model F.- Calculate (9s,o7)(38.9)(.OQ783)(7.38) log 98.07 = 1.9915 log 398.7 = 2.6006 log .0983 = 8.9926 X log 9.837 = 0.9929 12.5861 X log 38.9 =1.5899 log .00783 = 7.8938X log 7.38 =0.8681 12.3433 X log ans. = 2.5861 - 2.3433 = .2428 = log 1.749 1.749 Ans. EXERCISE CLXXXIII. Calculate : 1. (28.3)(.0587)(8.93)(1.1354). 2. (.0057)(.00342){.007893)(8496000). 3. (87.6)(83.9)(.000786)(508). 4. (98.7)(79.8)(9780)(.00789)(8.97). 5. (8.732)-\ 6. f8732. 3S4 ALGEBRA, 7. (c00768)(53.42)3: 1.358. 8. ('V^867':35.84)(l,537). 9. (34020)2(.0000842). 10 f(63.8)(47.02)(25.37). Cologarithms. 396. The computation of expressions involving division is much simplified by the use of cologarithms. The cologarithm (more exactly, the arithmetical complement of the logarithm) of a number is 10 minus its logarithm. It is computed by subtracting each digit of the logarithm from 9 except the last, and that from 10. Thus log 398.7 = 2.6006 and colog 398.7 = 7.3994 X. 397. If of two logarithms I^ and l^ the second were to be subtracted from the first, the same result, I, — l^, could be obtained by adding to l^ the cologarithm derived from /,, WITH AK EXCESS OF 10 0^ ACCOUNT OF THE COLOGAKITHM; that is, l^- l^ = {l^ + 10 - ZJ - 10. Thus in computing the expression in Model F we have performed two additions and one subtraction; the use of cologarithms Avould reduce this to one addition, as follows : • log 398.7 + log .0983 + log 9.837 - log 98.07 - log 38.9 - log .00783 - log 7.38 becomes log 398.7 + log .0983 + log 9.837 + 10 - log 98.07 -f 10 - log 38.9 + 10 - log .00783 + 10 - log 7.38; or log 398.7 + log .0983 + log 9.837 + colog 98.07 + colog 38.9 + colog .00783 + colog 7.38 398. The cologarithm of .00783 does not imply A^q" EXCESS OF 10, because 10 has already been added to the logarithm on account of the negative characteristic ; that is, 10 - (10 + log) = - log AVith these chane^es the calculation of Model F becomes : LOQAniTHMQ. 355 Model a. log 398.7 = 2.6006 log .0983 = 8.9926 X log 9.837 = 0,9929 1.9915 colog 98.07 = 8.0085 X 1.5899 colog 38.9 = 8.4101 X 7.8938 X colog .00783 = 2.1062 0.8681 colog 7.38 = 9.1319 X log ans. = 40.2428 XXXX = .2428 Ans. 1.749. The Logarithmic Schedule. 399. One of the most important things in all loga- rithmic computation is ordekly area]S"GEMEN^t; and the pupil is recommended to arrange his work in advance, by writing a schedule of the logarithms before opening his tables. Model H. (3.87)(3.087)(38.07) 30.08 The schedule would be made and filled out as follows (wherever a cologarithm is called for, the logarithm from which it is derived is written on the left of the schedule): log 3.87 =0.5877 log 3.087 = 0.4896 log 38.07 = 1.5806 1.4783 colog 30.08 =: 8.5217 X log fraction = 11.1796 X = 1.1796 log t^f raction = .3932 Ans. 2.473. 356 ALGEBRA. In-radius of a Triangle. 400. If the three sides of a triangle are represented by a, h, and c, the radius of the inscribed circle is determined by the formula r =A- ~ ~ -, where a -^ l -\- G 260, Model I.— Compute r for the triangle a = 253, i = c = 315. a = 253 log {s - - a) = 2.2068 6 = 260 log (s - - Z>) = 2.1875 c = 316 log {s - - c) = 1.9956 2s = 828 2.6170 colog s = 7.3830 X 5 = 414 log r^ = 3.7729 s — a = 2Ql log r = 1.8864 5 - 5 = 154 r = 76.98 s - C=: 99 EXERCISE CLXXXIV. Compute the radius of the inscribed circle for the triangles : a b c 1. 273 425 628 2. 175 527 600 3. 217 404 495 4. 255 407 596 5. 277 304 315 Roots of Proper Fractions. 401. In finding roots of numbers less than 1, where 10 has been added to the logarithm, the resulting logarithm will have an excess of some fraction of 10; thus LOGARITHMS. 367 would be too large by 3 J, and the subtraction of this excess would be a bother. So before dividing by 3 the excess is made 3 X 10, and the result turns out to be just 10 too large. Thus log .003987 = 27.6006 XXX log l/. 003987 = 9.2002 X 4/.003987 = .1586 EXERCISE CLXXXV. Find the value of: (8.732 00658 | 31.56 • N 708 ^ f .00937 |/.000532 83.7 ~Y ^' L38700000j • 6. (7 + colog 365)i 7. f 10 + log .000873. 8. V- log .07 4. V log 3.003 log 5.38 9. Find the value of logiV^ (log 8.53). 10. log 3.58 Evaluate the following formulce : \ogh wheniV^= 15.75, ^ = 21. If X log l = log iV, find iV^when x = 7.832 and d = 3.142. 11. Vs{s - d)(s - c){s - a) a = 19',b== 23.1 ; c = 37.7. p=: .6; P=:78Q; n = 60. F=1728; 7r = 3.1416. a = log 3; J = log 4; c = log 5. 12 13 14 15 * \ s{s — a) 3 3F 4;r a 358 ALGEBRA, 16. Find two approximate values that will satisfy the equation 3^'^^ + '^^^ — ^• 17. Insert five geometrical means between 8 and 128. 18. Find the value of V¥. 19. If the first term of a G. P. is 101, and the eleventh term is 800, what is the twelfth term ? 30. Solve (1- :.)>» = gl^j. Notation by Powers of Ten. 402. Where a numerical quantity is very large (such as the diameter of the earth's orbit expressed in feet, or the distance light would travel in a year, expressed in miles) or when it is very small (as the length of a wave of light ex- pressed in any ordinary unit) it is customary and very con- venient to express the number in question as a multiple of some power of ten. The distance liglit would travel in a year is an astronomical stand- ard of measure; thus tlie Dog-star is eiglit light-years away, and the pole-star is forty-seven. Model J. — To find the length of a light-year we must multiply 186,300, the number of miles light travels in a second, by the number of seconds in a year, 3600 X 24 X 365.25 log 365.25 = 2.5626 log 24 = 1.3802 log 3600 = 3.5563 log 186300 = 5.2702 12.7663 The number of miles in a light-year would therefore be 5.839 X 10^^ or 5839 X 10\ LOGARITHMS. 359 EXERCISE CLXXXVI. 1. Find the ratio of 374.8 X 10"^ to seven one-mil- lionths. 2. Find the 200th power of 2. /28!\ /2 3. Find the reciprocal of (- 180° 4. Find the ratio of 1'' to 71 ~\ioo 5. Find the 75th term in the expansion of j 1 (-f)" 6. A cubic foot of water weighs 62|^ pounds, and there are 2000 pounds in a ton; there are 5280 feet in a mile, and 3956.6 miles in the radius of the earth. The volume of the earth is given by the formula %7tR^, and it weighs 5.6 times as much as the same volume of water. Find the weight of the earth in tons. Other Logarithmic Systems. 403. A table of logarithms can be constructed with any number, commensurable or incommensurable, as a base. The system which has for a base the famous incommensur- able number e ^ 2.718281828 . . . called the Napierian system, is of great importance in the higher mathematics. 404. The logarithm of a number to a base other than 10 may be denoted by writing the base as a subscript. Thus, loge^; log^ {x^y\ log^ 32 = 5; because 2^ = 32. logo (i) = - i; because 9'^ = |. 360 ALGEBRA, 405. The base [b), the number (iV), and the logarithm {x) are connected by the equation (which is a definition) Z>^ = N, Model K. — What is the base of a system of logarithms in which the logarithm, of 16 is — 1^ ? b-h =2 bi=i Model L. — If the logarithm of ^ is 2, what is the loga- rithm of 16 in the same system ? b^ = ^ = 2-3 Z>^ = 16 = 2^ b = 2-§ 6^ ^ 2 " 2 _ 3a; 2" 2 =2^ r?; = — |. Ans. Since tlie logaritlim of a number in any system is easily calculated from the logaritlim of that number in the ordinary system, there is no need of extensive tables for bases other than 10. EXERCISE CLVXXXII. 1. If the logarithm of 4 is |, what is the base of the system ? 2. If logic {x -}- 5) — .75, what is the value of a; ? 3. What is the logarithm of 10 in a system whose base is .01 ? 4. When log?, 81 = — f , what is the value of ^ ? LOGAmTHMS. 361 5. When loga.as^ = — 1-5, what is the value oi x? 6. Find log25 (.008). 7. When p"" is base, what is the log ot p^^"^^ ? 8. A\^hen ^^"^ is base, what is the log of p^ ? 9. When 4 is base, what is the log of 2 ? of 8 ? of 4 ? of 1? of ^V? of .03125? of 0? 10. When ( — 16 is base, what is the log of \ — \h ? of I ~ 1 : of f^)'? of-? of M-'? of (- \yi y \yl \y _ ^ 11. If log 7 is 3, what is log 343 ? log Vl ? log 1^49 ? 12. If Z>^ = iVand 1) ^- 10% what is logio iV^? 406. Model M.— Find log, 7. 6^ = 7 iclog,„ 6 = log^„ 7 ^ log,„ 6 - .7782 log .8451 = 9.9270 X 9.8911 X colog .7782 = 0.1089 log X = .0359 X = 1.0863 Kote that in this example we use the logarithm of a logarithm ; thus 9. 9270 X = log .8451 = log log 7 ; and .0359 is not the logarithm we were asked to find, but the logarithm of that logarithm. EXERCISE CLXXXVIII. 1. Find the log of .01 in a system whose base is 20. 2. What is the base of a system of logarithms in which the logarithm of 20 is 20 ? _ 3. Find log 144 to the base 2^3"; to the base 6. 362 ALGEBRA. 4. For the system whose base is 2, write successively the characteristics of the logarithms for all the natural numbers from 1 to 10. 5. In the system where log 2 = 3, what is log 3 ? 6. Given log 2 = a; log 3 = Z>; find log 2881^3^ 7. Solve the equation 16^^ = 25. 57 8. Solve the equation 2^ = —. 9. Solve the equation 8^ + ^ = (201^17)3. 10. Solve the equation 11"^ = 2 log 22. 11. Insert six geometrical means between 10 and 1000. 12. In G. P. find a formula for n when a^ r, and I are given. 13. Eind the sum of six terms in G. P., of which the first is 3 and the last 701 , 14. The Volunteer Aid Association, desiring to equip a hospital ship, starts an '^endless chain ^' hoping to get $30,000. (This plan consists in sending out four letters, each numbered ^^1/^ and each asking the receiver to do three things: 1°, to make 4 copies, each numbered one more than the copy received; 2°, to mail them to friends; 3°, to mail 10 cents to the Association. The persons receiving letters numbered 20 are to make no copies.) Soon the number of replies becomes owerwhelmingly large, and notices have to be published to stop further contribu- tions. What should have been the last letter-number, to cover the necessary contribution ? "What would have been the amount received? [Assume that no letter fails.] 15. A sum of money invested in a certain business ia expected to double itself in 10 years. Excluding fractional rates of interest, what is the lowest rate that will accomplish this result, assuming interest to be compounded annually ? TABLE OF LOGAUUHMS, 363 TABLE OF COMMON LOGARITHMS. Except in the zero-colunm only the last three figures of every logarithm are given. The first figure of each logarithm is the same as the first figure in the same line of the zero-column, except ivhere an asterisk ai^pears, * Wliere the last three figures of the logarithm are pre- ceded by an asterisk, the first figure is the same as the first figure in the kext line of the zero-column. The computer will have to find for himself the value of D betiueen successive logarithms j but in the space to the right of the heavy vertical line are given the values of the fractions .ID, ,2D, .SD, ,J/-D, and ,5D corresponding to each value of D. This is the table of proportional parts. 364 TABLE OF LOGARITHMS. N 10 1 2 3 4 170 5 212 6 253 7 8 9 D 13 3 4 5 0000 043 086 128 294 334 374 43 4 9 13 17 22 11 0414 453 492 531 569 607 645 682 719 755 4a 4 8 13 17 21 12 13 0792 1139 828 173 864 206 899 239 934 271 969 *004 303 335 *038 *072 *106 367 399 430 41 14 1461 492 523 553 584 614 644 673 703 732 40 39 38 37 4 8 12 16 20 4 8 12 16 20 4 8 11 15 19 4 7 11 15 18 15 1761 790 818 847 875 903 931 959 987 ^014 16 2041 068 095 122 148 175 201 227 253 279 36 4 7 11 14 18 17 2304 330 355 380 405 430 455 480 504 5?,9 18 2553 577 601 625 648 672 695 718 742 765 So 4 7 10 14 18 19 2788 810 833 856 878 900 923 945 967 989 34 33 32 3 7 10 14 17 3 7 10 ia Iti 20 3010 032 054 075 096 118 139 160 181 201 3 6 10 13 16 21 22 3222 3424 243 444 263 464 284 483 304 502 324 522 345 541 365 560 385 579 404 598 3J 3 6 9 12 16 23 3617 636 655 674 692 711 729 747 766 784 »0 29 28 27 ?6 3 6 9 12 15 3 6 9 12 14 3 6 8 11 14 3 5 8 11 14 3 5 8 10 13 24 3802 820 838 856 874 892 909 927 945 962 25 3979 997 *014 *031 *048 *065 *082| ^099 *116 *133 26 27 4150 4314 166 330 183 346 200 362 216 378 232 393 249 409 265 425 281 440 298 456 ^5 2 5 8 10 12 28 4472 487 502 518 533 548 564 579 594 609 24 2 5 7 10 12 29 4624 639 654 669 683 698 713 728 742 757 28 22 21 2 5 7 9 12 2 4 7 9 n 2 4 6 8 10 30 4771 4914 786 800 814 829 843 857 871 "Oil 886 900 31 928 942 955 969 983 997 *024 "Uv3o 32 5051 065 079 092 105 119 132 145 159 172 20 2 4 6 8 10 2 4 6 8 10 2 4 5 7 9 33 5185 198 211 224 237 250 263 276 289 302 19 18 34 5315 328 340 353 366 378 391 403 416 428 17 2 3 5 7 8 2 3 5 6 8 35 36 5441 5563 453 575 465 587 478 599 490 611 502 623 514 635 527 647 539 658 551 670 15 2 3 4 6 8 37 5682 694 705 717 729 740 752 763 775 786 H 13 4 6 7 38 5798 809 821 832 843 855 866 877 888 899 13 13 4 5 6 39 5911 922 933 944 955 966 977 988 999 *010 12 11 12 4 5 6 12 3 4 6 40 6021 031 042 053 064 075 085 096 107 117 41 6128 138 149 160 170 180 191 201 212 222 42 6232 243 253 263 274 284 294 304 314 325 43 6335 345 355 365 375 385 395 405 415 425 44 6435 444 454 464 474 484 493 503 513 522 10 12 3 4 5 45 6532 542 551 561 571 580 590 599 609 618 46 6628 637 646 656 665 675 684 693 702 712 9 12 3 4 4 47 6721 730 739 749 758 767 776 785 794 803 48 6812 821 830 839 848 857 866 875 884 893 49 6902 911 920 928 937 946 955 964 972 981 g 12 2 3 4 50 6990 998 *007 *016 *024 ^033 *042 *050 *059 *067 51 7076 084 093 101 110 118 126 135 143 152 52 7160 168 177 185 193 202 210 218 226 235 53 7243 251 259 267 275 284 292 300 308 316 54 7324 332 340 348 356 364 372 380 388 396 TABLE OF LOGARITHMS. 365 N 1 2 3 4 5 6 7 8 9 D 1 3 3 4 6J 55 7404 412 419 427 435 443 451 459 466 474 8 12 2 3 4 56 7482 490 497 505 513 520 528 536 543 551 57 7559 566 574 582 589 597 604 612 619 627 58 7634 642 649 657 664 672 679 686 694 701 59 7709 716 723 731 738 745 752 760 767 774 7 112 3 4 60 7782 789 796 803 810 818 825 832 839 846 61 7853 860 868 875 882 889 896 903 910 917 62 7©24 931 938 945 952 959 966 973 980 987 63 7993 ^000 *007 *014 *021 *028 ^035 *041 *048 *055 64 8062 069 075 082 089 096 102 109 116 122 6 112 2 3 ^ 8129 136 142 149 156 162 169 176 182 189 66 8195 202 209 215 222 228 235 241 248 254 67 8261 267 274 280 287 293 299 306 312 319 68 8325 331 338 344 351 357 363 370 376 382 69 8388 395 401 407 414 420 426 432 494 439 500 445 506 70 8451 457 463 470 476 482 488 71 8513 519 525 531 537 543 549 555 561 567 72 8573 579 585 591 597 603 609 615 621 627 73 8633 639 645 651 657 663 669 675 681 686 74 8692 698 704 710 716 722 727 733 739 745 75 8751 756 762 768 774 779 785 791 797 802 76 8808 814 820 825 831 837 842 848 854 859 77 8865 871 876 882 887 893 899 904 910 915 78 8921 927 932 938 943 949 954 960 965 971 79 8976 982 987 993 998 *004 *009 *015 *020 *025 5 12 2 2 ~m 9031 036 042 047 053 058 063 069 074 079 81 9085 090 096 101 106 112 117 122 128 133 82 9138 143 149 154 159 165 170 175 180 186 83 9191 196 201 206 212 217 222 227 232 238 84 9243 248 253 258 263 269 274 279 284 289 85 9294 299 304 309 315 320 325 330 335 340 86 9345 350 355 360 365 370 375 380 385 390 87 9395 400 405 410 415 420 425 430 435 440 88 9445 450 455 460 465 469 474 479 484 489 89 9494 499 504 509 513 518 523 528 533 538 4 112 2 ^ 9542 547 552 557 562 566 571 576 581 586 91 9590 595 600 605 609 614 619 624 628 633 92 9638 643 647 652 657 661 666 671 675 680 93 9685 689 694 699 703 708 713 717 722 727 94 9731 736 741 745 750 754 759 763 768 773 95 9777 782 786 791 795 800 805 809 814 818 96 9823 827 832 836 841 845 850 854 859 863 97 9868 872 877 881 886 890 894 899 903 908 98 9912 917 921 926 930 934 939 943 948 952 99 9956 961 965 969 974 978 983 987 991 996 CHAPTER XYI. SUPPLEMENTARY PROBLEMS FOR PRACTICE AND REVIEW. 407. The problems in this chapter are selected, for the most part, from examination papers, set by colleges or in- stitutions of a similar grade. The arrangement of them is purposely left somewhat irregular, so that the attitude of the pupil towards his work may be more natural than when, in doing a set of carefully classified exercises, he knows be- forehand the kind of difficulty he is about to meet. This is especially noticeable in the ^^ concrete problems " which close the collection. In the absence of other directions, where an algebraic ex- pression is given, simplify it by performing the operations indicated : where an equation or a set of equations are given, solve them for what seem to be the unknown letters. Factor the follotuing expressions : 1. x^ -%x- 84. 2. (2^; + 3)2- {x - 3)1 3. ^2 _ ^ _ 2. 4. 324t^4Z>2 - Wb\ 5. x'^ + l^x - 85. 6. d{x-\-yY—^{x^—y'^)'-x{x-\-y). 7. x'^ — 2xy—xz-\-2yz. 8. x^ — 2xy — z^ -\- y^. 9. y^ + ^if' + %• 10. ^y — a;^ — y^ 4- 1. 11. a^ + a^ — «- — 1. 12. \2x^ - 5x - 2. 13. 12a;2 - x-1, u. {x + 1)* - 1. 15. 18x^-lSx^+4.x. 16. 2x^-3x^-U. 17. x^ - x - dO(j. 18. 8:^;^ - 39a: + 46. 19. {x^ - xf - 8. 20. x^^ - a^\ 366 SUPPLEMENTARY PROBLEMS. 367 21. Sex — 12r^ + '^ax — day. 22. a^ + 4^^ + 4a^^^ — c^ 23. 2am — b'^-\-m^-{-2bn-\-a'^—n^. 24. 3ax—bx—3ay-\-by, 25. ^^ — 20a;^ — 96?/^. 26. 2x^ + 5a;«/ — 12?/^. 27. «2 _ ^^J _ ^ - 1. 28. ^^^ - b'\ 29. «' + rt'^'' - 2I)\ 30. G^' -- l)(x - 2){x - 3) + (^ - 1)(:^ - 2) - :?; - 1. 31. a^y^-\-a^x^ — b^x^— h'^y^. 32. 6m^ — 38m — 28. 33. a^ + ^^ + ^^'^ — y^' 34. (^^ + ^^ — ^'^Y — 4:X^y^. 35. 4^;^ — (yyz — {9y^ + z^). 36. 9^^ + 4^:2; — {4:X^ -\- z^). 37. 5a;^ — 15^;^ — 90a;^. 38. a^-{-x^~{y^-]-z^)—2{yz—ax). 39. ^^ - b^. 40. a;2"* + i^"* + tV- 41. 9x^ - 37^:2 + 4. 42. 5a;^ — 15^;^ — 902;^. 43. {a-\-byx^ — {a'^—b^)x—ab, 44. x^ — x^ — X -{- 1. 45. («^ — V — c^y — 4cb^c^' 46. ^^ + (a + ^ + <^)^ + ^^ + ^^• 47. Having given 7n, the difference of the squares of any two consecutive numbers, find the numbers. Verify by substituting figures for letters. 48. Give three different forms of polynomials that are factorable, and illustrate the method of factoring by an example. 49. Find and factor the expression which added to the following will reduce it to zero: (b - a){a + b-c) + {c- b){b + c - a) and check the result by substituting a = ^; & = — f ; 50. Subtract Sx^ — 7x -{- 1 from 2x^ — 5x — 3, then subtract the difference from zero, and add this last result to 2x^ - 2x^ — 4. 51. X — [x — 2x — {x — 3x) — {x — 4a;)]. 52. (3a; - If - {2x' - l){3x + 1) - 3x[{l - 3xy - {2x' - 1) - 2{2a; - 1)]. 368 ALQEBBA. 53. If a; = — 1|, find the value of (1 -x){l -{- ^x) _ (1 + ^)(1 - 3^) 1 — ^x 1 -\-^x 54. Find fcur factors of 3(6^^ + bxf- 10{<6x^ + bx) - 8. 55. 2x - [a - (3Z> - 6c) + [x - (4a - b + x — c) + 3c] - [2b - 3a +Tc] ]. a{b — c){c — a){a — b) — b{c — b){a — c){b — a) ^^' {a + b){c-a){b-c) ' 57. What value of a will make 6x^ — 2x^-}-2ax^-}-2x + « an exact multiple of a^ — x -{- 1 ? b — 4: 58. Assign a value to b which will make equal to ~~" b-\-6 b-4' 59. Subtract the sum of the squares of ax + by and ay — bx from the product of a^ + y'^ and b'^ -\- xK 60. Supposing that a ~\- b — c = 0, simplify T2~r — • 4i2/ X I X 61. What fraction must be added to , —^7 — r—r-— , ^, {x — iy{x-{-l) (^-1)^ to make its value 1 ? 62. S{x + z) - {6a - z) - 2[x - {2a + z) - {a — 3z)], X 1 63. Express -^ ~ — as a single fraction, with X — y X — o (3 — a?)(3 + xy for denominator. 64. Find an expression such that when divided by lj^2a + 3a^ + 4:a^ it will give I — 2a -\- 3a^ - U^ for a quotient, and 5a;* — 6^;^ for a remainder. a^ — b^ . a — b ^^' a^ - 6ab + 6b^ "^ a'^ -+ 3ab - 10b'' ' 66. Find the value of x from the proportion 3^3 _ 4^2 _ 4a + 5 * 6as+7a2 - 13a - 15 "^ i^+^)'^' I SUPPLEMENTARY PROBLEMS. 369 67. Reduce to a single fraction in its lowest terms: -^—-(^ ^—\ X"" — X \x^ — 1 x^ -\- ll' 68. Find the square root of ^x^ - 20r^ + 21x^ + 22.^'3 - 2^x^ - ^x + 9. 69. Reduce 9^4 _ ^2 _p iQ^ _ 25 6:c^ - 2x^ + lx^ + x-b to its lowest terms as a single fraction. 70. Find the following expressions in their lowest terms : 2^ + 1 X -\-l Vix^ - lOx^ - 62; + 4 "~ 12x^ + 82:2 - 3:?; - 2 and 2:r + 1 ^ X + I l^x^ - lOx^ - 6.f + 4 * 12a;3 + 8:^^ - 3:?; - 2' 71. Reduce to their lowest common denominator and 12x^ - 22;2 _ 20a; - 6 40;^ - 6a:^ - 4a; + 6' and find, and reduce to their lowest terms, the difference and the quotient of these two fractions. 72. If -4— = «; — ^ - h] -4— = ^; find the y + z X ■\- z x-^ y value of — h ,— -— r + -7—, — . x^y + a;^f/^ ^^ — yH -f- :2/^^'^ 73. Find the value of ^--^ T 33 ^^^^ ^' = %• 370 ALGEBRA. 74. Find the value of {x -\- y){x — y)(x^ -\- y^) when the sum of the squares of x and ^ is 3, and the difference of their eighth powers is 21, y being greater than x. 75. li a = 0, h =- 1^ c = ^y d = — \, find the value of ' , Z>2 _ ^2 "+ :7 — ;7 + - c — dd — a^a — h' 76. [x^"^— x^— x"^ — x'^'iyx"'— x^ -\-^). 77. What value of x will make x^-\-lx^-^^x^-\-Sx-\-'ib^b an exact multiple of 2;^ — 7:?^ + 1 ? 78. Find the value of {a - h){a -\- I) - {a -{- bf when 3a + 2c = 4:5 and 3c + 2a = 15. 79. Find two expressions whose product is 12^;^ — 4:0x^ + 392; — 9 and whose quotient is 3x — 1. OA -El- ^ +-U 1 ,x + 2a.x + 2b , 4:ab 80. Fmd the value of — —^^ -\ —z— when x = — --r. X — 2a ^ X — 2b a + b 81. Find the value of ^{y + ^) + y[^ - {y + ^)] - ^[y - ^(^ - ^)] when X = 3, y = 2f z = 1, 2y_+jt^ 82. Substitute 1 — x^ for y in ^ .^ and simplify. X^ — V^ a I h 83. Find the value of o ■ 9 when x = 7 and x^ -{- y^ a — b a — b X *— a X ~~' b CL 84. Find the value of — when x = b a ^ a — b 85. Find the value of ^^^ " ^^ when ^ + ^/ = 8. 2x — y Ix — 2y 86. Show that the difference of the cubes of any two consecutive numbers is one more than three times their product. SUPPLEMENTAUY PROBLEMS, 371 x^ + ^xy x{x — 2i/) 87. What values has the expression —2—[—^f.„ when 3j/ ^ — 2^/ ? « U^ 88. What value must be assigned to - in order that y 2x- - xij = 2xy + 2/ ? 89. A snm of 1100 is put at compound interest at 4 per cent per annum for o:^ years; find a formula for the amount. 90. Wluit value must y have in order that when a; = 1 the equation 'dx -{- 2y ~ 1 = 2x -\- 6y — IS will be true ? 91. What value must be given to x in order that when a = f the following expressions may be equal ? 92. Find the sum of 7-, — , and -7 ; subtract be ca ao 2f- + - — -J from the result, and simplify the remainder. 93. Assuming that x^ - 60;^ + ^x^ + 2Sx^ - ^x^ - 54a; — 27 is a perfect cube, find what values of x will cause it to vanish. 94. Find the difference of the squares of the highest and lowest of any three consecutive even numbers, and express the result as a theorem. Is this theorem true of odd num- bers ? 95. If the quadratic Ix^ — 2a; + (2 — c) is a perfect square, what is the value of c ? 96. Prove that when a number of two digits is equal to four times the sum of its digits, one digit is double the other. 97. Two men, working separately, can do a piece of work in X days and y days, respectively ; find an expression for the time in which both can do it, working together. 372 ALGEBRA. 98. A is 20 years old, and B is — 2 years older ; what is the age of B ? 99. What are the values of x which satisfy the equation x^ = 32;? 100. Divide a\l) - c) + h^c - a) + c\a - h) by a-\-h + c and factor the quotient. 101. Find the H. 0. F. of Q^x^ - 2x^ + 9x^ + 9^; - 4 and 9x^ + ^^^^ ~~ ^ I what value of x will make both these expressions vanish ? 102. Find the H. 0. F. of 3x^-6x^+2 and 2x^-5x^+3. 103. Find the L. C. M. of VdaP{x^ - 3a^x + 2a^); Q6a^{x^ + ax - 2a^); and 25b%x^ - a^, 104. Find the G. C. M. of x^ - 93^ - 308 and x^ - 2l2;^+13l2;-231; and the L. C. M. of 12x^y{x^+y^)\ lSxy\x^ - y^); 2\x\f(x'^ + xhf + y^), 105. Find the H. C. D. of o;^ - 2x'' + 42^2-6:^ + 3 and x^ - 2x^ — 2x^ ^ 2. 146. 147. 2+ o 2 + i?; 2 + ' 2 -3a; 2-3 2 + :^; Z^^ a' 2-3^ 2 2 + _2_+_^ 2 - 3:r 2-3 24-a; 2 -3a; -1 + Z^2 «2^2 \rt2 _ ^2 - -T ^2 ^ J2/ 148. Substitute ^ = a2-Z>c; B = V-ac\ G=c^-ah: ^^^. + - + 149. 150. (/^ + q^){^ + ^)^ + ^(^^ - qy){q^ -yy) (x^ + 8)[a;(a; - 2) - 4] [^2 _ 2a; + 4][a;(a;'^ - 8) - 8]' 151. Express as a theorem the fact implied in the follow- ing equations : 9^ — 7^ = 4 X 8 ; 5^ — 3^ = 4 X 4 ; ^Z^ — 91^ = 4 X 92. Prove it, and then express the theorem in the most general way. 152. Eeduce 9.-^^ + 8 a; = 1; also when x bx^ + 4' and find its value when 376 ALGEBRA, 163. V{a:^ + U^){c' + d') - {ac + bdf- X X + 1 154. x-\ 155. X-\-\ X ^+1 1 i^ + 2 - x-\-\ ri 10 _ 6 n 3rt— 6 :?; + a—h x-\-'^ a — h- SU' a-^b 157. ^f -4bx'- 6 158. I I- + 2x a 12ax-4: \ 'dbx^—9 b' (i-*.^)-?„.|]. X a' X 159. 160. 161. 162. 163. 165. I — a/ '^ I -[- b '^ a -{- d'^' '!_ 2x^ + IW - 43:r - 24 _x 14a;^ -- 'dlx^ — 31:^; — ( G- a^6' + abc + Z>^6" «2^'2 + b' * 1 — X^ ( X -\- X x^ -\- y'^ — X -\- 1-/ .) 1 + g:^ 1 + a; 1 + a;" 1 + a; ' 1 - a;'^ "*" 1 - a; «^ 1 a _ 1_ 1 • 6 5 "^a « + a + b + 2 + ^t; 1 1 j4 2 — ^ *• 1 * a; + 2 1 T X ~ 2 1 Z>(aZ>(? -{- a + c) 8UPPLEMENTAEY PB0BLEM8. 377 166. Compute the value of the continued fraction 1 1 + 4+- 167. + \:{x—a){x—h) a{a — x)(a — h) h(b—x){b — ay 168. Divide — —7 — o , 70 ^J — ^ + 2 ir 169. Compute the value of the continued fraction 12 1 + ^^\ 170^ 171. ^ + 1 - ah X - 2 cd ¥ — 4^^ r(Z? + ^^^ * 173. X y z yz zx xy 174. 175 .a — Z> a-\-hr ^ \a — b ^ a «3 - b a' + ¥ + ab{a? + ¥) + a^Z>^ ^ ^'^ + ¥ + ab n^ - ¥ "^ a^ - //^ + &, 378 ALGEBRA. {a - by - ah{a - if - 2a%^ 176. (a - h){a^ - ¥) + 2a^Z>2 g^ + ^' Cl_-_b ^ ^4 _ ^2^2 ^ ^4 ^^■^^ a^ _ ^6 X ^. _j_ 5 "^ ^4 _|. ^2^2 _!_ ^4- 1 « — 5 1 rt — 6 ^''®- 9V/y _ 1\ ~~ ^f2 ^ ry^ i TTv + ^ 2(g - 1) d^ _ 7a _|_ 10 ^ 2 a^ - 9^ + 18* "^' t+yl + 2 "^ <^ + y)' 2' X , X — 1 X — ^ 180. . — , ow A + 181. x+^){x-l)^ X + 3)(2 -x) (2-^)(l -ic)' X ir 1 182- .Tlr:TT, + . 4- 3v/ «^ Va; — y a; + 2y/ «^ + ^^ ~ 2^*' 2^(y - ])(?/ - 2)(t/ - 3) + 1 183. 2 " — "U ~r -^ 1\' 2/"' - 3.«/ + 1 ,,2 ™3 184. • ^0 X ^3 'r- 7- ^ 1 + 1.^' ?: + i i+M a; o;^ y^ \x y I 185. 2:z: + 4 , a; + 5 ic'^ + 7ii: + 10 ~ ^2 + 4a; - 5 "^ :?;^ + :r - 2* 186. 3a; - 4^+2:^=1; 4a;-2?/+3^=9 ; 2a;-3?/+42;=8. 187. x-\-y — ^z=S\ y-}-z—3x=—4c', z-\-x—3y= — S. 188. 4a; + 2(3a; - ^ - i) = 1 + 3(y - 1); ^V(4a: + 3^) = g+l. SUPPLEMENTARY PROBLEMS, 379 189. — ^1^; 3 -i«^-h2/ '^^ ;l3 ^ 5 • 190. Find the values of a, b, and c in the following equations : ' 3a + |Z» = 30 - c |(^ -c) = d{2b - 7) 3 3 19 191. 13 x+2y+3 4.x-5y+Q' 6x~6y+4: ^x+2y+l' 2x — y bx 192. ^x — 5«/ _%. ^ + y :=^ll 7 4 193. 4(:r-22/+4)=2:.+3(^-i); 5(^+|-)~2(^+2) = ll 194. 8a; — ^ = 2^ + 2^ = 6. 195. - ~ 2 y =.*^x = y -- 2x = — 1, z " z 196. ?»x-y-\-2z=^-z-\-2x+2-^z-x+2y-b=ill. 197. x-b = \{y- 2); 4y - 3 = ^0-^+ 10). 198. 42; - 6y - 4 = 7^ + 2^ - 5 = - 2^; + 3y + 23. ^^^' Zx~^ by 9 ' 5:i; "^ 3^ 4 * 200. — =r + — ^ = 4; — =. + — ^ = 1. r^?; Vy Vx Vy 2(^2+ ^2) 201. x^ y X - y ^ab a^ - b^' 202. 2x + 4.y + 27z = 28; 7x - Sy - 15z = 3; dx - lOy - 33;2 = 4. 203. 5a;-32/+2;2=41; 2x+y-z=17; 6x+4:y-2z=d6. o^. o 1 .1 X. . ^^+^ 21-3a; 2(2a;+3) 204. Solve the equation 6 j— = — c, - # 380 ALGEBRA, 5a;-- 3 _ 3a; - 5 1 8(rg - 1) ^^^- 4 ~ 10 +20+ 5 ' 207. 1 + o;^ = 5i + ?!-Ilii. . 7r 208. (7 + i^)(8 _ a;) - 1 = y + 17a; - 23 ,3/1 1 \ a; - 1 209. .-..7—-^ + ^(^— t - ^J = ^^1- 23 10(a; 1 , 1 210. — — B + a; + 8 "^ o; + 4 a; + 2 "^ a; + 10" 5a; - 12i 3a; - 2^ _ 6a; + 7 2a; + 3 ^^^' 9 ^ 5 ~ 11 ~ 8 • 212. 3a; - 5 + VS{2x - 3)(a;+l)- (7a; - 5) (a; + 2) = 0. 213. ?^ + ^" = i[^+ 1(^+1)]. 18 ^ 6 a;2 + 1 , a;2 - 2 2a; 214. -o-^+ - a;^ — a; — 2 x ~{- V 215. (3a; - l)(4a; + 5) - (2a; + 3) (5a; - 2) = (a; - 2)(2a; + 1) + 12. 216. lo(a; + ^) - ^x[- - ^) = 23. 217. \a; 3 a; + 3 a; -- 1 __ 2a; X + 2"^a; + 1 a; — 1* ___ a; - 2 _ a; + 23 10 + 218. a; ~ 11-3-:==^-^-^^ 5a; + 2 219. ^^- 7a 220 -(-^)=5^'-(4-'+4 1 , 3 2_ _ SUPPLEMENTARY PROBLEMS, 381 222. 3a; - 4(2a; + 3)2 + 190 = 0. 223. x^ - 4.x -[-1 = 0, 224. ^^^ = ^^. 225. Vx^ + lQx+^ = 3a; - 5. 226. 4a;-5 — bx-^ -[-1 = 0. 1 _ 4 7 10 "" 2c?; - 3 5:?; - 6 ^ 8iz; - 9 11:?; - 12* 228. 11^2 = 2OO2; + 171. 229. Assuming that the first member of the following equation has three identically equal factors, find two values for a, and tell why you cannot find the other four: a^ - ISa^ + 114^4 - 288^3 -f- 228^2 - 72a + 8 = 185193. 1 Qy I A 230. Solve ^-^—^+^-^ = 2^ 231. What value must be given to i in the following equation in order that when a = % the value of x may be 4? 26^2 - h _ a? - %h 3a^b - x) _ b - X x + b ~^ 25-^^ ~" 232. 119 + V4tx^ + 2^ + 7 = 12a;2 + 6x. 233. 9.^?; - 7 + 2Vx^ _ 4:?; -f 7 = x^ + 6x. fx + 2 X - 2\ ^ . 236. 3(l + -^) = -4-(l+-i^,). \ 1 — XI 2;4-7\ X — 1/ 237. —^ + -^ + ^-=0. 4 — a; '« — 2 a; 382 ALGEBRA. ^2 238. ■^— ?>x = x-\- 3Vx^ - 8:^; + 9. • • .T + 1 240. x^ - 2x + 6Vx^ - 2a; + 5 = 11. 241. (i?;2 - 52^)2 - S{x'^ - 6x) = 84. 242. 4:X ^ = 14. 243. i^X^ — 3;^^ + 1 = X. X — I 244. ^x^y^ — 7 = xy; x -\- 4:xy = 9. 245. 3:?; ^ = — ; 7x — 3y = 10. X — 4/ o 246. X -\- y = 20; xy = 51. 247. x^ + y^=9; x + y = 3. 248. 3{x^ + xy) = 4:0y; x — y = 2. 249. X ^ y =^ x^; 3y — X = y^, 250. i^; — ^ — 5;-4--=-. X y 6 251. i?;^ — 2xy -\-3y^ = 3; x^ -{- 2xy + 3y^ = H. 252. 29 = ^^ + ^i^; 14 = ^^ + ^±^; 2^ = . -- - . ■,H = ^y- X ' y ^+y 253. x^-3y + ^^^, = (2:. + by- Z3){x' - 3y) = 0. 254. 3a; - 2^ = 10; x{2y + 3) = 6 255. -^; llx' = 2xy + 13y^ 256. x^ + dxy = 12 — xy = IQy^ — xy — x^, 257. 2x^ + 3xy = 12 — xy -\- x^ — IQy'^ — xy, 258. x^ — 2xy + 2/^ + ^(^ + ^) "~ ^ = a;(:c - 1/ - 1) + ^(a; - ^ + 1) = 0. 259. ^y -Vy'^^y'^- -^-^ — = 4. SUPPLEMENTARY PROBLEMS, 383 x\ _ xy^ — ^/ 3^ 1 / , :r\ xy^ — X _ 260. 5(a:y + -J=-.^ = 2 a: _ y _ 7 * y^ — 3 ~~ 3 — :?;^ ~~ ^^ — a;^' 262. ^!+l'^^^!:^'. 25^2 + 9^2,^450. 17 o 263. :2:2 + 2^^ - 4 = 7(2^2 _ 3^^) ^ 35. 264. ^_| = -4(2^-3:.) = 48. 265. '^x^ -\-bxy -'^ = llxy - 3^/2 + 1 = 20. 266. - + ^=|; c^2 + 2i/2 = 54. 267. xy — \^\ yz — ^0] xz = 18. 268. y{y - 2) = x{x + 2); y + 2 = X — 2. 269. a; + 2^ =r 8; ^2 _|_ 2^2 ^ 22. 270. 3 V(^ +y){^ + 1) = 3^ + ^ + 2 = 4. 271. a;2 -|- y2 _ ^ _ y — 7g. xy -\- X -{- y = 39. 272. 4.^2 + ^y == 6; 3a;y + ?/2 = 10. 273. 2:2 4- :^y -f- y2 — 52; a:y — ic2 — 8^ 274. ^2 + ^?/ — 2y^ = 7 ; x^ — 9^2 := 27. 275. {x - 2yy+ 3{x — 2y)+2=0; x^-2xy-3x+6y=l, 276. 2:^;^ + ^^ = 24; xy = 8. 277. {x+ l){y - 2) + (:z;+l)2^2; {y - 2)2 + 3(^ + l){y - 2) = 4. 278. a; + ^ = 12; x^ + y^ = 74. 279. + ^ = a; a;2 - 2/^ z= ^>2^ 280. — : —= 3 : 7; ^2 _ ^2 ^ 9^ 281. 2x^ + Sxy -3y^ +12 = 0; 3x + 5y + 1 = 0. 3Vx + 2Vy _ ^ . ^^ + 1 ^ ^^ - 64^ 4:Vx ^2 V~y 282. T-^ r-^ = 6; ^g 384 ALGEBRA. 283. 91.^2 — 2x = 45. 284. x^ — 21.^2 — 100. 285. x'^ ~ x^ = '7{x^ + 1). 286. x^ — 2x^ -\- X — 2 = 0. 287. {x + l){x - 2){x^ ^ Qx + 9) = 0, 2:r - 5 3 9x — 1 56 2; — x Form the quadratic equations whose roots are : 290. (a-i); (5 + f). [a + by ^ 291. ^^ ' — r-l ^ — «. a — b 292. - I; f 293. One root of the equation ^^ — 4:^; + c is 2 + Vs^; what is the other root, and what is the value of c? 294. For what value of m will the equation 2x'^ + dmx + 2 = have equal roots ? 295. Find the value of m which will make x^ — {^m — l)x + 2m exactly divisible by ^ — 1. ^x ~ a) 1 ^ \ a I 296. It — ■. — ( = 2 7 — ■ -2 b(x -\- a) a {x + af a ~ c X — a db(x — c) 29Y z=z - ' X — a a — c {a — c)(x — a)' 298. x^ + (2^2 + '^ab - 2^^2)2 ^ 5(^2 ^ ^2)^2^ SUPPLEMENTARY PROBLEMS. 385 299. x^ — hxy -\- y'^ = 15a^: -f^ = 1. X y ^ a h ah X ' y 301. Solve the equations separating the two sets of answers clearly from each other. 302. = 1 ; "^xy 4-9 = 0. X y ^„ hx a ~ I X y ^ 303. ^ i ^ -TT-; I = ^• - a b ay bx y — h" a ~ b X -{- a ~ 2 ' x-{- y X - y _ a^ - W 304. ^^-' - -—^ = -^-^; x' + y' = 2{a^ + b^^y X — y X -\- y ah ^ \«/ ( a{x -y) - b{x + y) = a!' - b{2a + b); ^"^- ( {a + b)x + (« - b)y = a{a + 2b) - b\ 306. One root of the equation 2a;^ + Sa;^ ~ 3a; — 2 = is 1. Find the others. 307. Find all the roots of the equation x^ = (125)1 308. What equation leads to the following answers : 1,2,-3,5? 112 1 11 309. = ^~ ? = T'- y X oa a — X a — y 4:a 310. (a — b)x = (a + ^)y', X -{- y = c. a; 1 1 + a ,1 311. - + - • — ^ + -z' a a X a^ 312. ctljx^ — (a^ + b'^)x + ab = 0. X y X y a ''' a-J^^'^ b+a=y 314. {p+q)x-{p-q)y=^pq; {p+q)x-{p-q)y =pq. ax , bx . r 316. 1 -\ = a + b. X — b X — a 386 ALGEBRA. X + a X — b 2{a + b) 316. — ; — y — . X — a X -}- b X ax^ -^ b a + bx^ _ 2(a^ + b^) ' ax -\- b~^ a — bx ~^ d^ — W ' 318. {a - b)y^ - {a + b)y + 22> = 0. 3:9. fil^ + A^ =: 1. 1 — px 1 -\- px 320. {ax — bf + 4:a{ax — b) = fa^. 1 , a a 321. -J- -\ j -, + X ^ r = 0. b * X + ab ^ 2x -]- ab 322. x^ — {a — b)x = [c — a){c — b). 323. 1 . - 2= 0. X — a ^ X — b bx -^ c ex + b ib -\- c){x + 2) 324. ^--7 = ^^ — /; . — ^• ax -{- c ax -\- b ax + b + c X y ^ X y 2 a b 3« ' 6^ 3 5:^; 1 /I \ , ^ d( 1 \ a; . Z» 826. [--\-x]+d = -[cx _ — _L._. c c\a J c\ adJ c ' c a b a — b 327. , = . X — X -[- a X 328 I ^^ + ''^'' + ^^ " ^^^ " ^""^^ ( (c + a)^ + {0 — a)y = 2ac. 329. {b — c)x^ + (^ — «^)^ -\- a — b — ^, 330. 52; + «5y = a2 + Z^2. ^2 + |2 = -2+p- 331. — 4- — = 2: by — ax — b^ — a^, X ^ y ^ 332. a^x — a^^ = 2«j«,; 2a^x + 2^^^ = 3^,^ — a^. ^{pq — a;(^ + q)\ C^p + $')5'^^ _ qx p'^q^ 333. i^ + s' i^Ci^ + s'? p {p + Qf 834. Z>:i: + 2/ = ^ + ^y = i(^ + ^) + !• SUPPLEMENTARY PB0BLEM8. 387 335. X -\' y = axy\ y -\- z — hyz\ z + x = cxz. 336. a — yz . X ; y{b — zx)-=z-\-x; z(c — xy) — y=x. 337. 338. 2a{a + b)'-I)^x bx — 2a 2 I c^\x 2al 4.a Ifx b ii-') ' -b 1 1 - b X a X b = 0. X + y \ct{x — y) x^ — y^i y :{lx - 2y) = {b - a) : {2a - 9b). [Substitute a = 6 and Z> = — 2 in the answers.] 6ax — 4:b\ i[2b{x + 1)]2 341. 342. 343. 345 ^bx^ -\- bax {a + 2b)x _ a — 2b s X 4cbx^ + ^^- ;)=o. a^ ___4^2 a — 2b x' f a{l + 2x) b{3x - 1)-| _ 1 + 3^ L^(l + 3x) a{2x + 1) X + 1 2__ ^ + 2 cx~~ ax — bx' *"*'** a^ 4" ^^ 1 /i?; - 3 1\ 2 ] = 0. c ax abx ^ d'^ — b^ 344. o , ,o — 1- (a''^+62)ic' i?;/ ; a^ic — 2 « Kd/^x — 2 i?; / 2^; — a^o;^* [Substitute a = — 1 in the answers.] 1 _ ^ ^ax? - Zb{x - 2) ^^^' X -"^ 2a{x^ + 1) + 36 • ^2x ■ 347. a^- x^2 1 = ^2-^ + ^ 348. ax ax -\-b 2x ' {b\l + x)x - ^2(1 - x)'\ = 6. 388 ALGEBRA. 349. 350. OKI / a - \x 2ax X - 4.1) hx — a 1 2b X Ix — a a)~2 ~ \2 " X — J' 2abx 2ax - b 2abx^ - {2a^ + b^)x + aV =1+-+- \ -\- a-\- X ax 352. ax ■ {2b + a)x + a _ P{x - 1) x^l 4a 353. 354. 355. X -\- a n^px ' X — 4:a 4:b — 7a m^p[x -\- a) ~ 2b — X — 2a _ bx ab — b'^ ax — bx 2a'— X — 19^ a — 2b — X bb — x ax — 2bx 356. 1 — a^ — 46^ ax + 2bx ^(x_ _ 2a^ - ?>x a \a X = 0. 357. {a + 1) — - 2 H V , . = 0. ^ ;T — 1 a + lL X a{x — l)J r 2x_ u{x + uy [jL 358 359. X- U a —U X + b ^ _x - 2a 3b 2a{a - 3b) _ = 0. 2a X — b 1-t'^ 2a - h X - b) 360. {x-{-a){x—b)- \x-\-a) b\x-b) _ 3aW x-\-b a {x—a)(x-[-b) 361. {x + 3y):{2x-y)=i^^-lyi', x^=^{xy + 3ay + lM^). [Substitute a = 2 and b = — 3 in the answers.] a 2b 1 r 3 1 362. i' {b-a)x L( y-\-^b x—y^ {b—a)x \_{a-\-b)y d^—V^_ [Substitute a — 3 and b —. — 1 in the answers.] = 0. 8UPPLEMENTABT PROBLEMS. 389 363. -i-f = 1; -+- = 4. a b X y 364. (3 + h^){x^ - a; + 1) = (362 + l)(;^2 + c^; + 1). 4^2 4^2 _ J2 J2 366. cc + 2 "^ a:(:c2 — 4) a; - 2* X y -^ X y 2a; + 1 3a; + 1 1/1 2 368. 369. ^ a x\b a, a b{2x + 1) :)• b{2x - 1) ^(r?;^ - 1) {2x - l){x + 1) ^ {2x'-l){x - 1) x + 3b , 3b a + 3b 370. o,.. .0..7. + " 8d^-12ab ' 4a^ — 9^''^ (2« + 3^>)(:?; - 3Z>)" 371. ax-{-by = p; cy -\- dz = q; ex -\-fz = r, X + 13a + 35 a-2b _ ^^^' 6a-3b-x ~ x + 2b ~ ' a + 1 a + 4: 373. {a + 2)x - {a + 3) {a + 6)x - {a + 6) b + 1 b + 4: ~ {b + 2)x -{b + S) {b + 6)x -{b + 6)* 5'^*- a + b+x a'^ b'^x' (JL V 375. - + -^- = 2 ; xy -^ ay — bx -\- db. X 376. ax ^by = 0\ px — qy = m, 377. If a, Z>, c, and cZ are in G. P., prove that {ac — bd){ab — c^) 2«c __ (ac + bd){ab + 6?^) "^ a^ + bd ~ ' ■ 390 ALOEBBA. 378. If «, h, c, d are in proportion, prove that d c -\- d ~ c^x -\- cdy + t?^2;* ^2 1 379. If fl^: &=^ : y, prove that «^+5^: a{a — b)— — ^ 380. If a, h, p, ^ are in proportion, prove that ma -f- "nh : ax — iy =^ pm -\- qn : px — qy \ if they are in coktii^ued proportion, prove that then 1. Show that if , ^, , = ^ — r^ = i ttt-' a _^ b c X -{- 2y -{- z ~ 2x -{- y — z ~ ix — 4:y -\- z' 382. If c : d = X : y, show that — =: ^^T — 5. •^ xy x'^ + ^ 383. Insert two geometric means between a^"" and J^^. 384. If b, c, and 2b — a are in G. P., show that ab, b'^, and c^ are in A. P. 385. Prove that if -^ = ,, ^7 ? then a, b, c, and J 22: — Zy 2a — 3£> are in proportion. 386. What relation exists between the values of x and y in the equation (12a; + y) ^ (H^ + ^) == 9 : 7 ? 387. The first and second terms of an H. P. are respect- ively 70 and 60; find the fifth and seventh terms. 388. An H. P. has the same first two terms as a Gr. P. whose common ratio is ^. Find the ratio of their third terms. SUPPLEMENTARY PROBLEMS, 391 389. Insert c arithmetical means between a and &. 390. Find the seventh term of 2; |; 1; . . . 391. Find the sixth term of 2; f ; 1^; . . . ; also the sum of the first six terms. 392. Given the series 3f + IJ + y^^ + . . ., find the sum of the first five terms and of the whole series. 393. Find the sum of nine terms of the series 2|^, 3f , 5 . . . 394. The first and third terms of a G. P. are 4 and 2i respectively; find the sum of the whole series. 395. Find the sum of n terms of the series (l - V^) + (2 + V^) + (3 + 3i/2") + • . • 396. Find the sum of n terms: 7 + 1 + ^ + . . . 397. Find the tenth term, the sum of ten terms, and the entire sum of the series whose second term is 5 and whose sixth term is y^-^. 398. Find the sum of n terms of an A. P. of which the first two terms are a and h] also of the G. P. which begins with the same two terms. 399. The sum of five numbers in continued proportion is 242, and the ratio of the third term to the fourth is 3 ; find the first. 400. If a, X, and y are in continued proportion, and if I I -A-n XI, X x + a y + a a, X -\- a, y -\- a are m A. P. , prove that a = — - — = — - — . o 401. How many terms of the series 18, 15, 12, . . . amount to 60 ? 402. If there are five numbers such that the first three and the last three are in A. P. , while the first, third, and fifth are in G. P., prove that the product of the first and fourth, added to the product of the second and fifth, will be double the product of the second and fourth. 403. The sum of the squares of the successive whole numbers beginning with 1 and ending with a certain number is 20 times that number; find the number. 392 ALGEBRA. 404. Find the sum of all multiples of 7 between 50 and 450. 405. Find the sum of all multiples of 3 between 1 and 3a + 1. 406. Sum the G. P. 27; - 9; ... to infinity. 407. Sum the series: I. 30 + 15 + 7i + 3f . . . to 10 terms. II. 30 + 15 + - 15 . . . to 10 terms. III. 2.7 + .09 + .003 + .0001 + ... to 10 terms. If each of these series be continued to infinity, which will have a finite sum ? If there is any such, find the sum. 408. Insert 2m — 1 arithmetic means between a and I. 409. Identify the series — =; =r ; ■ -\ and |/2 1 + |/2 4 + 3 V2 write its fourth term. 410. The first 121 integers can be arranged in five groups, each of which is a perfect square, thus: (1); (2, 3, 4); (5, . . . 13); (14, ... 40); (41, . . . 121). Prove this, and extend the property to a sixth group. 411. The second term of a G. P. is 54, and the fifth term 16; find the series, and its sum. 412. Find the sum of the first n natural numbers. 413. Find three geometrical means for 2 and 162. 414. Find the sum to infinity of the G. P. J; ^; -J; . . . 415. Find the sum of 18 terms of the series f; — 1; 92. 416. A person saves $270 the first year, $210 the second, $150 the third, and so on; in how many years will a person who saves every year $180 have saved as much as he ? 417. Insert three geometrical means between 3f and 18. 418. The first term of an A. P. is 2, and the difference between the third and seventh terms is 6. Find the sum of the first 12 terms. SUPPLEMENTARY PROBLEMS, 393 419. There are two numbers whose geometric mean is f of their arithmetic mean; and if the two numbers be taken for the first two terms of an arithmetic progression, the sum of its first three terms is 36. Find the numbers. 420. There are two numbers whose arithmetic mean is 33^^ greater than their harmonic mean. Find their ratio. 421. The sum of three terms in A. P. beginning with | is equal to the sum of three terms in G. P. beginning with 1^, and the common difference in the first case is equal to the common ratio in the second. What are the two series ? 422. Find the sum of ten terms of the G. P. in which the fourth term is 1 and the ninth term is ^J^. 423. Find the ratio of an infinite geometrical series of which the first term is 1 and the sum of the terms f . 424. Find the sum of the arithmetical series formed by inserting nine means between 9 and 109. 425. The first and ninth terms of an A. P. are 5 and 22 ; find the sum of 21 terms. 426. What is the sum of the first 200 odd numbers ? 427. Find the sum of 1 + (1 + ^) + (1 + 2^) + (1 + 3^) + . . . + (1 + nh) when ^ = 2, ^ = 11. 428. A and B start at the same time from the same point in the same direction. A goes at the uniform rate of 60 miles per day; B goes 14 miles the first day, 16 miles the second day, 18 miles the third day, and so on. At the end of 50 days who will be ahead, and how much ? 429. A traveller has a journey of 132 miles to perform. He goes 27 miles the first day, 24 the second, and so on, travelling 3 miles less each day than the day before. In how many days will he complete the journey ? 394: ALGEBRA, 430. If X — y is a mean proportional between y and y -\- z — 2Xy show that a; is a mean proportional between y and z, 431. Find three numbers in geometrical progression such that their sum shall be 14 and the sum of their squares 84. 432. What is the geometrical mean between 2a; — 3 and 2X^ -\- X^ — 4:X — 3? 433. If the arithmetic mean of two numbers is -y- and their geometric mean is |, find the numbers. ^ i_ 434. Eeduce a"^ and J** to equiradical surds, and find their sum, difference, product, and quotient. 435. Square root of 44 - 16 VY. 436. Square root of ^a - 1) + 2 V2a^ - 7a - 4:. 437. Find the value of (35 •l/«^+ 77 Va+ 63 Vb + 28 V^)( VJc - Va - VV) when ^ = 2, & = 3, c=5; and simplify the result. 438. Square root of 9a^V 3^^ + ^g , 2Vc 2^aW , 2^6 47 _J_ /y 439. Simplify V- a when \^J = Vl5. 440. When x= V% find the value of ^^ -r ^ 441. {x- If {x + 1)^* (l + |/2)( 1/6 - f 3)* 442. |/4 + c^ + 1^6 + 2:^ - i^6 - a; = 0. 443. i V'2a; + 8 + Vx + 5 = 1 ; verify your result. 444. ( Px^) X ( ^x^) X a;"^ ^ a;l 445. a; Va;2 + 12 + x Vx^ + 6 = 3. SUPPLEMENTARY PROBLEMS, 395 446. V^x~+T = V4:X + 5 - Vx - 4. 447. 2 I'iO + 3 1^108 + 1^500 - 1^3^ - 2 ^1372. 448. Square root of 449. Simplify Jl : ^|-; also {a^-^f : (-^) • 450 451. Simplify {«^^)^ ^; (~ «)2"(- a)2^+M 1^^; i/I08"+ 4/75"- 4/27; 5^ + 3 . 5'; (2^ . 2¥. [Suppose ^ to be an integer.] 452. Find the value of 1- — f ^ when x = .008. \jx 453. Vx -\- a -\- Vx + Vx — a = 0; explain the possi- bility of satisfying this equation, the connecting signs both being plus. 4S4. ■ -T- . 485. (3 - ^Q)y^_^_^- vr+Tf+TT/- 1^ _j_ 2 \ X 456. 11 + ^s]—^-— = 4vJ ■ ^; verify your results. 457. Square root of a-' + 2«-H2 - i-^) + Z^'^H- 4(1 - 1?-^^). 458. Square root of 49a^ - 2Sa^ - Yla^ + 6^^ + f . 459. iVx^^ - 21/3"= 4/3(7^+4). 460. ^^87 - 12^42^ 461. Find the value of (x\a + y^h){y'^a - x^^h) - {xyy{a^ - P) + x^ab when X = 2 and y = 3, 396 ALGEBRA. 463. Given (Vs - 1) : 4 - a = 4 + a : 3, find a. 463. Find the square root of -^ - na^ + QaW - Sab + 12b^ + %. ¥ ' ' a^ 464_ , .V ._...-...-», .„ i/x + 3+ \/x-3 (^,^^^^'. .65. Vx + ^ - Vx - 3 466. Square root of x^ + dx'^ — 4:X + lOa;^ — 12a;i. 467. (2 - 4/2"+ V3")(l + ^^ + ^6"). 468. Given x — Vd + xS/x^ — 3 = 3; find i?;. 469. Square root of a%'^- lOaJ-^- 10^-1^ + «-252+ 27. 470. ^^^.. 471. Value of (^y^32)-'. a^ — 0^ 472. Solve y + \=- -=-%x-\-h. 473. Eeduce 4/2^2 + «§ - 6^2 |/3^2 _|_ 5^^ _ 2^2 ^6^2— ll^?^ + 3^>2, and find its value when a = 3 and Z> = — 1. 474. Simplify x~^y + iz;^^"^ — 2;^;?/"^, expressing it as a single fraction. ml In 475. 4 ' 476. Vx — 4:+ Vx — 11 - V22: + 9 = 0. 477. Vx — Vy I Vx -\- Vy Vx 4- VIJ ^x — Vy 2 ^2 ^ y2 x-y^ a;2 - ^2 "*" 1_ 1 SUPPLEMENTARY PBOBLEMS. 397 478. Find the value oi 4:y — x when 2V3" V3~+ 1 1 + 2- V3' ^ V^ 479. |/5 - 22; + 4/15 + 3:?; == 4/26 - 5a;. 480. rt'^^'' X O?'^!)-^'' ~ (a3m^-2n >< |/p^). 481. Eationalize the denominator of |/ - 3 + 3 y _ 4 _ 2 |/3 .00 -E.- ^ ^1, 1 .21/7 -21/343 + 7 V28 , , 482. Find the value of r=^ to two 6|/63 decimal places. 483. /- 5 484. 4/3 + o; + Va; = —=z. Vx 485. Express with a rational denominator y + V^^— ^^ 486. State the value of 487. Find the square root of X — 2x^y~^ + Sax^ + y~* — Sax2/~^ + 16aV. 488. (7 - 4:V'd)x^ + (^ - i^3)^ = 2. 489. Find the value of 2aVl + x? when 2\y^ \ja)' 490. ^y ^ — 2a;*^ ^z ^-\-z ^ 491. 4 \^~x-\- Vx= 21; verify your answers. 398 ALGEBRA, 492. a"-* a^^ + * 493. (256§ ~ aZ> + aW^ - b^){ah + Z>^). x~^ 4- aix^ 4- a^)~* 508. / 7=--. Va? -\- x^ -\- aVx ^10 ■v'aA' /lOl 610. I — i-= I : X V 3 4/S V SUPPLEMENTARY PROBLEMS. 399 Find X and sim- 4 5a fa^ 9j- J4 f a^^»9 1^5 plify the answer. 511. Divide 63;*^+^^— 19ir'^+2''+20ii;'" + ^— 7i^;^— 4a;^ by 3a;2^ - 5:^;^ + 4. 512. (2x' - 1)* - (3^ + 1)* = (x- 4)*. 513. vl \^i 514. ^134 + 84^2. 516. V'? - i?: + 1/32; + 10 + -/iC + 3 = 0. 517. ^^ — a* = (i?; — Z>)^. 518. Which is the greater, VTo or f 46, and why ? 519. Square root of 75 — 12 t^2l. 520. Solve {x — af = {x - ^a){x^ + 4^2)1, 521. Square root of 41 + 12 V^. 522. Vx -\- ^ = Vx + |. 523. Keduce ^ + ^ ^ ^ to the form A + BV - 1. 1 + |/- 1 624. V2x - y = Vx-y + 1; x^ -\- 4:y^ = 17. 625. (- 1 + v^^y + (_ 1 _ v'^^y. 526. ( V3 + V~^^){Vd- V~^^). 527. Find the reciprocal oth^ + 2 + Vx"-2 )■ 528. Multiply 2ya - V -x by dV^T^ + 2^x. 529. Square root of 35 - 12 Vol 400 ALOBBRA, 530. , + ,— = 12. 531. Kationalize the denominators of ac ^ 1 7 + 2t^6 da^^h-^^' a^ + h^' 9-31/6' 532. Simplify 633. /ll +44/6: 534. ^5 - ^^"247 l^a + ^ |/« — a; 636, \x s la el; ^* ■>^- law 8 Ito 4 I a;3«/5 1 Va« 539. Find the value of ^^ + ^^ - _^^- a^» - Z>3 ^^ _ ^• 640. \/2 + VW. 541. Given a; = 3 - 1^11, ^ = 3 + j/ll ; find the values of the expressions xy; . 542. Vx + V^a + x = 2 f/^ + X. 543. (2 |/"=^ + 3 V~^^){4. V"^^ - 5 1^"^=^). 544. Find one value of ^17 12 1^2^ 545. Develop by binomial formula 8UPPLEMENTABT PROBLEMS, 401 546. Expand {a^ - U)\ 547. Convert , into an infinite series by the binomial theorem. 548. Write down the eighth term of {a — ly^, 549. Expand to four terms Va — x^, 550. Expand to five terms (1 + a)"^. 551. Expand -7- -—r into a series. (2a — 3)i 552. Expand to four terms {a + x)-^, 553. Expand by the binomial theorem Zb{2x — y)K 554. Write out the first five terms and the last five terms of {x — yY^, 555. Calculate the sixth term of 556. Find the fourth term of / Va _ V2 Y VvTi/d^ 3a* 'nj ' hVa _ 6|/^Y' \ 3 ~ a } ' aVa _\" /' 557. Find the sixth term of ( ^ 558. Find the last four terms of (^^ - 2^^)20. 559. Find the terms which do not contain radicals in / ^ ih^' the development of 1 (^-4)' 560. Find the prime factors of the coefficient of the sixth term of the nineteenth power of {a — l). What are the exponents in the same term, and what is the sign ? 561. Find the sixth term of {x — i/Y; also of / 6a^ ^Y \7b VI 'WaJ 402 ALomBA. 562. Find the sixth term of the nineteenth power of i^-i} 563. Find the tenth term of {x — yf^; also of \ vr Va) 564. Write out {x - yfK 565. Write out the first five terms and the last five terms of {x — yY^', then find and simplify the fifth term of 566. Find the sixth and twenty-fifth terms of the twenty- ninth power of {x — y); also the sixth term of the twenty- ninth power of \ b 2a J' of 567. Find the term which contains i^j* in (1 — ^x)^. 568. Find the middle term of {a - 2xy^. 569. Find the middle term of (1 + .t)^". 570. Write down the coefficient of x^ in the expansion 1 ?/(! + xY 571. Expand (1 — x^)~^ to six terms by the binomial theorem. 572. Find the sixth term and the ^th term in the ex- pansion of {a — xy^". 573. Find the ratio of the ^th term of 7- r- to the (1 — xy BUPPLEMENTAnr PROBLEMS, 403 \/[, y as expanded by the binomial theorem. 574. Find the eleventh term in the expansion of 575. Find the fifth term of {oT^ + 2ic"^)"^ 576. Expand (1 — y)'^ to five terms, and write down the (r + 5)th term in its simplest form. 577. If A is the sum of the odd terms, and B of the even terms, in the expansion of (x + «)^ show that 578. Write down the first four terms, the last four terms, and the middle term of {x — 2yy^. 579. Expand by the binomial theorem Ix -\ J 580. Expand (m~^ + 2n^y. 581. Find the fifth term of {x-'^ - 2y^yK 582. Give the first, third, and fifth terms in the expansion of fx Vy + :r^V' 2Vx/ 583. Find the first term with a negative coefficient in 11 the expansion of (1 + «) . 584. Fifth term of 585. Fourth term fit _ i^V 1^ n 3'bJ ' ° \{/^i Say ' /^ 3 Vf V \2Vf «' /' 586. Fourth term of 587. Find the fourth term of (ps^ — 3)^^ 404 ALGBBBA. 688. Find the fourth term of the eleventh power of \3 ^ • bVh) \ 'da' 2 ' 589. Find the eighth term of ( IZz. — _aVa. h~ Z 590. Write out (x — yY^ ; then find, in its reduced form, the fourth term of this expansion, when X — — -= and v = -— — . |/^ ^ Za 591. Find the eleventh term in the expansion of (28 -j- 2^^)^. 592. Given log 1.3287 ^ .1234269 and log 1.3288 = .1234596; find log .00132874. 593. Given log 3.8795 = .5887758 and log 38796 = 4.5887870; find log ( |/."0387957)( 1^387.954). 594. Find the value of 7 log, if + 5 log., ff + 3 log, %\. 595. Find the number whose logarithm is — 2.4211522, having given log 379.18 = 2.5788454 and log 379.19 = 2.5788569. 596. Find the logarithm of 6561 to the base 3 i^3. 597. Given log 3000 = 3.4771213, solve the equation r.Ol)^ = .0009. 598. Given log 2 ::= .30103 and log 3 = .47712; find the values of log 1^376 and log, 1^376. 599. Given log 2.5 = .39794 and log 2.25 = .35218; find log 2.7 to four places. 600. Given log a = 2, log 5 = 3; find log^ R 8VPPLBMENTART PBOBLBMS. 405 Practice in Forming Equations. 408. In forming the equation for any problem where the selection of a meaning for x seems difficult, it is a good plan to write down a list of the different numbers which would be mentioned in an explanation of the problem, but whose numerical value is not given in the statement of the prob- lem. Then let x stand for one of these, and see if abbrevi- ations for the others can be readily derived ; perhaps it will be necessary to have more than one letter. It may be that two different expressions may be found which represent the same number; in that case those two expressions would form an equation. When there are several facts given in the statement of the problem, it will be found that some of the facts are used in forming the different abbreviations; when they are not all so used, the remaining facts, expressed in algebraic form by means of the abbreviations, will form the equa- tions needed. Model A. — A baby gets 9 steps away from her mother be- fore the mother starts after it ; the baby takes 7 steps while the mother takes 5, but 1 of the mother^s steps is equal to 2 of the baby^s. How many steps must the mother take to catch the baby ? Here, as in many problems, it pays to draw a simple dia- gram. A B G Let ^ represent the mother's position, B the place the baby gets to before the mother starts, and C the place where the mother catches it. The numbers that we need to speak of in explaining the problem would be the number of baby's steps from B to 0; and the number of mother's steps from A to (7. 406 ALGEBRA, If we let X represent either of these, an abbreviation for the other may be derived from the fact that ' ' the baby- takes 7 steps while the mother takes 5 " ; but for the answer asked for, it would be more convenient to let x = the num- ^ X ber of mother's steps from A to G, Then — =: the num- D ber of baby's steps from B to C. Now we have the dis- tance from ^ to ^ given equal to 9 steps, and that from Ix B to Coequal to — steps; but the distance from A to C, which is equal to x steps, is expressed in a different unit, namely, mother's steps; using the third fact of the prob- lem, we can reduce the x mother-steps to 2x baby-steps, and we get the equation 2^ = 5+9. 5 Model B. — A and B run a mile. First A gives B a start of 44 yards, and beats him by 51 seconds; at the second heat A gives B a start of 1 minute 15 seconds, and is beaten by 88 yards. Find the time in which A and B can run a mile separately. Drawing diagrams for the separate heats, . 44 . 1716 yds. . . 1672 yds. . 88 . X B Y Z AW we see that in the first heat B ran only 1716 yards; in the second heat all the facts stated refer only to J.'s running 1672 yards. The numbers that we need to speak of in ex- plaining the problem are the number of seconds A takes to run a mile ; the number of seconds B takes to run a mile ; the number of seconds A takes to run 1672 yards ; the number of seconds B takes to run 1716 yards. SUPPLEMENTARY PROBLEMS. 407 If we let X and y stand for the first two of these, we have : X = the number of seconds A takes to run a mile ; y = tlie number of seconds B takes to run a mile ; 1673a5 = the number of seconds A takes to run 1672 yards ; "T^z— = the number of seconds B takes to run 1716 yards. 17d0 Eeducing these fractions to lowest terms we have 601. The perimeter of a right triangle is 11 times as long as the shortest side. What is the ratio of the two sides containing the right angle ? 602. Divide 33 into three parts so that the first may be to the second in the ratio 1.5, while the second is to the third as \ is to \, 603. The area of an oblong floor is 175 square feet, and its perimeter is 53 feet. Writedown and solve the quadratic equation which gives the dimensions of the floor. 604. A vote was taken in a debating society, and the mo- tion was carried, 5 to 3; on reconsideration, 50 of the affirmative votes deserted to the negative; if 60 more had deserted the motion would have been lost, 3 to 4. What was the vote on reconsideration ? 605. A business man starts to walk to his house, a mile away; at the same moment his son starts from the house to run to the office and back. The son being able to run in 4 minutes a distance that the father walks in 9 minutes where will he meet his father, and where will he overtake him? 606. The number of months in a man^s age on his birth- day in the year 1891 is exactly \ of the number denoting the year in which he was born. In what year was he born ? 408 ALGEBRA, 607. A man who can row a miles an hour in still water rows down a stream which flows h miles an hour and back to his starting-place. Show whether the time he takes is longer or shorter than the time required to row the same distance in still water. 608. An agent for a type- writing machine wishes to ad- vertise that he will give one machine to any one selling a certain number. What must that number be in order that the agent may make an average profit of $5, when the machines cost the agent $20 and are retailed for 130 ? 609. A train goes from Oxford to London, 63 miles, 14 minutes faster than a train which travels 3 miles an hour slower. Speed of each train ? 610. How much rice at 8 cents a pound must be mixed with 20 pounds at 11 cents in order that the mixture may be worth 10 cents a pound ? 611. A fast express train makes its entire run at an aver- age speed of 51 J miles an hour; the up-grades at an aver- age speed of 50 miles an hour, and the down-grades at an average speed of 54 miles an hour ; there are 70 miles more up-grade than down-grade; what is the length of the run ? 612. A man has 20 coins^ of v/hich some are half-dollars and the rest are nickels; if he should change the halves for dimes, and the nickels for cents, he would have 100 coins. How much money has he ? 613. A miner bought 20 dogs, some at San Francisco and some at Juneau; they cost him on an average $10 apiece. If he had been able to get them all at San Fran- cisco, he would have saved $160; if he had had to get them all at Juneau, he would have lost $160. How many were bought at each place ? 614. A certain number exceeds twice the product of its two digits by 73, and 3 times the sum of its digits by 61; find the number. SUPPLEMENTARY PROBLEMS, 409 615. Two trains start from opposite ends of a two-track road, and pass each other 6 hours later; one completes the run in 5 hours' less time than the other. Find the running time for each train. 616. Two pedestrians start together on a certain course, and one walks twice as fast as the other, over the whole course and back again; but the slower one walks only | of the course and back again, being passed by the faster man I mile from the winning-post. Find the length of the course. 617. The difference in the expense of fencing two square fields is ^ of the difference in value of the fields themselves; and the total expense of fencing both fields is twice the difference in value of the fields. Supposing the cost of fencing per yard happens to be equal to the value of the land per square yard in either field, find the area of the fields. 618. A man buys 570 pulleys, some at 16 for a dollar and the rest at 18 for a dollar; he sells them all at 15 for a dollar and gains $3. How many of each sort does he buy ? 619. The difference of two numbers multiplied by their product is 30, and the difference of their cubes is 98. Find the numbers. 620. My income is a third less than my brother^s, but my expenses are 40 per cent, less than his ; and we each save $100. Income of each ? 621. A dozen pair of boots and a dozen pair of shoes can be had for $46; and a dozen pair more of shoes can be had for $105 than of boots for $100. Price of each per pair? 622. Find three numbers which are to each other as 2:4:5, and such that the sum of the greatest and least ex- ceeds the other by 21. 623. The sum of the squares of the two numbers formed 410 ALOEBBA. by the same pair of consecutive digits is 585. Find the numbers. 624. My brother and I can do a piece of work in 6 days ; you and I would take 9 days to do it. In what time could you and he do it, supposing that he works twice as fast as I? 625. A sailor on shore leave spent ^ of his wages in presents for his friends, \ for carriage-hire, \ for circus- tickets, had \ stolen, and retained $1.50 when he returned to the ship. How much had he at first ? 626. A large sum of money passes through the hands of an importing agent, a custom-house official, and a lawyer, each of whom takes from it a different percentage of the value it has when he receives it; if the sum remaining is $1200, and the sum taken by the last of the three men is $300, $400, or $600, according to the different orders in which the money can be successively submitted to them, what was the original sum ? 627. A and B run a mile race. In the first heat B re- ceives 4 seconds start, and is beaten by 32 yards; in the second heat B receives 64 yards start, and wins by 2 sec- onds. Find the time each takes to run a mile. 628. A and B worked together for 8 days, and completed half of a definite task that had been put before them ; B then left the job, and A, after working with C, a new workman, for 6 days, left the job also; then C worked on alone, and completed the task in 2 more days. What C accomplished in the last two days was less than what A would have done by the exact amount that B would have done in one day. Find the time in which each workman could do the work alone, and the proportions in which they should be paid. 629. An egg-dealer bought a certain number of eggs at 32 cents per score, and 5 times that number at $1.50 per SUPPLEMENTARY PROBLEMS. 411 hundred. He sold them all at 20 cents per dozen, gaining $648 by the transaction. How many eggs did he buy ? 630. A number of soldiers drawn up in a hollow square 4 ranks deep can also be formed in a solid column of which the number of ranks will be equal to the number of men in each rank ; the number of men in the front rank being 25 greater in one formation than in the other. Find the num- ber of men. 631. A had :J^ as much money as B had; and since then he has paid B 16. The money A now has bears to the sum he originally had the same ratio that the money B now has bears to 115. How much had each at first ? Explain the negative answer. 632. A man walks away from home a miles per hour, rests h hours, and then walks back at the rate of c miles per hour, being gone from home h hours in all. How far away did he get ? 633. On what day of a year which is not a leap-year are the ^^ days past" and the ^*days to come" of a calendar consecutive squares ? Note that the *'days past " for any given date include that day. 634. A letter-carrier is delayed half the time during the first 45 minutes of his trip of 6 miles; then he finds that ho must go 2 miles per hour faster to get through on time. How many miles per hour is he expected to go? 635. At what time between 1 and 2 o'clock is the minute- hand as far from the hour-hand as the latter is from 12 ? Generalize this problem; discuss the solution for different hours, and the possibility of realizing the solutions on an actual clock. 636. A clock is set right at noon, January 1; when the correct time is 1:15 p.m. the same day, the hands of this clock are 6 minutes apart. When will it be an hour slow ? 637. A certain number divided by another gives a quotient 412 ALGEBRA. 3 and a remainder 2; if 9 times the second number be divided by the first, the quotient is 2 and the remainder 11. Find the numbers. 638. A number consists of two figures whose product is 21; if 22 be subtracted from the number, and the sum of the squares of its figures added to the remainder, the order of the figures will be inverted. What is the number ? 639. A fraction becomes f by the addition of 3 to the numerator and 1 to the denominator. If 1 be subtracted from the numerator and 3 from the denominator, the frac- tion becomes ^. Find the fraction. 640. Divide 111 into three parts, such that the products of the several pairs may be in the ratios 4:5:6. 641. A and B run a race of 480 feet. The first heat A gives B a start of 48 feet, and beats him by 6 seconds; the second heat A gives B a start of 48 yards, and is beaten by 2 seconds. How many feet can each run in a second ? 642. A and B set out at the same time to walk to a place 6 miles distant and back again. After walking for 2 hours, A meets B coming back. Supposing B to walk twice as fast as A, find their respective rates of walking. 643. A man bought a certain number of eggs for $2. If he had paid 5 cents more per dozen, he would have received 2 dozen less for the same money. How many dozen did he buy, and what did he pay for them ? 644. A boy spent his money in oranges. If he had bought 5 more, each orange would have cost him |- cent less; if 3 less, | cent more. How much did he spend, and how many did he buy ? 645. A number is made up of three figures whose sum is 17. The figure of the hundreds is double that of the units. When 396 is subtracted, the order of the figures is reversed. What is the number ? 646. The smaller of two numbers divided by the larger is SUPPLEMENTARY PROBLEMS, 413 .21, with a remainder .04162; the greater divided by the smaller is 4, with .742 for a remainder. What are the numbers ? 647. A number is made up of three figures whose sum is 17. The figure of the units is f that of the hundreds. When 297 is subtracted, the order of the figures is inverted. What is the number ? 648. A and B together can do |- of a piece of work in 6 days. If B can do -J- of it in 18 days, how long will it take A to do i^ of it ? 649. A person sets out at the rate of 11 miles in 5 hours; 8 hours after, another person sets out from the same place, and goes after him at the rate of 13 miles in 3 hours. How far must the latter travel to overtake the former ? 650. A certain number of persons were divided into three classes, such that the majority of the first and second to- gether over the third was 10 less than 4 times the majority of the second and third together over the first; but if the first had 30 more, and the second and third together 29 less, the first would have outnumbered the last two by 1. The whole number was 34 more than 8 times the majority of the third over the second. Find the number in each class. 651. A person has a certain sum, half of which he lends at 5 per cent interest, and half at 4^ per cent interest. The first loan yields him $60 more interest than the other. What is the amount of his capital ? 652. From the two formulas in simple interest, a = p~\- i and i = trp, find what p equals in terms of a, t, and r. Express in words the truth you have expressed in letters. Make an application of this truth in solving a problem with numbers in simple interest. 653. The width of a rectangular garden is 2 rods less than its length, and | of its area is equivalent to 8 rods d:14: ALGEBRA. less than | of the square of its width. How many rods long and wide is the garden ? 654. If the side of a square wei-e 5360 centimeters longer it would contain 65536 square metres. What is the length of a side ? 655. Thirty feet more than ^ of the present height of the highest of the Egyptian pyramids is equal to \ of its original height. Three hundred feet less than 6 times its present height is equal to 5 times its original height. Find the original and the present height. 656. One of the parallel sides of a trapezoid is 3 inches longer than the other; and the altitude is 1 inch greater than the shorter of these. The area is 68 square inches. Find the altitude and the length of each base. 657. The product of the sum and difference of two num- bers is a, and the product of the sum of their squares by the difference of their squares is 7na, Find the numbers. 658. How long will it be before the hands of the clock will again assume the same relative position that they have at this moment ? 659. On a certain street railway two sizes of cars are used. What is the seating capacity of each, if 14 more persons can be seated in 3 large cars than in 4 small ones, and 2 more persons in 2 large cars than in 3 small ones? 660. A rectangular lawn 20m. 5 dm. long and 8m. 5dm. wide has a path of uniform width around it. If the area of the path equals 62 ca., what is its width ? 661. A takes 3 hours longer than B to walk 30 miles; but if he doubles his pace he takes 2 hours less time than B. Find their respective rates of walking. 662. Divide 20 into three parts such that the products of the three pairs may be in the ratios 6 : 10 : 15. 663. The first digit of a number is 3 times the second ; and SUPPLEMENTARY PROBLEMS. 415 if the number, increased by 3, be divided by the difference of its digits, the quotient is 16. Eequired the number. 664. Find the number whose cube root is \ of its square root. 665. A and B can do a piece of work together in 8 days. A works alone 4 days, and then both finish it in 5 days more. In what time could each have done it alone ? 666. The sum of two numbers is 16, and the sum of their reciprocals is \. What are the numbers ? 667. Find two numbers such that their product, their sum, and the difference of their squares shall be equal to each other. 668. A certain number of two digits is equal to twice the Bum of its digits, and the number got by interchanging the digits is equal to the sauare of the sum of the digits. Find the number. 669. Find three numbers of which the first is greater than the second by as many units as the second is greater than the third* the sum of the squares of the three being 66. 670. Find two numbers whose product is 78, such that if one be divided by the other the quotient is 2 and the re- mainder 1. 671. Divide 1152 into three parts, such that 9 times the sum of the first and second shall be equal to 7 times the sum of the second and third; and if 8 times the first be subtracted from 8 times the second, the remainder shall be equal to the sum of the first and third. 672. Find the number whose square added to its cube is 9 times the next higher number. 673. Two travellers set out from two distant towns and go towards each other. When they meet they find that one of them has gone 30 miles more than the other, and can complete his journey to the other town in 4 days, while the 416 ALGEBRA, other will need 9 days to complete his journey. How far apart are those two towns ? 674. A cask contains 12 gallons of wine and 18 gallons of water, and another cask contains 9 gallons of wine and 3 gallons of water; how many gallons must be drawn from each cask so as to produce by their mixture 7 gallons of wine and 7 gallons of water ? 675. A deer fleeing from a tiger gets 60 jumps away from a certain tree before the tiger passes that tree; she makes 6 jumps while the tiger makes 5, but 7 of the tiger's jumps are equal to 9 of the deer's. How many jumps from the tree will the tiger go before he catches the deer ? 676. Two workmen were employed at different wages and paid at the end of a certain time. The first, who had worked all the time, received $26.25, and the second, who worked 6 days less, received $19.80. If the second had worked all the time, and the first had omitted six days, their wages, taken together, would have amounted to $48.75. How many days did each work, and what were the wages of each ? 677. A dealer sells two kinds of goods, 8 yards more of the second kind than of the first, and receives $100 from the sale. He then finds that he has left just as much of the first kind as he has sold of the second, and that the quantity of the first kind thus left is worth $100; and that he has left of the second kind just as much as he has sold of the first, and that the quantity left of the second kind is worth $16. Find the number of yards sold of each kind, and the price of each per yard. 678. Two horsemen start at the same time, on the same road, from two places 15 miles apart. At the end of 10 hours the second horseman overtakes the first, and on com- paring their rates they find that there has been a difference SUPPLEMENTARY PROBLEMS, 417 of 5 minutes in the time of going every 7 miles. Find their rates, and the distances they have gone. 679. Two pipes which supply the same reservoir fill it in 4 hours and 12 minutes when both run together; but the first pipe alone can fill it in one hour less than half the time in which the second pipe alone can fill it. Find the time for each pipe alone. 680. Find a number such that if it be multiplied by 4 and the product increased by 3, the result shall be the same as if it were increased by 4 and the sum multiplied by 3. 681. Compare the rates of two boat^s crews, one rowing 10 miles down-stream in an hour, the other 5 miles down and 5 miles back again in an hour; supposing the current to run 2 miles an hour. 682. A sets off from Boston to New York, and B at the same time from I^ew York to Boston, and they travel uni- formly; after they have met on the way, it takes A 16 hours to reach New York, and B 36 hours to reach Boston. Find in what time each performed the journey. 683. A man hires a certain number of acres of land for $336. He cultivates 7 acres for himself, and, by letting the rest for $4 an acre more than he paid for it, gets his own patch rent free. Find the number of acres he hired. 684. A laborer having built 105 rods of stone wall found that if he had built 2 rods less a day he would have been 6 days longer in completing the job. How many rods a day did he build ? 685. A and B have together f as much money as C; B and C have together 6 times as much as A; and B has $680 less than A and C have together. How much has each ? 686. A certain man has $1400, which he separates into two portions, and puts at interest at different rates, but so that the two portions produce equal returns. If the first portion had been lent at the second rate of interest, it would 418 ALomnA. have brought in $18 per annum; and if the second portion had been lent at the first rate, it would have brought in 132 per annum. Find the rates of interest and the two portions of the principal. 687. On a certain road the telegraph-poles are placed at equal intervals, and their number is such that if that num- ber were less by one, each interval between two poles would be increased by 2|| yards. Find the number of poles and the number of intervals in a mile. 688. Two workmen, A and B, working together on the same job, complete it in 15 days. It is found, on the com- parison of their work, that A, if working alone, would have required 16 more days for doing the whole job than B would have needed for it, working alone. Find the time in which each workman would have done the job if working alone. 689. A certain share of the profits of a farm is divided equally every month among the hands. In July the num- ber of hands was 8 more than in June, and each hand re- ceived $6 less; the sum divided being $700. In August the number of hands was 2 less than in June, and each hand received $1 less ; the sum divided being $540. Find the sum divided in June. 690. A certain cistern is supplied by three pipes, A, B, and C, of which A, when running, discharges into the cistern 1000 gallons per hour. Pipe B, running alone, would fill the cistern in 8 hours less than A alone; pipe C, running alone, would require 5 hours more than A alone; while A and B, running together, would fill the cistern in -^-^ the time re- quired by C alone. Find the number of gallons which the cistern is capable of holding. 691. Two men, A and B, had a money-box, containing $210, from which each drew a certain sum daily; this sum being fixed for each, but different for the two. After six weeks the box was empty. Find the sum which each man StlPPLBMBNTAMT PROBLEMS, 419 drew daily from the box ; knowing that A alone would have emptied it 5 weeks earlier than B alone. 692. A party of friends went on a pleasure excursion, the expense of which they shared equally. If the number of the party had been decreased by 1, and if the total expense had been $150, the assessment for each person would have been $1 more than it was; but if the number of the party had been increased by 8, and if the total expense had been $160, the assessment for each person would have been $1 less than it was. Find the number of the party, tlie assessment for each person, and the total expense of the excursion. 693. A battalion of soldiers, when formed into a solid square, present 16 men fewer in the front than they do when formed in a hollow square 4 deep. How many men are there in the battalion ? 694. Two mowers, working steadily together, mow a cer- tain field in 3 hours. On comparing their several shares of the work done, they find that A would have required, to do B^s share, 2|^ hours more than B would have required to do A^s share. Find the time in which each mower, working alone, would have mowed the whole field. 695. Two workmen, A and B, are employed on a certain job at different wages. When the job is finished, A receives $27, and B, who has worked 3 days less, receives $18.75. If B had worked for the whole time, and A 3 days less than the whole time, they would have been entitled to equal amounts. Find the number of days each has worked, and the pay each receives per diem. 696. Several friends on an excursion spent a certain sum of money. If there had been 5 more persons in the party, and each person had spent 25 cents more, the bill would have amounted to $33. If there had been 2 less in the party, and each person had spent 30 cents less, the bill 420 ALGEBRA, would have amounted to only $11. Of how many did the party consist, and what did they spend ? 697. A man walks 2 hours at the rate of 4|- miles per hour; he then adopts a different rate. At the end of a certain time he finds that if he had kept on at the rate at which he had set out, he would have gone three miles further from his starting-point; and that if he had walked 3 hours at his first rate and |- hour at his second rate, he would have reached the point he has actually attained. Find the whole time occupied by the walk, and his final distance from the starting-point. 698. A reservoir, supplied by several pipes, can be filled in 15 hours, every pipe discharging into it the same fixed number of hhds. per hour. If there were 5 more pipes, and every pipe discharged per hour 7 hhds. less, the reser- voir would be filled in 12 hours. If the number of pipes were 1 less, and every pipe discharged per hour 8 hhds. more, the reservoir would be filled in 14 hours. Find number of pipes and capacity of reservoir. 699. A man bought a certain number of railway shares when they were at a certain rate per cent discount for $8500; and afterwards, when they were at the same rate per cent premium, he sold all but 20 of them for $9200. How many did he buy, and what did he give for each of them ? 700. A and B can do a piece of work in 18 days; A and can do it in 45 days ; B and C in 20 days. Find the time in which A, B, and C can do it, working together. 701. A man bought a certain number of sheep for $300; he kept 15 sheep, and sold the remainder for $270, gaining half a dollar a head. How many sheep did he buy, and at what price ? 702. A hires a certain number of acres for $420. He lets all but 4 of them to B, receiving for each acre $2.50 SUPPLEMENTARY PROBLEMS, 421 more than he paid for it. The whole amount received from B is $420. Find the number of acres. 703. A man walks at a regular rate of speed on a road which passes over a certain bridge distant 21 miles from the point the man has reached at noon. If his rate of speed were half a mile per hour greater than it is, the time at which he crosses the bridge would be an hour earlier than it is. Find his actual rate of speed and the time at which he crosses the bridge Explain the negative answer. 704. A man setting out on a journey drives at the rate of a miles per hour to the nearest railway station, distant 1) miles from his house. On arriving at the station he finds that the express for his destination has left c hours before. At what rate should he have driven to catch the express ? Having obtained the general solution, find what the an- swer becomes in the following cases; (1)0=0; {2)c=J-; (3) ^ = - ^. In case (2) how much time does the man have to drive from his house ? In case (3) what is the meaning of the negative value of c? 705. A landowner laid out a rectangular lot containing 1200 square yards. He afterwards added 3 yards to one dimension of his lot, and subtracted 1^ yards from the other, thereby increasing the area of his lot by 60 square yards. Find the dimensions of the lot before and after the change. How do you explain the negative answer ? 706. A vessel is half full of a mixture of wine and water. If filled up with wine, the ratio of the quantity of wine to the quantity of water is 10 times what it would be if the vessel were filled up with water. Find the ratio of the original quantity of wine to that of water. 707. Three students, A, B, and 0, agree to work out a 422 ALOEBBA. series of difficult problems, in preparation for an examina- tion; and each student determines to solve a fixed number every day. A solves 9 problems per day, and finishes the series 4 days before B; B solves 2 more problems per day than C, and finishes the series 6 days before C. Find the number of problems, and the number of days given to them by each student. 708. A certain whole number composed of three digits has the following properties: 10 times the middle digit exceeds the square of half the sum of the digits by 21; if 99 be added to the number, the order of the digits is inverted; and if the number be divided by 11, the quotient is a whole number of two digits, which are the same as the first and last digits of the original number. Find the number. 709. A certain manuscript is divided between A and B to be copied. At A^s rate of work, he would copy the whole manuscript in 18 hours; B copies 9 pages per hour. A finishes his portion in as many hours as he copies pages per hour; B is occupied 2 hours more than A upon his portion. Find the number of pages in the manuscript, and the number of pages in the two portions. 710. Two casks, of which the capacities are in the ratio of a to h, are filled with mixtures of water and alcohol. If the ratio of water to alcohol is that of m to n in the first cask, and that of ji? to g in the second cask, what will be the ratio of water to alcohol in a mixture composed of the whole contents of the two casks ? Eeduce the answer to its simplest form. What does the answer (in its simplest form) become if m = q = ] and what is the simplest statement of the question in this case ? 711. A boat's crew, rowing at half their usual speed, row 3 miles down a certain river and back again, in the middle of the stream, accomplishing the whole distance in 2 hours SUPPLEMENTAUY PROBLEMS. 423 and 40 minutes. Find (in miles per hour) the rate of the crew when rowing at full speed, and the rate of the current. (Notice both solutions of this problem.) 712. A and B have 4800 circulars to stamp for the mail, and mean to do them in 2 days, 2400 each day. The first day. A, working alone, stamps 800 circulars, and then A and B together stamp the remaining 1600; the whole job occupying 3 hours. The second day A works 3 hours, and B 1 hour; but they accomplish only y\- of their task for that day. Find the number of circulars which each stamps per minute, and the length of time that B works on the first day. 713. A broker sells certain railway shares for 13240. A few days later, the price having fallen $9 per share, he buys for the same sum 5 more shares than he had sold. Find the price, and the number of shares transferred each day. 714. At 6 o'clock on a certain morning, A and B set out on their bicycles from the same place, A going north and B south, to ride until 1:30 p.m. A moved constantly north- wards at the rate of 6 miles per hour. B also moved always at a fixed rate; but after a while he turned back to join A. Four hours after he turned, B passed the point at which A was when B turned; and at 1:30 p.m., when he stopped, he had reduced, by one-half, the distance that was between them at the time of turning. Find B's rate, the time at which he turned, the distance between A and B at that time, and the time at which B would have joined A if the ride had been continued at the same rates of speed. Find the answers for both solutiofis. 715. Two travellers, A and B, go from P to Q at uni- form but unequal rates of speed. A sets out first, travel- ling on foot at the rate of 20 minutes for every mile. B PQ follows, going 1 mile while A traverses the distance -— . B oO 424 ALGEBRA. overtakes and passes A, 8 miles from P; and when B reaches Q, he is 9 miles ahead of A. Find the distance PQ, and B's rate of speed in minutes to the mile. 716. Tristram is 10 years younger than Launcelot; and the product of the ages they attained in 1870 is 96. Find the ages they attain in 1888. 717. A certain librarian spends every year a fixed sum for books. In 1886, the cost of his purchases averaged $2 per volume; in 1887, he bought 300 more volumes than in 1886; and in 1888, 300 more volumes than in 1887. The average cost per volume was 30 cents lower in 1888 than in 1887. Find the number of volumes bought each year, and the fixed price paid for them. 718. Two tanks, A and B, are discharging water; A at the rate of x barrels per hour, and B at the rate oi x -\-100 barrels per hour. At a time {1 -\- y) hours after noon, A contains 480 barrels less than at noon ; and at a time {1 — y) hours after noon, B contains 400 barrels less than at noon. Find the rate at which eacli tank is discharging water ; and the times {l-\- y) and (1 — y) hours after noon. 719. A certain railway runs due east and west, P, Q and E being successive stations on the road from east to west, and so situated that PQ = 216 miles, PR = 240 miles. Two trains, on parallel tracks, pass P simultaneously at noon. The rate of motion of train No. 2 from east to west is 8 miles per hour less than that of train Iso. 1 ; and train No. 2 passes Q 1| hours later than train No. 1 passes E. Find the rate and direction of motion of each train ; and find the hour at which train No. 1 passes E, and the hour at which train No. 2 passes Q. 720. Two wheelmen, A and B, are riding eastward over the same road, B's eastward rate exceeding A^s by 4 miles an hour. A certain milestone, M, is passed by A at noon, and by B 15 minutes later; a second milestone, N, 6 miles SUPPLEMENTAET PROBLEMS. 425 east of M, is passed by both wheelmen at the same instant. Find the rate of each wheelman in miles per hour, and the time of their passing N^. 721. A and B start at the same time from two towns and travel towards each other. When they meet, B has travelled a miles more than A; and it will take A d hours longer, and B c hours longer, for each to reach the town the other has left. Find the distance between the towns. 722. A gentleman has two horses and one chaise. The first horse is worth a dollars less, and the second horse h dollars less, than the chaise. If f of the value of the first horse be subtracted from that of the chaise, the remainder will be the same as if J of the value of the second horse is subtracted from twice that of the chaise. Find the value of each horse and that of the chaise. What are the answers ii a = — 50; ^ = 50 ? 723. Divide the number a into two such parts that if the first be multiplied by m and the second by 7i, the sum of the products is d. In what case would the terms of the fractional values of the unknown quantities become zero ? and how could they, then, satisfy the conditions of the problem ? 724. A courier started from a certain place n days ago, and makes a miles daily. He is pursued to-day by another making Z> miles daily. In how many days will the second overtake the first ? In what case would both the terms of the fractional value of the unknown quantity become zero ? and how could this value be a solution ? 725. A wine merchant has two kinds of wine: the one costs a shillings per gallon, the other 1) shillings. How must he mix both these wines together in order to have n gallons worth c shillings per gallon ? In what cases would the values of either of the unknown quantities be nega- tive ? Why should this be the oase, and could the enun- 426 ALGEBRA. elation be eorrected for this ease ? In what eases would the value of one of the unknown quantities be zero^ and what would this value signify ? 726. The owners of a certain mill make a dollars a day each, sharing equally. If the number of owners were h less, they would make c dollars each. Eequired the number of owners and the total daily profit. What are the answers if « = 80, ^ = - 3, c = 50 ? 727. Two trains are running westward on parallel tracks, at the rate of a and l miles an hour respectively. At noon they are m miles apart. When are they together ? When is the unknown quantity positive ? When does it become zero ? infinite ? indeterminate ? negative ? Explain the signification of each result. 728. A certain sum of money will amount to a dollars in m months, and to h dollars in n months. Find the princi- pal and the rate of interest. Find the answers when a = 1837.50, b = 1890, m = 10, n = 16. 729. A banker has two kinds of coin. It takes a pieces of the first, or b pieces of the second, to make a dollar. If a dollar is offered for c pieces, how many of each kind must be given ? 730. I have a counters distributed unequally among three cups. Taking some of the counters from the first cup, I double the number of counters in the second and third ; next, taking from the second, I double the number in the first and third; and lastly, taking from the third cup, I double the numbers in the first and second. Then I find that the second cup contains twice, and the third cup three times, as many counters as the first. How many in each cup at the beginning ? 731. A merchant who had two brands of flour sold a barrels of the first and b barrels of the second at an average price of $c per barrel; and at the same rates, he sold m SUFPLEMEl^TAET PROBLEMS, 427 barrels of the first and n barrels of the second at an aver- age price of $^ per barrel. Find the price of each brand. 732. A person has a hours at his disposal. How far can he ride in a coach at the rate of b miles an hour, so as to get back in time, walking at the rate of c miles an hour ? 733. A cask of wine contains a gallons; b gallons are drawn off, and the cask filled up with water. After this has been done n times, how many gallons of the original wine are in the cask ? INDEX. Abbreviation of explanations, 1 of rules, 41 Addition, 61 see also Summation. Age problem, 12 Algebra, three uses of, 1, 41, 88 Algebraic scale, 132 Alternation, 300 Analogy, 303 between terms and factors, 241 Answers alike, 107 apparently different, 108 infinite list of, 135 interpreted, 110, 285 meaningless, 115 suggest related problems, 113 tabulated, 172 to quadratic pairs, 170 Antecedents, 299 Approximate square roots, 52 Arithmetic complement, 354 Arithmetic mean, 318 Arithmetic progression, 310 Arrangement in multiplication, 81 Associative law, 59, 60, 72 Augmented logarithms. 353 Auxiliary quadratic, 182 Axes, coordinate, 133 Axiom, 15 Axiom A, 104 Base, 349 Binary equation, 174, 176 Binomial, 70 formula, 341, 343 quadratic surd, 256 theorem, 334, 344 Binomial theorem, any one term, 340 Boundary between + and — num- bers, 287 Brackets, see Parentheses. Calculation by logarithms, 353 Cautions for H. C. F., 211 Changing signs, 63, 241 see also Law of signs. Characteristic, 351 Circular race problem, 39 Circulating decimal, 321 Cistern problem, 36. 289 Clearing of fractions, 218 Clock problem, 38 Coefficients, 15 detached, 328 irregular, 337 of powers, 333 symmetrical, 336 Coin and bill problem, 5 Cologarithms, 354 Combination, 141, 170 Common difference, 310 logarithms, 349 ratio, 315 roots, 205 Commutative law, 59, 60 Comparison, 185 Complete list of answers, 135 Completing the square, 96 Complex binomials, 258 fraction, 230 Composition, 301 and division, 301 Compound interest, 55 Conditions, 88, 137, 151, 153 Conjugate surds, 252 429 i30 INDEX, Consequent, 299 Consistent equations, 151 Continued equation, 155 fractions, 232 multiplication, 69 proportion, 305 Conventions, 4 Convergent series, 321 Corresponding values, 131, 306 Cross-multiplication, 77 Cross-products, 77 Cube roots, 267, 270 Current problem, 25 Cyclic order, 221 Z>, 350 d, 350 Day's work problem, 36 Degree, 166 Demonstration of theorems, 88 Detached coefficients, 328 Diagrams for linear equations, 133-136 for quadratics, 138, 169 Difference of like powers, Ex. 11, p. 198 of the cubes, 90 of the squares, 89 of unknown numbers given, 8 Digit problem, 22 Discussion of equations, 292 of problems, 285 of the quadratic, 294 Distributive factoring, 73 law, 11, 72, 248 Divergent series, 321 Dividend, 82 Division, 83, 194 (in proportion), 301 Divisor, 82 Double -cross-products, 97 e, 359 Elimination, 141 by combination, 141 by comparison, 185 by detached coefficients, 332 by substitution, 166 generalized, 292 finding a binary, 176 first method, 131 for H. C. F., 200, 203 irregular devices, 183 more than three letters, 158 Elimination, numerical terms, 176 second method, 163 three letters, 153 where it fails, 150 Entire surd, 246 Equal factors, 69 Equation, 3 binary, 174 continued, 155 degree of, 166 factorable, 104 homogeneous, 174 how to form, 405 identical, 88 of condition, 88 simple, rule for, 15 with radicals, 258 Equations, inconsistent, 151 independent, 151 linear and quadratic, 163 of the New Set, 154 solved like quadratics, 262 vs, expressions, 217 Equilateral triangle, 42 Equiradical surds, 246 Expansion, 335 Exponent, see Index Expression, 3, 217 Extremes, 299 Factor, 69 Factorable equations, 104 Factorial, 338 Factoring by completing square, 99 by cross-multiplication, 79 by parts, 126 by theorem A , 93 distributive, 73 for the L. C. M., 215 higher equations, 125 literal quadratics, 278 quadratic pairs, 173 Factors of an equation, 104 test for simple, 197, 235 Falling bodies, 42 First method of elimination, 131 use of algebra, 1 Formula, 41 binomial, 341 for area of triangle, 50 for in -radius, 356 INDEX. 431 Formula for interest, 54 for solving quadratics, 122 for square root, 51 for X, 166 Formulae, application of, 46 for /and s, 810, 315 involving square root, 53 translated into rules, 41 Four answers (pair of quadrat- ics), 170 Four-place tables, 349 Fourth proportional, 304 roots, 272 Fractional equations, 28, 217 indices, 244, 345, 348 Fractions, 190 complex, 230 continued, 232 laws of, 192 Fundamental laws, 72 ^,43 Generalization, 274 linear equations, 292 quadratic equation, 294 the binomial theorem, 338 Geometric mean, 304, 318 progression, 315 Graphical method, 132 G. C. D., G. C. M., see H. C. F. Grouping of factors, 60 of terms, 59 Harmonic progression, 323 H. C. F., 198, 203, 208 as related to L. C. M., 236 by detached coefficients, 331 of three expressions, 207 Highest and lowest terms, 201 History of equations, 3 Homogeneous equation, 174, 176 Homologous terms, 302 Hyperbola, 170 i = |/^. 118 Identities, 88 Imaginaries, 119, 254 Imaginary unit, 254 Impossible problems, 151. 293 Inconsistent equations, 151 Independent equations, 151 Indeterminate equations, 152 Index, 69, 239 distributive law, 248 fractional, 244 Index negative, 240 sign of, 240 zero, 240 Induction, 344 Infinite geometric series, 319 Infinite list of answers, 135 Infinity, 288 In-radius, 356 Interest formula. 54 Interpolation, 350 Intersections, 140 Inverse variation, 307 Inversion, 300 Irrational quantity, 177, 246 Irregular coefficients, 337 elimination, 182 Known numbers, symbols for, 42, 374 Law, Associative, Commutative, and Distributive, 72, 248 of signs, see Signs. Laws of algebra, 72 of fractions, 192 of imaginaries, 254 of indices, 239 of proportion, 299 L. C. M , 212 by factoring, 215 by mems of H. C. F., 236 of factorable equations, 234 of three expressions, 237 Least Common Multiple, see L. C. M. Light-year, 858 Like powers, sums and differ- ences of, Exs. 11 and 12, p. 198 Limiting values, 285 Linear equation, 166 and quadratic, 163, 17i Literal equation, 275 quadratics, 278, 384 simultaneous equations, 281 Logarithm, 348 Logarithmic schedule, 355 Long division, 84 Lowest Common Multiple, see L. C. M. Mantissa, 352 Mathematical induction, 344 Mean proportional, 304 Meaning of negative answers, 110 4:32 INDEX. Meaningless answers, 115 Means, 299, 319 Member of an equation, 3 Minus, see Negative Mixed number, 226 surd, 246 Mixture problem, 19 Modified reduction of fractions, 225 Modulus, 341 Monomial, 71 factors, 73 Multiple, 212 Multiplication, 69 distributive principle, 11 indicated without sign, 3 of fractions, 193 of polynomials, 75 of proportions, 302 Napierian logarithms, 359 Negative characteristic, 352 fractions, 33 index, 240, 345 quantities, 56 terms, 57 Nests of parentheses, 66 New Set, 154, 158 New work on old patterns, 28 Non-algebraic conditions, 3 51 Notation by powers of 10, 358 Numerical terms eliminated, 176 Order of factors, 60 terms, 59 Pairs, linear-quadratic, 163 of quadratics, 169 Parabola, 170 Parentheses, 34, 64, 67 Partial products, 76, 81, 126 Pattern expansions, 337 7f, value of, 41 Polynomials, 71 Positive terms, 57 Power, 69, 239 Powers, arrangement br, 81 of 10, 348, 358 Principal dividing line. 230 signs of a fraction, 220 Problems put into equations, 405 Products, 77 Progressions, 310 Proof of binomial formula, 343 of theorems, 88 Proportion, 299 Proportional parts, 351 Quadratic, 78, 108. 166 and linear, 163 completing the square, 99 expression, 78 form, 262 formula, 122 generalized, 294 literal, 278 pairs, 170 standard form, 121 surd, 246 with three letters, 187 with zero terms, 295 Quotient, 82 Radical equations, 258 Radicals, 246 Ratio, 191 Rational expression, 246 Rationalizing a quadratic surd 250 a term, 250 factors, 253 Rearranging denominators, 221 factors, 60 terms, 59 Reciprocal, 58 of a power, 240 Reciprocals in fractional equa- tions, 227 solving for, 148 Reduction of equations, 11 of fractional expressions, 225 of fractions, 193 of fractional products. 194 Removing parentheses, 65 Root, 69, 239 Roots, common, 205 finding, 51, 265, 356 of an equation, 124, 384 of proper fractions, 358 of a quadratic, 294 Rule for addition in algebra, 61 for literal equations, 275 for simple equations, 15 for subtraction, 63 of signs, see Signs. of Three, 300 Rules, abbreviation of, 41 application of, 44 translated into formulae, 43 INDEX. 433 s, 50 Scale, algebraic, 132 Scales of notation, 24, 330 Schedule, logarithmic, 355 of steps, 3 Second method of elimination, 163 Second use of algebra, 41 Series, 310 by expansion, 345 convergent, 321 for cube root, 271 for square root, 266 of equal ratios, 303 Shortages, 12 Sign of identity, 88 Signs in division, 83 in multiplication, 76 in subtraction, 63 of a fraction, 220 of indices, 241 of parentheses, 65 Similar surds, 247 terms, 15 Simple equation, rule, 15 Simplest form of a surd, 247 Simplification due to logarithms, 348 Simultaneous equati ns, 139 literal equations, 281 values, 306 Solving simple equations, 15 Sphere, 42 Square of a polynomial, 89 of the difference, 89 of the sum, 89 root, 51, 265 root of a binomial surd, 256 Squares by Theorem A, 91 Standard form, 121 Straight products, 77 Substituting for a?, 164 Substitutions, 86 Subtraction, 62 Sum of cubes, 90 of like powers, Ex. 12, p. 198 of unknown numbers given, 19 Summation, 60 of fractions, 213 Supplementary problems, 366 Surd, 177, 246 Symmetrical elimination, 179 Symmetry in binomial expan- sion, 336 not obvious, 184 Systems of logarithms, 359 Table of logarithms, 363 of powers of 10, 348 Tabular difference, 850 Tabulated answers, 172 Terms of an expression, 3 Tests for simple factors, 197, 235 Theorem A, 89 Theorems, 88 of fractions, 193 of proportion, 299 Third method of elimination, 185 use of algebra, 88 Three equations, one quadratic, 187 means, 324 principal signs, 220 unknown letters, 153, 187 uses of algebra, 1 Transformations, 56 Translating formulae into rules, 41 identities into theorems, 90 rules into formula?, 43 the sign — , 58 Transposition, rule of, 15 Triangle, area of, 50 in- radius, 356 Trinomial, 70 Two answers (pair of quadrat- ics), 172 (one quadratic), 104 Unit, imaginary, 254 Unknown letters, more than two, 152 numbers, symbols for, 4 Using logarithm tables, 349 Valuation problem, 5 Value of It, 41 Variables, 68, 284, 305 Variation, theorem of, 308 Verifying expansions, 345 Zero, 287 coefficients, 295, 329 index, 240 MATHEMATICS Gillet's Elementary Algebra. By J. A. GiLLET, Professor in the New York Normal College, xiv + 412 pp. i2mo. Half leather. $1.10. With Part II, xvi + 512 pp. i2mo. $1.35. Distinguished from the other American text-books covering substantially the same ground, (i) in the early introduction of the equation and its constant employment in the solution of problems ; (2) in the attention given to negative quantities and to the formal laws of algebra, thus gaining in scientific rigor without loss in simplicity ; (3) in the fuller development of factoring, and in its use in the solution of equations. James L. Love, Professor in Harvard University : — It is un- usually good in its arrangement and choice of material, as well as in clearness of definition and ex- planation. J. B. Coit, Professor in Boston University : — I am pleased to see that the author has had the purpose to introduce the student to the reason for the methods of algebra, and to avoid teaching that which must be unlearned when the student moves on into higher studies. William A. Francis, Phillips Academy, Exeter, N. H.: — I find many things in it to admire, es- pecially the shortening of alge- braic processes. F. F. Thwing, Manual Train- ing High School, Louisville, Ky.: — Two features strike me as being very excellent and desirable in a text-book, the prominence given to the concrete problems and the application of factoring to the so- lution of quadratic equations. E. F. Lohr, Duluth {Minn) High School: — What I especially like is the author's early intro- duction of the equation and the prominence given to drill in prob- lems. A. L. Baker, Professor in Uni- versity of Rochester, N. V.: — The author's diagramatic method of multiplying and factoring should be a valuable help to the student. His treatment of undetermined coefficients is particularly good. The early introduction of syn- thetic division is to becommended. I cannot speak too highly of his treatment of negative quantities. It is an excellent book in every way, and thoroughly up to date in many little ways that only a practical teacher can appreciate. Pomeroy Ladue, Professor in New York University : — The book is an excellent one, free from most of the imperfections of many American text-books, and very well adapted for beginners. . . . indicates that it is written by a successful teacher. J. G. Estill, Hot ch kiss School, Lakeville, Conn.: — The order in which the subjects are taken up is the most rational of any algebra with which I am familiar. Mary C. Noyes, Lake Erie Fe- male Seminary, Painesville, O.: — The book seems to be particularly clear in its statements and expla- nations. I have noticed particu- larly the explanation of the theory of exponents. Mathematics Gillet's Euclidean Geometry. By J. A. GiLLET, Professor in the New York Normal College. 436 pp. i2mo. Half leather. $1.25. This book is " Euclidean " in that it reverts to purely geo- metrical methods of proof, though it attempts no literal repro- ductions of Euclid's demonstrations or propositions. Metrical applications and illustrations of geometrical truths are inter- spersed with unusual freedom. *^ Originals " are made an integral part of the logical development of the subject ; that is, they include theorems and problems which are ordinarily found in the regular series, and thus, by devoting the pupil's inventive energies to these instead of to merely incidental ^'originals," he secures the necessary power at a smaller ex- penditure of time. Percy F. Smith, Professor in Yale University : — The return of the *' spirit of Euclid" should be much appreciated, and ft will be interesting to watch the workings in the class room of the two alter- native methods of Book V. Con- sistency and rigor are carefully maintained in both works, and I shall take great pleasure in using and recommending them. Dr. Charles A. Pitkin, Thayer A cadeniy. South Braintree^ Mass. : — I find it to be a very attractive book, both in its general appear- ance and in the clearness with which the demonstrations are set forth. The somewhat unusual arrangement of the order in which the author gives the earlier prop- ositions seems to me, in general, a good one. The problems and exercises are well selected. C. W.Green, Northwestern Uni- versity, III.: — The book is of supe- rior merit, especially in the sim- plicity and completeness of its demonstrations. E. P. Thompson, Professor 171 Miami University , O. : — An ex- cellent G^omftlvy for the pupil, and that is certainly what we want. It is written with sufficient fulness and is suitable for the kind of pupils we have, possessed of hu- man nature. It is direct and simple and easily intelligible. John R. French, Professor in Syracuse University : — I am pleased with it. It seems to me admirably adapted to use in the recitation room. Daniel Carhart, Professor in the Western University of Pennsyl- va7iia : — The definitions, often ac- companied by illustrations, are made clear ; the exercises seem to be judiciously chosen, and the queries introduced occasionally are well selected. The chapter on Conic Sections will prove help- ful. The work as a whole is com- mendable. E. L. Caldwell, Morgan Park Academy, III. : — I find in them the best results of modern research combined with rigid exactness in definition and demonstration. Mary C. Noyes, Lake Erie Sem- inary, Painesville, O. : — It seems to me far better to give, as Gillet does, some of the propositions in the form of original work for the pupil instead of giving full demon- strations for all of them. The metric exercises given are calcu- lated to give pupi.s clear ideas of the subject and of its applications. Mathematics Keigwin's Elements of Geometry. By Henry W. Keigwin, Instructor in Mathematics, Norwich (Ct.) Free Academy, iv -4-227 pp. i2mo. $1.00. This little book is a class-book, and not a treatise. It cov- ers the ground required for admission to college, and includes in its syllabus the stock theorems of elementary geometry. It is, however, out of the common run of elementary geometries in the following particulars : 1. The early propositions, and a few difficult and funda- mental propositions later, are proved at length to furnish models of demonstration. 2. The details of proof are gradually omitted, and a large part of the work is developed from hints, diagrams, etc. 3. The problems of construction are introduced early, and generally where they may soon be used in related propositions. Oren Root, Professor in Hamil- ton College, N. V. : — I like the book, especially in that it gives " inventional geometry" while giving the fundamental propo- sitions. Geometry is taught very largely as if each proposition were an independent ultimate end. Pupils do not grasp the interlock- ing relations which run on and on and on unendingly. Mr. Keig- win's book, compelling pupils to use what they have learned of re- lations, must help to prevent this. C. L. Gruher,Fo. Normal Sc/iool, Kutztown : — The method of the book is an excellent one, since it gradually leads the student to de- pend in a measure upon himself and consequently strengthens and develops his reasoning powers in a manner too often neglected by teachers of the present day. It gives neither too little nor too much. W. A. Hunt, High School, Den- ver, Colo. : — It does not do for the pupil what he should do for him- self. With strong teaching, the book is just what is needed in preparatory schools. Miss Emily F. Webster, State Nor77ial School, Oshkosh, Wise: — At the first I looked upon the book as very small, but I now con- sider it very large, for it is per- fectly packed with suggestions and queries which might easily have extended the book to twice its present size had the author seen fit to elaborate, as so many authors do ; but in not doing so lies one of the finest features of the book, as much is thus left for the student to search out for him- self. The original exercises are fine and in some cases quite un- usual. The figures are clear and the lettering is economical, some- thing which is by no means com- mon, and much valuable time is wasted by repeating unnecessary letters in a demonstration. Dem- onstrations are made general, which is an advantage, for it is often difficult to induce pupils to do so when the author has failed to set them the example. George Buck, Dayton {0,)High School . — I am highly pleased with it and commend its general plan most heartily. Mathematics Newcomb*s School Algebra. By Simon Newcomb, Professor of Mathematics in the Johns Hopkins University, x 4- 294 pp. i2mo. 95 cents. (A'