UC-NRLF H- $B 531 551 ■ ^H I ■ 1 '■^' ill mjKt"* 1 i M LIBRARY OF THE University of California. GIFT OF Class Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/firstyearalgebraOOmilnrich FIRST YEAR ALGEBRA BY WILLIAM J. MILNE, Ph.D., LL.D. M PRESIDENT OF NEW YORK STATE NORMAL COLLEGE ALBANY, N.Y. NEW YORK :• CINCINNATI •:• CHICAGO AMERICAN BOOK COMPANY MS MAY 12 1911 GIFT Copyright, 1911, by WILLIAM J. MILNE. Entered at Stationers' Hall, London. FIRST TEAR ALGEBRA E. P. I PREFACE This book has been written to meet the growing demand for a High School Algebra that contains only the first year's work. While the order of topics resembles in general that found in the author's other algebras, yet a number of changes have been made, for the purpose of simplifying the work and deferring difficulties until the pupil is able to cope with them. One of the hardest ideas for the young student to grasp is that of negative numbers; and the common practice of pre- senting them at the very beginning of the book results not only in the bewilderment but also in the discouragement of the stu- dent. In this book, therefore, the pupil is first taught the sym- bols and the fundamental operations as applied to positive numbers, and not until he has become thoroughly familiar with these is he introduced to negative numbers. He can thus con- centrate his entire attention on the one new idea, and it becomes a pleasure to him to extend his knowledge by applying the principles he has already learned to the new concept. Again, the troublesome operation of removing and inserting signs of aggregation is deferred until the pupil's gain in power of manipulating algebraic numbers renders the work compara- tively easy. On the other hand, in order to arouse from the first the interest of the pupil, simple problems to be solved both arithmetically and algebraically, as well as easy solutions of simultaneous equations and of quadratic equations by factoring, are presented very early in the course, while the more difficult phases of these subjects are discussed later. Throughout the work, indeed, the greatest emphasis is placed on equations and problems, which furnish the most apt illustrations of the practical uses of algebra. 4 PREFACE The treatment of every principle is based on the pupil's knowledge of arithmetic. This close correlation of the two subjects not only illuminates both of them, but adds further to the simplicity of the book. The problems are based on interesting facts gathered from a variety of sources, including physics, geometry, and com- mercial life. A few problems of the older stjde are included for the purpose of familiarizing the pupil with them and for their disciplinary value. Graphs are presented in a simple and comprehensive man- ner, but the chapters are introduced in such a way as to render practicable their omission, without disturbing the continuity of the course. Factoring is thoroughly taught, and the study is greatly simplified by the careful classifying and summarizing of the various cases. New terms are illustrated or defined wherever they are needed, the object of this plan being to prevent the confusion that results in the pupil's mind from the massing of large col- lections of definitions at the beginning of each chapter. For- mal definitions of all terms are placed at the end of the book in a glossary arranged in alphabetical order. Abstract and concrete work is- well balanced, so that the drills in algebraic processes and representation are as plentiful as the exercises for the development of the reasoning faculties. Accuracy is secured by the numerous checks, tests, and verifications that are required of the student, and thorough- ness is acquired through the frequent and exhaustive reviews. In the preparation of the "work, careful consideration has been given to the courses of study outlined by the Regents of the State of New York and by educational authorities else- where. The book will be found to meet the requirements of these courses in every particular. WILLIAM J. MILNE. CONTENTS PACK Intboductiox .... 9 Algebraic Solutions 12 Problems 13 Factors, Powers, and Polynomials 19 Numerical Substitution 22 Review 23 Fundamental Operations (Positive Numbers) . . . .26 Addition 25 Subtraction 28 Multiplication 30 Division . .34 Equations and Problems 39 Algebraic Bepresentation 43 Pi'oblems 44 Review 48 Positive and Negative Numbers 49 Addition and Subtraction 51 Sum of Two or More Numbers 51 Difference of Two Numbers 54 Transposition in Equations 58 Algebraic Bepresentation 61 Problems 02 Multiplication 64 Special Cases in Multiplication 68 Simultaneous Equations 74 Problems 77 Division 80 Special Cases in Division 84 Parentheses .86 Equations and Problems 92 Literal Equations 93 Algebraic Bepresentation 94 Problems 96 Review 98 5 C CONTENTS PAGE Factoring 101 Monomials 101 Binomials 103 Trinomials 104 Polynomials 109 Special Applications and Devices 113 Review of Factoring . .117 Equations solved by Factoring 120 Problems 123 Fractions . . . 125 Signs in Fractions 125 Reduction of Fractions 128 Addition and Subtraction of Fractions 135 Multiplication of Fractions 139 Division of Fractions 141 Complex Fractions 143 Equations and Problems 146 Clearing Equations of Fractions 145 Algebraic Bepresentation . . . ■ . . . . 148 Problems 149 Review 151 Simple Equations . 153 One Unknown Number .153 Numerical Equations . . 154' Literal Equations 165 Problems 167 Solution of Formuloe 164 Simultaneous Simple Equations 168 Two Unknown Numbers . 168 Elimination by Addition or Subtraction .... 170 Elimination by Comparison 171 Elimination by Substitution 172 Literal Simultaneous Equations 175 Problems 176 Three Unknown Numbers 182 Graphic Solutions 184 Simple Equations 184 Review . .^ . . 195 Involution 197 The Binomial Formula '. 200 CONTENTS 7 PAGE EVOLCTION 203 Square Root of Arithmetical Numbers 209 Roots by Factoring 212 Radicals 213 Reduction of Radicals 218 Addition and Subtraction of Radicals 222 Multiplication of Radicals 224 Division of Radicals 226 Involution and Evolution of Radicals 227 Rationalization 229 Radical Equations 232 Review 236 Quadratic Equations 237 • Pure Quadratic Equations 237 Problems . 239 Formidoi 239 Affected Quadratic Equations 241 Literal Equations 246 Radical Equations 247 Problems 249 Formulce 265 Graphic Solutions 256 Quadratic Equations — One Unknown Number .... 256 Equations in Quadratic Form 260 Simultaneous Quadratic Equations 264 Problems 272 Graphic Solutions 275 Quadratic Equations — Two Unknown Numbers .275 Simultaneous Quadratic Equations 279 Ratio and Proportion 284 Ratio 284 Properties of Batios 285 Proportion 286 Properties of Proportions 287 Problems 292 General Review 296 Factors and Multiples 305 Highest Common Factor 305 Lowest Common Multiple 308 Glossary 311 FIRST YEAR ALGEBRA INTRODUCTION 1. In passing from arithmetic to algebra, the student will not find the change a very marked one. He will meet signs, definitions, principles, and processes with which he is already- familiar. The fundamental principles of arithmetic and algebra are identical, but in algebra their application is broader. Algebra uses the same number symbols as arithmetic, namely, 1, 2, 3, 4, 5, etc., but from time to time, as need for them arises, various new symbols will be introduced. While arith- metic, to a limited extent, uses letters to represent numbers, their use is a distinctive feature of algebra. The terms, addition, subtraction, multiplication, division, fraction, etc. ; the associated terms such as addend, subtrahend, multiplier, quotient, numerator; and the signs, +, — , X, -^, =, have the same meanings that they have in arithmetic ; but it will be seen that algebra gives to some of them additional meanings. In algebra, multiplication is also indicated by the dot (•) or by the absence of sign ; thus, ax b, a - b, and ab all mean the same. Division is often indicated by a fraction ; thus, o -*- 6 and ^ have the same meaning. 10 INTRODUCTION EXERCISES 2. Read, and tell the meaning of each of the following algebraic expressions : 1. 2 + 3. 8. W -f- V. , e 15. 7>l 2. a + &. - 9. 4.5. 71 3. 8-5. 10. x-y, 16 a& 4. x-y. 11. pq. 3a; 5. 2x5. 12. ab — rs. 17. ? + -• 6. m X n. 13. 3v-|-5^. a — V 7. 8-4. 14. a + m-n. 18. 7^ 1 » Indicate the 19.^ Sum of 5 and 2 ; of a; and ?/. 20. * Difference of 9 and 6 ; of m and n. 21. Product of 3' and 4 in two ways ; product of r and s in three ways. 22. Quotient of 8 divided by 5 in two ways ; quotient of p divided by q. 23. Sum of 5 times d and 2 times c. 24. Difference of a times b and 2 times 4. 25. Product of 3 m and n. 26. Quotient oi v — iv divided by c times d. 27. Product of 2 a; + 7 and 3 2/ - 2. The product of 2 jc + 7 and 3 y — 2 is indicated thus : (2x + 7)(3y-2). Note. — Parentheses, ( ), are used to group numbers, when the num- bers in each group are to be considered as a single number. 28. Product of a — 6 and 5 m + 2. 29. Product of a and a + b divided by the product of b and a — b. 30. A boy had a apples and his brother gave him b more. How many apples had he then ? *In this book, the ' difference ' of two numbers means the first mentioned less the second. INTRODUCTION 11 31. Edith is 14 years old. How old was she 4 years ago ? a years ago ? How old will she be in 3 years ? in b years ? 32. At J? cents each, how much will 5 oranges cost ? 33. If ijcaps cost 10 dollars, how much will 1 cap cost ? 34. At y cents eachXnow many pencils can be bought for x cents ? 35. George won a race by running the distance in t seconds. Kepresent Elmer's time, if he took 2 seconds longer. 36. James weighs p pounds. Represent Edward's weight, if he weighs 10 pounds less than James. 37. A boy who had p, marbles lost q marbles and afterward bought r marbles. How many marbles did he then have ? 38. If m represents the number of miles a boy can walk in a certain time, indicate the distance his father, who walks twice as fast, can walk in the same time. 39. Mary paid c cents for a pin and half as much for a belt. Represent the number of cents she paid for the belt. / 40. What two whole numbers are nearest to 9 ? to x, if x is a whole number ? to a, if a is a whole number ? 41. li y is an even number, what are the two nearest even numbers ? the two nearest odd numbers ? 3., Unite terms as indicated by their signs : 20 2 tens 2x10 2t 2x '2z + 40 +4tens +4x10 +4« +4a; +42 + 30 +3 tens +3x10 +3^ +3x +32 90 9 tens 9x10 9t 2^ + 4^ + 3^ = 9^. 2x-\-4:X-]-Sx=? 22 + 42 + 32 = ? Such terms as 2 i», + 4 a;, and + 3 a; are called like, or similar, terms because they have the same unit, x. The multipliers, 2, 4, and 3 are called coefficients of x. Such terms as 2 ^, + 4 a;, and 3 z are unlike, or dissinmar, terms because they have different units, t, a;, and z. 12 INTRODUCTION ALGEBRAIC SOLUTIONS 4. The numbers in this chapter do not differ in character from the numbers with which the student is already familiar in arithmetic. The following solutions and problems, however, serve to illustrate how the solution- of an arithmetical problem may often be made easier and clearer by the algebraic method, in which the numbers sought are represented by letters, than by the ordinary arithmetical method. Letters that are used for numbers are usually called literal numbers. 5. Illustrative Problem. — A man had 400 acres of corn and oats. If there were 3 times as many acres of corn as of oats, how many acres were there of each ? Arithmetical Solution A certain number = the number of acres of oats. Then, 3 times that number = the number of acres of corn, and 4 times that number = the number of acres of both ; therefore, 4 times that number = 400. Hence, the number = 100, the number of acres of oats, and 3 times the number = 300, the number of acres of corn. Algebraic Solution Let X = the number of acres of oats. Then, Sx = the number of acres of corn, and 4 a; = the number of acres of both ; therefore, 4 a; = 400. Hence, x = 100, the number of acres of oats, and 3a; = 300, the number of acres of corn. Observe that in the algebraic solution x is used to stand for ' a certain number' or 'that number,' and thus the work is abbreviated. 6. A statement of the equality of ty^ numbers or expres- siMs is called an equation. J^ 6 X = 30 is an equation. *^ INTRODUCTION 13 Problems 7. Solve the following problems : 1. A bicycle and suit cost $54. How much did each cost, if the bicycle cost twice as much as the suit ? 2. Two boys dug 160 clams. If one dug 3 times as many as the other, how many did each dig ? 3. Find a number whose double equals 52. 4. If 3 times a number equals 75, find the number. 5. A certain number added to 3 times itself equals 96. What is the number ? 6. The average length of a fox's life is- twice that of a rabbit's. If the sum of these averages is 21 years, what is the average length of a rabbit's life ? 7. The battleship fleet that sailed for the Pacific consisted of 20 ships. The number of warships was 4 times the num- ber of the auxiliary ships. How many warships were there ? 8. The water and steam in a boiler occupied 120 cubic feet of space, and the water occupied twice as much space as the steam. How many cubic feet of space did each occupy ? 9. One year the United States exported 24 million pounds of butter and cheese. If this included twice as much butter as cheese, how many pounds of each were exported ? 10. Porto Rico and the Philippines together produce 400,000 tons of sugar each year. If the latter produces 3 timers as much as the former, how much does Porto Rico produce? 11. Canada and Alaska together annually export furs worth 3 million dollars. If Canada exports 5 times as much as Alaska, find the value of Alaska's export. 12. The poultry and dairy products of this country amount to 520 million dollata' a year, or 4 times the value of the potato crop. What is the value of the potato crop ? 14 INTRODUCTION 13. At Portland, Oregon, recently vessels were loaded with 25 million feet of lumber for home and foreign ports. Find the foreign shipment, if it was 4 times that to home ports. 14. In constructing the Galveston sea wall 10,000 loads of sand and crushed granite were used. If there were 3 times as many loads of sand as of granite, how many loads of each were used ? 15. The Weather Bureau of the United States yearly saves the country 30 million dollars, or 20 times its cost. What is the annual cost of the Weather Bureau ? 16. One year in continental Europe 6 million watches were made, and this number was i of a million more than twice the number made in the United States. How many were made in this country ? Suggestion, 2 x = 6 — ^. 17. Probably Ceylon has the oldest tree in the world, and its age is about 2200 years. If this is 70 years more than 6 times the age of the Powhatan Oak in Virginia, find the age of the latter. 18. The value of the King's Cup, the challenge trophy for yachting, is twice as great as that of the Bennett Cup, the prize for long-distance balloon racing. If the difference in value is $ 2500, find the value of each. 19. The owner of a piano found that' the annual cost of keeping it in tune and insuring it against fire was $12.50, and that the cost of keeping it in tune was 9 times the cost of insupiug it. Find the cost of each item. 20. The Quebec bridge that collapsed was 1800 feet long, and twice the length of the Forth bridge was yV ^f the length of that at Quebec. Find the length of the Forth bridge. 21. One year 1500 violins were made in the United States. Twice as many were made in New York as in Massachusetts, and these two states made half of all that were made in this country. How many violins were made in New York ? INTRODUCTION 15 22. The sides of any square (Fig. 1) are equal in length. How long is one side of a square, if the perimeter (distance around it) is 36 inches ? Fig. 1. Fig. 2. Fig. 3. 23. The length of each of the sides, a and h, of the triangle (Fig. 2) is twice the length of the side c. If the perimeter is 40 inches, what is the length of each side ? 24. The opposite sides of any rectangle (Fig. 3) are equal. If a rectangle is twice as long as it is wide and its perimeter is 48 inches, how wide is it ? how long ? 25. Divide 21 into three parts, such that the first is twice the second, and the second is twice the third. Suggestion. — Let x = the third part ; then, 2 x = the second part, and 2 • 2 x = the first part ; that is, a; + 2 a; + 2 . 2 x =21 . 26. Three newsboys sold 60 papers. If the first sold twice as many as the second, and the third sold 3 times as many as the second, how many did each sell ? 27. The battleship) Connecticut has twice as many 8-inch as 12-inch guns, and the sum of the two equals the number of its 7-inch guns. If it has in all 24 guns of these sizes, find the number of each. 28. One winter the Borough of Richmond had four falls of snow amounting in all to 16^ inches. The second and third falls were each 4 times the first. Find the depth of the fourth fall, if it was twice the first. 29. In a recent year Massachusetts produced twice as many barrels of cranberries as New Jersey, and New Jersey 6 times as many as Wisconsin. Find the production of each of these states, if their total crop was 400,000 barrels. 16 INTRODUCTION 30 A plumber and two helpers together earned $ 7.50 per day. How much did each earn per day, if the plumber earned 4 times as much as each helper ? 31. James bought a pony and a saddle for $60. If the saddle cost J as much as the pony, find the cost of each. Suggestion. — Let x = the number of dollars the saddle cost. 32. Separate 72 into two parts, one of which shall be J of the other. 33. Separate 78 into two parts, one of which shall be i of the other. 34. A skating rink accommodated 4000 persons. If there were -J as many skaters as spectators, find the number of each. 35. The total production of sulphur averages 625,000 tons per year. How much is produced by the rest of the world, if it is I the amount produced by Sicily ? 36. The average height of the land above sea level is -^ as great as the average depth of the ocean, and the sum of the two is 13,000 feet. Find the average height of the land and the average depth of the ocean. 37. The first issue of Christmas stamps by the Delaware Red Cross Society was ^ as much as the second, which was i as much as the third. If the three issues amounted to 350,000 stamps, how many were there in each issue ? 38. Sand and clay road costs ^ as much per mile as macadam. If the former costs $ 400 per mile, find the cost of the latter. Solution Let X = the number of dollars macadam costs per mile. Then, ' ^ x = 400. Therefore, x = 6 times 400 = 2400. Hence, macadam road costs $ 2400 per mile. 39. The gold output of the United States for a recent year as 110 million dollars, or \ that of i was the world's output for that year ? was 110 million dollars, or \ that of the entire world. What INTRODUCTION 17 40. A man in New York rented Ms house and lived in an apartment costing him $ 2000 a year. This was ^ as much as the rent of his house. For how much did his house rent ? 41. The Pennsylvania Railroad station in New York is 780 feet long, and this is 404^ feet more than ^ the length of the Capitol at Washington. Find the length of the Capitol. 42. A basketball team won 16 games, or J of the games it played. Find the number of games it played. Solution Let X = the number of games it played. Then, I x = 16, and ^x = S. Therefore, x = 24, the number of games it played. 43. The largest thermometer in the world has a glass tube 16 feet loug. Find the length of the thermometer, if the tube is ^ of the entire length. 44. What is the annual rainfall of Hawaii, if at least 56 inches, or ^ of it, passes off without rendering any service ? 45. Of the inhabitants of Guam, -3%, or 8100, can read and write. What is the population of the island ? 46. The average annual fire loss in Berlin is ^ of that in Chicago. If the fire loss in Berlin is $150,000, what is the fire loss in Chicago ? 47. The largest stone ever quarried in the South was dressed down to weigh 60,000 pounds. If this was | of its weight as originally blocked out, find its original weight. 48. Find the amount of lumber on hand in San Francisco at the time of the earthquake, if | of it, or 36 million feet, were consumed by the fire that followed the earthquake. 49. The manufacturing industries of Great Britain use 150 million tons of coal per year. If this is f of the total amount used, what is that country's annual consumption of coal ? MILNE's IST YR. ALG. 2 18 INTRODUCTION 50. The number of German-speaking people in the world is 75 million, or f the number that speak English. What is the number of English-speaking people ? 51. The United States sent to Germany one year 135,000 pairs of shoes. This was f of the number sent the next year. How many pairs of shoes were sent the second year ? 62. If I of a number is added to the number, the sum is 12. What is the number ? Suggestion. x -\- ^x = 12 ; that is, | a; = 12. 53. , If ^ of a number is subtracted from twice the number, the difference is 35. What is the number ? Suggestion. 2 a; — ^ x = 35 ; that is, | a: = 35, 54. The total cost of the Pennsylvania Capitol was 13 million dollars. If the furnishings cost 2i times as much as the construction, what was the cost of each? 55. The retail dressmaking trade each day uses -J of the total daily output of spool silk. If the manufacturing trade uses the remainder, or 16,000 miles, how much does the dress- making trade use per day ? 56. Out of the average daily output of stamped envelopes ^ are plain stamped. The remainder, 2,800,000, bear the return address. What is the daily output ? 57. In one year, 5600 tons of dynamite were required for the Panama Canal. If the amount for the Culebra division was If as much as that for the rest of the canal, find the amount required for the Culebra division. 58. In the first twenty-one hours after the institution of regular wireless service, 6|- times as many words were sent to Europe as were received, and the number sent was 11,000 more than the number received. Find the number sent; the number received. 59. The Pacific battleship fleet carried twice as much ham as it did salt pork, and 2^ times as much beef as it did ham. The weight of the beef was 800,000 pounds more than that of the salt pork. Find the weight of each. INTRODUCTION 19 FACTORS, POWERS, AND POLYNOMIALS 8. Since the product of 2 and 6 or of 3 and 4 is 12, each of the numbers 2, 6, 3, and 4 is a factor of 12. So also, each of the numbers 3, a, b, 3 a, 3 b, and ab is a factor of ^ a6. 9. In algebra, as in arithmetic, such a product as 2 x 2 x 2 x 2, called a power of 2, may be more briefly written 2*. The small figure 4, placed at the right of, and a little above, the 2 to indicate the number of times 2 is used as a factor, is called an exponent. Since a^ means the same as a, the exponent 1 is usually omitted. a^ is read ' a second power ' or ' a square ' ; a^ is read ' a third power ' or * a cube ' ; a^ is read ' a fourth power,' ' a fourth,' or ' a exponent 4.' The terms ^coefficient' and ^exponent' should be distin- guished. Thus, 5 a means a-ha-{-a-\-a-\-a, but a^ means a a • a- a- a. EXERCISES 10. Kead, and tell the meaning of : 1. a-«. 4. Q^y\ 7. 3 2wl iq. 9a6Vd*. 2. y^. 5. a^b\ 8. 4/)Y- H- ^pVs't^. 3. 2*. 6. r's-. 9. 2mV. 12. 7 x^ym^n*. Express in abbreviated form by using exponents : 13. 2 . 2. 16. 3 aaa. 19. 2 . 2 . 2 . a; . «. 14. 3-3.3. 17. 8 nil 20. 7 '7 'Z .z 'Z -z. 15. 5 • 5 • 5 • 5. 18. 9 ssrrr. 21. SS'Sa-a-bbb. 22. What is the coefficient of a; in 3 a; ? in ax? in 3 aa; ? Note. — A coefficient is numerical, literal, or mixed according as it is composed of figures, letters, or both. When not otherwise specified ' coefficient ' means numerical coefficient. Since la means the same as a, the coefficient 1 is usually omitted. 23. What is the literal coefficient of f in at- ? in gt^ ? in nH^ ? of 2^ in nf ? in arf ? in bnf ? 20 INTRODUCTION Name the various factors of : 24. ax. 26. a^. 28. 6n. 30. pqrs. 25. Smn. 2t. 5r^^. 29. 15 z\ 31. 24 v«. 32. In each of the exercises 24-31, name the factors in sets such that the product of the factors in each set shall equal the given number. 11. An algebraic expression is called a monomial, binomial, or trinomial according as it has one, two, or three terms. Thus, 3 a is a monomial ; 2 ic + y^, a binomial ; and x^ + 2xy ■{■ y'^, a trinomial. The name polynomial is often applied to any algebraic expression of more than one term. EXERCISES 12. From the algebraic expressions given below select the : 1. Binomials. 3. Monomials. 5. Similar terms. 2. Trinomials. 4. Polynomials. 6. Dissimilar terms. 2ax', Sa^y, 2a4-36; 3a; + 26; Sax-{-2y^; 6a — c-\-d; 3 a^x^ — 4:ax-\-2d — y^; 2x^y — xy + a V. 7. Find the value of 3 + 4-2 + 3; of 3x4^2x3. Solutions. 3+4-2 + 3 = 7-2 + 3 = 5 + 3 = 8; 3x4-2x3 = 12 -2x3 = 6x3 = 18. When only + and — occur in any expression^ or only x and -4-, the operations are performed in order from left to right. Find the value of : 8. 3-2-1 + 8-3 + 4. 10. 10--2x8--4--2x6. 9. 5_j_i_4_|.3_2 + 6. 11. 35-f-7--5x3x4--2. 12. Find the value of 7 + 10 - 6 ^ 3 X 4. Solution. 7 + 10 -6-3x4 = 7 + 10 -2x4 = 7 + 10 -8 = 9. Unless otherwise indicated, as by the use of parentheses, when x, -^, or both, occur in connection with +, — , or both, the indicated WiUUi^tv^a- tions and divisions are performed first. « INTRODUCTION 21 Find the value of : 13. 5x10-7. 18. 6 + 2x8-4h-2. 14. 5x(10-7). 19. (6 + 2) X 8-4-2. 15. 2x5 + 3x4. 20. (6 + 2x8-4)^-2. 16. (25-13)-4x2. 21. 6 + 2x(8-4)-2. 17. 16-2x2xl2-f-4. 22. 6 + 2 x(8h-4 --2). Kead, and tell the meaning of each of these polynomials : 23. 2x^ + y\ 26. a-hd{ax-y). 29. Sa^-\-2y-3z. 24. 3x-4.y. 27. 3-\-A(y-3z). 30. a^x- - 3 xy + 2 z-. 25. 4a6-c3. 28. c(P-{-e)-^d. 31. 5 6"?/ + ary + 5 c;^^ Represent algebraically : 32. The sum of five times a and three times the square of x. 33. Three times b less twice the fifth power of a. 34. The product of a, b, and a — c. 35. Three times x, divided by five times the sum of a, 6, and c. 36. Seven times the product of x and y, increased by three times the cube of z. 37. Six times the square of m, increased by the product of m and n. 38. The product of a used five times as a factor, multiplied by the sum of 6 and c. 39. Twelve times the square of a, diminished by five times the cube of b. 40. Eight times the product of a and b, divided by four times the seventh power of c. 41. Six times the product of a second power and n, increased by five times the product of a and the second power of 7i. 42. The fourth power of the sum of a and b, increased by three times the product of the square of a and the square of b, diminished by the cube of d. 22 INTRODUCTION NUMERICAL SUBSTITUTION 13. When a particular number takes the place of a letter, or general number, the process is called substitution. EXERCISES 14. 1. When a = 2 and 6 = 3, find the numerical value of Sab; of a*. Solutions. 3 a& = 3 • 2 • 3 = 18 ; also, a* =2 ■ 2 ■ 2 • 2 = 16. When a = 5, 6 = 3, c = 10, m = 4, find the value of : 2. 10 a. 6. 5 m". 10. am\ 14. ^ab\ 3. 2ab. 7. 2a^b. 11. (abf. 15. ^bm. 4. 3 cm. 8. 3 6m^ 12. a^b\ 16. i-abc, 5. 6 be. 9. 4a^6. 13. a^c. 17. 362cml 18. When m = and n = 4, find the value of 3 m^n. Solution. 3 m^n = 3 . O^ . 4 = 3 • • 4 = 0. Note. — When any factor of a product is 0, the product is 0; there- fore, any power of is 0. When a = 4, 6 = 2, r = 0, and s = 5, find the value of : 19. 7 6V. 21. 3s''6''. 23. fa»6s. 25. 2«6VV1 20 i^ 22 'Jl^ 24 ^^. 26 ^^ ^'^'^° ' s6 * ' abs'-' ' b'^a' ' Ga'b's' 27. When x = S and ?/ = 2, find the value of (a5 + 2/)^; of x'^2xy + f. Solutions (x + yy = (3 + 2)2 = 5 . 5 = 25. .r2 + 2 x?/ + 2/2 = 3 . 3 + 2 . 3 . 2 + 2 . 2 = 9 + 12 + 4 = 25. 28. Show that 2x-\-Sx = 5x when a; = 2; when a; = 3. Giving X any value you choose, find whether 2x-{-Sx = 5x. 29. Show that m (a -f 6) = ma + m6 when m = 5, a = 4, and 6 = 3. Find whether the same relation holds true for other values of m, a, and 6. REVIEW 23 30. Showthat (a-6)2 = a--2a6 4-&^wheua = 4 and 6 = 2. Find whether this is true for other values of a and b. When a = 5, 6 = 3, m = 4, ?i = 1, find the value of : 31. a' + b\ 33. 71^-1. 35. m»-^ 32. (a + bf. 34. (n-iy. 36. (6??i)*- 37. a6 — bn -\- mb^ -i- 3 mn^. 38. (a6 — 6/1 4- m6^) -5- 3 m?i^. 39. 2"wi^w- — abmn -^ 4 &?i — ??i^ri^. 40. a^nbn^ — 1 6^??^ + | m V — | m^. REVIEW 15. Bead the following ; classify each expression according to the number of terms it contains ; find the number repre- sented by each expression when t = 10. 1. 6^. 4. f. 7. t^. 2. 7t. 5. ^2 + 2^ + 1. 8. t^-\-2t--{-5t + 4:. 3. 9^ + 9. 6. 3^2^6^4-5. 9. 5f-^St^+ St-{-6. 10. Write 25 as a polynomial in t^ t representing 10 ; letting t represent 10, and, using exponents to represent powers of t, express in polynomial form : 732 523 893 4867 6248 12,mn 11. What does 2 a denote ? a^? Illustrate the difference in meaning between 2 a and a^ when a = 1 ; when a = 2 ; when « = 3 ; when a = ^ ; when a = J. For what value of a are 2 a and a^ equal ? 12. Which is the greater, 2^ or 3^? 4^ or 2*? 2« or 5^? 13. Compare also 2« and 2^; (if and (|)2; 1^ and 1^ 14. Find, for x = 1, the value of : 3iB 4if2 6ar^ 8;r'- 4.T^ + 2ar^-;v-h 5 Name the exponent of x in each term that contains x. 15. Name the coefficient of n in each of these monomials : 2 w It \ n bn 3 b''n a-b^n 24 REVIEW 16. Write three similar monomials; four dissimilar mo- nomials. 17. If 71 is a whole number greater than 1 and a is any num- ber, what is the meaning of a" ? Find the value of each of the following expressions when a = 5, & = 4, c = 3, d = 2, e =1, and n = 3. 18. 6ab; 2cd; 4w&d; ^ea; ndJ'+K 19. Sd'h) 3ab'; 3{ahf', d'n^; (diif-, d''-^ 20. a + b^d — n-^e. 22. 10-^d + S^n — e. 21. a(b — d)-\-a — 7i^c. 23. 10 ^ (d -\- 3) -\- ac ^ n. 24. c*+c^ + 2c^-2c2-3c + 3. 25. d^-hd«-+-3d«-5(i* + 2d3-4c?2 + 8c?-l. 26. For what value of a; is 12 a; equal to 72 ? Write ^ 12 07 is equal to 72 ' as an equation. Solve the equation. Express in algebraic form ; solve equations when you can : 27. Three times a certain number, x, is 21. 28. The sum of a certain number and three times the num- ber is 40. 29. Six times a number, less 4 times that number, is 13. 30. The distance around a square lot, each side a feet long, is 1280 feet. 31. Half of a certain number is 17. 32. Twice a certain number, less J of the number, equals 15. 33. Mary had m books and James had twice as many, the two together having 18 books. 34. John had 50 cents, spent c cents, and earned d cents. How much money had he then? 35. I bought 2 bottles of olives at b cents per bottle, 3 packages of crackers at p cents per package, and a small cheese for c cents. How much did I expend for all? How much money had I left out of a dollar ? FUNDAMENTAL OPERATIONS 16. In this chapter the student will use numbers he has used in arithmetic and letters to represent such numbers. He will notice that the processes of addition, subtraction, multipli- cation, and division here are performed as in arithmetic. ADDITION 17. To add monomials. 1. How many are 2 plus 5? How mauy times a number are 2 times the number plus 5 times the number ? 2. If n stands for a number, how many times n are 2 times n plus 5 times w ? 2 ?i + 5 n = ? 3. 2x-\-6x=z? 4. 2r + 5r=? 5. 2« + 6« = ? 6. How many are 3 4- 4 -f 6 ? 7. How many days are 3 days + 4 days + 6 days ? 8. 3d + 4d + 6d = ? 9. 3y + 4y + 6y=? EXERCISES 18. 1. Add 4 a and 3 a. PROCESS Explanation. — Just as 3 a's and 4 a's are 7 a's, so 3 a A +4a = 7a; that is, when the monomials are similar the sum 3a la may be obtained by adding the numerical coefficients and annexing to their sura the common literal part. Add: 2. 3 3. 3 a; 4. 7 5. 7 m 6. 3y 6 6a; 1 m 4y "" "26 26 FUNDAMENTAL OPERATIONS Add: 7. 2n 5n 8. 3 a; 8a; 9. 4:xy 1 xy 10. 3mw2 9mn^ 11. 5r 2r 4r 12. 9^i 4n 6n 13. 2a6 4a6 14. Gc^d^ 8 c^c?-^ Perform the additions indicated : 15. 8a-h2a + a + 3a+a + 7c(. 16. 52/-|-32/ + 82/ + 10.y + 6^ + y + 22/. 17. 8 m + 3 m + 5 m + 2 m + 6 7;i 4- 4 m. 18. 7 6c + 6c + 4 6c + 5 6c + 8 6c' + 3 6c. 19. 4 a;22/2 ^ 5 ^^^2 _^ 3 ^.y _,_ ^^2 _^ ^0 x^?/-' + 6 a;^^^^^ 20. 3(a6)2 + 9(a6)2 + (a6)2 + 7(a6)2 + 9(a6)2 + 2(a6)2 + {abf. 21. 5(a;+.v)+2(a;+?/)H-3(a;+2/)+8(a; + 2/)+2(a;+.y) + (a;+2/). 22. 4(a + 6)^ + 11 (a 4- 6)^ + 7(a + 6)^ + 2(a + 6)^ + 5(a + 6)^. Only similar terms can be united into a single term. Dis- similar terms are considered to have been added when the addition is indicated. 23. Add 6 a, 5 6, 2 a, 3 6, 2 c, and a. Solution. — Sum =6a + 2a+a + 56 + a6 + 2c = 9a + 86+2c. Add : 24. 2 a;, 4 a, 3 a;, and a. 27. 5 r, f t, 2 r, and \ t. 25. m, 3 c, 6 m, and 4 c. 28. ^p, f Q', i p, and | Q'. 26. 4 w, -u, 3 w, and 10 v. 29. c?, .4 6, .5 d, and .6 6. 30. 2 m, nm, n, 2 m^i, 3 7n, 4 n, and 5 mn. 31. 3 6, 2 a, 2 6, 2 c, 2 d, 0, c, 6, 4 d, and 3 c. 32. rs, 3 r-s, 4 rs^ 2 ?-s, rs^, 4 r% 2 rs^ and 3 rs. 33. 3 a;.v, 2 pg, 7 cd, j9g, 2 cd, 8 pg, 4 cd, and 2 a;?/. 34. a;2, 4 xy, 7 2/^ 2 xy, 3 2/', 6 a;^, /, xy, 5 a;2, and 4 /. FUNDAMENTAL OPERATIONS 27 19. To add polynomials. EXERCISES 1. Add X -^ 2 y -}- 3 z, x-\-y, and x-\-^y -\-z. PROCESS Explanation. — For convenience, similar terms x-^ ^ y + 6 z jj^g^y ^jg written in the same column. ^-\- y The sum of the first column is 3 x, of the second x + iiy -\- z Ty-, of the third 4 z ; the sum of these dissimilar 3 a; 4- 7 w 4- 4 2 terms is then indicated. Add: 2. 2a4-46 3. 4r+3s 4. .r'-h2a«/-f^ 6a-^2b r-h s x^ +y- a+36 3r-\-2s Sxy + y^ 5. Add 2 c + 5 d, 7 c -h d, d -f 4 c, and 2 d + c. 6. Add 6m4-4n, 2m-h3n, 5n-f7m, and 2 n -h 3 m. 7. Add ab + a^c+5, 3 a6 + 3 a2c + 7, and 2a2c-|-2 a6 + 3. Express in simplest form: 8. 2a + 26-|-3c-f 46 + 4a-|-6a4-2c. 9. Sw-\-4:X-{-7 y-{-2v + 2tv + x-^3y-^4:V+3x + 4:W + v. 10. a^z -{- 5 xz^ -\-7 xy -\- 6 xz^ + 2 x^z -\- 4:xy -\- 4 x^z -^ xz^ -\- xy. Add: 11. 6 m + 8 n 4- X* + 2/, 2 ?>i + 2 ?i + 3 a; + 4 1/, and m + x-\-y. 12. 3 a; -h 7 2/ 4- 4 «4- 6 w, 7 z 4-4 a; -f 2 y-hw, and a;4-t/4-2; 4-w. 13. a^ + 2xy-{-y^ 2x'+xy-\-y', ^ ^ xy + y\ 3xy + y'' + x', 2o?-\-3xy + 'if,x'-\-xy + 2f,2in^2xy + 3x' +4:f. 14. 2c+7d4-6n, Ilm4-3c4-5ri, ln-\-2d+%c, d4-3m4-10c, 4 fZ 4- 3 ?i -f 8 m, m-\-^n, and 2 m 4- 3 cf . 15. 3 a;"» 4- 2 y"", 4 a;"* 4- 5 y", 2 a;*" 4- 7 iT, and 2 x^ 4- y". 16. 42/-4-2« + t(A y^-{'2w^-\-3z^, 5^'4-3^/;^ 2y«4-w;*, and 28 FUNDAMENTAL OPERATIONS SUBTRACTION 20. 1. How many are 8 less 3 ? How many times a num- ber are 8 times the number less 3 times the number ? 2. Letting n stand for a number, how many times n are 8 times n less 3 times n? Sn — 3n = ? 3. 8z-3z = ? 4. 8s-3s=? 5. Sa-3a = ? EXERCISES 21. 1. From 10 a subtract 4 a. Explanation. — Just as 10 a's less 4 a's are 6 a's, so 10 a 10 a — 4a=6a; that is, when terms are similar their 4 ^ difference may be obtained by subtracting the numerical 6 a coefficients and annexing the common literal part. 2. 3. 4. 5. 6. 7. From 9 9x 7 1 ah 18 m^ 20 xy Take 4 4x 3 Sab 13 m^ 16 xy 8. 9. 10. 11. From 16 ax" 14^^53 8 xYz 21 (a + 6) Take 9ax^ Tr'^ 6 x^yh 11 (a + 6) 12. 13. 14. 15. From 3i)-f8r/ 4? + 2« 9x + ly r 5r4-8s Take 2p + 4g 41+ t 2x + 3y r 2r + 5.«* 16. 17. 18. From n-^5n'-h2 'n^ 3r4-2s + i 8a2 + 2a&+36=^ Take n+ n^-\- n^ 1 '+ s-\-t 5a2-f a6 + 2 62 19. From 12x + l y subtract 8 a; + 3 2/. 20. From 10 a& -|- 3 c subtract 5ah -\-2c. 21. From 7r-\-5s-\-6t subtract 3r4-2s + 5^. 22. From 9 a^ + 8 2/^ -f 6 a;?/ subtract 5 a;^ + 3 2/^ + 2 a??/. FUNDAMENTAL OPERATIONS 29 23. From 5mH-7n4-8Z-f-6 subtract 5m + 4n + 4Z4-5. 24. From 7 ar^+ 32/ + 62; + 4v subtract 3a^ + 22/-f50 + 3v. Subtract : 25. 2a:3_^^_^3^from 4a^^72/» + 5r3. 26. 4a6 + 262 + 2cdfrom6a6 + 362 + 6cd. 27. 3a:2v_|.a^^5 from 9a:22^-f6x^H-a^ + 8. 28. 2 ?;%^ 4- 2 vw + 4 i^^ from 12 -v^ + 9 v^w;^ +6 vw. 29. 5 m2?ia^ + abd from 18 m^na^ + 12 a^feV + 4 abd. 30. 4 a;*" + 2 rC^y" + 5 2/*' from 7 a;"* + 2 oTy'' + 9 y"- 31. 6 m' -f- 11 m'n* 4- 5 n' from 10 m' -h 1 1 m'n' -f 8 w'. 32. a"*^" -h 3 6'"-" + 7 c^" from 3 a"'^-" + 5 ft'""" + 9 c^". 33. 10 (m + n^) -f- 5 (m^ -\- n) from 12 (m + 7i-) + 8 (m^ + n). Simplify, adding or subtracting in order as signs indicate : ' 34. 9a;-4xH-6a;. 39. 8 r- 6r + 5 r-2 r. 35. 5n + 3n— 7ri. 40. 7y-\-Sy—6y-^7y. 36. 8a — 6a — 3a Al. 5z-\-7z — 2z — 4:Z. 37. 2s + 8s — 5s. 42. 9v — 3v + 2v — 5u 38. 3 6— 26 + 7 6. 43. 7n — 2w — 3 w + 4 w. 44. 8a; + 7a;-3a: + 4a; — 2«-3a;4-6a:. 45. 2?/ + 3?/-2/ + 7?/-32/ + 92/ + 22/-6?/. 46. 9z-5z-\-(jz-3z-\-4.z + 2z-7z + Sz. 47. 5v-{-6v-\-2v — 5v-\'4:V — 6v-{-9v — ^v — Sv. 48. 7 m + 6 71 — 3 m + 5 ?i 4- 7 m — 4 »i + 3 wi + 4 ?i + 5 m. 49. 9r+8s+7r-2s-f9s — 3rH-2r— 7s + 6s— 5 r+4 r. 50. 2Z-|-9i+3Z-/4-3«+2<+8Z-5«-f-9Z-6Z+2^-4«+7f. 51. 10 (a — X) 4- lo(a — x)-\- 7 (a — x)— lS{a — x) — 12(a — x). 30 FUNDAMENTAL OPERATIONS MULTIPLICATION 22. Product of two monomials. In algebra, as in arithmetic, the product of two numbers contains all the factors of both numbers, arranged or grouped in any way we please. Then, since o? = aa and a^ = aaa, or ' a^ = (aa) (aaa) = aaaaa = a^. That is, a^ ' a^ = a^+^ = a^. (Add exponents) Similarly, 3 a^ • 5 a^ = (3 • 5) (a' • a^) = 15 a\ (Multiply coefficients) Again, 3 a=^6 • 5 a'b' = (3 • 5) (a' - a') (6^ • b') = 15 a'b^ Hence, for multiplication : 23. Law of exponents. — TJie exponent of a number in the product is equal to the sum of its exponents in multiplicand and multiplier. 24. Law of coefficients. — The coefficient of the product is equal to the product of the coefficients of multiplicand arid multiplier. EXERCISES 25. Tell products quickly: 1. 7 a 2. 3i» 3. 5 m 4. Sab^x^ 3 a 4 iy 2 m^n b*cx* 21 a^ 12 xy 10 m'n 8 ab^cx^ 5. 3,7 6. 4a;2 7. Sav 8. 12 a^bc 4y 7fl^ 3 aw Sa'b'd' 9. ab' 10. 3xf 11. 2 ax 12. 16 c'd^m a'b 9xz 2 by 2(^d'n 13. xSj 14. 7^97^ 15. Sc'd 16. 2axft xy p'q 4rdh 6 a^yzh FUNDAMENTAL 0PP:RATI0NS 31 26. To multiply a polynomial by a monomial. Multiplying as in arithmetic, we have : 1. 43 2 86 321 3 963 40+3 2 4 tens + 3 units 2 4^ + 3?/ 2 2. 80 + 6 300 + 20 + 1 3 900 + 60 + 3 8 tens + 6 units 3 x + y + z a ax + ay-^ az 27. The product of a polynomial by a monomial is equal to the sum of the partial products obtained by multiplying each term of the polynomial by the monomial. EXERCISES 28. 1. Multiply : a^ + 2 2. ax^-\-y 4 ax 4x^-^ Sx" 3. 5mh-\-2t 3 St' 4. m^ + n^ 5. x^-{-2xy-\-y- mn xy 6. xy-\-yz-\-xz xyz 7. l4.2.x + 6ar^ + 4a^ 8. x' ^6a:2 + 2ic + l In exercise 7, the multiplicand is arranged according to the ascending powers of x-, in exercise 8, according to the descending powers of x. Arrange according to the ascending or descending powers of some letter and perform the multiplications indicated : 9. ab(6a' + a* + l + 4.a''-[-4:a). 10. 2a;?/(8a^?/ + 2.T^ + 2?/* + 12a^2/^ + 8a;/). 11. a26c(3 a' + 16 6* + 2 a¥ + 4 a«6 + 5 a-b'). 12. 8 t^^ (t^ + 6 «s^ + 20 t^^ + 15 t\^ + s« + 15 ^.a.a-^-i^^^^.^ or ?1^' = — a«-363-2 ^ 7 fj2^i ^ 7 ^25 3 a'b' 3 Hence, for division : 32. Law of exponents. — TJie exponent of a number in the quo- tient is equal to its exponent in the dividend minus its exponent in the divisor. Since a number divided by itself equals 1, a^ -^ a^* = a^~^ = a^ = !• that is, a number whose exponent is is equal to 1 . 33. Law of coefficients. — TJie coefficient of the quotient is equal to the coefficient of the dividend divided by the coefficient of the divisor. FUNDAMENTAL OPERATIONS 35 EXERCISES 34. Tell quotients quickly : 1. 5)53 2. 7 c^d ^)35 c*d' 3. 2 d^ )a\v 5=^ Sc^cif ^a'x 4. 22)2f 5. 3^-3^ 6. 4«m «)4^mV 12 g^ft- g 18a;V ^ 21 a6V * 4a62 * * 8af^^ ' ' 7 b^ ' ^Q 28 a^6-c ^^ 16a^^ ^2 24^y?!. 4 a6c 4 a;2/'^2; 8 a;^2^ 20a^6y ^4 36' + 4a*6^)^4a^63. 13. (3o?yz''-{-l^xY^-\-^x^y^ + l%xYz)^3;x?yz. 38. To divide a polynomial by a polynomial. EXERCISES 1. Divide 3 ar^ + 35 + 22 .t by a; + 5 ; test the result. PROCESS TEST 3a;2 + 22aj + 35 a; 4-5 60-6 3 X times (a; + 5) 33.-2 + 15 0; 7a.' + 35 3a; + 7 = 10 7 times (x + 5) 7a; + 35 Explanation. — For convenience, the divisor is written at the right of the dividend and the quotient below the divisor. Both dividend and divisor are arranged according to tlie descending powers of x. Since the dividend is the product of the quotient and divisor, it is the sum of all the partial products formed by multiplying each term of the quotient by each term of the divisor. Hence, if 3 x^, the first term of the dividend as arranged, is divided by x, the first term of the divisor, the result, 3 a;, is the first term of the quotient. Subtracting 3 x times {x + 5) from the dividend, leaves 7 x + 35, the part of the dividend still to be divided. V FUNDAMENTAL OPERATIONS 37 Proceeding, then, as before we find, 1x-r-x = l, the next term of the quotient. 7 times {x + 5) equals 7 x + 35. Subtracting, we have no remainder. Hence, the quotient is 3 x + 7. Test. — When x = 1 , the dividend equals 60 and the divisor 6. The quotient then should equal 60 -h- 6, or 10. On substituting 1 for x, we find that the quotient is equal to 10. Presumably, then, the result is correct. 2. Divide 3.-3+ 6 a;- + 12 a; + 10 by a; + 2. PRO a;3 + 6ar^ + 12a; + 10 CESS a; + 2 TEST 29-r-3 4a;^ + 12a; ^+4. +4+^^^ =H 4x^4. 8a; 4a; + 10 4a;+ 8 \ As in arithmetic, the whole of the undivided part of the dividend is not brought down for each division, but only so much of it as may be needed each time. The remainder 2 is written over the divisor in the form of a fraction which is then added to the quotient as in arithmetic. Rule. — Airange both dividend and divisor according to the ascending or the descending powers of a common letter. Divide the first term of the dividend by the first term of the divisor, and ivrite the result for the first term of the quotient. Multijyly the whole divisor by this term of the quotient, and sub- tract the product from the dividend. Tlie remainder ivill be a new dividend. Divide the new dividend as before, and continue to divide in this way until the first term of the divisor is not contained in the first term of the new dividend. If there is a remainder after the last division, write it over the divisor in the form of a fraction, and add the fraction to the part of the quotient previously obtained. 38 FUNDAMENTAL OPERATIONS Divide, and test each result : 3. x--{-2x + lhj x-\-l. 6. 3 -h7/ + 2?/* by 2/2-1-3. 4. a'-h5a-\-6hj a-\-2. 7. 6c(^ -\-af -{-7 hy x^ -]-l. 5. 5r + r2 + 4by r + 4. 8. G^^.,. 20^ + 23 by 3^ -|- 7. 9. .v^-h3/ + 32/ + lby2/ + l. 10. (jz'-h4. + 10z^-\-4.:^hj 4:z'-\-2. 11. b^ + Cyb^-\-b'-{-9b^-\-4.b-\-Shjb^-^4:. 12. Dividea*-h6a'^ + 27a2-h54a-f81 by a^ -h 3 a -1- 9. PROCESS TEST a4^(5^3^27a2-f-o4a-h81 a^ -h 3 a + 9 169-13 a* + 3a^+ 9a^ a^ + 3a + 9 = 13 3a' + lSa' + 54.a Sa^-\- 9a2 + 27a 9a- + 27a-|-81 9a2 4-27a-f-81 Divide, and test each result : 13. x'-\-A^i-12:t^-j-16x+16hy x^-\-2x-{-4:. 14. 4:l'-\-4.l' + 13l* + 6l'-j-9hj2l' + P + 3. 15. 42/^-|-52/' + /4-ll2/H-3/ + 6 by .v^ + 3.v + 2. 16. 6r^ -f-26r2 + 18-hl5r+7r3by2r-f-3r2-h3. 17. X' -h .^y^-{-5xy-\-4.y^-\-:ihf. 21. Add2a-3&, 2b -3c, 5 c-Aa, 10a-5h, and76-3c. 22. Add x + y-\-z, X — y-i-z, y — z — X, z — X — y, and x — z. 23. Add 4:a^-2x'-7x-\-l, a^-{-3 3(^-\- 5 X- 6,4: x^-Sa^-\-2 -6x,2s(^-2x^ + Sx-\-4:,2ind2x^-Sx'-2x-{-l, 24. Add5a;-32/-2;2, 4i/-2a;+6 0,3a-2a;-4 2/, 4 6-22 — 5y, a— 5b, 5y — 6x, Sx-{-2y — 5a — 2b, and 6x — y -2 2 + 4 6. 25. Add .12a.'3-4a^4-aJ + 2, .4^72 - 4 a? + . 4 - ic^, 3ia;-.6 4- 3 a.-^ -f 2 a.-^, and 1 - i a? + 1 . 2 a;'^ + II a^. 26. Add 20 aj^" — 4 a;"'?/'* + 36 2/2% 4 a^'"?/'* — 15 ?/2« — 12 a;-'", 3 /" + 3 af"^, 4 a;"'?/* — 11 a;^"' — 16 y-"", and a.--"* — 2/-'*. Difference of Two Numbers EXERCISES 64. On account of the extension of the scale of numbers below zero (§ 51), subtraction is always possible in algebra. When the subtrahend is positive, algebraic subtraction is like arithmetical subtraction, and consists in counting backtoard along the scale of numbers, as illustrated in § 58. Subtract the lower number from the upper one : 1. 6 6 6 6 6 6 6 3 4 5 6 7 8 9 2. Observe that subtracting a positive number is equivalent to adding a numerically equal negative number. -3 -3 -3 -4 -5 -6 -7 1 2 3 4 5 6 POSITIVE AND NEGATIVE NUMBERS 55 When the subtrahend is negative, it is no longer possible to subtract as in arithmetic by counting backward. 3. Subtract — 2 from 8. Explanation. — If wepe subtracted from 8, the result would be 8, the minueBd itself. 8 The subtrahend, however, is not 0, but is a number 2 — 2 units below in tlie scale of numbers. Hence, the differ- 8 _j_ 2 = 10 6nce is not 8, but is 8 + 2, or the minuend plus the sub- trahend with its sign changed. Or, — 2 is subtracted from 8 by beginning at 8 in the scale of numbers and counting 2 units in the direction opposite to that indicated by the sign of the subtrahend, arriving at 10. Remark. — Notice that any number is added by counting along the scale of numbers in the direction indicated by its sign; and any number is subtracted by counting in the direction opposite to that indicated by its sign. Subtract the lower number from the upp6r one: 4. 4444579 -1 -2 -3 z:± zlL nl. 5. _5 _5 _5 _5 -1 -4 -6 -1^ -2 -6^ -3 -1_ -^ Observe that : Principle. — Subtracting any number {imsitive or negative) is equivalent to adding it with its sign changed. Subtract the lower number from the upper one : 6. 10 7. 12 8. 20 9. 16 10. 40 11. 16. 4 17. 4 18. —4 19. -9 20. — 7 4-4488 — 2 5 -6 -4 -8 12. -3 13. -7 14. 10 15. -5 -2 -6 4 -5 10 56 POSITIVE AND NEGATIVE NUMBERS 21. Subtract 12 from —1. 23. From subtract —3. 22. Subtract - 4 from 14. 24. From — 3 subtract . 25. From subtract —7; from the result subtract —4; then add — 2 ; add — 3 ; add 7 ; subtract 11 ; and add — 6. 26. AVhich is greater and how much, 3 or — 5 ? — 2 or -5? 6or8-3? _ 2 + (-8) or -2 - ( -8)? A weather map for January 16 gave the following minimum and maximum temperatures (Fahrenheit) : Chicago DXTLUTH Helena MONTBEAL New Orleans New York Minimum Maximum 24° 30'' -6° 2° -12° -4° -12° 18° 64° 76° 20° 42° 27. The range of temperature in Chicago was 6°. Find the range of temperature in each of the other cities. 28. The freezing point is 32° F. How far below the freez- ing point did the temperature fall in Montreal ? 29. How much colder was it in Duluth than in Chicago ? in Montreal than in New York? in Helena than in New Orleans ? 30. An elevator runs from a basement, — 22 feet above the first floor, to the tenth story, 105 feet above the first floor. Express its total rise from the basement to the tenth floor; from the tenth floor to the basement. 65. From the work of this chapter, the student will have discovered that negative numbers give the definitions of addi- tion, subtraction, sum, and difference a wider range of mean- ing than they had in arithmetic. In algebra addition does not always imply an increase, nor subtraction a decrease. In §§ 20, 21, the student learned how to subtract one literal expression from another, all the terms being positive and the subtrahend being less than the minuend. This is arithmetical subtraction. He will now apply the broader algebraic idea of subtraction to literal expressions. POSITIVE AND NEGATIVE NUMBERS 57 EXERCISES 66. 1. From 10 x subtract 15 x. PROCESS ^^^ Explanation. — Since (§ 64, Prin.) subtracting any 15 X number is equivalent to adding it with its sign changed, — 15 ic may be subtracted from 10 x by changing the sign H 5^ of 15 X and adding — 15 x to 10 x. 2. 3. From 5 a 5 a; Take 2_a -2x 7. From 8 a; — 3 y subtract 5x — 7y. PROCESS Explanation. —Since (§ 64, Prin.) subtracting Sx — Sy any number is equivalent to adding it with its sign 5 a; — 7 y changed, subtracting 5 x from 8 a: is equivalent to I adding — 5x to 8a-, and subtracting — 7 y from Q I j^ — Sy ia equivalent to adding + 7 y to —By. Rule. — Change the sign of each term of the subtraJiend, and add the result to the minuend. After a little practice the student should make the change of sign mentally. 4. 5. 6. 9 am — Smn 3a,V 21 am — 4m?i -lOx'y' 8. 9. 10. 11. From 9a + 7b 5r-10s 10x-\-2y 3m-3n Take 2a + 36 7r+ 4s 6x-4ty 2m-5n 12. From 5x — 3y-{-z take 2x — y + Sz. 13. From 3 a^b -j-b^-a^ take i a^b - S a^ -{- 2 b^ 14. From 13a2 + 5&2_4c2 take 8a2 + 952^10c2. 15. From 15x — 3y + 2z subtract Sx-^Sy — 9z. 16. From a"-ab- b^ subtract ab-2a^-2 b\ 17. From m^ — mn -f n* subtract 2m^—3 mn + 2 n^. 18. From 4 a;^ _|_ 3 ^^ ^ ^2 subtract 2 a;^ _ 5 ^.^ ^ 2 y^ 58 POSITIVE AND NEGATIVE NUMBERS 19. From S ab-\-a^-\-b^ subtract a^ + 4 a6 + 61 20. From 6 oc^ -{- 4: xy — 3 i/ subtract 4:y- — 3xy -{-6 ^. 21. From the sum of 3 a^ — 2 a6 — 6^ and 3 a6 — 2 a^ subtract a^— ah — b'\ 22. From 3x — y-\-z subtract the sum of x — 4:y-\-z and 2x + Sy-2z. 23. From a-{-b + c subtract the sum of a—b — c, b — c — a, and G — a — b. 24. Subtract the sum of m^n — 2mn^ and 2m~u — 7n^ — 7i^ + 2 mn^ from m^ — n^. 25. Subtract the sum of 2c — 9a — 36 and 36 — 5a — 5c from 6 — 3 c + a. 26. From the sum of 3 0?"*+ 4?/'* +2;"'+" and 2 z'^+'' -\- 2x^^—3 y'' subtract 4 ic"* — 2 2/" + z'^+\ Ux = a^-hb\y = 2ab,z = a'- b\ and v = a^ - 2 a6 + 6^, 27. a? + ?/H-2; + v = ? 29. a; — y + s; — 'y=? 28. x — y — z-\-7^ = l 30. 2/ — ^— '^ + 2; = ? TRANSPOSITION IN EQUATIONS 67. In the solution of equations the student has used certain principles, stated in § 40 and § 42 and summed up in § 44. They are usually stated in somewhat broader terms as in the following section and are so simple as to be self-evident. Such self-evident principles are called axioms. 68. Axioms. — 1. If equals are added to equals, the sums are equal 2. If equals are subtracted from equals, the remainders are equal. 3. If equals are midtipUed by equals, the products are equal. 4. If equals are divided by equals, the quotients are equal. In the application of axiom 4, it is not allowable to divide by zero or any number equal to zero, because the result cannot be determined. POSITIVE AND NEGATIVE NUMBERS 59 EXERCISES 69. 1. Solve a; — 6 = 4 by adding 6 to both members (Ax. 1). 2. Solve the equation a; -f- 3 = 11 by employing Ax. 2. 3. Solve I ic = 10 by employing Ax. 3. 4. Solve 7 a; = 21. Explain how Ax. 4 applies. 5. Solve !»= 16 in two steps, first finding the value of ^x by Ax. 4, then the value of x by Ax. 3. Solve, and give the axiom applying to each step : 6. 2.^ = 6. 13. ic-h2 = 10. 20. fm = 9. 7. 5x = 5. 14. IV — 5 = 11. 21. 1 71 = 8. 8. 4?/ = 8. 15. i(;-}-l = 12. 22. fa; = 10. 9. 3?/ = 9. 16. .s-7 = 10. 23. fa; = 21. 10. iz = 5. 17. 9 + .s = 12. 24. |z = 20. 11. iz = 2. 18. r)-\-y = lo. 25. |« = 15. 12. \v = S. 19. 10 + .y = 12. 26. iw = 49. 70. 1. Adding 7 to both members of the equation a- - 7 = 3, we obtain, by Ax. 1, a; = 3 + 7. • — 7 has been removed from the first member, but reappears in the second member with the opposite sign. 2. Subtracting 5 from both members of the equation X + T) = 9, we obtain, by Ax. 2, .« = 9 — 5. When plus 5 is removed, or transposed, from the first mem- ber to the second, its sign is changed. 3. Explain the transposition of terms in each of the following : 2a;-l=5; 2a; = 5 + l. 3.^• + 2 = ll; 3a;=ll-2. 4a; = 14-3a; 4 a; 4- 3a; = 14. 71. Principle. — Any term may be transposed from one member of an equation to the other, provided its sign is changed. 60 POSITIVE AND NEGATIVE NUMBERS EXERCISES 72. 1. Solve the equation 6 — 5 ic + 18 = 6 -f 3 a; — 30. Solution By Ax. 2, ^- 5*- + 18 = ^ + 3a;-30. Transposing, § 7 1, - 5 a; - 3 x = - 30 - 18. Uniting terms, — 8 a; = — 48. Changing signs, 8 x = 48. Dividing by 8, Ax. 4, x = 6. Verification. — Substituting 6 for x in the given equation, 6 _ 5 . 6 + 18 = 6 + 3 • 6 - 30, or - 6 = - G. Hence, 6 is the true value of x ; that is, the value 6 substituted for x satisfies the equation. Suggestions. — 1. By Ax. 2 the same number may be subtracted, or canceled, from botli members. 2. By Ax. 2 the signs of all the terms of an equation may be changed, for each member may be subtracted from the corresponding member of the equation = 0. Solve and verify : 2. 3 = 5-0;. . 10. 8 + 7a = 5a+20. 3. 9-5a;==-l. 11. 2 + 13/^ = 50-9. 4. 10 + ^ = 18-^. 12. 50-n = 20 + n, 5. 2z-\-2 = S2-z. 13. 3x- 23 = a; -17. 6. 7x-\-2 = x-\-U. 14. 4a; + 12 = 2a; + 36. 7. 3p + 2=p + 30. 15. 2a; + Ja; = 30-|-a;. 8. 6y-2 = 3y + 7. 16. Sx-^x^SO + lx. 9. 5 m — 5 = 2m-f 4. 17. 5iv — 10 = ^w + 16. Simplify as much as possible before transposing terms, solve, and verify : 18. 10 a; + 30 -4 a; -9 a; 4- S3 + 12 a; = 90 +12 -4 a;. 19. 16a; + 12-75 + 2a;-12-70 = 8a;-50-25. 20. lls-60 + 5s + 17-2s + 41-3s = 2s + 97. 21. 10 2 - 35 - 12 2 + 16 + 32 = 4 ;2 - 35 + 10 ;2 + 32. 22. 14 w - 25 = 19 - 11 n + 4 + 16 - 10 n + w + 136 - 16 n. POSITIVE AND NEGATIVE NUMBERS 61 Algebraic Representation 73. 1. How much does 8 exceed S-\-2? z exceed 10 + y? 2. What number must be added to 5 so that the sum shall be 9 ? to m so that the sum shall be 4 ? 3. George rode a miles on his bicycle ; then b miles on the cars; and walked 3 miles. How far did he travel ? 4. A man bought a house for m dollars ; spent n dollars for improvements ; and then sold it for s dollars less than the entire cost. How much did he receive for it ? 5. If 40 is separated into two parts, one of which is Xy represent the other part. 6. A man made three purchases of a, b, and 2 dollars, respectively, and tendered a 20-dollar bill. Express the num- ber of dollars in change due him. 7. Represent three times a number plus five times the double of the number. 8. What two integers are nearest to 8 ? to a;, if x is an integer ? to a -j- 6, if a + 6 is an integer ? 9. What are the two even numbers nearest to 8 ? What are the two even numbers nearest to the even number 2n? 10. Express the two odd numbers nearest to the odd num- ber 2 ?i + 1 ; the two even numbers nearest to 2 n -f- 1. 11. There is a family of three children, each of whom is 2 years older than the next younger. When the youngest is x years old, what are the ages of the' others? When the oldest is ?/ years old, what are the ages of the others ? 12. A boy who had 2 dollars spent 25 cents of his money. How much money had he left ? If he had x dollars and spent y cents of his money, how much money had he left ? 13. The number 876 may be written 300 + 70+6. Write the number whose first digit is x, second digit 2/j and third digit z. 62 POSITIVE AND NEGATIVE NUMBERS Problems 74. If 3 a; = a certain number and a; + 10 = the same num- ber, then, 3ic = a; + 10. This illustrates another axiom to be added to the list that is given in § 68. Axiom 5. — N^tmbers that are equal to the same number , or to equal numbers, are equal to each other. This axiom is useful in the solution of problems, for its application is always involved in writing the equation of the problem. 76. The student is advised to review the general directions for solving problems given on page 45. 1. The Borough of Manhattan contains 22,000 elevators. If 2000 more are for freight than for passengers, how many freight elevators are there ? 2. The total height of a certain brick chimney in St. Louis is 172 feet. Its height above ground is 2 feet more than 16 times its depth below. How high is the part above ground ? 3. There are 3141 of the Philippine Islands, of which the number that has been named is 195 more than the number that is nameless. Find the number of each. 4. The value of the toys made in Germany one year was ^22,500,000, or $100 more than 4 times the amount purchased by the United States. Find the value of the latter's purchase. 5. The Canadian Falls in the Niagara Eiver are 158 feet high. This is 8 feet more than \^ of the height of the Ameri- can Falls. Find the height of the American Falls. 6. The summer bird population of Illinois is estimated at 30,750,000. and the number of English sparrows is 19,750,000 less than the number of other birds. Find the number of sparrows. 7. The porch of a temple. in India is 876 feet in perimeter, and \ of its length is 6 feet more than its width. Find its length and width. POSITIVE AND NEGATIVE NUMBERS 63 8. With the machines of the present time a pin maker can turn out 1,500,000 pins a day, or 60,000 more than 300 times the daily output of a pin maker of early times. How many pins did the early pin maker turn out per day ? 9. The cost of dressing the fur of a beaver is 2 cents more than 8 times that for a muskrat. For a muskrat the cost is 9 cents less than for a mink. If the cost of dressing all three furs is 71 cents, find the cost of dressing a beaver's fur. 10. The daily consumption of water per person in New York City is 22 gallons less than that in Boston. The daily consumption in Pittsburg is 250 gallons, or 30 gallons less than that in New 'York and Boston together. Find the daily con- sumption per person in Boston. 11. A carpenter, a plumber, and a mason together earn $12.70 a day. If the carpenter earns $1.70 less than the mason, and he and the plumber together earn $ 7.50, how much does each earn? 12. A letter sent from Indianapolis to Point Barrow, Alaska, travels 6800 miles. It goes 900 miles more by train than by steamer and 200 miles more by dog sleds than by train. How far does it travel by each ? 13. In the first three years of excavation, 313,356 cubic yards more were taken from the Panama Canal than from the New York Barge Canal. The amount taken from the former was 6,364,484 cubic yards less than twice that from the latter. Plow much was excavated from each ? 14. Of the wood used for pulp in New York State one year, 500,000 cords were supplied by the state. The amount im- ported was f of that used by Maine. If New York used twice as much as Maine, how many cords were used by each ? 15.. At one time the coffee stored at the docks of Havre, France, was | of the total yearly production of the world and ^ that of Brazil. If Brazil produces 3^ million bags more than all the rest of the world, find the amount stored at Havre. 64 POSITIVE AND NEGATIVE NUMBERS MULTIPLICATION 76. Primarily multiplication is the process of taking one number as many times as there are units in another. Thus, 3 X 5 = 5 + 5 + 5 = 15. In this section and the next, the sign x is to be read ' times.' Even in arithmetic multiplication is extended to cases that cannot by any stretch of language be brought under the original definition. Thus, strictly, in 3| x 4, 4 cannot be taken 3f times any more than a revolver can be fired 3| times. So in algebra there are still other cases, like-— 3 x 4, that do not come under the original definition. What we are concerned with, however, is the method of find- ing the product (consistent with the laws of operation used in arithmetic) and the interpretation of the results obtained. 77. Sign of the product. 1. Just as in arithmetic 3 times 4 are 12, so 3 times 4 posi- tive units are 12 positive units; that is, +3x+4 = + 12. (1) 2. Also, 3 times 4 negative units are 12 negative units ; that is, ■ +3x-4 = -12. (2) 3. Just as in arithmetic 4x3 = 3x4, so — 3x4=:4x— 3; and since — 12 = 4 x — 3, -3x4 = -12. (3) 4. Again, since 6 — 4 = 2, . by Ax. 3, -3(6-4) =-3x2 = -6. Also, §27, -3(6-4) = -3x6-3x-4 = -18-3x-4. Now, since both — 18 — 3x— 4 and —6 are equal to —3(6—4), by Ax. 5, _i8-3x-4 = -6. Transposing — 18, § 71, _3x-4 = -6 + 18; that is, _3x -4 = 4-12. (4) POSITIVE AND NEGATIVE NUMBERS 6b 5. The preceding conclusions may be written as follows: From (1), + a X + 6 = + a&, from (2), -\-ax — b = -ab, from (3), — a X + & = — a6, and from (4), — a x — 6 = + a6. Hence, for multiplication : 78. Law of signs. — Tfie sign of the product of two factors is -\- when the factors have like signs, and — when they have unlike signs. EXERCISES 79. 1. Multiply each of the following by + 2 ; then by — 2 : 3, 5, -6, 10, -8, -9, -12, a, x, -b. 2. Multiply -8 9 6 4-2 By _6 1 Zl5 nI 10 3. Multiply a —6 —x —y n By 1 _J?. jzl Zll -12 80. When there are several factors, by the law of signs, — a X —b = -{-ab', — a x—bx—c = -{-abx — c = — abc ; — ax — bx—cx—d = — abc x — d = + abed ; etc. Hence, The product of an even number of negative factors is positive; of an odd number of negative factors, negative. Positive factors do not affect the sign of the product. EXERCISES 81. Find the products indicated : 'l. (_1)(_1)(-1). 6. (-!)(_ 2) (-3) (-4). 2. (-2)(-a)(-6)c. 7. (_a)(-6)(+c)(-rf). 3. (-l)(x)(3,)(-36). 8. (-x)(-y){-l)zi~v). 4. (-3)(2)(-2a)6. 9. (-x)(y)(z)(-v)(-io). 6. (-2a){-3b)(-c)x. 10. (-r)(-s)(-0(x)(-3y). milnk's 1st yr. alo. — 6 66 POSITIVE AND NEGATIVE NUMBERS 82. To multiply when the numbers are either positive or nega- tive. Having learned to multiply when the numbers are posi- tive, §§ 22-29, and having just learned about the sign of the product when there are negative factors, the student is now prepared to multiply whether the terms of the factors are posi- tive or negative. EXERCISES 83. 1. Multiply -4 aar^ by 2 aV. Explanation. — Since the signs of the monomials are unlike, the sign of the product is — (Law of Signs, §78). 4-2=8 (Law of Coefficients, § 24). a . a^ = «! . aS = «i+3 _ ^4 (Law of Exponents, § 23). cc2 . a;4 = x2+4 = 0^ (Law of Exponents). Hence, the product is — 8 a^oifi. PROCESS — 4aa^ 2aV -8aV Multiply : 2. 10 a' 5a« — 5 m^n^ Smn 6. xy 10. -2x 2a^ 14. -Zn^ 6 b' 18. 2a-+i 3a' 22. d*-« ^2s+n 7. -2rq^x 11. -Ga'c'x -4.a'bn 15. 4.a'bY Sa'b'y 19. - 2 a'-b'' Ian'" 23. 8 7--" 3 r"s«-2^ 4. 8. 12. — 4 abc 2a^b -Sab -1 -Sab 2ba 16. — 1 -1 5. 3 a^b(^ - 7 abH 9. -5aV ~2ajx^ 13. -2a^y? - 4aa;* 17. -hm'^d' - 2 m'^'cd' 20. x'Y xy 21. 4a;"-i -2a;"+i 24. -a^- — a?" 25.' 2/""" ym-n+l 26. Sx^-2xyhj 5xf. 27. 3a'-6a'bhj -2b. 28. TTv^n^ — 3 m/i'* by 2 mn. 29. j^V— 2i)§^by — pg. 30. 4 a^ - 5 ft^c _ c^ by aftc^. 31. — 2 ac -f 4 aa? by —5 aca;. POSITIVE AND NEGATIVE NUiMBEKS 67 Expand, and test each result : 32. (a-f-&-fc)(a + 6-f c). 33. {a^-ah + y'){a? + ah + h-). 34. (a^ + 3a-6 + 3a62-t-63)(tH-6). 35. {a-h){a + h){a'-^h''){p> + b'). 36. (»^ — ^y-\- ^]f- — x\f-\- y*) (x + y). 37. (a2 + 62 + c2 + d2)(a2_62-|.c2-cF). 38. (x'- xy-\-y'-{-x-{-y-^l)(x-{-y + l). 39. (a-^ + 3 a'b -\-3ab^-\- b^) (a" H- 2 a6 + b^. 40. (a2_a6 — «c4-&^-&c + c2)(a + 6-f c). Arrange both multiplicand and multiplier according to the ascending or the descending powers of the letter involved; multiply, and test each result. 41. x-\-a:^-{-l-\-x^hy x—1. 42. a^ + 10-7x-4:X^hjx-2. 43. U-9x-6x^-\-x^hj x-^1. 44. a»-30-lla + 4a2by a-1. 45. 4a2_2a3-8a + a*-3by 2 + a. 46. 2m-3 + 2m3-4m2by 2m-3. 47. x + x^-5hj x^-3-2x. 48. 62 _^56-4by -4 + 252-36. 49. 4n8 + 6-2n*H-16n-87i2 + n«by n + 2. 50. l + aJ4-4a^ + 10a^ + 46.r* + 22ic + g)(i> + g). 11. (505-1-2!/. 21. (a^-^by. 2. (r-\-s)(r + s). 12. (2a-{-xy. 22. (a^ + b^f. 3. (a-\-x)(a-{-x). 13. (ab + cdf. 23. (a'^ + ft**)". 4. (a; + 4) (a; + 4). 14. (5x + 2yy. 24. (af« + ^«)2. 5. (a-h6)(a + 6). 15. (7z-^Sc)\ 25. (Sa'-^5by. 6. (2/ + 7)(y-f-7). 16. (Sb + 4:x)\ 26. (l-\-2a'by. 7. (z + l)(z + l). 17. (2m-f-3n)2. 27. {S xy^ + 4: a^yf. 8. (c + 9)(c-{-9). 18. (3c-f-7d!)2. 28. (9 a-'"-f-2 62«)2. 9 (v-\-S)(v-hS). 19. (8 s + 5 ty. 29. (4 aj^r 4. y^+iy 10. (w-\-o)(w-\-5). 20. (5w-{-3uy. 30. (a;-^ + 1/""^^. POSITIVE AND NEGATIVE NUMBERS 69 87. The square of the difference of two numbers. 1. Multiply a — bhya — b', find the square of x — y. a—b x—y a—b x—y a?—ab a? — xy -ab-\-b^ —xy + f a'-2ab-^b^ x'-2xy-\-f 2. How is the first term of the square of the difference of two numbers obtained from the numbers ? How is the second term obtained ? the third term ? 3. What signs have the terms of the result ? 88. Principle. — The square of the difference of two numbers is equal to the square of the first number, minus tivice the product of the first and second, i)lus the square of the second. EXERCISES 89. Expand by inspection, and test each result : 1. {x-m){x-m). 14. {2a-xy. 27. (3a;-2)«. 2. {m-n){m-n). 15. {Sm-nf. 28. {2x-5y)\ 3. (a;-6)(a;-6). 16. {^x-y)\ 29. (5m-3w)*. 4. (p_8)(i)-8). 17. (m-4n)2. 30. {^p-bq)\ 5. (9-7)(g-7). 18. {p-^qf. 31. (a* -6"/. 6. (a - c) (a - c) . 19. (a - 7 bf. 32. (af - y^'f. ' 7. (a -a;) (a -a;). 20. (4 a -3)2. 33. {a^-2by. 8. (a;-l)(a;-l). 21. (5 a; -4)1 34. (f-6xy. 9. (6 -5) (6 -5). 22. (ab-Sy. 35. (ab-2(^\ 10. (st-2)(st-2). 23. (2a-Sby. 36. (4:0(^-5yy. 11. (a: -4) (a; -4). 24. (2z-7yy. 37. (xf-^'^y^y. 12. (2 -3) (2! -3). 25. (Sx-5yy. 38. (mx^-ny^y. 13. («;-9)(w-9). 26. (9^v — 2vy. 39. (af-^-t/**" V- 70 POSITIVE AND NEGATIVE NUMBERS 90 1. The square of any polynomial. 1. Find the square of a + & -f- c. a-{-b-\-c a-\-b-\-c a^+ ab-{- ac + ab i-b'-\- be + ac + bC-^(^ a" -\- 2 ab -^2 ac -^ b^ + 2bc -i- c" That is, {a-\-b + cy = a^-\-b^-{-c^-{-2a.b ^2ac + 2bc. 2. Show by actual multiplication that = a^-\-b^ + c'-\-d^-\-2ab-{-2ac-]-2ad-]-2bc-\-2bd + 2cd. 3. Similarly, by squaring any polynomial by multiplication, it will be observed that : 91. Principle. — The square of a polynomial is equal to the sum of the squares of the terms and twice the product of each term by each term, taken separately, that follows it. When some of the terms are negative, some of the double products will be negative, but the squares will always be positive. For example, since (- 6)2 = + &2^ (a - & + c)2 = a=2 +(_ 6)2 + c2 + 2 a(- 6) + 2 ac + 2(- 6)c = a2 + 62 + c2 - 2 a6 + 2 ac - 2 6c. EXERCISES 92. Expand by inspection, and test each result : • 1. (x + y-[-zy, 3. (x-y-zf. 5. (x-\-y-Szy. 2. (x + y-zy. 4. (x-y-\-zy. 6. (x-y-{-3zy. 7. (a-26 + c)2. 13. (Sx-2y + 4.zy. 8. (2a-b-cy. 14. (2a-5b + 3cy. 9. (rs + st-rty. 15 (2m-47i-r)2. 10. (qb-pc-rdy. 16. (12-2x + 3yy. 11. (ax — by -\- czy. 17. (a -^ m -\- b -{- ny. 12. (xy — Sc — aby. 18. (a — m + b — 7if. POSITIVE AND NEGATIVE NUMBERS 71 93. The product of the sum and difference of two numbers. 1. Find the product of a -|- 6 and a — 6 ; of x — y and x + y. a — b -ah-b^ a' -b' x-y ^■¥y x^ — xy 2. How are the terms of the product of the sum and differ- ence of two numbers obtained from the numbers ? 3. What sign connects the terms ? 94. Principle. — The product of the sum and difference of \wo numbers is equal to the difference of their squares. EXERCISES 95. Expand by inspection 1. (x-\-y)(x-y). 2. ia + c)(a-c). 3. (P-^Q)(P-Qy 4. (p + 5)(i)-5). 5. (x-\-l){x-l). 6. (a^ + l)(^_l). 7. (a^ + l)(a^-l). 8. (x'-l){x^ + l). 9. (a^-l)(x^ + l). 10. (x- + f)(af-f). 11. (ab-\-5)(ab-5). 12. (cd-\-dr)(cd-cP). 13. (ab-c^(ab + c^. 14. (^y + z^(4.y-z^. 15. (Ix — 5 m) (^a; + 5 m). and test each result : 16. (2x + ^y){2x-^y). 17. (3m + 4n)(3m-4n). 18. {12 + xy){\2-xy). 19. {ab-^cd){ab-cd). 20. (3m2n-6)(3m2n-(-6). 21. (2a?+5y^(2o^-5f), 22. (3a^ + 2/)(3a^-2/). 23. (2 a^ 4- 2 62) (2 a^- 2 62). 24. {-5n-b){-5n + b). 25. {-x-2y){--x + 2y). 26. (3 af' + 7 y") (3 a;« - 7 y"). 27. (maf + 2 2/'') (maf - 2 2/*). 28. (a''6"'-f a'^6'*) (cC^b"^- a'^b''). 29. (xT-^ + y""^^) (xT - 1 — 2/"+^). 30. (5 a^b"" + 2 x') (5 a%^ - 2 af ). ) 72 POSITIVE AND NEGATIVE NUMBERS 96. The product of two binomials that have a common term. Let a; + a and a; 4-6 represent any two binomials having a common term, x. Multiplying a; -f- a by a? + 6, x-\-a x-\-h o(f-\-ax bx-i-ab x^+ (a + b)x-^ab 97. Principle. — The product of tivo binomials having a common term is equal to the sum of : the square of the common term, the product of the sum of the unlike terms and the cominon term, and the product of the unlike terms. EXERCISES 98. 1. Expand (x-\- 2)(a5-f 5) and test the result. Solution The square of the common term is x^ ; the sum of 2 and 5 is 7 ; the product of 2 and 6 is 10 ; .-. (a; + 2)(x + 5) = a;2 + 7a;4-10. Test. — If x = 1, we have 3 • 6 = 1 + 7 + 10, or 18 = 18. 2. Expand (a + 1) (a — 4) and test the result. Solution The square of the common term is a^ ; the sum of 1 and — 4 is — 3 ; the product of 1 and — 4 is — 4 ; • ' .-. (a + l)(a-4) = a2-3a-4. Test. — If a = 4, we have 5 • = 16 - 12 - 4, or = 0. 3. Expand (n — 2)(n — 3) and test the result. Solution The square of the common term is n^ ; the sum of — 2 and — 3 is — 5 ; the product of — 2 and — 3 is 6 ; .-. (n-2)(ri-3) = n2-5n + 6. Test. — If n = 3, we have 1 . = 9 — 15 + 6, or = 0. POSITIVE AND NEGATIVE NUMBEKIS 73 Expand by inspection, and test each result : 4. (x-{-5)(x + 6). 18. (af'-5)(af + 4). 5. (a; + 7)(a;-f-8). 19. (af»-a)(af -2 a). 6. (x-7){x-{-S). 20. (y-2b){y-\-Sb). 7. (x-{-7)(x-S). 21. (z-4:a)(z + 3a). 8. (aj-5)(a;-4). 22. (2 x-{-5)(2 x + S). 9. (a;-3)(a;-2). 23. (2 a;- 7) (2 a; + 5). 10. (a;-5)(a;-l). 24. (3 y-lXSy-^2). ^ .11. (a;H-5)(a; + 8). 25. (-ia^ -\-l)(-ia^ -7), 12. (29-4)(/) + l). 26. (a6-6)(a6 + 4). 13. (r-3)(r-l). 27. (o^^- a)(a:2/ + 2 a). 14. (n-S)(n-12). 28. (3xy -hf)(y'-xy). 15. (ri-6)(n-hl5). 29. {h'^c? + ef)(h^(? -ef). 16. (a^ + 5)(a:2-3). 30. (5 ce& + 2c2)(5a6 - 2 c^). 17. (a;3-7)(x3 + 6). 31. (3 ar» + 2 2/=^(3ar^- 2 1^^. By an extension of the method given above, the product of any two binomials having similar terms may be written. • 32. Expand (2 a; - 5) (3 a; + 4) . PROCESS Explanation. — The product must have a term 2 a; — 5 in r^, a term in x, and a numerical, or absolute, term. XThe x*-term is the product of 2 x and 3 x; the x-term is the sum of the partial products — 5 • 3 x and 2 x • 4, *^ '^ "*" called the cross-products ; and the absolute term is the 6 ar^ — 7 a; — 20 product of - 5 and 4. The process should not be used except as an aid in explanation. Expand by inspection, and test each result : 33. (2a^4-5)(3a; + 4). 38. (2dH-5 6)(5 a + 2 6). 34. (3a;-2)(2a;-3). 39. (J n' -2p)(2n^ -7 p). 35. (3a-4)(4a + 3). 40. (aft^ _ m^ (aft^ _ 4 m^. 36. {3x-y)(x--3y), 41. (4 ?•"• - 3 s") (2 ^-^ - 5 s**). 37. (72-a)(32 + 2a). 42. (a^-i + 5y)(2a^-i-3y). 74 POSITIVE AND NEGATIVE NUMBERS SIMULTANEOUS EQUATIONS 99. If 4 bananas and 9 oranges cost 35 ^, and 4 bananas and 6 oranges cost 26 ^, and it is required to find the cost of 1 of each, we may simplify the problem thus : 4 bananas and 9 oranges cost 35 ^ (1) 4 bananas and 6 oranges cost 26 ^ (2) Subtracting, 3 oranges cost 9 p (3) By thus eliminating the cost of the bananas, we have obtained a relation, (3), more simple than either of the two given rela- tions, (1) and (2), for it involves only one unknown cost. Or, let X represent the number of cents 1 banana costs, and y the number of cents 1 orange costs. Then, 4 bananas will cost 4 x cents, 9 oranges 9 y cents, etc. 4a; + 9?/ = 35 (1) 4a; + 6y = 26 (2) Eliminating the a;'s, 3y= 9 (3) y= 3, or 1 orange costs 3 ^. Since y = S,9y in the first equation is equal to 27. Substituting 3 for y in the first equation, 4 a; + 27 = 35, (4) x= 2, or 1 banana costs 2 ^. 100. In eliminating the aj's in the preceding section, equal numbers, 4 a; + 6 i/ and 26, were subtracted from the members of (1). Hence, Ax. 2, the results are equal, giving a true equa- tion, 3 2/ = 9. This method of elimination is called elimination by subtrac- tion. 101. How must the equations 2x-^3y=16 and 5x—S y—19 be combined to eliminate the y^s ? 2x + 3y = 16 5a;-3y = 19 Adding, Ax. 1, 7 a; = 35 This method, of elimination is called elimination by addition. POSITIVE AND NEGATIVE NUMBERS 75 102. Equations like those discussed in § 99 or in § 101, in which the same unknown numbers have the same values, are called simultaneous equations. EXERCISES 103. 1. If 2x-{-3y = lS and 2 x + 2/ = 10, what is the value of each unknown number ? Solution 2 a; + 3 y = 18 (1) 2x+ y = lO (2) Subtracting, Ax. 2, 2y= 8; :.y = i. Substituting 4 for 2/ in (2), 2 a: + 4 = 10; :.x = 3. Test. — Substituting 3 for x and 4 for y in (1) and (2), (1) becomes 6 + 12 = 18, or 18 = 18 ; (2) becomes 6 + 4 = 10, or 10 = 10. Note. — The value of y may be substituted in either of the given equa- tions. Solve, and test results: = 22, 21. 3. 6. 7. 8. 9. (5x-{-2y = [5x-{- y = \3x-\-2y = 19. r4a; + 52/ = 32, [2x-\-5y = 26. (7 x-2y = 22, \Sx-\-2y = lS. i6x + 7y = lS, \6x-\- y = 7. I9x-2y = n, [7x-2y^Sl. '5x-^Sy = 16, 2 a; + 3 2/ = 10. (6x + 2y=:22, [6x-7y = 4:. 10. 11. 12. 13. 14. 15. 16. 17. Sx-4:y = 16, 5 a; 4- 4 y = 48. (6x-^5y = 70, 1 x — 5y = 0. ' 5 » - 4 ?/ = 8, 3 ic — 4 2/ = 0. (Sx-^5y = lS, [Sx-\-8y = 14:. 7x-3y = 39, 5x-3y = 27. 5 y — 4 a; = 9, . 6 2/ -h 4 a; = 46. (Sx-3y = 39, Sx 36. 42/ I 7 a; + 5 y = 83, l7a;-42/ = 47. 76 POSITIVE AND NEGATIVE NUMBERS 18. If 2 ic -f- 3 2/ = 16 and 5 a; -f 4 ?/ = 33, find x and y. Solution 2x + Sy = lQ, (1) 6a; + 4y = 33. ' (2) We may eliminate either x or y. If we choose to eliminate x, we must first prepare the equations, so that x may have the same coeflBcient in each. Multiplying both members of (1) by 5, and both members of (2) by 2, 10 a: + 15 y = 80 (3) and 10 a; + 8^ = 66 (4) Subtracting (4) from (3), 7y = U; /.y = 2. Substituting 2 for ?/ in (1), 2 a; + 6 = 16 ; ..x = 6. Test. —These values, in (1) and (2), give 10 + 6 = 16 and 25 + 8 = 33. Note. — To eliminate y instead of x, proceed as follows : Multiplying (1) by 4, Sx + 12y = 64. Multiplying (2) by 3, 16x-}-12y = 99. Subtracting the upper equation from the lower, thus avoiding negative coeflBcients, 7 ic = 35 ; .. a; = 5. 10 + 3!/ = 16; .'.y = 2. Substituting 5 for a: in (1), Solve, and test results : 19. 20. 21. 22. 23. 24. 25. (9x-\-2y = 20, [Sx+ y = l. r6a; + 52/ = 28, |2aj + 32/ = 12. |aj-2/ = 2. '5a; + 22/ = 49, 3a;-22/ = 23. 4 a; -2/ = 27, x - 2/ = 3. {2x+ 2/ = 13, \ a; + 42/ = 17. r42/-3a: = 30, l52/-6£c = 33. 26. 27. 28. 29. 30. 31. riO.'B + 32/ = 62, I 6a; + 4?/ = 46.. lla;4-82/ = 37, 5a; + 6 2/=18. r2/ + 2a; = 18, [y-2x=.2. 2 y ~ 3 X = 5, 5?/ + 4a; = 93. 4:X-7y = 12, 3 x + 5y = 50. 8a; + 72/ = 37, 4a;-32/= -1. lla;-52/ = 29, 3aj + 22/ = 18. POSITIVE AND NEGATIVE NUMBERS 77 Problems 104. 1. The sum of two numbers is 8 and their difference is 2. Find the numbers. Solution Let X = the larger number. and y = the smaller number. Then, x-\-y = S, (1) and x-y = 2. (2) Adding (1) and (2), 2x = lO, .■.x = 6. Subtracting (2) from (1), 2y = 6; .-.^ = 3. Hence, the numbers ire 5 and 3. Find two numbers related to each other as follows : 2. Sum = 14; difference = 8. 3. Sum of 2 times the first and 3 times the second = 34 ; sum of 2 times the first and 5 times the second = 50. 4. Sum = 18 ; sum of the first and 2 times the second = 20. 5. A cotton tent is worth ^10 less than a linen one of the same size, and 3 cotton ones cost $2 more than 2 linen ones. Find the cost of each. Solution Let X = the number of dollars a linen tent costs, and y = the number of dollars a cotton tent costs. Then, x-y = 10, (1) and 3y— 2x = 2. (2) Multiplying (1) by 2, 2x-2y = 20. . (3) Adding (2) and (3), y = 22. (4) Substituting (4) in (1), X - 22 = 10 ; .-. X = 32. Hence, a linen tent costs $ 32 and a cotton one 1 22. 6. A steam train took 10 minutes longer to pass through the Simplon tunnel than an electric train. What was the time of each, if the steam train lacked 8 minutes of taking twice as long as the electric train ? 78 POSITIVE AND NEGATIVE NUMBERS 7. During one month tlie number of arrivals and departures of vessels at the port of Seattle was 183. There were 5 more arrivals than departures. Find the number of each. 8. At one time the United States Navy had 17 coaling sta- tions on the Atlantic and Pacific coasts. If the Atlantic had had 1 less, it would have had 3 times as many as the Pacific coast. How many coaling stations were there on each coast ? 9. The length of the Grand Canal in China is 13 times the approximate length of the Panama Canal, and the difference in their lengths is 600 miles. Find the length of each. 10. A steam train is 25 tons heavier than an electric train that carries as many passengers. If 9 such steam trains weigh as much as 14 of the electric ones, find the weight of each train. 11. If at Christmas time 3 dozen carnations and 2 dozen orchids cost $30, and 2 dozen carnations and 3 dozen orchids cost $ 40, find the cost of each per dozen. 12. Small goldfish are worth $4 per hundred less than large ones. If 3 hundred of the former and 2 hundred of the latter together cost $ 18, find the cost of each per hundred. 13. A stock car will hold 45 more sheep than hogs. If the number of animals in 16 cars of sheep is the same as in 25 cars of hogs, find the number in a car load of each. 14. One sugar factory employs 2200 men in the factory and fields. If the number of field hands is 100 more than twice the number of factory workers, find the number of each. 15. The Eoosevelt dam in Arizona is 17 feet lower than the Croton dam, and twice the height of the latter plus 3 times the height of the former is 1434 feet. Find the height of each. 16. The duty paid on 7 pianos entering Italy was $173.70. If the duty paid on each upright piano was $17.37, or i- that on each grand piano, how many pianos of each kind were imported? POSITIVE AND NEGATIVE NUMBERS 79 17. Prospect Park is 326| acres smaller than Central Park, and twice the area of the former is 1891 acres more than the area of the latter. Find the area of each. 18. A woman picked 5 crates of Brussels sprouts, containing in all 192 quarts. If the crates hold 32 and 48 quarts respec- tively, how many crates of each size did she pick ? 19. One year a jeweler had 693 broken watch springs brought to him to renew. The number broken in summer lacked 39 of being twice the number broken in winter. How many were broken in summer ? in winter ? 20. An engine at Sharon, Pa., weighed 40 tons less than 5 times as much as two of its castings. The weight of the whole engine, minus twice the weight of the castings, was 314 tons. Find the weight of the whole engine and of the castings. 21. In a typewriting contest in Paris a woman in a given tiitie wrote 500 words less than a man, and twice the number that the man wrote is 15,500 less than 3 times the number that the woman wrote. Find the number of words written by each. 22. In United States money, 2 marks, German money, and 3 francs, French money, are valued at $1,055, and 1 mark and 5 francs at $ 1.203. What is the value in United States money of a mark ? of a franc ? 23. The expense of running a small automobile is estimated at 51 ^ a week more than the expense of keeping a horse and carriage. The former can be run for 3 weeks for $ 2.22 less than the latter can be kept for 4 weeks. What is the weekly expense of each ? Suggestion. — Both equations should be expressed in terms of cents, or both in terms of dollars. 24. It cost 42 cents to stop a certain train and get it back to its former speed. Another train of less speed cost 35 cents to stop and start. If in all both trains made 5 stops, at a cost of $ 1.89, find the number of stops made by each. 80 POSITIVE AND NEGATIVE NUMBERS DIVISION 105. Sign of the quotient. Since division is the inverse of multiplication, the following are direct consequences of the law of signs for multiplication given in (§ 78) : + a X H- 6 = + «& ; . *. -\- ab ^ -{- a = -{- b. -\-ax—b = — ab', .'. — ab-r--\-a = — b. — ax-{-b = — ab; .•. —ab-. — a = -\-b. — a X — & = + a&; .*. +ab-. — a = — 6. Hence, for division : 106. Law of signs. — The sign of the quotient is + ^chen the dividend and divisor have like signs, and — whe7i they have unlike signs. EXERCISES 107. 1. Divide each of the following numbers by 2: . 6, -6, 10, -10, 14, -12, -18, 22, -8. 2. Divide each of the foregoing numbers by — 2. Perform the indicated divisions : 3. 7)-l- 4 -4). 4. 8. -3)15 22-(- 2). 5. -3)- -12 6. 10. -1)9 7. 4-5-(- 9. -l- (-1). -6h-3. .1. 9^(- ■3). 12. -21- 3. 13. 45^(- -5). 14. -8-f-2. '■!■ 16. 28 -7 ■'■ =f- 18. -20 -5 -T?- 20. 48 -6 -^- 22. 72 8* 108. To divide when the numbers are either positive or negative. Division, when the numbers involved are positive, was treated in §§ 30-38. The student is now prepared to divide whether the numbers are positive or negative, since the only new point involved is the matter of signs, just discussed. POSITIVE AND NEGATIVE NUMBERS 81 EXBRCISBS 109. 1. Divide - 8 aV by 2 aV. Explanation. — Since the signs of dividend and PROCESS divisor ai-e unlike, the sign of the quotient is — (Law of 2aV )-8aV Signs, § 106). — 4aa^ 8-=-2 = 4 (Law of Coefficients, § 33) . a* -f- a^ = a*-^ -a} = a (Law of Exponents, § 32). x6 ^ a:4 _ a^-4 -^a (Law of Exponents). Hence, the quotient is — 4 ax^. Divide : 2. 30 m^n^ by 5 mn. 0. 42 n^x* by - 6 w«. 3. -24a:22^Vby 8x2y. 7. -12pVby 12 p^. 4. 21 aa^?/ by — 7 ay. 8. - 20 7-V by - 10 r^. * 5.-9 db<^ by — 3 abc. 9. 40 mnx^ by — 8 mnv. 10. Divide 4 a^ft - 6 a*6* + 4 a6» by 2 a6 ; by - 2 a6. PROCESS PROCESS 2a6)4a»6-6a262^4a6» ^2 ah) 4a»6-6a^62-|-4a^» 2a2 -3a6 +2 6^ -2a^ -h3a6 -26* Test of Signs. — When the divisor is positive, the signs of the quotient should be like those of the dividend. When the divisor is negative, the signs of the quotient should be unlike those of the dividend. Divide : 11. a^y -2afhj ay, by - ay. 12. 9ar^/ + 15a;3/2by 3xy; by -3a2/».. 13. —x^ — ^x^-{- Q?7? by — icz ; by a». 14. 3ar»-6ic» + 9a;'-12a^by 3ic2. by _3a^ 15. 30r3s3 + 15?V-45rs< + 75rby 15r; by -15r. 16. — ^u — t*uv + tu^v — ^wV +^w*V by tu] by — ^m. 17. af + 2af'+^-5ic«+2-a^+« + 3a^+*by aj«; by -»•. 18. a?" - a;"-^ + a;''^ _ jb»-3 + a;"-< - a;""* by ar»; by — a^. 19. 2/"+^ — 2 2/**+2 + 2/"+^ — 3 2/**+* + 2/"-^' by y""^^ ; by — y**+\ 20. a{x -h 2/)''- a&(a; + yf^ o:'h\x + y)* by - a ; by a{x + y)\ MILNE'S 1st TR. ALO.-— 6 82 POSITIVE AND NEGATIVE NUMBERS 21. Divide 81 + 9 a' + a* by a^ _ 3 ^ + 9 ; test the result. PROCESJ 5 TEST a'-\-9a^-\-Sl a2_3a + 9 91 H- 7 a*-3a^-\-9a' a24-3a + 9 = 13 3a« + 81 3a3-9a2 4-27a 9a'-27a + Sl 9a2_27a+81 Note. — The test is made by substituting 1 for a ; similarly, the result may be tested by substituting any other value for a, except such as gives for the result -r- or any number divided by 0, because we are unable to determine the numerical value of such results. Divide, and test : 22. a* + 16 + 4a2by 2a + a2 + 4. 23. a^-61x-60hyx^-2x-S. 24. a'^-41a-120by a2+4a-f5. 25. 25ar^-a^-8ic-2a;2by 5ar^-4a;. 26. a» + a« + cfc* + a2 + 3a-l by a + 1. 27. 4/-92/'-l+62/by3i/ + 2/-l. 28. 2a*-5a^b-\-6a^b^-4.ab^'\-b'hy a'-ab + b^ PROCESS TEST a2_ ab-j-b^ 0-1 2a^_3a6 + 62 =0 2 a' - 5 a^b -^6 a'b^ -4.ab' + b' 2a^-2a^b-{-2a'b' - 3 a^ft + 4 a'b^ - 4 ab^ -Sa^-^-Sa'b^-Sab^ a'b''- ab'-^b* Note. — It will be observed from the test that -?- 1 = 0. In general, -^ a = ; that is, zero divided by any number equals zero. 29. Divide aa^ — aV — by?-\- IP- by ax — b. 30. Divide 20 a^y -25 x^ -IS f -^27 xf hj 6y - 5 x. 31. Divide a* - 4 a^a; -f- 6 a V -^aa^-\-x* by a^-2ax-{-x'. POSITIVE AND NEGATIVE NUMBERS 83 32. Divide c^ — 8 by c + 2. PROCESS c»-8 c»4-2c2 ~2&- -8 -2&- -4c 4c -8 4c + 8 c + 2 f -i-d -L (?-2 -16 c -f-^ -h c + 2 35. m' — n* by m + n. 36. m* + n* by m -f w. -16 Divide, and test results : 33. ar* + 32 by a; + 2. 34. a^-?/«by ar' + Z. 37. a;'4-2aj«-2a;*H-2a^-lby a; + l. 38. 2r' + 32/* + 5?/3 + 32/2 + 3?/ + 5by.7-fl. 39. 2n*-4n*-3n3 + 7n2-3n + 2by w-2. 40. i/*+7y-102/2_2^^15 by /-2y-a 41. 7a,'8 + 2a;<-27ar^ + 16-8a5by ar^ + 5a;-4. 42. 28a^+6a:8^gaj2_ga._2by 2-|-2a;+4««. 43. 25v2_20i;8+3'y*-M6v-6by 3'y2-8i;4-2. 44. 4-18a;+30«2_23aj3^.6a^by2a^-5a; + 2. 45. 32a^ + 24aj*-25a;-4-16ar^by 6»»-a;-4. 46. 1 by 1 -f- a; to five terms of the quotient. 47. 1 by 1 — a; to five terms of the quotient. » 48. a3-6a2-hl2a-8-6'by a-2-6. 49. 2/^ + 32a:«by 16a;*-hy*-2a!y»-8a5»yH-4a:«2/2. 50. 2 - 3 n' + 13 n^ + 23 w*' - 11 n^ + 6 n«' by 2 + 3 n«. 51. 6a2« + 5a2«-i_i0a2»'-2 + 20a2— 3-16a2"»-* by 2 a"* + 3a'»-i-4a'^-2. 84 POSITIVE AND NEGATIVE NUMBERS Special Cases in Division 110. 1. By actual division, {x^-y^)-^{x-y)=x + y. {3?-f)^{x-y)=s? + xy-{-f. {:(^-y'^)^{x-y)=Q(? + x^y + xf-\-i^. Observe that the difference of the same powers of two num- bers is divisible by the difference of the numbers. ' Divisible ' means ' exactly divisible.' 2. By actual division, {x'^-y^)^(x-\-y)=x-y. (ar'-y')-J-(ic + 2/) = ar^-a;?/ + 2/^ rem., -2f. ^__ (a;* - y*)^{x + y)^x^-ay^y + xy^ - f. (x^ — y^) H- (a; 4- 2/) = ic* — x^y + a^y^ — xy^ + y\ rem., — 2 ?/*. Observe that the difference of the same powers of two num- bers is divisible by the sum of the numbers only when the powers are even. 3. By actual division, (o^ + 1/2) -J- (a; - 2/) = ^ + y, rem., 2 y^. (a^ + /) -^(x — y)=x'^-+- xy -f y^ rem., 2 f. (x* -\- y^)^(x - y)= a^ + x^y -{- xy^ + f, rem., 2y*, Observe that the sum of the same powers of two numbers is not divisible by the difference of the numbers. 4. By actual division, (x^ + y^^(x + y)=x-y, rem., 2y^. (x^^ -\-y^)^(x -{.y) = x' - xy -\- y\ (x* + y*)^{x -^ y) = a^ - x^y -\- xy^ - f, rem., 2y*. (x^-^f)-i-(x + y)=x* — x^y+a^y^~xf-\-y*. Observe that the sum of the same powers of two numbers is divisible by the sum of the numbers only when the powers are odd. 111. Hence, the preceding conclusions may be summarized as follows, n being a positive integer : POSITIVE AND NEGATIVE NUMBERS 85 Principles. — 1. a;* — y" is always divisible by x — y. 2. of — y"" is divisible by x + y only when n is even, 1 3. aj" -f y" is never divisible by x — y. |4. a;** + y" is divisible by x-\-y only when n is odd. 112. The following law of signs may be inferred readily : When x — y is the divisor, the signs in the quotient are plus. When x-^y is the divisor, the signs in the quotient are alter- nately plus and minus. 113. The following law of exponents also may be inferred : In the quotient the exponent of x decreases and that of y increases by 1 in each successive term, EXERCISES 114. Find quotients by inspection : gs — 68 'fR^ — n^ g^ — 8 a — b m — n a — 2 4. Devise a rule for dividing the difference of the cubes of any two numbers by the difference of the numbers. Find quotients by inspection : 5. t±^. 6 TLjtJ^. 7 t±IL. a+b ' m+n ' c+3 8. Devise a rule for dividing the sum of the cubes of any two numbers by the sum of the numbers. Find quotients by inspection : 9. t^l?^. 12. r^izl. 15. ^+«' 10. ^^ • 13. f . 16. n-f-4 71 — 1 11. Vl±^. ■ 14. ^3i. 17. 1 + a ar'- 32 X- -2 a*- ■81 a + 3 86 POSITIVE AND NEGATIVE NUMBERS PARENTHESES 115. The student has seen how parentheses, ( ), are used to group numbers that are to be regarded as a single number. Other signs used in the same way are brackets, [ ] ; braces, \ \ ; the vinculum, ; and the vertical bar, \ . Thus, all of the forms, (a + 6)c, [a + fe]c, {a + h]c, a-\-h-c, and a + 6 c, have the same meaning. These signs have the general name, signs of aggregation. When numbers are included by any of the signs of aggregation, they are commonly said to be in parenthesis^ in a parenthesis^ or in parentheses. 116. Removal of parentheses preceded by -f- or — . EXERCISES 1. Remove parentheses and simplify 3a + (6 + c — 2a). Solution The given expression indicates that (6 + c — 2 a) is to be added to 3 a. This may be done by writing the terms of (6 + c — 2 a) after 3 a in suc- cession, each with its proper sign, and uniting terms. .-. 3a+(&-f-c-2a)=3a-f6 + c-2a = a + 6 + c. 2. Remove parentheses and simplify 4 a — (2 a — 2 6). Solution The given expression indicates that (+2 a — 2 6) is to be subtracted from 4rt. Proceeding as in subtraction, that is, changing the sign of each term of the subtrahend and adding, we have 4 a - (2 a - 2 5) = 4 a - 2 a + 2 6 = 2 a + 2 6. Principles. — 1. A parenthesis preceded by a plus sign may be removed from an expression without changing the signs of the terms in parenthesis. 2. A parenthesis preceded by a minus sign may be removed from an expression, if the signs of all the terms in parenthesis are changed. POSITIVE AND NEGATIVE NUMBERS 87 Simplify by removing parentheses : 3. a -h (6 — c). 9. a — m + (n — m). 4. x — (y — z). 10. a—b — {c — d). 5. x-{-y-^z). 11. 5a-26-(a-26). 6. m — n — (— a). 12. a — (b — c-\- a) — {c- b). 7. m-(7i-2a). 13. 2a^ + 3/-(a^ + a;i/-2/2). 8. 5x—(2x + y). 14. m4-(3m— w) — (2w— m)+ri. When an expression contains parentheses within parentheses, they may be removed iw succession, beginning with either the outermost or the innermost, preferably the latter. 15. Simplify 6a;-j3a + (9 6-2a)-f-4x-10 65. Solution 6 a; _ {3 a + (9 6 - 2 a) + 4 X - 10 6} Prin. I, =6a:-{3a + 96-2a + 4x-106} Uniting terms, =6x — {a — b + ix} Prin. 2. =6x-a + 6 — 4x Uniting terms, = 2 x — a + ft. Simplify : 16. Aa-\-b — \x-{-4:a-\-b — 2y — (x — y)\. 17. ab — ]ab-\-ac — a — (2a — ac)-\-2a — 2ac\. 18. a + [y-\5-{-4:a-6y-rS\-(7y-4:a-l)']. 19. 4 m — [p -H 3 71 - (m + n) + 3 — (6 p — 3 n — 5 m)]. 20. ab — \5 -{• X -{b + c — ab -\- x)\ -{-[x — (b — c — 7)]. 21. t- X - \1 - X - [1 - X- (1 - x) - {x -1)'] - X + 1\. 22. Simplify a^-f a(6-a)-6(26-3aV Solution The expression indicates the sum of a^, a(h — a), and — 6(2 6 — 3 a). Expanding, a(6 — a)=ab^a^ and — 6(2 6 — 3 a) = — 2 6^ + 3 a6. Therefore, writing the terms in order with their proper signs, o8 + a(6-a)-6(2 6-;Ja)=a-^ + ab - a^ - 26=^ + 3a6 = 4a6 - 26^. 88 POSITIVE AND NEGATIVE NUMBERS Simplify : 23. x^ + xijj-x), 26. ct?-.f-{x-y)\ 24. c'-dc-d). 27. c(a-6)-c(a + 6). 25. 5-2(a:-3). 28. a^ -h'' -'6ah{a-h). 29. -2{a^y-xy'')-5{xy^-x'y). 30. (3a-2)(2a-3)--6(a-2)(a-l). 31. (3m-l)(m4-2)-3m(m + 3)-h2(m4-l). 32. {a-by-2(a^-b^-2a(-a-b)-4:b\ 33. (a:^ + 2aryH-2/2)(a:2_2a;y-h2/^-(x' + i/^(a^ + 2/^. 34. f-l2x^-xy{x^y)-f^ + 2(x-y)(a^^xy + y^. 35. When a = - 2, 6 = 3, c = 4, find the value of a'-'(a-c){b-\-c)-\'2b. Solution a2_(a_c)(6 + c) + 26=(-2)2_(-2-4)(3 + 4)+2.3 = 4-(-6)(7) + 6 = 4_(_42)+6 = 4 + 42 + 6 = 52. When aj = 3, 1/ == — 4, 2 = 0, m = 6, n = 2, find the value of: 36. m(x — y) + z\ 39. (x -\- y) (m — n) -[- 3 z. 37. 2-fm2-(/-l). 40. (m-\-xy-(n-yy-y\ 38. a:^ — ?/^ — 771^ + n^. 41. xyz — n (x — my — (nxy. 42. 3 m (m — w) + 4 n (y — a;) — 7 (?/ + 2). 43. x^y^(m — ny(m-i-n)-i-(m-\-ny(m--n). 44. Sm{x — y — ny — (y — n — x)(n — x — y). 45. (a; + 2/ + 2!)^- a;^ (y + 2: - a;) (a? + 2 - 2/) - 2; (a; + 1/ - 0). 46. (2 a; + yy -(0^-2 yy - (m + ny (x-^y + z).^ 47. {m+n+xy—{m-\-n — xy — {m—n+xy{—m-\-n-\-xy. 48. Show that {a-b-^-cy^a" -\-W -{-c^ -2ab + 2ac~2hc, when a = 1, 6 = 2, and c = 3 ; when a = 4, 6 = 2, and c = — 1. POSITIVE AND NEGATIVE NUMBERS . 89 117. Grouping terms by means of parentheses. It follows from § 116 that : Principles. — 1. Any number of terms of an expression may he inclosed in a parenthesis preceded by a plus sign without chang- ing the signs of the terms to be inclosed. 2. Any number of terms of an expression may be inclosed in a parenthesis preceded by a minus sign, if the signs of the terms to be inclosed are changed. EXERCISES 118. 1. In a^ 4- 2 a6 + 6^, group the terms involving 6. Solution a2 + 2 «6 + 62 = a2 + (2 a6 -f &2). 2. Irv a^ — x^ — 2 xy — y^, group as a subtrahend the terms involving x and y. Solution a2 _ a;2 - 2 xy - y^ = rt* - (a;2 + 2 ary + y2). 3. In a.i^ -f- a5 -t- 2 a^ + 2 6, group the terms involving x*, and also the terms involving 6, as addends. 4. In a'' + 3 a^b -h 3 a6^ + 6', group the first and fourth terms, and also the second and third terms, as addends. In each of the following expressions group the last three terms as a subtrahend : 5. a''-b^-2bc-c\ 7. a'-\-2ab-\-b''-c^ + 2cd-d\ 6. a^-b^ + 2bc-c\ 8. a^ -2ab + b^ -(?-2cd-d\ 9. In a'^ + 2aZ;4- 6^ — 4 a — 4 6 + 4, group the first three terms as an addend and. the fourth and fifth as a subtrahend. Errors like the following are common. Point them out. 10. a?-'S?-\-x-'l={3?'-l)-ia?-irx). 11. a*~2/^-h22/2-22_a^_(y2^.2y2-20' 90 POSITIVE AND NEGATIVE NUMBERS 119. The use of parentheses in grouping numbers enables us to extend the application of certain cases in multiplication. Thus, in § 94 and in § 97, one or both numbers may consist of more than one term. EXERCISES 120. 1. Expand (a + m — 7i)(a — m + w). Solution a-}-m — n = a+(m — n) and a — m + n = a — (m — w). .-. [a + m — njla — m + «] = [a + (w — n)][a — (m — w)] § 94, = a2 - (m - ny § 88, = rt2 _ (7^2 _ 2 m« + n2) = a^—m^-j-2 mn — nP: Expand : 2. (r-\-p-q){r-p + q). 5. {3^-^2x+l){o?+2x-l). 3. {r-^p + q){r-p-q). 6. (a^+2a;-l)(a^-2«4-l). 4. (a;4-6 4-w)(a;-6-n). 7. (a^+3 a!-2)(a^-3 aj+2). 8. [(a+6)+(c + d)][(a + 6)-(c + d)]. 9. (a+2>+ a'4-2/)(<^ + ^ — a? — 2/)- 10. (a + 6 4-m — n)(a 4-& — '^ + w)- 11. (x — m + y— n)(x — m — y -\- n). 12. (a — m — b — n)(a + m—b-\-n). 13. Expand (a; + ?/ + l)(aj + 2/ — 3). Solution (X 4- 2/ + l)(a; + 2/ - 3) =(^+1/ + l)Cx+l/ - 3) §97, =(x + yy-2{x-^y)-S § 86, = a;2 + 2 xy + y2_ 2 X - 2 2/ - 3. Expand: 14. (x-y-2)(x-y-S). 17. (^*- 2 ^2_5)(^4_ 2i2_|_ 2). 15. (x''-^x-l)(x^ + x-}-S). 18. (2s4-3r + 4)(2s-}-3r-3). 16. (m~7i + 2)(7nr-w-4). 19. (2 a+5b-\-6)(2 a+5 b-S). POSITIVE AND NEGATIVE NUMBERS 91 121. Collecting literal coefficients. EXERCISES Add: 1. ax 2. bm 3. —ex 4. (t-hr)x bx — cm -dx (t-{-2r)x (a + b)x (b — c)m — (c + d)x (2t-h3r)x 5. ax 6. cy 7. —mp 8. (a -h b)x nx -dy -np (2a-hc)x Subtract the lower expression from the upper one : 9. mx 10. dy-\-az 11. ax — by nx ey — bz 2x — cy 12. a^x-\-aby 13. mx — ny 14. {2r — s)y b^x 4- aby nx — my (r 4- 2 s)y 15. Collect the coefficients of x and of y in ax— ay— bx— by. Solution. — The total coeflBcient of x is (a — b); the total coefficient of y is (— a — 6), or — (a + &). .'. ax — ay —bx — by ={a — h)x — (a + 6)y. Collect in alphabetical order the coefficients of x and of y in each of the following, giving each parenthesis the sign of the first coefficient to be inclosed therein : 16. ax—by — bx—cy + dx—ey. 20. x^+ax—y^-{-ay. 17. 5 ax-\-3 ay — 2 dx-\-ny —5 x—y. 21. a?— ay— ax— y^. 18. cx—2bx-{-lay-\-3ax—lx—ty. 22. bx—cy—2ay-\-by. 19. bx-\-cy — 2ax-\-by—cx—dy. 23. ra;— ay— sa;H-2cy. Group the same powers of x in each of the following : 24. av? + b^ — cx-\-e^ — dx^ —fx. 25. a^-\-3a^-{-Sx-ax^-3a3(^ + bx. 26. a^ — abx — a^—ba^ — ex — mnx^ -\- dx. 27. ax* — ai^ — ax^ -{- x^ + a^ — x —' aba^ 4- x^. 92 POSITIVE AND NEGATIVE NUMBERS EQUATIONS AND PROBLEMS 122. 1. Given 6(2 a; - 3) - 7(3 a; -f 5) ==- 72, to find the value of X. Solution 6(2 X- 3) _7(3a; + 5) = -72. Expanding, 10 a; - 16 - 21 x - 35 = - 72. Transposing, 10 x - 21 x = 15 + .36 — 72. Uniting terms, — 11 x = — 22. Multiplying by - 1, 11 x = 22. .•.x = 2. Verification. — Substituting 2 for x in the given equation, 5(4 - 3) - 7 (6 + 6) = - 72. 5-77 =-72. Hence, 2 is a true value of x. Find the value of x, and verify the result, in : 2. 2 = 2a;-5-(a;-3). 4. 1 = 5(2 a;-4) + 5 a; + 6-. 3. 10 a? -2(a;- 3) =22. 5. 7(5-3a;) = 3(3- 4a;) -1. 6. 4 a; — ar^ = a;(2 — a;) 4- 2. 7. 7(2a;-3)=2-3(2a; + l). 8. 3(2-4a;)-(a;-l) = -6. 9. ^x^-^{y?-^-\-x-2)=^y?. 10. 5 + 7(a;-5)==15(a;4-2-36). 11. 2(a;-5) + 7 = a; + 30-9(a;-3). 12. (a?-2)(a;-2) = (aj-3)(a;-3) + 7. 13. (a;-4)(a; + 4) = (a;-6)(a;4-5)+25. 14. a;2_(2a; + 3)(2a;-3)-f(2a;-3)2=(a; + 9)(a;-2)-2, 15. 3(4-a;)2-2(aj + 3) = (2a;-3)2-(a; + 2)(aJ-2)+l. 16. 20(2-a;)+3(a;-7)-2[a; + 9-3S9-8 + 28n=23. 17. (2a;-4)2-25a;-6-3a;(4 + 5)5=4(a;-f 2)2 + 71 18. 3(aj~7)~(a?-9) + 136 = a;-2[3a; + 4-(2a; + 6-f-9a?)]. POSITIVE AND .NEGATIVE x\ UMBERS 93 Literal Equations 123. 1. Find the value of x in the equation bx — b^ = cx — c^. Solution bz-b^ = cx-c^. Transposing, bx-cx= b'^ - c2. Collecting coefficients of x, (6-C)X=62_c2. Dividing by 6 — c, 62 _ f 2 = b-\-C' c 2. Find the value of a? in the equation x — a^=2 — ax. Solution X — a' = 2 — ax. Transposing, ax + x = a' + 2. Collecting coefficients of x, (a 4- l)x = a' + 2. Dividing by a + 1, x = ^^-±-? = a2 - a + 1 + —^ • a+1 a+ 1 Find the value of a; in : 3. S(x-a-2b) = 3b. 7. a^ - ax-\-5x = 7 a-10. 4. 56=3(2a;-6)-46. 8. 2 m» - ma; + '^a; - 2 7i» = 0. 5. cx-<:^-(JP-\-dx = 0. 9. a^-oa;— 2a6 + 6a;+ft'^=0. 6. x—l—c=:cx — €^ — c*. 10. 2?i'^-|-5?i+a;=7i^— 72a;— 2. 11. 3a6-a2-26a; = 262_aa;. 12. a2a;-a» + 2a2 + 5a;-oa + 10 = 0. 13. aa;-26j; + 3ca; = a6-26- + 36c. 14. ca;-c*-2c'-2c2 = 2c-a; + l. 15. 9a^ + 4:mx = -(3ax-16m^). 16. a; + 6n*- 471^ = 1 -3na; + 2?}-n«. 17. Ti^a; — 3 wi V + na; 4- 3 7?i2 + a; = 0. 18. a;-362-1926V-4ca;-}-16c2a; = 0. 19. r(x — sa; — 1) -f 7^(x — r + s^)=—l — x — r(sx + 1) + r's*. 20. a<-c-aa;-6a;4-cx-6*c = 2a262 + c(a;-l)-6Vl-f c). 94 POSITIVE AND NEGATIVE NUMBERS Algebraic Representation 124. 1. Find the value of x that will make ^x equal to 48. 2. Indicate the product when the sum of x, y, and — d is multiplied by xy. 3. If a man earns a dollars per month, and his expenses are h dollars per month, how much will he save in a year ? 4. Indicate the sum of x and z multiplied by m times the sum of X and y. 5. From x subtract m times the sum of the squares of (a 4- 6) and (a — 6). 6. A number x is equal to {y — c) times (d -f c). Write the equation. 7. What is the number of square rods in a rectangular field whose length is (a + h) rods and width (a — h) rods ? 8. At a factory where N persons were employed, the weekly pay roll was P dollars. Find the average earnings of each person per week. 9. How many seconds are a; days +c hours -^-d minutes? 10. Express in cents the interest on y dollars for x years, if the interest for one month is z cents on one dollar. 11. If it takes h men c days to dig part of a well, and d men e days to finish it, how long will it take one man to dig the well alone ? 12. Find an expression for 5 per cent oix\ y per cent of z. 13. A train ran M miles in ^ hours and m miles in the suc- ceeding h hours. Find its average rate per hour during each period and during the whole time. 14. A farmer has hay enough to last m cows for n days. How long will it last (a — 6) cows ? 15. A dealer bought n 50-gallon barrels of paint at c cents per gallon. He sold the paint and gained g dollars. Find the selling price per gallon. POSITIVE AND NEGATIVE NUMBERS 95 Problems 125. Solve the following problems and verify the solutions : 1. I bought 40 stamps for 95 cents. If part of them were 2-cent stamps and part 3-cent stamps, how many of each did I buy ? Solution Let X = the number of 2-cent stamps. Then, 40 — x = the number of 3-cent stamps. .-. 2a; + 3(40-a;)=95. Solving, X = 25, the number of 2-cent stamps, and 40 — x = 15, the number of 3-cent stamps. Verification. — The results obtained may satisfy the equation of the problem and still be incorrect, because the equation may be incorrect. If, however, the results satisfy the conditions of the problem^ the solution is presumably con-ect. 1st condition : The whole number of stamps bought is 40. 26 + 16 = 40. 2d condition : The total cost of the stamps = 95 ^. The cost of 25 stamps @ 2 ^ -|- 15 stamps @ 3 ^ = 95 ^. 2. A certain paper mill produces 350 tons of paper from sawdust each week. Of this 50 tons more is used for news- papers than for wrapping paper. How many tons are used for each? 3. The roadway of the Connecticut Avenue concrete bridge in Washington, D.C., together with two sidewalks, is 52 feet wide. How wide is the roadway, if it is 8 feet less than twice the combined width of the sidewalks ? 4. One year the box factories of New England used 6,000,000 feet of boards. The amount of white pine used less 1,200,000 feet was 3 times that of the other timber. How much white pine was used ? 5. It costs 2i^ more a day to feed an immigrant than it does to feed a United States private soldier. If it costs as much to feed 44 immigrants as it does to feed 49 privates, find the cost of the daily rations of each. 96 POSITIVE AND NEGATIVE NUMBERS 6. Of the 160,000 inhabitants of Hawaii, twice as many were Japanese as Chinese. The rest of the inhabitants, or J of the total, were Americans and Europeans. Find the num- ber of Chinese. 7. The combined capacity of two ice factories is 264 tons a day. If the capacity of the smaller one is increased 57 tons, its capacity will be half that of the larger one. Find the capacity of each. 8. It cost a man 60^ to send a telegram at *30-2', that is, 30^ for the first 10 words and 2^ for each additional word. How many words did the message contain ? Suggestion. — Let a; be the number of words in the message. Then, a; — 10 -will represent the number of words in excess of 10 words. .-. 30 + 2(x~10) =60 9. How many words can be sent by telegraph from New Haven to New York for 75 t at the day rate, ' 25-2 ' ? 10. A long-distance telephone message cost me $1.25. The rate was 50 f for the first 3 minutes and 15 ^ for each additional minute. How long did the conversation last ? 11. The day rate for a telegram between New Orleans and New York is '60-4' and the night rate is '40-3.' A message of a certain number of words cost 25^ less to send, at night than in the daytime. Find the number of words. 12. A boy was twice as old as his sister 4 years ago. Now his sister is | as old as he is. Find the age of each. 13. During one month the Dead Letter Office received 1,000,000 pieces of mail matter. If the number remaining in the office was \ as many as the number returned to the senders, how many pieces were returned ? 14. An eighteen-hour train between New York and Chicago was late 91 times during its first year's run. It Avas late at Chicago 10 times more than 50 % as many times as it was late at Jersey City. How many times was it late at Jersey City ? POSITIVE AND NEGATIVE NUMBERS 97 15. In China, one \^oinan earned 3^ and another 8^ a day by embroidering. The former worked 28 days on a piece of work, and then the two finished it. If the labor cost ^5.02, how long did each work ? 16. The shed that sheltered an airship was 544 feet in perimeter. If twice its length was 52 feet more than 4 times its width, what was its width ? its length ? 17. The average life of 5-dollar bills is | of a year longer than that of 1-dollar bills, and | as long as that of lO-dollar bills. If a 10-dollar bill lasts If years longer than a 1-dollar bill, find the average life of a bill of each denomination. 18. A farmer's net receipts from hens in a year were $ 90.15. The eggs sold for $92.55 more than the chickens, and the ex- penses were $ 72.65 less than the selling price of the eggs. What did the eggs sell for ? the chickens ? 19. Upon the floor of a room 4 feet longer than it is wide is laid a rug whose area is 112 square feet less than the area of the floor. There are 2 feet of bare floor on each side of the rug. What is the area of the floor ? of the rug ? 20. A party of 8 traveled second class from London to Paris for $5.70 less than twice the amount paid by a party of 3 traveling first class. If a first-class ticket cost $4.15 more than a second-class ticket, find the price of each. 21. A military cable and telegraph system between Seattle and Alaska covers 4044 miles. The length of the submarine cable is 272 miles less than twice that of the land telegraph. The land telegraph is 12 miles longer than 13 times the wire- less. How long is the wireless ? 22. The United States has 280 life-saving stations, 1 being situated at the falls of the Ohio River. Of the remainder, the Atlantic coast has 11 J times as many as the Pacific. Find the number on the Pacific coast, if it lacks 2 of being ^ the num- ber on the Great Lakes. MILNU^S 1st YR. ALQ. — 7 98 REVIEW REVIEW 126. 1. What are positive numbers? negative numbers? In the following expression point out the positive numbers ; the negative numbers. Perform the indicated operations : 3ax-]-7by — 9 bx-\- 10 by — 4: ax — 3 bx-\- A ax — 2 ax— 12 by. 2. What two meanings has the minus sign in algebra? If distance north is positive, what is the meaning of — 150 miles ? + 75 miles ? 3. Distinguish between arithmetical numbers and algebraic numbers. 4. Instead of subtracting a number (positive or negative), what may be done to secure the same result ? Illustrate by- subtracting — 7 from + 12. What is the absolute value of each of these numbers ? 5. What is transposition ? Give the principle relating to transposition. 6. State the law of signs for multiplication ; for division. . 7. What is the sign of the product of an even number of negative factors ? of an odd number of negative factors ? 8. In what respect do (a — b) and (b — a) differ ? Expand (a — by and (b — af and compare the results. 9. For what values of n is ic" + ?/" divisible hj x +y? by x — y? When is x"" — ?/" divisible hj x + y? hy x —y? 10. State the law of signs for the quotient when x"" -f- y"" or of — 2/" is divided by x-^y or x — y -, the law of exponents. 11. What must be added to cc^ — 10 x to make it the square of x — 5? to a^ + b^ to make it the square of a -f 6 ? to X* 4- icy + 2/* to make it the square of x^ -f y^ ? 12. How may a parenthesis preceded by a minus sign be removed from an algebraic expression without changing the value of the expression ? REVIEW 99 13. Add 3 a + 56 -lie, 6-2a+c, 2c-\-8a—b, 7c—b+6a, 5 6 — 4 a — 2 c, 6 — a, c-\-b ~a, and c — 4 a. 14. Subtract the sum of x — 2y-}-Sz — 5w and 7 a; -|- i« — 2 2 from 10 X — y-\-z — Sw. 15. If a; = 7-^ + rs — S-, y=2r^+4rs-|-2s^, smd z=r^—Srs—^j find the value of a; + 2/ ~ ^• Expand, and test each result : 16. (r^-hTr's-Sr^-\-2s^(r^-\-2rs-\-^. 17. (3P + 6Z2'-m-12Z'-m2-f 3m3)(4^ + 3^m + 2m2). 18. (x*4-y'-4:xf-{-5a^y'-hSa^y)(a^-\-Sa^y + 3xy'-hf). Expand by inspection, and test each result: 19. (Sa-hTby. 23. (7 r + 4s)(7r-4s). 20. (9w-2vy, 24. (Sx-5y + zy. 21. (x-{-2y)(x-2y). 25. (2c +d)(3c + 2d). 22. (a-3)(a + 10). 26. {oa-3b)(2a + 2b). Divide, and test each result : 27. 2Z«-f-5ZV-3^V-6Z»r3H-3Zr^ -?•« by 2^-36- -h?-2. 28. 3x^-{-Sy^-10yz-Sa^z-Sz^-\-10a^y by ar-\-2y-3z. 29. 4 aj^" - 25 aj^Y" - 10 a^V* - 2/*" by 2 ar^" - 5 xy - fy Find quotients by inspection : 30. 2/ + 1 32. ^'^^^. a + 2 3^ c3 + 125fr^ c + 5(i 31. .^3-64 a;-4 33. ^^-^^y\ 3x + 4:y __ 243 -a;» 35. 3 — a; 36. Simplify 17 a;- -i3y + 4.z-[z + 5 a + a;-3a-22/]j. 37. Simplify « + 26-[4c + 2(a + 2&)-6 + 4c-a] + 6. 100 REVIEW When a; = 2, 2/ = — 3, and 2 = 5, find the value of : 38. xz — (x + y-{-z). 40. x^ — 3x{y + z)-\-y^— z. 39. 3(x-y)+2(y-x)-zy. 41. (x-y)(y + z)-z\y -z). 42. When a = 2 and 6 = 3, prove that b(ab + b -2 d)= ab'' -{-b' -2 ab. 43. Collect similar terms within parentheses : aoc^ — cy-^CLX — 2 ax^ -\-2cy^ — ax — cy^ -f ax^ + cy. 44. Collect the coefficients of x and of y in 7 ax — 8 by — 22 a^x -^ o ay — 17 bx -^ cy — 4tx -^ 13 y. Solve for x, and test results : 45. 3«H-7(a;-2)-13 = 12-3a;. 46. 20 = 7-5(3-a;) + 9(a; + 2). 47. x^ — 1 =x{oi^ — x) -{- X + 3 -{- x^. 48. (x-A)(x-\-3) = (x-{-6y-3 + 2x. 49.^ Aa — 3{b + x)— 5 a = 7 b + 4:a — 5(x + a). 50. (a — x) (b — x) — b\x — (a — x) — xl = a^ — 2 bx -{- ab. Solve, and test : 51. '3x-{-2y = 13y 2x + y = S. gg^ (7x + Ay = 39, I a;-?/ = 4. -2/ = 3, 2.7=13. 53. I3aj- 54. 55. 56. ' 5x-\-7y = 24:, 3x-2y = 2. 3x+5y = 13, [2x-3y = -4.. (9x-2y = W, [5x-\-7y = 57. Supply the missing coefficients in the following equations 57. 3a-*bi-6a + Bb-*xy = *a-\-b-2xy. 58. x'-\-2xy-{-3f-[2a^-h*f-]=*o'^-\-*xy. 69. 6m2-f9mri-3n2-[3m2+*mn]+w2 FACTORING 127. In multiplicatwriy we find the product of two or more given numbers ; in factoring, we have given the product to find the numbers that were multiplied to produce it. These numbers are called the factors of the product. 128. A number that has no factors except itself and 1 is called a prime number. MONOMIALS 129. To factor a monomial. While in factoring it is usually the prime factors that are sought, this is not ge;ierally true in the case of monomials, be- cause the factors of a monomial, except those of the coefficient, are evident. Thus, 2rt852 shows its prime factors as well as though written 2 . a • a • a • 6 • 6, but 81 a^h"^ should be written 3* a^h'^ to be considered in factored form. However, it is often desirable to separate a monomial into two factors, one being given or both being specified in some way. EXERCISES 130. 1. In each of the following, if ocy is one factor, find the other : 6 sc^y, 15 x^y"', 2 ory, a^x-b-y^, — mnxy, — xy. 2. In each of the following, if ahc is one factor, find the other: c^hc, ah\ ahc\ - a-tV, - a'hc, -\ahc. 3. Find two equal positive factors of a;'; of 9aV; of 64?7i*. 4. Find two equal negative factors of 25 a^ ; of 16 a^ ; of 9 a^ 101 102 FACTORING 131. A factor of two or more numbers is called a common factor of them. 132. One of the two equal factors of a number is called its square root. Every number has two square roots, one positive and the other negative. The square root of 25 is 5 or — 5,'for 5 • 5 = 25 and (— 5)(— 5) = 25. In factoring, usually only the positive square root is taken. 133. To factor an expression whose terms have a common monomial fagtor. EXERCISES 1. Find the factors ot Sxy — 6x^y + 9 xy^. Explanation. — Bv examining the PROfPSS terms of>Jihe expression, it is seen that S xy — 6 x"^!/ -\- 9 xy^ the monomial Sxy is a, factor of every = Sxy(l —2x-\- 3y) term. Dividing by this common factor gives the other factor. Hence, the factors of Sxy — 6 z^y + 9 xy'^ are 3 xy and 1 — 2x + Zy. Test. — The product of the factors should equal the given expression ; thus, Sxy(l-2x + 3y) = Sxy - 6x^y + 9xy^. Factor, and test each result : 2. 5af-5a^. 12. a^2 _^ a^i + a^i^ _ a^. 3. Sx^ -\-2x\ 13. ac — be — cy — abc. 4. Sx^-6x\ 14. 3a^f-3x^y' + 12xy. 5. 4a2-6a6. 15. 3m^- 12mV + 6 mn^ 6. 5m2-3mn. 16. 9 a% - 18 a^V + 24 a^^y. 7. Sa^y^-Sa^f. 17. 12 x'yh^ - W x^yh' - 20 x^y^^. 8. 4:a%-6a^b\ 18. 25 c^da^ + 35 c«d V - 55 c^c? V. 9. 5m*7i-10mV. 19. 16 a^dV - 24 a^W + 32 a^^V. 10. 3a^--9a^-6a;^. 20. 14 a^mw^ - 21 a^mV - 49 a^wl 11.3 a> - 2 a% + aK 21. 60 m*nV - 45 m^^iV -f- 90 mM^r^. FACTORING 103 BINOMIALS 134. To factor the difference of two squares. By multiplication, § 94, (a + 6) (a - 6) = a^ - h\ Therefore, a^ - 6^ = (a + b) (a - b). Hence, to factor the difference of two squares, Rule. — Find the square roots of the tico terms, and make their sum one factor and their difference the other. EXERCISES 135. Factor, and test each result : 1. x'-m^ 3. a^-1. 5. n^-4?. 2. a^-y\ 4. a^-^\ 6. 1--- 7. Factor aV- 4 c^. Solution a2x2-4c2 = (aa;)2_(2c)2 = (ax + 2 c) (ax -2 c). Factor, and test each result : 8. a2_81. 12. a^-h\ 16. l-lUm\ 9. 6* -49. 13. m'-n^. 17. Q^a^-a'c^. 10. 25 2/2-1. 14. 81-icy- 18- 81 a^- 100. 11. m^-16ri2. 15. 9 a^- 49 61 19. 121 ?r- 367^. 136. To factor the sum or the difference of two cubes. By applying the principles of §§ 111-113, ^f!±^ = a^ _ a6 + 6^ and ^^^^=^' = a^ + a.6 4- 6'. a-\-b a — b Then, § 30, a' + b' = (a-\-b) (a' -ab + b% and a^-b^ = {a - b)(a^-\-ab + b"). By use of these forms any expression that can be written as the sura or the difference of two cubes may be factored. 104 FACTORING EXERCISES 137. Factor, and test each result : 1. a^ + iyl 3. m^-l. 5. 7^ + 2^ 2. a?-f. 4. l + m\ 6. --1. 8 7. Factor x^ + /. Solution «« + ys = {x^y + (2/"^)3 = (x2 + y^) (x* - xV + 1)' 8. Factor a»- 8 61 Solution a9 -8&^=(a3)3 -(26)3 =:(a3-26)(a6 + 2a3& + 462). Factor, and test each result : 9. o?-f. 11. a^63-27. 13. aj^/g^+l. 10. 8 7^ + s^ 12. v^-\-(jU\ 14 w^-lOOO. By applying §§ 111-113, as in § 136, any expression that can be written as the sum or the difference of the same odd powers of two numbers may be resolved into two factors. Thus, a^ + 6-5 = (a + 6) (a* - a'^b + a^-h'- - ab^ -\- 6*), and a5 - 65 = (a - 6) (a* + a^b + a^-b^ + ab^ + 64). Factor : 15. m^ 4- n^ 17. a^ + 32. 19. 1 -f s*^. 16. m^-7i*. 18. 32 -a*. 20. x^-y^^. TRINOMIALS 138. To factor a trinomial that is a perfect square. By multiplication, § 85, (a-^b)(a + b) = a'-\-2ab-hb^ Then, a^^2ab + b' = (a + b)(a + by Also, § 88, (a -6) (a - 6) = a2 _ 2 a6 + h\ Then, a^ - 2 a6 + 62 = (a - 6) (a " 6). FACTORING 106 These trinomials, o? + 2 ab 4- b^ and a^ — 2 a6 + ^S are perfect squares, for each may be separated into two equal factors. They are types, showing the form of all trinomial squares, for a and b may represent any two numbers. Hence, to factor a trinomial square, Rule. — Connect the square roots of the terms that are squares with the sign of the other term, and indicate that the result is to be taken twice as a factor. 139. Factor, and test each result : 1. x'-\-2xy-}-y\ 3. (^ -h 2 cd -h d'. 5. ar'-2a;4-l. 2. p^ — 2pq-\- q\ 4. t^ —2 tu + u^ 6. x^ -\-2x + l. 140. From the forms in the preceding discussion and exer- cises it is seen that a trinomial is a perfect square, if these two conditions are fulfilled : 1. Two terms, as -\-a^ and -\-b^, must be perfect squares. 2. The other term must be numerically equal to twice the product of the square roots of the terms that are squares. Thus, 25 a-2 - 20 xy + 4 y"^ is a perfect square, for 26x2 = (6 ic)2, 4 y* = (2 «/)2, and - 20 xy = - 2(5 «) (2 y). EXERCISES 141. Discover which of the following are perfect squares, factor such as are, and test each result : 1. a^ + 6x-\-9. 8. Sx^-\-Sxy-{-2y\ 2. 4-4a + al 9. 16p2_24p + 9. 3. 7-2- 8 r 4- 16. 10. x*-\-2xY-h?A 4. m^-mn-\-n\ 11. 9 + 42 />» + 49 &«. 5. 1+46 + 461 12. l-6m* + 9m«. 6. l-ea'-f-Oa^. 13. 4: x^y^ - 20 xy -{- 25. 7. m*-f m-fj. 14. 4 .t^ -f 12 a;]/« -h 9 .?/V. 106 ^ FACTORING 142. To factor a trinomial of the form jr^ +/?jr -f q. By multiplication, § 97, Then, Jr^ + (a 4- b)x + ab= {x -\- a)(x + 6). This trinomial consists of an a;^-term, an oj-term, and a term without X, that is, an absolute term. Therefore, it has the type form x^ -\- px -\- q. Hence, if a trinomial of this form is factorable, it may be factored by finding two factoids of q (the absolute term) such that their sum is p (the coefficient of x), and adding each factor of q to x. Thus, x--\-Sx + 15 = (x-}-8)(x -\-5), a:^-Sx-{-15 = (x-8)(x-5), x' + 2x-lo^(x~S)(x-{-5), r^-2x-lo = (x-\-3){x-5). EXERCISES 143. 1. Resolve ar' — 13 a; — 48 into two binomial factors. Solution. — The first term of each factor is evidently x. Since the product of the second terms of the two binomial factors is — 48, the second terms must have opposite signs ; and since their alge- braic sura, — 13, is negative, the negative term must be numerically- larger than the positive term. The two factors of — 48 whose sum is negative may be 1 and — 48, 2 and — 24, 3 and — 16, 4 and — 12, or 6 and — 8. Since the algebraic sum of 3 and — 16 is — 13, 3 and — 16 are the factors of — 48 sought. ... a;2_13x-48=(x + 3)(ic-16). 2. Factor 72 — m^ — m. Solution. — Arranging the trinomial according to the descending powers of wi, 72 - m2 - wi = - m2 - m + 72 Making m^ positive, = - (m^ -f- »w — 72) = -(m-8)(m + 9) = ( - wi + 8) (m + ft) = (8 -m)(9-f- m). FACTORING 107 Separate into factors, and test each result by assigning a numerical value to each letter: 3. a^-f7a; + 12. 13. x^ -{- 5 ax -{- 6 a^. 4. 2/2-72/H-12. 14. x^-6ax-\-5a\ 5. p'-S2J + 12. 15. y' -4.by- 12 b\ 6. r2 + 8r + 12. * 16. f-3ny-2Sn\ 7. 15-\-2a-a\ 17. z'-anz-2a^nK 8. b^-\-b-12. 18. -a^+25a;-100. 9. 30-?*^ + r. 19. X* -{- 19 ca:^-\- 90 c^. 10. c^-c- 72. 20. a.-^ 4- 12 aa^ + 20 a^. 11. c2-5c-14. 21. a;i«-116V + 24&*. 12. aj2-a;-110. 22. n^a:^ _ i;|^ ^^^ _,_ 3q yj 144. To factor a trinomial of the form ax^ 4- 6x -f- c. EXERCISES 1. Factor 3 a^ 4- 11 a; - 4. Solution. — If this trinomial is the product of two binomial factors, they may be found by reversing the process of multiplication illustrated in exercise 32, page 73. Since 3 x^ is the product of the Jirst terms of the binomial factors, the first terras, each containing x, are Sx and x. Since — 4 is the product of the last terms, § 78, they must have unlike signs, and the only possible last terms are 4 and —1,-4 and 1. or 2 and - 2. Hence, associating tliese pairs of factors of — 4 with 3 x and x in all possible ways, the possible binomial factors of 3 sc^ -|- 11 a: — 4 are : 3 a; + 4 1 3.r-ll 3x-4 1 3a;+11 3a--f2| 3x-21 jB - 1 j ' a! + 4 J ' a- + 1 I a; - 4 J .r - 2 j ' a; + 2 J Of these we select by trial the pair that will give + 11 a; (the middle term of the given trinomial) for the algebraic sum of the * cross-products,' that is, the second pair. .-. 3 a;2 + 11 X - 4 = (3 ar - l)(x 4- 4). 108 FACTORING By a reversal of the law of signs for multiplication and from the preceding solution it may be observed that : 1. When the sign of the last term of the trinomial is 4-, the last terms of the factors must be both -f- or both — , and like the sign of the middle term of the trinomial. 2. When the sign of the last term of the trinomial is — , the sign of the last term of one factor must be +, and of the other — . Factor, and test each result : 2. 2?/2 + 3?/ + l. 10. 2x^-{-x-15. 3. 5x^-\-9x-2. 11. 5a^-\-lSx-\-6. 4. Sx'-Tx-G. 12. 3a^-17a; + 10. 6. 4r2_^3y._|_3 13 ex^-nx-S5. 6. 6x^-7x-{-2. 14. 15x;^ + 17x-4:. 7. 2a^-5x-12. 15. Wx'-Ux-S. 8. lOt^' + t-S. 16. 2x'-\-3xy-2y\ 9. en^-lSn + Cy. 17. 3x^-10xy -\-3y\ When the coefficient of a:^ is a square, and when the square root of the coefficient of x^ is exactly contained in the coeffi- cient of X, the trinomial may be factored as follows : 18. Factor 9 a;2 + 30 a; + 16. Solution = (3a:)2+10(.3a:)+16 = (3a; + 2)(3x + 8). Separate into factors, and test each result : 19. 9a^-9a; + 2. 25. 4.9x^-i2x-55. . 20. 4.2(^-4.x-W. 26. 25a^4-25x-24 21. 9a^-42a; + 40. 27. l(ja^-S2x-hl5. 22. 25 a^ + 15 a; -f- 2. 28. 64 ar^-~ 32 a? -77. 23. 16a:2 + 16a; + 3. 29. 100a^ + 40x + 3. 24. sear' -36 a: + 5. 30. 81 a:^- 108 a: +35. FACTORING 109 POLYNOMIALS 145. To factor a polynomial whose terms may be grouped to show a common polynomial factor. EXERCISES 1. Factor ax -\- ay -{- bx -{- by. Solution ax + ay-h bx + by = (ax + ay) + (bx + by) = a(x -^ y) -{- b(x -\- y) = (a + 6)(x + i/). 2. Factor ax -\- by — ay — bx. Solution ax + by — ay — bx=:ax— ay — bx + by § 117, Prin. 2, = (aa; - ay) - {bx - by) = a{x-y)-h{x-y) = (a-b)(x-y). Remark. — The given polynomial must be arranged and grouped in such a way that after the monomial factor is removed from each group the polynomial factors in all the groups will be alike in every respect. Factor, and test each result, especially for signs : 3. am — an 4- fnx — nx. 14. ar — rs — ab-\- bs. 4. be — bd -^ ex — dx. 15. ar^-j-x^-f a; -}- 1. 5. pq — px — rq -\- rx. 16. y' + y* — 3 ?/ — 3. 6. ay — by — ab -\- bK 17. xr^ + x"^ -{- a^y + y. 1. a^ — xy — 5x-{-5y. IS. 2 — n^—2n-hn\ 8. b^ — bc-\-ab — ac. 19. ax — x — a + x^. 9. x^-^xy-ax-ay. 20. 12a' -Sb - Sa^ + 2ab, 10. c^ — 4c + ac — 4a. 21. Sax-\-6ay — 4:bx — 3by. 11. l-m-\-n-mn. 22. 3a^-15x-{-10y -2x^y. 12. 2x-y + 4:Qi^-2xy. 23. 3r2^- 9 r^^^ar-Sa^ 13. 2p-{-q + ()p^-{-8pq. 24. ax— a — bx-\-b-'CX-\-c. 110 FACTORING 146. To factor by the factor theorem. Zero multiplied by any number is equal to 0. , Conversely, if a product is equal to zero, at least one of the factors must be or a number equal to 0. If 5 flj = 0, since 5 is not equal to 0, x must equal 0. If 5 (x — 3) = 0, since 5 is not equal to 0, x must have such a value as to make ic — 3 equal to ; that is, a; = 3. If 5 (a; — 3), or 5 a? — 15, or any other polynomial in x reduces to when a; = 3, a; — 3 is a factor of the polynomial. Sometimes a polynomial in x reduces to for more than one value of X. For example, x^ — 5 a; -f- 6 equals when aj = 3 and also when aj = 2 ; or when a; — 3 = and a; — 2 = 0. In this case both a; — 3 and a; — 2 are factors of the polynomial. ... «2_5^_^6=(a;-3)(a;-2). / 147. Factor Theorem. — If a polynomial iyi x, having positive nntegral exponents, reduces to zero when r is substituted for x, the polynomial is exactly divisible by x — r. XThe letter r represents any number that may be substituted for x. EXERCISES 148. 1. Factor ar^-a;2- 4 aj + 4. Solution. — When x = 1, a;^ — x^ — 4a-. -f-4 = l — 1 — 4 + 4 = 0. Therefore, a: — 1 is a factor of the given polynomial. Dividing x* — a;^ — 4 x + 4 by x — 1, the quotient is found to be x'^ — 4. By § 134, a;2 _ 4 ^(a; 4- 2)(x - 2). ... yfi - x^ - ix + i =(ix - l)(x + 2)(ix- 2). Suggestions. — 1. Only factors of the absolute term of the polynomial need be substituted for x in seeking factors of the polynomial of the form x - r, for if X - r is one factor, the absolute term of the polynomial is the product of r and the absolute term of the other factor. 2. In substituting the factors of the absolute term, try them in order beginning with the numerically smallest. 2. Factora53-|-a;2-9a;-9. Suggestion. — When x = 1, x^ + x'^ — 9 x — 9 = — 16. Therefore, x — 1 is not a factor of the given polynomial. Whenx=-1, x^ + x2-9x - 9 = 0. Therefore, x— (-1), or x4- 1, is a factor of the given polynomial. FACTORING 111 Factor by the factor theorem : 3. 13a^-6x-S. 10. a^- 19 a; + 30. 4. a^-jx-^6. 11. .x-3-67a;-126. 5. x'-9x--\-23x-W. 12. m^ + 7m2+2m-40. 6. aj3-4a^-7 aj+lO. 13. x^ - 25 x' -\- 60 x - 36. 7. 0^-6x^-9 x + U. 14. a;* + 13 ar^ — 54 a; -h 40. 8. a^-lla^+31a;-21. 15. ;r* -f- 22 ar' 4- 27 a; - 50. 9. aj3- 10^2 ^29 a; -20. 16. a;^-9a;3+ 21 a^+x- 30. 17; Factor2a:3 4-a^i/-5a;?/2 + 2/. Suggestion. — When x = y, 2 3fi-\- x^y-bxy'^ + 2 j/S = 2 y^ + r/3 - 5 ys + 2 y^ = o. Therefore, a; — y is a factor of 2 x^ -\-xhf — b xy'^ + 2 y^. Factor by the factor theorem : 18. af»-13a;2/'^-|-122/3. 20. a;*-9 a^/ + 12an/»-42/*. 19. 3?-3\xy^-30f. 21. x^ -9 o^y^ - 4: xf ■\- 12 y\ MISCELLANEOUS EXERCISES 149. In the exercises under the preceding cases, except those under the factor theorem, the expressions given have been completely factored by one application of a single case, but frequently it is necessary to apply two or more cases in suc- cession or one case more than once to factor the given expres- sion completely. Monomial factors should usually first be removed, as they often disguise a familiar type form. 1. Factor a:3^3aJ2_;[Oa;. Solution By § 133, a:8 + 3a;2_i0x = a;Cx2-h3ar-10) By §142, =.r(ar + 6)(x-2). 112 FACTORING 2. Factor a;^ — 2/". Solution Writing the expression as the difference of two squares, we have, §134, =(x3 + 2,^)(x3-2^3) § 136, =(.x + y) (x2 -xy + y"^) {x - yXx^ + xy + y^). 3. Factor a^ — y\ Solution By successive applications of § 134, a^-y»=(x'i + yi){Xi-y*) = ix^ + y')(x^ + y^)(x +y){x- y). Factor completely, and test each result : 4. x^-xy^, 6. a^-h\ 8. w;*-81. 10. ^ -1. 5. m — 7n*. 7. a^— 1. 9. 3^ — 3^. 11. 5c^—5. 12. 4a-4a2 + a^ 16. ?i^ + 2 71 -3 w^- 6. 13. 2x^-^r)xy + 2y\ 17. 7^^ + 2 r' + 5 7^ -\- 10. 14. 10 .^•2- 20^+10. 18. 18a26 + 60a52-f 50 6^. 15. lla-a; — 55afl7 + 66 a;. 19. lo^ -\- uw^ -\- vw^ + uvw. 20. a;*' — /. 23. 4: ax +2 aoiy^ — 4S a. 26. 27w + n^ 21. Z^-16Z. 24. 18m=^-3m-36. 27. 64-2 a*. 22. Sr'-3s\ 25. y -\- 10 b'y-h 25 b*y. 28. m^+mV. 29. 21a2-a-10. 35. aj'-12 a^ + 41 a;- 30. 30. -a2_4a + 45. 36. 30 -m-6 m^ -\-m\ 31. 16 a:^ _f_ 20 a; -66. 37. 3 ^-^s - 9 rs^ -j. 6r - 3 6s. 32. 36a;2-48a;-20. 38. 15tP-9r-n-S5tl+21ln. 33. a^-21a;?/2+20?/3^ 39. w*-w^-7w'^ + w + 6. 34. 3a26a^-3a26a;-6a'''6. 40. m*- 15 m2 + 10m + 24. FACTORING 113 SPECIAL APPLICATIONS AND DEVICES 150. The method of factoring by grouping the terms of an expression in certain ways is very important. Polynomials may often be arranged in some one of the type forms already studied, and even many of these types themselves may be factored by grouping to show a common polynomial factor. 151. To factor a polynomial that may be grouped to form the difference of two squares. Just as, § 134, a^ -62 = (fl + 6) (a -b), so a^ - (6 + c)- = [a + (6 + c)]la - (ft -h c)] = (a + 6 H- c)(a — b — c). EXERCISES 152. Factor: 1. (a + 6)2_c2. 3. x'-(y-z)\' 5. (a-bf-i^, 2. )^-(s + ty. 4. (l-\-my-n\ 6. l-{v-\-w)\ 7. ¥siGt0TX*-(3x''-2yy-. Solution a:4_(3a;2_2?/)2 = [x-2+ (S x"^ - 2 y)][x^ - (3 x^ - 2 y)} = (x^ + Sx^ -2y)(x^ - Sx^ + 2 y) = (4x2-2y)(2y-2x-2) = 2{2x^-y)2{y-x^) = 4(2x^-y)iy-x'^). Test. — When x = 2 and y = 3, a^_ (3 a;2_ 2 yy^ = 2* - (3 • 2^ - 2 • 3)2 = 16 - (12 - 6)2 = 16 - 36 = - 20, and 4(2 x^ - y){y - x2) = 4(2 ■ 22 - 3) (3 - 22) = 4(8 - 3) (3 - 4) = - 20. Factor, and test each result : 8. 4c2-(6 + c)2. 12. 49r2-(5r-4s)2. 9. (2a-irhY-b\ 13. ?;Qz^ - {3z-l yf. 10. 9 «2 -(2^-5)2. 14. (6?()-3A')2-64A:2^ 11. 25a2-(ft4-c)-. 15. (3m4-8?0'-16m2. milne's 1st yr. alg. — 8 114 FACTORING 16. Factor a^ -|- 4 - c* - 4 a. Sl'ggestion. — The given expression contains three square terms and a term that is not a square. The latter may be the middle term of a trino- mial square. If so, it contains as factors the square roots of two of the square terms and these are the other terms of the trinomial square. Then, arranging and grouping terms, we have ^2 + 4 - c2 - 4 a = (a2 - 4 a + 4) - c2 = (a - 2)2 - c2. Note. — It will be observed that the term that is not a perfect mono- mial square furnishes a key to the grouping. Factor, and test each result : 17. w'-2ax + x'-n\ 21. c" - a"^ -b^ -2ab. 18. h^-{-2by + y^-n\ 22. b' -x" - y^ + 2xy. 19. 1 — 4 g -f 4 ^2 _ ^2 23. 4 c^ — oc^ — f — 2 xy. 20. r2-2ra + aj2-16^2 24. Oa^ _ 6a6 + 6^ _ 4^2^ 153. The principle by which the difference of two squares is factored may be extended to expressions that may be written as the difference of two squares by adding and subtracting the same monomial perfect square. EXERCISES 154. 1. Factor a^ + a^ft^ + 6^ Suggestion. — Since a^ + a'^h'^ -{■ h^ lacks -\- a'^h'^ of being a perfect square, and since the value of the polynomial will not be changed by add- ing a2&2 and also subtracting a%^^ the polynomial may be written a* + 2 am -H 6* - a262, or {a^ + 62)2 _ ^^252. 2. Factor a;< - 21 a^ + 36. Suggestion. — This expression has two square terms, but in order that it shall be a perfect trinomial square it must fulfill another condition, namely (§ 140), the other term must be numerically equal to twice the product of the square roots of the terms that are squares ; that is, the middle term must be either -t- 12 x"^ or — 12 x\ Hence, the number to be added and subtracted is either 33 a;2 or 9 a;2, but the former will not give the difference ^1 two squares, for 33 x"^ is not a perfect square ; then, 9 x2 is the number to be added and subtracted, giving a;4_ 12x2 + 36-9x2, or (x2 - 6)2 - 9x2. FACTORING 115 Separate into prime factors, and test each result : 3. x*-{- x^f- 4- 2/*- 8. a;^ + a:^ + 1. 4. a^+a'h^ + h\ 9. /i^ + n^ + 1. 5. 9x^ + 20Q^y^-J^lQy\ 10. 16 a;* + 4 it-^i/^ + ?/*. 6. 4a* + lla262^-9 6^ 11. 25 a^ - 14 a^ft^ ^ fts^ 7. 16 a*- 17 aV 4-0^. 12. 9 a* + 26 a^d^ + 25 6^ 13. Factor a* + 4. Suggestion, a^ + 4 = a* + 4 a2 ^. 4 _ 4 ^a _ (aj2 _^ 2)2 - 4 a^. Factor completely, and test each result : 14. a* + 4 h\ 16. a:^ - 16. 18. m* + 4 mn* 15. m*4-64. 17. 4 a* + 81. 19. a^y^ -\- A^ xy\ - 155. The method of factoring by grouping to show a common polynomial factor applied to cases already solved by other methods. The student has learned how to factor several type forms by special methods. He will now see how many of these forms may be factored by the method of § 145, which is of impor- tance because its application is so general. EXERCISES 156. Tlie forms a^-\-2ah + h^ and a?-2ab + b\ 1. Factor ar* + 2 a?!/ 4- y^ ; also 9 a%2_ 6^^1 + 1. Solution Solution x^ + 2xy + y'^ 9 a^m^ - 6 am + I = x^-\-xy + xy + y^ =9 a^m^ - 3 am - 3 am + 1. = x(x -^ y) + y (x -\- y) = Sam(Sam- l) — {Sam— 1) = (x + y)(x-hy). =(3aw-l)(3am-l). Factor by separating and grouping ; test each result : 2. ?2 + i4;4.49. 4. 4ar' + 12a;2/4-92/*. 3. /-^-ISr-fSl. 5. 16a=^-24a6 + 96l 116 FACTORING The form a^-h\ 6. Factor x^-y^-, also 9 r^ - 4 s^. Solution = x(x-y)-[-y(x-y) = (x + y)(x^y). Solution 9r2-4s2 = 9r2 - 6 rs + 6 rs - 4 s2 = 3r(3r-2s)+2s(3r-2s) = (3r + 2s)(3r-2s). Factor by grouping ; test each result 7. a' z\ 9. n^-\, 8. x^-\. 10. 9a^-25. 2%e form oi^ -\- px •\- q. 13. Factor a;^ + 8 a; + 15 ; also a^ - 2 a; — 15. 11. 3622-25i;2, 12. 49n2-100;2. Solution = a;2 + 5ic+3a; + 15 = a:(x+5) + 3(x + 5) Solution a;2_2a;-15 = a;2_5a; + 3a;_ 15 = a;(a:-5)+3(ic-5) = (a; + 3)(x-5). Factor by separating and grouping ; test each result : 14. a^ + 12a; + 20. 16. 2/^+8?/-20. 15. ?7i2_^9m + 18. 17. ^2-57^-14. The form aa? -{- hx '\' c. 18. Factor 2 a^H- 11 aj-j- 12; also 2 a;'^ + a; - 15. Solution 2a:2+lla; + 12 = 2x2 + 8 a: + 3 a; + 12 = 2a;(a; + 4) + 3(x + 4) = (2x + 3)(x4-4). Solution 2x2 + a;-15 = 2x2 + 6ic-5x- 15 = 2a;(x+3)- 5(a; + 3) = (2ic-5)(x + 3). Factor by separating and grouping ; test each result : 19. 2ar^ + 3a; + l. 21. 3m^-{-5m-22. 20. 92/'-h2l2/ + 10. ' 22. 10 7^-3rs-lS s\ FACTORING 117 Factor by separating and grouping : 23. 12n2 + 31n + 9. 25. 20 x'-^-lSxy-lBy^. 24. 5^24.482-77. 26. Ua'-23ab-30b\ REVIEW OF FACTORING 157. Summary of Cases. — In the previous pages the student has learned to factor expressions of the following types : Monomial Factors I. Of monomials ; as a-b'c. (§ 129) II. Of expressions whose terms have a common factor ; as /7jr H" /»/ H- nz. (§ 133) ' Binomials III. Difference.of two squares ; as a--b'. (§§ 134,161,155) IV. Sum or difference of two cubes ; as a' + b'ova^-b'. (§136) V. Sum or difference of same odd powers ; as a" 4- b" or a" - 6" (when n is odd). (§ 137) Trinomials VI. That are perfect squares ; as a' + 2ab + b' and a' - 2 ab 4- b'. (§§ 138, 155) VII. Of the form x-+px-\- q. (§§ 142, 155) VIII. Of the form ax' -hbx-\-c. (§ § 144, 155) Polynomfals IX. Whose terms may be grouped to show a common poly- nomial factor ; as ax + ay-\-bx + by. (§ § 145, 155) X. Having binomial factors (Factor Theorem). (§ 146) 118 FACTORING 158. General Directions for Factoring Polynomials. — 1. Re- move monomial factors if there are any. 2. Then endeavor to bring the polynomial under some one of the cases II-IX. 3. Whe7i other methods fail, try the factor theorem. 4. Resolve into prime factors. Each factor should be divided out of the given expression as soon as found in order to simplify the discovery of the remaining factors. EXERCISES 159. Factor, and test each result: 1. 2/4-1. 8. l + aj'2 15 8-27aV. 2. 1— a^. 9. y-a^y. 16. 32« — 2a5^. 3. x^'^-1. 10. x^y-f. ^ 17. 6 6^ + 24. 4. x^-1. 11. a^^-ab^ I 18. a^ + 27al 5. a -a'. 12. 64.-2 f. 19. 6^-196. 6. 6^ + 6. 13. In' -In. 20. 450 - 2 a^. 7. j94-t-4. 14. 4ic* — 4 a;. 21. Ty^ — 175. 22. a^-xy-lS2y\ 32. a:^-ax-72a^. 23. aar^ — 3 aa; — 4 a. 33. n^ — an— 90 a^. 24. a!3-l-5a^-6a;. 34. a-b^ -{■ ab — 56. 25. 3a^ + 30a; + 27. 35. a^n _ 2 a"6« + 6^ 26. 128 a^- 250 a^. 36. 25 a;^ + 60 a;^/ + 36 /. 27. 6 a^- 19 a; + 15. 37. 6 aa^ -\- 5 axy —6 ay\ 28. a;2« + 2a;"?/P + 2/^^. 38. 169 a;^ - 26 aa;« + a V. 29. 7a;2_77a^_84 2/2. 39. a*c^ + a^b^c^ -{- b\ 30. 2/'-25 2/a; + 136a;2. 40. 16 a:^ -f 4 a^y + ^^ 31. 9:x^-24.xy-\-16y\ 41. 6*c - 13 6^c + 42 c. FACTORING 119 Factor, and test each result : 42. 9'j^-{-21x-{-10. 55. b' + by + y*. 43. 5 a^ - 26 xy -\- 5 y\ 66. xy — Sy + x — Sf. 44. 2/^-hl6a2/ — 36 al 57. ast? — axy — ax -\- ay. 45. 8a2-21a6-962. 53. 9 c2_aj2_y2_^2a^. 46. 9 a:^ — 15 ic — 50. 59. ir^ — a-a; — 4 fe^a; _ 4 a&a?. 47. 30a^-37a;-77. 60. b& - 9 a'b - b^ - 6 ab\ 48. 207^ + 28^2 _,_eg^ 61 a62_4a«-12a2c-9ac2. 49. a2^62_^_2a5. 62. a^- ca;+ 2da;-2cd. 50. aiB^-f 10 aa;- 39 a. 63. a^y + 4 a^2/ - 31 a;y - 70 y. 51. n^ + n'a'b'^a'b^* 64. ar^ - 3 aa; + 4 6a; - 12 a6. 52. aV + aV + a*. 65. aa» - 9 aar' + 26 aa; - 24 a. 53. a^"*- 16 a*" -17. 66. 12aa;-86a;-9 ay + 6 ft?/. 54. aV-4aa;4-3. 67. 25 af- 9 2^-24^2-16 22. 68. 2bH-^ab^ + 2btx-^dbx, 69. a:^y-f-14ar'y + 43ary-f 3O2/. 70. arV - 15 a;22^ + 38^2/ -24 2/. 71. ab3i?-\-^(iboi? — abx — ^ab. 72. 3 bmx + 2 6m — 3 anx —2 an. 73. 20 aa;3 _ 28 aa;2 4- 5 a^a;- 7 a^. 74. (a + 6)3 — 1. 79. a:^ — a.-2 — a;* + a;^. 75. a^ — 2 a^-^-l. 80. a^ — xy — a^y -\- y^. 76. 6^-462 + 8. 81. 12 a;3^3 aj2_8^_2. 77. 3a;^ + 96a;. 82. 2 a;^ _j_ ^Lq ^ _j. ^a, _,_ 5 ^ 78. 8a;*-6a;2-36. 83. m^ -\- m^ - wm - mn^. 84. Factor 16 4- 5 a; — 11 o^ by the factor theorem. 85. Factor a;^ — 6 6ar + 12 6^a; — 8 6^ by the factor theorem. 120 FACTORING EQUATIONS SOLVED BY FACTORING 160. Equations thus far solved have been such as involved, when in simplest form, only the first power of the unknown number. Such equations are called simple equations. 161. A valuable application of factoring is found in the solution of equations that involve the unknown number in powers higher than the first. An equation that in simplest form involves the second, but no higher, power of the unknown number is called a quadratic equation. Thus, x2 = 4 and x'^ + 2 a; + 1 = are quadratic equations ; but x^-\-^x = ic2 + 4 is a simple equation, for in its simplest form, 3 x = 4, it has only the first power of x. 162. An equation that contains a higher power of the unknown number than the second is called a higher equation. 163. To solve quadratic equations by factoring. EXERCISES 1. Find the values of x that satisfy a^ + 1 = 10. Solution X2 + 1 = 10. (1) Transposing so that all terms are in the first member and uniting terms, a;2 _ 9 = 0. (2) Factoring the first member, § 134, (x-3)(x + 3) = 0. (3) Since the product of the two factors is 0, one of them must equal ; that is, the equation is satisfied for any value of x that will make either factor equal to @. Ifx-3 = 0, x = 3; ifx + 3 = 0, a;=-3. Hence, the values of x that satisfy (3) and therefore (1) are 3 or — 3. Verification. — When x = 3, (1) becomes 9 + 1 = 10, or 10 = 10. When x = - 3, (1) becomes 9 + 1 = 10, or 10 = 10. FACTORING 121 Solve, and verify results : 2. i»2 4-3 = 28. 6. or' 4-3 = 84. 3. a;^4-l = 50. 7. ar^-24 = 120. 4. x'-5 = 59. 8. a:2 + ii=,i80. 5. a:2_7^29. 9. a:2_ii==iio. 10. Solve 07^ 4- 4 a; = 45. Solution Transposing, x=* -H 4 a; — 45 = 0. Factoring, § 142, (x - 5) (x + 9) = 0. Hence, x— 5=:0orx-}-9 = 0; whence, x = 6 or — 9. Solve, and verify results : 11. a^-6x = ^0. 16. 2/2 4-42 = 132/. 12. a^-8a; = 48. 17. t^-\-63 = mt 13. ar' - 5 a; = — 4. 18. v^ — 60 = 11 v. 14. a;2 4-4a;4-3 = 0. 19. ar^-7a; = 18. 15. r2 4-6r4-8 = 0. 20. a?^ 4- 10 a; = 56. 21. Solve 6ar^4-5a;-21=0. Solution 6x-2 + 6x-21 = 0. Factoring, § 144, (2 x - 3) (3 x 4- 7) = 0. Hence, 2x-3=0 or 3x4-7=0; whence, x = | or — J. Solve, and verify results : 22. 3a;^4-2a;-l=0. 27. 2v^-9r-3^ = 0. 23. 5a^4-4a;-l=0. 28. 6y^ - 22 y -^20 = 0. 24. Sf-\-y-10 = 0. 29. 622_ll2_21 = 0. 25. 7a:2 4.6«-l=0. 30. 4ar^- 15a;4- 14 = 0. 26. 23^*4- 9a;-18 = 0. 31. 53^^-483;-- 20 = 0. 122 FACTORING Solve, and verify results : 32. a.-2-21 = 4. 41. 32-ar^ = 28. 33. i»2-56 = 8. - 42. 65-0^=16. 34. a^-9aj = 36. 43. a.-^ -f- a- - 132 = 0. 35. a?-{-llx = 26. 44. S2 = 4:W-{-w\ 36. a^-12a; = 45. 45. 3s = 88-s2. 37. 2/2_i52/ = 54. 46. 160 = a^-6a;. 38. /-2l2/ = 46. 47. 42/ = /-192. 39. 3/-42/-4 = 0. 48. 3a^ + 13a-30 = 0. 40. 42/' + 92/-9 = 0. 49. 4a^ + 13a;- 12 =0. 50. (2» + 3)(2a;-5)-(3a;-l)(a;-2) = l. 51. (2a;-6)(3aj-2)-(5aj-9)(a;-2) = 4. Other methods of solving quadratics will be given in §§ 336-348. 164. To solve higher equations by factoring. Any higher equation may be solved by the method just given for quadratic equations, whenever the expression ob- tained by transposing all of its terms to one member is factorable. EXERCISES 165. 1. Solve the equation a^ — 2a^ — 5a; + 6 = 0. Solution Tactormg, § 146, (x -l)(x- 3) (a; + 2) = 0. Hence, a:-l = 0ora;-3=0oric + 2 = 0; whence, aj = 1 or 3 or — 2. 2. x^-15x'-{-71x-105 = 0. 4. a^-12x-\-16 = 0. 3. iB3 4-10aj24-ll«-70 = 0. 5. a^- 19aj-30 = 0. 6. a;*-|-a^-21a^-a; + 20 = 0. 7. a;*-7a^-f a;24-63a;-90 = 0. 8. x^-10x^ + 35x'-50x + 24. = 0. FACTORING 123 Problems 166. 1. A sealing fleet carries 4000 men, and the number of men on each ship is 40 less than 8 times the number of ships. Find the number of ships and the number of men on each ship. Solution Let X = the number of ships in the fleet. Then, 8 a; — 40 = the number of men on each ship, and x(S x — 40)= the total number of men with the fleet. .-. a;(8 X - 40) = 4000. Expanding, dividing by 8, and transposing, a;2 _ 5 a; _ .500 = 0. Factoring, (x - 26) (x + 20) = 0. Hence, x - 25 = or x -|- 20 = ; whence, x = 26 or — 20, and 8 X - 40 = 160 or - 200. The second value of x and of 8 x — 40 is evidently inadmissible, since neither the number of ships nor the number of men on a ship can be negative. Hence, there are 26 ships in the fleet, and 160 men on a ship. Solve the following problems and verify (§ 125) each solution : 2. The gold mined in a recent year would fill a square room, the height of which is 1 foot less than its length. If the area of one wall is 90 square feet, find the dimensions of the room. 3. Certain wooden paving blocks are twice as long as they are wide and the thickness of each is 4 inches. Find the length and width, if the volume of each block is 128 cubic inches. 4. A rectangular swimming tank on board a ship is 3 times as long as it is wide. If it were divided into 3 square tanks, the area of each would be 225 square feet. Find the dimen- sions of the tank. 5. A man bought as many tons of crude borax as it is worth dollars a ton, and crude borax is worth -f as much as refined borax. If the same amount of refined borax would be worth {$ 2800, find the value of crude borax a ton. 124 FACTORING 6. A farmer keeps his chickens in a rectangular lot that is 4 times as long as it is wide. If its area is 2500 square feet, find its length and width. 7. At a luncheon with Menelik of Abyssinia there was a pile of bread containing 448 cubic feet. Its height was twice its width and its length was 14 feet. Find its height and width. 8. A large rectangular freight station in Atlanta, Georgia, covers an area of 41,750 square feet. If the length is 16.7 times the width, what is the width of the station ? 9. An automobilist paid $ 3.60 for gasoline. If the num- ber of cents he paid per gallon was 2 less than the number of gallons he bought, find how many gallons he bought and the price per gallon. 10. St. Louis has the largest steam whistle in the world. The number of times it is blown each day is. 3 more than the number of dollars it costs to blow it once. How many times is it blown a day, if the cost for 12 days is $48 ? 11. In the Panama Canal Zone a washerwoman washed as . many dozen pieces as she received dollars a dozen for her labor. If she had washed 2 dozen more, she would have received $ 15. How much did she receive a dozen ? 12. The number of pounds of duck feathers that a man bought was the ^ame as the number of cents that he paid a pound for them. If he had bought 10 pounds more, they would have cost $ 20. How many pounds did he buy ? 13. The area of one of the largest photographic prints ever made is 180 square feet. Its dimensions are 18 times those of the picture from which it was enlarged. Find the dimensions of the picture, if its length is 2 inches greater than its width. 14. The cages holding canaries imported into this country are arranged in rows in crates. - The number of rows in a crate is 2 less than 5 times the number of cages in a row. If there are 231 cages in a crate, how many rows are there ? FRACTIONS 167. lu algebra, an indicated division is called a fraction. The fraction - means a -i- b and is read ' a divided by b.' h 168. The dividend, written above a line, is the numerator; the divisor, written below the line, is the denominator ; the nu- merator and denominator are called the terms of the fraction. An arithmetical fraction also indicates division, but the arithmetical notion is that a fraction is one or more of the equal parts of a unit ; that is, in arithmetic, the terms of a fraction are positive integers, while in algebra they may be any numbers whatever. 169. The student will find no difficulty with algebraic frac- tions, if he will bear in mind that they are essentially the same as the fractions he has met in arithmetic. He will have occa- sion to change fractions to higher or lower terms ; to write in- tegral and mixed expressions in fractional form ; to change fractions to integers or mixed numbers ; to add, subtract, mul- tiply, and divide with algebraic fractions just as he has learned to do with arithmetical fractions, except that signs must be considered in dealing with positive and negative numbers. Signs in Fractions 170. There are three signs to be considered in connection with a fraction; namely, the sign of the numerator, the sign of the denominator, and the sign written before the dividing line, called the sign of the fraction. In — ^ the sign of the fraction is — , while the signs of its terms are -h . oz 126 126 FRACTIONS 171. An expression like — — indicates a process in division, — b in which the quotient is to be found by dividing a by 6 and prefixing the sign according to the law of signs in division ; that is, -a_ a + a a + 6 "^6 — a a + b~ ,6' + a_ a -b b By observing the above fractions and their values the follow- ing principles may be deduced : 172. Principles. — 1. TJie signs of both the numerator and the denominator of a fraction may be changed without changing the sign of the fraction. 2. The sign of either the 7iumerator or the denominator of a fraction may be changed, provided the sign of the fraction is changed. When either the numerator or the denominator is a polynomial, its sign is changed by changing the sign of each of its terms. Thus, the sign of a — 6 is changed by writing it — a + 6, or & — a. EXERCISES 173. Keduce to fractions having positive numbers in both terms : x 1 ZL?. *^. ±JlLi. 5 (^a-b ^ 7 -2-m -4* '- 2x ^,f;s' c + d ' ' 2 + n 2. A.. 4. -^' . 6. -■ -^ . S,. -Ha + b) ^ -5 -b-y p-a-y ^r^Mj-x-fy) 174. In accordance with § 80, " / K f f^— / -J^ y Principles. — 3. The sign of either term of a fra^^i^H^' changed by changing the signs of an odd number of its factors. 4. The sign of either term of a fraction is not changed by changing the signs of an even 7iumber of its factor's. ^^ ^ _ / FRACTIONS 127 EXERCISES 175. 1. Show that (° - '') (f -' 1 = (" - ^K« - '^ . (c — a) (6 — c) (a — c)(6 — c) SoLUTiox OR Proof Changing (d — c) to (c — d) changes the sign of one factor of the nu- merator and therefore changes the sign of the numerator (Prin. 3). Similarly, changing (c — a) to (a — c) changes the sign of the denomi- nator (Prin. 3) . We have changed the signs of both terms of the fraction. Therefore, the sign of the fraction is not affected (Prin. 1). 2. Show that {b-a)(d-c) ^ _~ia-b){c-d) . (c-b)(a-c) (h-c)(a-c) Solution or Proof Changing the signs of two factors of the numerator does not change the sign of the numerator (Prin. 4). Changing the sign of one factor of the denominator changes the sign of the denominator (Prin. 3). Since we have changed the sign of only one term of the fraction, we must change the sign of the fraction (Prin. 2). 3. Show that -^ — may be properly changed to h—a a—b 4. From derive by proper steps. 6-a + c a-b-c ■ ^ ^ 5. Prove that — ^= —: that - ^ ^ 6. Prove that 1 —X x — 1 4 — ar^ ar^ — 4 (b-a)(c-b) (a-b)(b-c)' 7. Prove that (m-n)(m4-n)^ -m^ + n^ . (a — c)(b — a) (a — c)(a — b) 8. Prove that {a-b)(b-aj-_c)_ ^ (a-b)(a-b-c) , (y-x)(z-y)(z-x) (x-y)(y-z)(x-z) 128 FRACTIONS REDUCTION OF FRACTIONS 176. The process of changing the form of an expression without changing its value is called reduction. 177. An expression, some of whose terms are integral and some fractional, is called a mixed number, or a mixed expression. n h X^ Cfi 1 ' Thus, a , 2 H — , and a — h -\ are mixed expressions. c a2 a;2 ab 178. To reduce a fraction to an integer or a mixed expression. This change in form is made in algebra in the same manner as in arithmetic. EXERCISES 179. 1. Reduce to a mixed number : -; ^^i±^. 4 X PROCESS PROCESS 15 = 13^4 = 3 + - = 3i. ^^^±^ = (aa.4-6)-^aj = a+-- 4 4 . a; x Explanation. — Since a fraction is an indicated division, by perform- ing the division indicated the fraction is changed into the form of a mixed number. Reduce to an integral or a mixed expression : 9 ' \hxy a 136* ' 9c * * 6a; 8. Reduce ^^^^ — a —a-\- ^^ ^ mixed number. Solution. «^-3a^-a + l ^ « _ 3 4. - «^ + 1 = ^ _ 3 _ o^li . Note. — It is not necessary to write the step («8 — 3 a^ — a + 1) -^ a^. The division should be continued until the undivided part of the numera- tor no longer contains the denominator. a + 3 4 a^ 4-12 a3_a^4-34 2a3 + 5 a2 + 3a& -5&2-f6c a- -26 ^-Ix"- - 4 a; -H 40 FRACTIONS 129 Reduce to an integral or a mixed expression : ^ 4ar'-8ic2-f2a;-l ,_ a^ + O a^^ 24 a -f 18 9. — - — — • ly. 2a; 10 CLb — bc — cd + d^ 2^ 6 , , a V — aa;^ — a; -- 1 oi 11. ^* /6l. ax 12 ^ZI^nl?. 22. a;-4 x^-3 ' ^ x^-2xy-f oo «' + 3a^6 -a5^ + a6 lo. = • Ao. ; • x — y a^ + h 2xy ' x-\-y ,, a^-6x'-\-Ux-9 ^^ ar^-3ar-^'*4-5 af*-3 15. • *o. • x-2 a~-l '^- — ^33 ^^- — r=^7 ,^ a;3_^5^2_^33._g :x?-\-x'y-y''-xf-Z 17. — • -o/. • a; -f 2 x — y .T^ + 2a:^-a.-^-f5 3 a;y-12 a:^4- 2 y^- 3 ar' + ar^ * " 3x^ + 2 180. To reduce a fraction to its lowest terms. T 4. 3 9 9 3 Just as- 4 = 1^°'^ 12 = 4' ■ SO - = — or — = - • 1 hat is, h hm bm b 181. Principle. — Multiplying or dividing both terms of a frac- tion by the same number does not change the value of the fraction. 182. A fraction is in its lowest terms when its terms have no common factor. MILKENS IST YR. ALG. — 9 130 FRACTIONS 183. The product of all the common prime factors of two or more numbers is called their highest common factor (h. c. 1). Thus, to find the h.c.f. of two expressions, as 9 ax^y -^ S bx^y Siud 6 x*y^^6 xV» we first factor them. Since 9 ax^y-\-Sbx^y=S ■ x^ • y (^Sa-{-h) and 6 x:^y^ — 6 a;V = ^ - ^ ■ ^'^ - y^ (x + y){x — y), their common factors are 3, x, x, and y ; hence, their h. c. f. = 3 x'^y. (See also §§ 407-411.) Note. — The number of literal factors in a term determines its degree, and the term of an expression that has the greatest number of literal factors determines the degree of the expression. Thus, 2 a^h is of the third degree and is higher than 5 ah, which is of the second degree; the expres- sion 2 ^25 _|. 5 qJ)^ then, is of the third degree. EXERCISES 184. 1. Reduce to lowest terms : — ; ?L^^. 24' SOa^xz ^ PROCESS 20 2.2.5 5 24 2.2.6 6 PROCESS 21 g Vy ^ 3-7 a Vy ^ 7 xy 3(}a?xz ^' 10 a^xz 10 az Explanation. — Since a fraction is in its lowest terms when its terms have no common factor, a fraction may be reduced to its lowest terms by- removing in succession all common factors of its numerator and denomi- nator ; or by dividing the terms by their highest common factor. Reduce to lowest terms 2 ^^ awrrC h^xy^ 6. 16 m^nxh^ 40 am^y^ 210 h(?d 750 ab'c' -25affz'' - 100 xy ' 8. 9. 10. - 7 a'bcd^ 42 ab^cd' ' St 2x 4:X^ — 6 ax 11. Reduce -— to its lowest terms. b' Solution. § 174, Prin. 4, bx — ax _ x{b — g) a2-62 ~ {a + b){a-b) _ —x(a — b) {a + b){a—b) a + b X a + b FRACTIONS 131 Reduce to lowest terms : 12 ^^-^' 23 ^-^^^-^^ • d'-{-2ab-^b'' ' 2x^-2 -ix+e 1 o ^^1 ^^ > ^ . 24 4a^-9ar^ 05 20-21 a^ + a;' ' 8a3 + 27a^' " a;^-26a^ + 25' J 6xy-3a^y 2^ 3 a^H-4 aa;-4 a^ * x^-Sxy' ' 9a^-12ax-h4:x'' Sa'b-Sb' 2^ ar^ + l + Sar' + Sa; * 2b*-2a'b' . ' 4:-h4:X-x^-Q^ 17 ^V-f^y±Y. 28. ^ — ^^^ — ^^ + ^^ . ^,6 _ ^ ^jj _ ^^ ^ ^2 -_ ^^ 18. A — ^f + f gg am — a?t — m + yt a;* + 2/* a?n — a?i4-m — n 19 .^^16^1+5^. 3Q a3-6«-3a^6 + 3a&^ ' a^ + 2ic=^-35a;* * 3a62-3a26- ^^ 7a;-2a^-3 ,, a:» + 5 x2-9a;-45 4-7a;-2a;2 a:3^3^_25a;_75 2^ ^2^-2^*j-^ ^ l-I/V ,> gg 2aa;-av-4 6a;4-2&v x^ — x^ " * 4: ax — 2 ay — 2 bx -^ by ^^ a^-^2a'b + ab'' ,^ a^ + 2a^-23 a;-60 a' -2 a^b^-^ab* ar^ - 11 a^ - 10 a; + 200 185. To reduce a fraction to an equal fraction having a given denominator or a given numerator. In order to change f to a fraction whose denominator is 12, both terms must be multiplied by 12 ^ 4, or 3 ; similarly, to change - to a fraction whose denominator is ns^, both terms z must be multiplied by nz^-i-z, or 7)z. 132 FRACTIONS EXERCISES 186. 1. Eeduce ■ ^ to a fraction whose denominator is a + b PROCESS (a^-b") -^(a-}-b) = a-b. Then g ^ a(a-b) ^ a'-ab ' a-^b (a-^b)(a-b) d'-b^' Explanation. — Since the required denominator is (a — h) times the given denominator, in order that the value of the fraction shall not be changed (§ 181) both terms of the fraction must be multiplied by (a — &). 5 a 2. Eeduce — to a fraction whose denominator is 42. 6 3 X 3. Eeduce to a fraction whose denominator is 55 b. 11 b 4. Eeduce to a fraction whose denominator is 84 a;^. 14 a; 4 a^ 5. E-educe to a fraction whose denominator is 20 'tf. 5y 6. Eeduce ~' to a fraction whose denominator is («— 1)1 a; — 1 2x—5 7. Eeduce — to a fraction whose denominator is (2 a;H-5)^. 2x-{-5 8. Eeduce ^ to a fraction whose numerator is 3 a -j- a-, S — a 9. Eeduce ^~-^ to a fraction whose numerator is a;^ — /. 2x-i-y 10. Eeduce ^^ — to a fraction whose denominator is 4 — x^. a; — 2 11. Eeduce ^ to a fraction whose denominator is ?>- — 9. 3-5 12. Eeduce a;— 5 to a fraction whose denominator is a;-f-5. 13. Eeduce 3 ^+2 Uo a fraction whose denominatQi: is 2 Z— .3 1. FRACTIONS 133 187. Reduction to lowest common denominator. In algebra, as in arithmetic, it is frequently desirable to reduce fractions that have different denominators to respec- tively equal fractions that have a common denominator. 188. In algebra, lowest common denominator corresponds to least common denominator in arithmetic. The word 'lowest' has reference to the degree of the denominator. 189. It is not always easy to discover by inspection the lowest common denominator (1. c. d.), that is, the lowest common multiple (1. c. m.) of the given denominators. However, it may be found, as in arithmetic, by factoring the denominators, for it is the product of all their different prime factors, each factor used the greatest number of times that it occurs in any denominator. (See also §§ 412-414.) Thus, if the given denominators are ax — hx, a^— 6*, and a* — 2 a6 f 6^, on factoring we find : ax—hx = x{a — h)', a"^ — h^ z= (jot, -\- 6) {a — h)\ and a2-2a6-H62 = (a-6)(rt-6). Then, the factors of the 1. c. d. are or, a + 6, a — 6, and a — h. Hence, the 1. c. d. = a5(a 4- h) (a - by. 190. 1. Keduce to fractions having their lowest common denominator: - and -; -^ and — ^• 6 8' 3 6c 6 a6 PROCESS PROCESS l.c.d. = 24 l.c.d.=6a6c 5 = 5.x4^20 j>_^^x2a ^ 2a^ 6 6x4 24 She 3bcx2a (5 abc 3^3x3^^ _c_^_±xc ^ c^ 8 8 X 3 24 6ab (yab xc 6abc Explanation. — The 1. c. m. of the given denominators is found for the 1. c. d. in accordance with § 180. Then, each fraction is reduced to an equal fraction having this denominator, as in § 185. Note. — All fractions should first be reduced to lowest terms. 134 FRACTIONS 2. Reduce 2 m and ^"^^ to fractions having their lowest m— n common denominator. Suggestion. — First write 2 m as a fraction with the denominator 1. Reduce to fractions having their lowest common denomi- nator : 8. J_, Zl^ J_. 9 ^ ]jL ^. 8 d-c 4 Wc a?hc^ 3. ^"<^¥• 4. %^ and 3 X. 5 5. 6. -5La„d-5_. 2 icy 4 a?/ 7. iMand^S? cx 3 62/ 13. -.^„,^^, 10. m — nf) a ? ^> ■ • a m-\-n 11. x + y x-y x'-y' 2 ' 4 ' 6 12. X^ X X a^-1' x + l'x-l 2a a* -16' a2a.4'4_^2 2a -2a Suggestion. — By § 172, Prin. 1, 14. 15. 16. 17. 4-a^ a2 - 4 4a Sb 1 b— a a-\-b o? — b^ a X —ax 1 — ax' 1 + ax' ax — 1 1 1 Q^j^l x + l(^' x^-\-x-2' x' + ^x-b 3a; x-1 0^ + 3 a:2_3^_^2' x^-^x-^6' a:^-4x + 3 a-f-5 a — 2 a + 1 a2-4a-+-3' a^-Sa + lS' a^-Ga + S \ FRACTIONS 135 ADDITION AND SUBTRACTION OF FRACTIONS 191. The method of adding and subtracting fractions is the same in algebra as in arithmetic. In algebra, however, sub- traction of fractions practically reduces to addition of fractions, for every fraction to be subtracted is really added with its sign changed (§ 64, Prin.). Justas l + ^ = -i + -?- = i±^, 3^4 12^12 12 ' a c ad . be ad -\- be so — -|- — — — — -f- — ^ • b d bd bd bd 1_1^±_ 3 ^ 4-3 3 4 12 12 12 ' a c _ad be __ ad — be b d bd bd bd EXERCISES 192. 1. Add ^,1^, and g. Solution. — Since the fractions have unlike denominators, they must be reduced to fractions having a common denominator. By § 189, the 1. c. d. = 60. 3a; 7x . 5g_45g 42a; 25g 4 10 12 60 60 60 Also as so _ 46a; + 42a; + 2og _ 87x + 26g 2. Subtract ^::i^ from 5^^+-- 7 8 4 Solution 5x-l a; a;- 2 _ 35a; -7 14a; 8a;-16 8 4 7 66 66 66 ^ 35a;-7 + 14a;-(8a-- 16) 56 _ 36a;-7 + 14a;-8a; + 16 _ 41a; + 9 66 66 Suggestion. — When a fraction is preceded by the sign — , it is well for the beginner to inclose the numerator in a parenthesis, if it is a poly- nomial, as shown above. 136 FRACTIONS Rule. — Reduce the fractions to similar fractions having their lowest common denominator. Change the signs of all the terms of the numerators of fractions preceded by the sign — , then find the sum of all the numerators, and write it over the common denominator. Reduce the resulting fraction to its lowest terms, if necessary. Add : Subtract : _ bm r 4m 7. — — from 6 3 8. -— from — -. 9 2 9. -^from ?. 3 y 10. ^±±i,om^^. 3. 2; and ^^. 5 2 ■ 4. 4a , 6b 3 ""S • 5. 2a ^ , 3a 36 ""S6- 6. -'and -2 7x Sx Simplify : 11 2a; + l a; — 2 a; — 3 . 5-x ,- a;-2 x-4: , 2-3a; 2a; + l ^^' ""6 9""^"^ 12~' a; — 1 a; — 2 4a; — 3 ■ 1 — a; ^^' ~3~ 18 27 "^ 6 * 2 — 6a; , 4a; — 1 5a; — 3 1 — a; 14 . 5 2 6 3 ,^ a;4-3 x-2 , a;-4 a; + 3 '^- ~4 5~ + l0 6~* ,^ l-2a , 2a-l 2a-a^-\-l 3 + a;-a;^ l-a; + a;^ l-2a;-2a;^ 4 6 3 FRACTIONS 137 5 a^ 4- 52 18. Reduce „ ,„ — 2 to a fraction. a^—b^ Solution 6 gg 4. ?)2 2 ^ 5 g^ + &^ - 2(q2 - &2) a:i--b-^ 1 g-^-6-2 _ 5 gg + ft2 _ 2 g2 + 2 &2 g2 - 6-^ Reduce the following mixed expressions to fractions : 19. a -I- 20. X- -V. 2 21. .b-c 33 l^i^^^_2. X l-\-x Sa'-4a^ ab be 28. a-\-b a — b a—b a+6 29. x-y 30. x x-2 x-2 x-\-2 31. x—1 32. m f-n, 34. 3 a - 2 a; 35. 3a-\-2x 1 1 2 a: — 1 a; -f 1 x- 36. -t i-+ 2a a + 6 a — 6 a'^ — b- o — a; a-{-x a^ — xr m — n ax 38. 3.'B + -^-(^2a;4-— \ V - <^7 138 FRACTIONS 39. -^ ^^^=^4- ^ 2 a + 2 4-a2 Suggestion. — By § 172, Prin. 1, — — 40. ^±1+-^+ 4a a— 1 a+1 1 41. i::U[L^ + a;2_4 ^ x-2 2-x 42 ^(a+^_3a^^-^^4^^ 43. a — x X— a a a 2ab 4a6^ a — b a-\-b a^ + b^ a*-\-b* Suggestion. — Combine the first two fractions, then the result and the third fraction, then this result and the fourth fraction. .. a-{-b a — b 4 a6 , Sab^ 44. — ■ -f- a-b a + 6 a' + b^ a'-\-b* ^^ 1 1 26 , 2b' 45. — - o ■ ,o + a-b a + 6 a' + b' a*-\-b* 4g a^ + y y + g J g-ha? (y-z)(z-x) (x-z)(x-y) {y-x){z-y) Solution Sum= '^±y + y + ^ + ?-±^ {y-z){z-x) (z-x){x-y) {x-y){y'-z) = ^- = 0. ix-y)iy -z)iz-^) (6-c)(a-c) (c-a)(a-6) (6-a)(6-c) 48. ^+i + ^-±1 + " + 1 - (a — b){a — c) (6 — c) (6 — a) (a — c)(b — c) FRACTIONS 189 MULTIPLICATION OF FRACTIONS 193. Fractions are multiplied in algebra just as they are in arithmetic. Thus, 7X0= :; — n ' ' 4 2 4x2 In general, 1X3 = 7^- That is, a oa Principle. — The product of two or more fractions is equal to the product of their numerators divided by the product of their denominators. EXERCISES X 194. 1. Multiply -^-^ by a^-25. x-\-o Solution r2^-^ = (x-6)2 = a;2-10a;+25. 2. Multiply ?^ by 1 + x-\-2 x-\-l Solution \x + 2)\ x + \) x + 2\x + l x + lj ^x + Z Z"^ jM^. x + 1 x + l' General Suggestions. — 1. Any integer may be written with the denominator 1. 2. After finding the product of the numerators and the product of the denominators the resulting fraction may be reduced to lowest terms, in many cases, by canceling common factors from numerator and denomi- nator. It is, however, more convenient to remove the common factors before performing the multiplications. 3. Generally, mixed numbers should be reduced to fractions. 140 FRACTIONS Multiply : 3. i ^^ I 4. >^h ■ S. 2y 3 a' 6. ^^■1 by 2ae ^ Sax 7. 10 c^ ^ 3 be a' 8. 9. 10. 11. 12. 4mr? 2 ax bv 15 bx 16 m2 12 by ^ x" a by a + 6 x]f- 20-8« a — 6 25 - 10a; by a^y l-6a; + 5a.'^ b ^~^ a^-3a; + 2 "^ l-a;* Simplify each of the following : \ 13. 15. 18. 20. 21. a + 5 ,4 ^4 - a6 a' - M 14. ^;;i£:x4±^„x^ aa; + ^ a^ + ar^ 4a 2x + y Aa^ — ab 2a (a + xy 4:X^ — y^ 16 i> + 2 3a^-27 4 • a._3 2jp2_8 ^a;-f3p 17. ^llZ^v-^^^X 7>^ {p-qf lf~\-pq f + (f a^ + 8 a^ + 2a + 4 a3_8 a2_2a + 4* 19. ^^l±_^^!^!dt^ X a* — aa^ a^ + 4 a^ — ax -f a^ a^ + a + l a^ + a^ + l a2 + 2a + 2 4^2-4 lOr + lOa 5r2 + 10rs + 5s2^ Sr^-Ss^ FRACTIONS 141 22. 23. fl + 7 a; + 11 y. _ 17 a; -11 \ 24. - 25. 26. 27. 4 aar^ — 4 a?/^ ^ 3 aa^ — 3 oary + bxy — by^ 3 a^ + 3 axy -j- 6a;?/ -f by- 5 aar^ — 10 axy + 5 ai/^ ar3_5a;2_^8a.'-4 a^-lQ 3^4-33 a;-3G a^-Sx'-{-19x-12' x'-6x^-\-nx-6 ' x*-Sa^-2Sx^ + 75x-50 ^ ar^ - 10 x* + 29 a; - 20 a.*_5ar»-21a;2 + 125a;-100 * ar^ - 12 ar^ + 45 a; - 50 ' a^ -\-ab + ac-\-bc ^ a^ — ax 4- ay — xy ^ a^—a;( ?/ — «)— ay ax — ay — x-+xy d^ -\- ac + ax -^ ex a^ — a{y — b)—by DIVISION OF FRACTIONS 195. The reciprocal of a fraction is the fraction inverted, or 1 divided by the fraction. 2 ^ X z ni \ The reciprocal of - is - ; of - is - ; of m, or — , is — • S 2 z X 1 m 196. The reciprocal of a number is 1 divided by the number. m. Justas M = fx| = |^^, a c a.d ad so --j-- = -X- = — • b d b c be Principle. — Dividing by a fraction is equivalent to multiply- ing by its reciprocal. EXERCISES 198. Write the reciprocal of : a b' 3. rs. 5. Ztu. 7. 4 ab' . 3m P be 6. 1 . 3m 8. a — x h-y 142 FRACTIONS 9. Dmde-^by_^ Solution a;2 - 4 a; + 2 ^ Ca>K2r)(a; - 2) x'^ -I x — 1 (x + 1) (a>-^T; >-r2 X + 1 Simplify : 10 5 ^^ . ^Q ^^^ ^K g* — 6^ . a^ + 6 ^ ' 6bx ' 3 ax'' ' a^-2ab + b^ ' a'-ab' 11. ^-^^-^abx. 16. ^±£^^!±a±l!. i^ 7 x' — y' X— y (m + z/)2 m^-?/2 \ yj \ ^y a + b b \b J \b^ J Suggestion. — Reduce the dividend to a fraction. 24. .2 _ />2 ar^ + 2ar^-19a;-20 ' a^ + lO a;^ ^29a; + 20' FRACTIONS 143 Complex Fractions 199. A fraction one or both of whose terms contains a frac- tion is called a complex fraction. EXERCISES 200. 1. Simplify the expression y a - b a . X a ,v ay Solution. — =--^-=-x^ = -*. £: y b X bx y Simplify : 2 -^. 3. t+« 8. Simplify the expression ^ — ^ + - + 1 r 2/ Solution. — On multiplying the numerator and denominator of the fraction by y'^, which is the 1. c. d. of the fractional parts of the numera- tor and denominator, the expression becomes ^ —xy + y' #x. a;2 + xy + y2 ^- -> 144 FRACTIONS Simplify: x^ — 1 jc^ + y^ x^ + y^ X xu 2y a; 4-1 ar — .r// + y^ ^ _^ a."^ icy y X 1 1 1 1 1 10. T-^T— 12. "" + : • 14. -i— ^. - — — — -J 1 g ^ 2/ + 2 a + 1 . a-1 1 1 1 ic+l , a;-hl 1-a; 15. 1 1 1 + a; 1 — a; 1 + a; ^4-^ 16. ?_A+£^ 1 ^^b'-^(f-a' 17. Simplify the expression a b-\-c 2 be 1 1+-^ X Solution. — By successive reductions and divisions, 1 1 1 x+1 x+1 1+_J_ i+-JL_ 1 |. ^ x+li-x 2a;+l 1 + 1 £±I ^ + 1 X a; Simplify : ' 1 2 18. 20. x-\- 2 l+-^ + l 3-a? 2-x 1 x-2 19. 7- 21. a-\ ^- a; — 2 . 1 a;-l a a; — .6 FRACTIONS 145 EQUATIONS AND PROBLEMS 201. Since the student has learned how to perform opera- tions when fractions are involved, he is now prepared to solve certain equations that heretofore he could solve only by a roundabout method, and others that he could not solve at all. Clearing Equations of Fractions 202. The process of changing an equation containing frac- tions to an equation without fractions is called clearing the equation of fractions. 203. 1. Solve the equation ^ = 10- 1 Solution ?=10_?. 2 3 Since the first fraction will become an integer if the members of the equation are multiplied by 2 or some number of times 2, and since the second fraction will become an integer if the members are multiplied by 3 or some multiple of 3, the equation may be cleared of fractions in a single operation by multiplying both members by some common multiple of 2 and 3, as 6, or 12, or 18, etc. It is best to multiply by the 1. c. m. of the denominators, that is, by the Led. of the fractions, which in this case is 6. Multiplying by 0, Ax. 3, 3 a; = 60 - 2 x. Transposing, etc., § 71, 5 a: = 60. Hence, Ax. 4, a; = 12. VerificatioiJ. — When 12 is substituted for x, the given equation be- comes Q = (S \ that is, the equation is satisfied for x = 12. «cii i-i, i.- x — 1 x — 2 2 x — 3 2. Solve the equation — — - = Z o o 4 Suggestion. — Multiplying both members of the equation by the I. c. d., which in this case is 12, we obtain 6(.r - 1)- 4(x - 2) = 8 - 3(a; - 3). MILNE's IST YR. ALG. — 10 146 FRACTIONS To clear an equation of fractions : Rule. — Multiply both members of the eqimtion by the lowest common denominator of the fractions. Cautions. — 1. To insure correct results in solving equations : Before clearing, reduce all fractions to lowest terms, and unite frac- tions that have like denominators. Test results and reject such as do not satisfy the equation. 2. If a fraction is negative, the sign of each term of the numerator must be changed when the denominator is removed. Solve, and verify each result : - ^^+l=f • - l^l=f 4. ?+io = i3. ■ 6. ri-f|=f 7. XX a; 3a; 5 a; „ 2 3 4 10 12~ 8. 25 a; 5 a; 2 a; ^^_2 18 9 3 6 9. 72! + 2 12-2 2 + 2 g 6 4*2 10. w-3 it + 5 w + 2 . 7*3 6 11. y-1 y-2 2/-3_52/-l. 2 ' 3 ' 4 6 12. a; — 5 2a; + 2 a;— l_a; + 4 3 8 4 6 13. 1.07 a; +.32 =.15 a; + 10.12 + .675 a;. Suggestion. — Clear of decimal fractions by multiplying by 1000. 14. .604 a; -3.16 -.7854 a; + 7.695 = 0. .2a; .la; .la; . .4a; _ .3 * "T X "2~'^T~~14' FRACTIONS 147 14 6 a; + 2 d6 14 Suggestion. — The equation may be written, ^ ' 14 14 6 X + 2 56 56 56 ' Qx + 2 14 ,^ 3a;-2 , 3a;-21 6a;-22 18. 19. 2a; -5 5 10 4a; + 3 ^ 8a;4-19 7a;-29 9 18 5a;-12 6p^-{-p 2p-4 ^ 2j>-l Wp 7p-13 5 20. Solve the equation ^^ + ^^^ = ^^ + ^— ^ • x — 2 x — 7 x—i) x — 3 Solution. — It will be observed that if the fractions in each member were connected by the sign — , and if the terms of each member were united, the numerators of the resulting fractions would be simple. The fractions can be made to meet this condition by transposing one fraction in each member before clearing of fractions. x-1 x—2 x-^ x—6 Transposing, Uniting terms. 2 x-S x-Q X -1 -1 a;2-5a;-f-6 x^-lSx + 42 Since the fractions are equal and their numerators are equal, their denominators must be equal. Then, a;2 - 5 a; + 6 = a;2 - 13 a; + 42. .-. x = il 21. 22. 23. X -1 X -7 X — 5 X -3 X -2 +^ -8" X — ^ +x -4 X -S X -7 X — 8 X -4 X -4 + x -8" X- ^ +^ -5 V ^2 V' + 3 v + 5 V- 4-6 v + l v + 2 v + 4 v-h5 148 FRACTIONS Algebraic Representation 204. 1. What part of m — n is j9 ? 2. Indicate the sum of I and m divided by 2, and that result multiplied by n. 3. Indicate the product of s and (r — 1) divided by the nth power of the sum of t and v. 4. A boy who had m marbles lost - of them. How many marbles had he left ? 5. By what number must x be multiplied that the product shall be 2 ? 6. Indicate the result when the sum of a, 6, and — c is to be divided by the square of the sum of a and b. 7. It is t miles from Albany to Utica. The Empire State Express runs s miles an hour. How long does it take this train to go from Albany to Utica ? 8. A cabinetmaker worked x days on two pieces of work. For one he received v dollars, and for the other w dollars. What were his average earnings per day for that time ? 9. A train runs x miles an hour and an automobile x— y miles an hour. How much longer will it take the automobile to run s miles than the train ? 10. Indicate the result when h is added to the numerator and subtracted from the denominator of the fraction - • c 11. A farmer had - of his crop in one field, - in a second, ^ X y and - in a third. What part of his crop had he in these three z fields ? 12. A student spends — of his income for room rent, — for m n board, - for books, and - for clothing. If his income is x s r dollars, how much has he left ? FRACTIONS 149 Problems 205- Solve the following problems and verify each solution : 1. If a large lemon grown in Mexico had weighed 2^ pounds less, its weight would have been | of its actual weight. What was its actual weight ? 2. The crew of the Lusitania numbers 800. If this is 200 less than \ the number of passengers and crew that may be accommodated, what is the passenger capacity ? 3. The box of a Chinese sedan chair is 1^ feet higher than it is long and the area of ^fclrc floor is 4 square feet. Find its three dimensions, if the capacity of the box is 14 cubic feet. 4. The sum of the heaviest loads that can be carried by a man, a horse, and an elephant is 2900 pounds. The elephant can carry 10 times as much as the horse, and the horse If times as much as the man. What load can each carry ? 5. The first issue of Tlie Sun devoted \ of its columns to advertisements and \ to miscellaneous news. The rest of the paper, 5 columns, was devoted to poetry, finance, and shipping news. How many cohimns did it contain ? 1 "^^ 6. Of the world's supply of rubber one year. South America produced \ and Africa \. How much was produced by each, if the rest of the world produced 26,600 tons ? 7. A large ear of seed corn exhibited at the Iowa Experiment Station sold for ^150. At the same rate, if it had weighed 8 ounces less, it would have sold for $ 90. How much did it weigh ? 8. The feathers that a Toulouse goose yields in a year are valued at $2.80. If it yielded 4 ounces more, they would be worth f 3.50. Find the weight of the yield of feathers. 9. The number of vessels entering at New York in one year was 11,399. If \ of the number of steamships was 449 more than \ of the number of sailing vessels, how many of each were there ? ' i 150 FRACTIONS 10. Find the world's production of nickel in a year when the United States and Canada together produced ^ of it, Eng- land ^ of it, and the rest of the world 3800 tons. 11. In a recent year, the United States produced 2 times as much aluminium as Germany, 1^ times as much as France, and 6200 tons more than England. If all these countries produced 19,800 tons, how much did the United States produce ? 12. The largest log in a shipment of mahogany sent to New Orleans weighed 14,000 pounds. If the weight of the rest of the shipment had been 1 ton more, the weight of this log would have been ^^ of the total weight of the shipment. Find the weight of the shipment. 13. In constructing the Hall of Records building in New York City, 600,000 pounds of copper were used. The dome lacks 5250 pounds of having ^ as much as the rest of the build- ing. How many pounds of copper are there in the dome ? 14. The freight charges on a car load of hay were ^ as much as on a car load of apples. If there were 10 tons of hay and the charges on each ton were f 2 less than ^ of the charges on all the apples, find the charges on each car load. 15. The number of pound cans of salmon in a case is 4 more than Jg of the number of cans that can be packed in a minute in a Washington cannery. If 1000 cases can be packed in an hour, how many cans are there in a case ? 16. A boy sold from his garden a certain number of bunches of beets. If he had sold 7 bunches more he would have re- ceived $ 11 for them. If he had sold 5 bunches less he would have received $ 9.80. How many bunches did he sell and at what price ? 17. if the number of pounds of alligator teeth sold in a given year had been 50 less, the approximate number of teeth would have been 14,000 ; if 200 less, the number of teeth would have been 3500. Find the number of pounds sold and the average number of teeth in a pound. REVIEW 151 REVIEW 206. 1. What three signs are to be considered in connection with a fraction ? What is the sign of a fraction ? 2. Under what conditions may the sign of the numerator or of the denominator of a fraction be changed ? 3. Show that ^ ~^ 4. What is the effect of changing the sign of an odd num- ber of factors in either term of a fraction ? an even number ? 5. Show that (^-y)(^-y) = -^^"ixy-f ^ (z — w)(u — z) {z — w)(z — u) 6. When is a fraction in its lowest terms ? What principle applies to the reduction of fractions to higher or lower terms ? 7. Reduce to lowest terms : 4aV + 12a6ar^-f9 6V 36a^-1962^ 8. In reducing a fraction to an equal fraction having a given denominator, how is the number found by which both terms are to be multiplied ? Q 7, 9. Reduce to a fraction whose denominator is a-2b a2_4a6+462. 10. Define highest common factor ; lowest common multiple. Illustrate by finding the highest common factor and the low- est common multiple of ax -\-ay, a^ — y^, and a^ -\- 2 xy -{- y^. 11. What is the reciprocal of a fraction ? of any number ? Give the reciprocals of -, ^"*" , and x, y x-y 12. Define complex fraction and illustrate by writing one. Simplify the one you have written. 152 REVIEW Reduce to an integral or a mixed expression: 13 a^'-9/H-7 ^^ 4a'+20a'b-\-27 ab'+9b^ x-3y ' ' 2a+36 Reduce the following mixed expressions to fractions : 15. x'-xy + f---^^. 16. !^^1±^' - ,1 - m. x-^y m-\-7i ■ Simplify : 17. iiJ + 'lui. ^+^' 2rl ' 4.rl Arl{s-{-ty 18 <^ + ^ .(^ — x 2(x~ — 2 a) xT2 2^^ x^-4: 19. ( ^'-f ^^ ^ + ^y \.f ^''y + ^y^ x ^'^'^ \ \xy + y' x-y J ' \x^ + 2 xy '-\- y^ y' J • [a'-2ab-\-b' a-bj^\ 2b '2a-2bj 21. What is meant by clearing an equation of fractions ? State the axiom upon which it is based. 22. What precautions must be taken to secure correct re- sults in solving equations that involve fractions ? Solve, and verify each result: 23. ^-1^ = 6. 25. £±J = Bi^ + l. 2 4 ^2 24. 20a: + i^ = ^. 26. ^-^^ 1 1:!?^* = ^11^. 3 6 10 o 15 x^-3 X 12 x-S 27. 4 24 4 28. ^±I^^±_§^ 3a:-1.5 _ a; + 2^ 3 9 29. • 2.04 a; - 3.1 - 2.95 x = 8.12 - 5 a; + 1.05. SIMPLE EQUATIONS ONE UNKNOWN NUMBER 207. The student already knows what an equation is; he has solved several different kinds ; and he knows some of the kinds by name. In this chapter and the next he will meet some of the same kinds with the treatment extended to a few new forms and some additional methods of solution. 208. An equation that does not involve an unknown num- ber in any denominator is called an integral equation. X + 5 = 8 and f- 6 = 8 are integral equations. Though the second o equation contains a fraction, the unknown number x does not appear in the denominator. 209. An equation that involves an unknown number in any denominator is called a fractional equation. 8 2x x + 6 = - and = 7 are fractional equations. X X— 1 210. Any number that satisfies an equation is called a root of the equation. 2 is a root of the equation 3 x + 4 = 10. 211. Finding the roots of an equation is called solving the equation. 212. Two equations that have the same roots, each equation having all the roots of the other, are called equivalent equations. X -f 3 = 7 and 2 x = 8 are equivalent equations, each being satisfied for X = 4 and for no other value of x. 163 154 SIMPLE EQUATIONS Numerical Equations 213. By applying axioms to the solution of equations, the endeavor is made to change to equivalent equations, each sim- pler than the preceding, until an equation is obtained having the unknown number in one member and the known numbers in the other. Solve, and verify each result : 1. 8^ = 24. 5. 11+0^ = 15. 9. 4/1 + 3 = 7. 2. 9r=54. 6. 20+a; = 30. 10. 6 r - 7 = 5. 3. ^r = 1.5. 7. 72/-5 = 2. 11. 16 + 3 = 8. 4. ia; = 2.5. 8. 22 + 3 = 9. 12. ia; + 2 = 6. 13. 8aj-7 = 3 + 6a;. 18. 17 ^ + 5(2 - 3 i) = 18. 14. 7a; + 6 = 6a; + 8. 19. 5 a;- (4 - 6 a;- 3) =26. 15. 5a;-10 = 2a; + 20. 20. (2w-l)=^ = 4(^-3)1 16. 4r-18 = 20 + |r. 21. 21x-\-{x-4.f==(p + x)\ 17. 5n-(2n + 3) = 12. 22. (12 a; + 6) H-3 = 9-3a;. 23. (aj + l)(a; + 2) = ll + .'«2. 24. \x-4. + \x = lQ + \x-10. 25. fl;(a5 + 5)-6 = a;(a;-l) + 12. 26. 3(2-a;)-2(a; + 3)=6-2a;. 27. aj-(2 + 4a;) = 13-5(aj + 5). 28. 2Sa;-2a;-2S=3ja;-(3x-3)J. 29. 6a;-13-9a5 + a; = 4x-12 + 3a:-6a;-13. 30. 36 +5 aj- 22 - (7a;- 16) = 5 a; + 17 - (2 a; + 22). 31. 2(r-5)(r-4)=(r-4)(r-3) + (r-2)(r-5). 32. 12a;-(6a;-17a;-15-a;) = 15-(2-17a;+6aj-4-12a;). SIMPLE EQUATIONS 165 33. 3.T-? = 14. 34 ^-f = f o 3 6 4 2x 7 X 5x X _4 ' 3 8 18 24~9' 36. 3^-5 "^t-lS ^o t-\-3 4 6 2 37. r(2_r)-^(3-2r) = ^'^^^ 38. 2^ ' 4^ '6 6r + 3 3r-l ^ 2r-9 15 5r-25 5 39 ^ + 1 , g + 6^s + 2 g + 5^ s + 2 s-f7 .s-i-3 s + 6* Literal Equations 214. 1. Solve the equation ^11:^ = ^11^ for x. ah Solution a ~ h ' Clearing of fractions, hx—h^ = ax — a^. Transposing, etc. , ax—hx = a^ — 6«. (a-6)x = a3-63. Dividing by (a - 6) , x = a^ + ab + 62. Verification. — Since a and b may have any numerical value, let a = 2 and 6 = 1; then x = a'^ + ab-\-b^ = i + 2 + l = 7, and the given equation becomes ■ ~ = ' ~ , or 3 = 3; consequently, the equation is satisfied for a; = a^ + a6 + bK 156 SIMPLE EQUATIONS Solve for a;, and verify each result : _ c^—x.n- 1 ^ X x + 2h a o z. 1 — _-. a. = - — c). nx ex c , hah _-, ah _1 49 ^ x-a , 2x r ^ ^^ 6. 1 — • y. ■ — 1 = o-\ X- ah ahx ha a 4. rx-\-s' = 9^ -sx. . 10. 6-\-l-2x = l(x-2). 5. a^x-h' = h^x-a\ 11. c'x + d'' = c^' -^ d'x. a^-\-h^ a — h __h -„ x — 2 a x _ a--^h^ 2hx 2 hx^ X a h ah ^ 2x-a x-o- ^^ ^^ a' h^ ^ a + h 3(a4-6) X— a x + a ' ' hx ax ah x 14. 6x + lS(l — ^a) = a{x — a). 15. a:\a — x) = ahx + h\h-\- x). 16. h(2x-9c-14:h) = c(c-x). 17. a(x-a-2h)-^h(x-h)-^c{x + c) = 0. 18. (a — x)(x—'b)-\-(a-\-x)(x — h) = (a — hy. 19. (?/i H- a;)^ + (7>i -f- a;)(?i — ic) = (m + ?i)^ 20. (a — h)(x — c) — (h — c)(x — a) = {c — a)(x — h). a — h-{-c h — a-{-c 21. 22. 23. 24. x—1 a—1 a? — a^ a — 1 x — 1 {a — l){x—l) 1 2 mr^ m x — n a-\-x 2x x^(x — a) _t a a-\-x a(a^ — x^ 3 Suggestion. — Simplify as much as possible before clearing the equa- tion of fractions. 25 ^(^-^) , ^Q>-^) _ ^-^ ^Q • 12Q,2_^^-^ h{x-b) h' SIMPLE EQUATIONS - 157 Problems 215. Review the general directions for solving problems given on page 45. 1. What is the weight of a turtle, from which 6} pounds of tor^ise shell is taken, if this is ^^^ of the turtle's whole weight ? 2. The powder and the shell used in a twelve-inch gun weigh 1265 pounds. The powder weighs 15 pounds more than J as much as the shell. Find the weight of each. 3. One day three lace makers earned 80 cents. The be- ginner earned \ as much as the expert maker, and the average worker earned 3 times as much as the beginner. How much did each earn ? 4. One ton of coal will make 8.7 tons of steam. If the Lusitania requires 1200 tons of coal a day for this purpose, how many tons of steam are required an hour ? 5. A grocer paid $8.50 for a molasses pump and 5 feet of tubing. He paid 12 times as much for the pump as for each foot of tubing. How much did the pump cost? the tubing? 6. In lighting a hall a certain number of 16-candle power electric lamps and twice as many 20-candle power lamps were used. The total illumination amounted to 224 candle power. Find the number of lamps of each kind used. 7. At the waterworks 2 large pumps and 4 small ones de- livered 4800 gallons of water per minute. Each of tlie large pumps delivered 4 times as much water as each small pump. How many gallons per minute did each pump deliver ? 8. The crew of a United States battleship in target practice made 11 hits in less than a minute. If J of the number of shots fired was 9 times the number of misses, how many shots were fired ? 9. The courtyard of a palace is 101 feet longer than it is wide. If its width were decreased 25 feet, its length would be twice its width. Find the dimensions of the courtyard. 158 - SIMPLE EQUATIONS 10. In making 5000 pounds of brass there were used 81 times as much copper as tin, and twice as much tin as zinc. How many pounds of each metal were used ? 11. A merchant bought 62 barrels of flour, part at $4f per barrel, the rest at $5^ per barrel. If he paid $320 for the flour, how many barrels of each grade did he buy ? 12. A dealer paid $ 185 for 25 boxes of candles. If he paid $ 9 a box for part of them and $ 6.50 a box for the rest, how many did he buy at each price ? 13. A merchant purchased an assortment of bath robes for $ 480. By selling \ of them at $ 6 each, i of them at $ 7 each, ^ of them at f 5 each, and the rest, or i of them, at $ 8 each, he gained $ 128. How many did he sell at each price ? 14. In a certain balloon race, the sum of the distances covered by the Lotus II and the United States was 1025 miles. The distance covered by the former was 50 miles more than \ of that covered by the latter. How far did each travel ? 15. A newspaper reporter saved \ of his weekly salary, or $ 1 more than was saved by an artist on the same paper, v/hose salary was $ 5 greater but who saved only ^ of it. How much did the reporter earn per week? the artist? 16. During a year of 365 days one locality had 6 days less of ^ clear' weather than of 'cloudy' weather, and 4 days more of 'clear' than of 'partly cloudy' weather. Pind the num- ber of days of each kind of weather during the year. 17. The bark from a cork tree lost \ of its weight by being boiled. The boiled cork was then scraped, its weight thus being reduced \. How much did the cork weigh before and after these two operations, if the entire loss was 16 pounds ? 18. At a certain depth a diver saw the sun as a reddish disk. At a depth 25 feet more than twice this depth it could still be faintly seen. If darkness occurred on descending 100 feet more, or at a total depth of 325 feet, at what depth did the sun appear as a reddish disk ? SIMPLE EQUATIONS 159 19. Find a fraction whose value is f and whose denominator is 15 greater than its numerator. 20. Find a fraction whose value is | and whose numerator is 3 greater than half of its denominator. 21. The numerator of a certain fraction is 8 less than the denominator. If each term of the fraction is decreased by 5, the resulting fraction equals ^. What is the fraction ? 22. An acre of wheat yielded 2000 pounds more of straw than of grain. The weight of the grain was .3 of the total weight of grain and straw. How many 60-pound bushels of wheat were produced? 23. The total diameter of a large wooden fly wheel is 30 feet. The number of inches in the thickness of the rim is 2 less than the number of feet from the center to the rim. How thick is the rim ? 24. A shipment of 83,000 postal cards in two sizes weighed 472 pounds. The smaller cards weighed 5 pounds per 1000 and the larger ones weighed 6 pounds 3 ounces per 1000. Find the number of cards of each size in the shipment. 25. I paid 18^ more for a screen door, 7 feet by 3 feet, than for 3 window screens, each 2| feet by 3 feet. Find the price per square foot in each case, if it was 3^ less for window screens. 26. A grocer bought a box of soap containing 72 cakes for $4.50. Some of the soap he sold at 3 cakes for 25^, and the rest at 10^ a cake. This gave a profit of $1.90. How many cakes did he sell at each price ? 27. It costs 3.6 ^ less to travel 100 miles on the Swiss rail- roads under public management than it did to travel 90 miles when they were under private management. The average fare per mile under private management was 1.9^ more than it is under public management. Find the rate per mile under each. 28. Two orange pickers together earned $4.50 a day, and one of them picked 20 boxes more than the other. If the slower one had picked twice as many as he did, they would have earned $ 6.50. How much did each receive a box ? 160 SIMPLE EQUATIONS 29. A can do a piece of work in 8 days. If B can do it in 10 days, in how many days can both working together do it ? Solution Let X = the required number of days. Then, - = the part of the work both can do in 1 day, \ = the part of the work A can do in 1 day, yiy = the part of the work B can do in 1 day ; ...1 = 1 + 1. X 8 10 Solving, X = 4f , the required number of days. 30. A can do a piece of work in 10 days, B in 12 days, and C in 8 days. In how many days can all together do it ? 31. It takes a man 6 days to make a Panama hat, and a boy 7 days. How long would it take them, if they could work to- gether ? 32. The average amount of coal blasted out by a keg of powder can be mined by one man in 2 days and by another in 3 days. How long would it take them to mine it if they worked together ? 33. A and B can dig a ditch in 10 days, B and C can dig it in 6 days, and A and C in 7^ days. In what time can each man do the work ? Suggestion. — Since A and B can dig -^^ of the ditch in 1 day, B and C I of it in 1 day, and A and C j^ of it in 1 day, iiy + i + i^j is twice the part they can all dig in 1 day. 34. A and B can load a car in 1| hours, B and C in 2^ hours, and A and C in 2^ hours. How long will it take each alone to load it ? 35. In a certain year. New York State furnished 158.9 mil- lion pens, or 54 % of all that were made in the United States. How many pens were made in the United States ? 36. The per cent of copper contained in an ancient die found in Egypt was 2^ % more than 3 times the per cent of tin. If these metals formed 92^ % of the die, what per cent of each did it contain ? SIMPLE EQUATIONS 161 37. Of the population of Mexico at one time the per cent of whites was I that of Indians. Mixed races formed 5% more than the per cent of Indians. Find the per cent of each. 38. Crude oil when refined produces 2 J times as much kero- sene as it does gasoline, and the remainder, which is 65%, is fuel oil. If a certain refinery produces 2250 barrels of kero- sene a day, what is its daily capacity of crude oil ? 39. The units' digit of a two-digit number exceeds the tens' digit by 5. If the number increased by 63 is divided, by the sum of its digits, the quotient is 10. Find the number. - Solution Let X = the digit in tens' place. Then, a; -f- 5 = the digit in units' place, and 10 a; + (z+ 6) = the number ; . 10 a; + (x-f5) + 63 _.^Q. 2x + 5 ' whence, x = 2, and X -f- 6 = 7. Therefore, the number is 27. 40. The tens' digit of a two-digit number is 3 times the units' digit. If the number diminished by 33 is divided by the difference of the digits, the quotient is 10. Find the number. 41. The tens' digit of a two-digit number is J of the units' digit. If the number increased by 27 is divided by the sum of its digits, the quotient is 6J. Find the number. 42. An officer, attempting to arrange his men in a solid square, found that with a certain number of men on a side he had 34 men over, but with one man more on a side he needed 35 men to complete the square. How many men had he ? Suggestion. — With x men on a side, the square contained x^ men ; with (x + 1) men on a side, there were places for (x -f 1)^ men. 43. A regiment drawn up in the form of a solid square was reenforced by 240 men. When the regiment was formed again in a solid square, there were four more men on a side. How many men were there in the regiment at first ? milne's 1st yr. alg. — 11 162 SIMPLE EQUATIONS 44. At what time between 6 and 6 o'clock will the hands of a clock be together ? Solution Starting with the hands in the position shown, at 5 o'clock, let x represent the number of minute spaces passed over by the minute hand after 5 o'clock until the hands come together. In the same time the hour hand will pass over ^^ of x minute spaces. Since they are 25 minute spaces apart at 5 o'clock, a; _ £- = 25 ; 12 ,'.x — 27^^, the number of minutes after 5 o'clock. 45. At what time between 1 and 2 o'clock will the hands of a clock be together ? 46. At what time between 10 and 11 o'clock will the hands of a clock point in opposite directions ? 47. At what two different times between 4 and o o'clock will the hands of a clock be 15 minute spaces apart ? 48. Mr. Reynolds invested $800, a part at 6 %, the rest at 5%. The total annual interest was $45. Find how much money he invested at each rate. Suggestion. — Let a; = the number of dollars invested at 6%. Then, 800 — a; = the number of dollars invested at 5% ; 49. A man put out $ 4330 in two investments. On one of them he gained 12%, and on the other he lost 5%. If his net gain was $251, what was the amount of each investment? 50. Mr. Bailey loaned some money at 4 % interest, but re- ceived $ 48 less interest on it annually than Mr. Day, who had loaned -f as much at 6 % . How much did each man loan ? 51. A man paid $80 for insuring two houses for $6000 and $ 4000, respectively. The rate for the second house was \ % greater than that for the first. What were the two rates ? simplp: equations 163 52. United States silver coins are ^-^ pure silver, or * ^^ fine.' How much pure silver must be melted with 250 ounces of silver ^ fine to render it of the standard fineness for coinage? Suggestion. — Let x = the number of ounces of pure silver to be added. Then, ^250) -\-x= the number of ounces of pure silver after the addition. Also, 1^^(250 + x) = the number of ounces of pure silver after the addition. 53. In an alloy of 90 ounces of silver and copper there are 6 ounces of silver. How much copper must be added that 10 ounces of the new alloy may contain | of an ounce of silver? 54. If 80 pounds of sea water contain 4 pounds of salt, how much fresh water must be added that 49 pounds of the new solution may contain If pounds of salt ? 55. Four gallons of alcohol 90 % pure is to be made 50 % pure. What quantity of water must be added? 56. Of 24 pounds of salt water, 12 % is salt. In order to have a solution that shall contain 4 % salt, how many pounds of pure water should be added? 57. A man rows downstream at the rate of 6 miles an hour and returns at the rate of 3 miles an hour. How far down- stream can he go and return within 9 hours ? 58. An airship traveled 11 miles with the wind in the same time as 1 mile against it. If it traveled 55 miles and returned in 12 hours, what was its rate against the wind? with the wind ? 59. A train went 905.4 miles in a certain length of time. Another train with a speed 3 miles greater per hour covered 54 miles more in the same length of time. What was the speed of each train ? 60. An express train whose rate is 40 miles an hour starts 1 hour and 4 minutes after a freight train and overtakes it in 1 hour and 36 minutes. How many miles does the freight train run per hour? 164 SIMPLE EQUATIONS Solution of Formulae 216. A formula expresses a principle or a rule in symbols. The solution of problems in commercial life, and in mensura- tion, mechanics, heat, light, sound, electricity, etc., often de- pends upon the ability to solve formulae. EXERCISES 217. 1. The circumference of a circle is equal to tt (= 3.1416) times the diameter, or C = 7rD. Solve the formula for D and find, to the nearest inch, the diameter of the wheel of a locomotive, if the circumference of the wheel is 194.78 inches. Solution From C = ttD, ttD = C. .•.i)=^ = lM:I§ = 62.0H-. TT 3.1416 Hence, to the nearest inch, the diameter is 62 inches. 2. Area of a triangle = \ (base x altitude), or A = \bh. Solve for 6, then find the base of a triangle whose area is 600 square feet and altitude 40 feet. 3. The area of a trapezoid is equal to the product of the altitude and half the sum of the bases ; that is, A = h-\{b + bi). The bases are h and 6'. 6' is read ' &-prime.* Solve for li, then find the altitude of a trapezoid whose area is 96 square inches and whose bases are 14 inches and 10 inches, respectively. 4. The volume of a pyramid = \ (base x altitude), or Solve for J5, then find the area of the base of a pyramid whose volume is 252 cubic feet and altitude 9 feet. SIMPLE EQUATIONS 165 5. The charge (c) for a telegram from New York to Chicago, 40^ for 10 words and 3^ for each additional word, may be found by the formula, (? = 40 + 3(/i-10), in which n stands for the number of words. Find the cost of a 16-word message. Solve for n, then find how many words can be sent for $ 1. 6. In the formula, i = p - — -^ • f , i denotes the interest on a principal of p dollars at simple interest at r% for t years. Solve for t, then find the time $300 must be on interest at 5 % to yield $ 60 interest. Solve for r. At what rate of interest will $4500 yield $900 interest in 5 years ? Solve for p. What principal at 3^% will yield $210 annually? 7. The formula for the space (s) passed over by a body that moves with uniform velocity {v) during a given time {t) is 5 = vi. Solve for v, then find the velocity of sound when the condi- tions are such that it travels 8640 feet in 8 seconds. 8. The formula for converting a temperature of F degrees Fahrenheit into its equivalent temperature of C degrees Centi- grade is ^-''- ''' ^=^{^+243)'°^^- 34. Solve ^ = ^stl\ for E; for r, e r 35. Solve i = - + ifor f^: for f,. 36. Solve ^Wl = ^, for W; for S; for:?^. c c SIMULTANEOUS SIMPLE EQUATIONS TWO UNKNOWN NUMBERS 218. In the equation x + y = 12, X and y may have an unlimited number of pairs of values, as x = l and 2/ = 11 ; or x = 2 and ?/ = 10 ; etc. For since y = 12 — x, if any value is assigned to x, a corresponding value of y may be obtained. An equation that is satisfied by an unlimited number of sets of values of its unknown numbers is called an indeterminate equation. 219. Principle. — Any single equation involving two or more unknown numbers is indeterminate. = 10| = 15 J 220. The equations 2x-{-2y = 10 and 3 a; 4- 3 2/ express but one relation between x and 2/; namely, that their sum is 5. In fact, the equations are equivalent to x + y = 5 and to each other. Such equations are often called dependent equations, for either may be derived from the other. 221. The equations x + y = 5] x — y = l} express two distinct relations between x and y, namely, that 168 SIMULTANEOUS SIMPLE EQUATIONS 169 their sum is 5 and their difference is 1. The equations cannot be reduced to the same equation ; that is, they are not equivalent. Equations that express different relations between the un- known numbers involved, and so cannot be reduced to the same equation, are called independent equations. 222. Each of the equations x — y = li is satisfied separately by an unlimited number of sets of values of X and y, but these letters have only one set of values in both equations, namely, a; = 3 and y = 2. Two or more equations that are satisfied by the same set or sets of values of the unknown numbers form a system of simultaneous, or consistent, equations. 223. The equations y have no set of values of x and y in common. Such equations are called inconsistent equations. 224. The student is familiar with the methods of solving simple equations involving one unknown number. The general method of solving a system of two independent simultaneous simple equations in two unknown numbers, as x + y = 5) x-y=^S) is to combine the equations, using axioms 1-5 (§§ 68, 74) in such a way as to obtain an equation involving, x alone, and another involving y alone, which may be solved separately by previous methods. The process of deriving from a system of simultaneous equa- tions another system involving fewer unknowa numbers is called elimination. x-\-y = 5] 170 SIMULTANEOUS SIMPLE EQUATIONS Elimination by Addition or Subtraction 225. Elimination by addition or subtraction has been dis- cussed and applied to the solution of simultaneous equations in §§99-103. EXERCISES 226. 1. Solve the equations 2 a; + 3 ?^ = 7 and 3 a; + 4 ?/ = 10. Solution |2a;+ 3y=7, . (1) \sx-\- 4y = 10. (2) 8a;+ 12^ = 28. (3) 9x + 12?/ = 30. (4) X=:2. (5) 4+ Sy = 1. .'.y = l. To verify, substitute 2 for x and 1 for y in the given equations. . (1) X 4, (2) X 3, (4) - (3), Substituting (5) in (1), Rule. — If necessary, multiply or divide the equations by such numbers as will make the coefficients of the quantity to be elimi- nated numerically equal. Eliminate by addition if the resulting coefficients have unlike signs, or by subtraction if they have like signs. Solve by addition or subtraction, and verify results : 3 d 4- 4 ?/ = 25, 4d + 32/ = 31. 5^ + 6g = 32, 7p-3q = 22. f 3 a + 6 2 = 39, [9a-4.z = 51: 6x-5y = SS, 4 a; -f- 4 2/ = 44. 4. 5. '7ic — 5?/ = = 52, .2x-\-5y = = 47. 'Sx-h2y = = 23, .x + y = S. 'Sx-4.y = = 7, .x-\-10y = 25. (2x-10y = 15, 2x-4.y = :18. 7. SIMULTANEOUS SIMPLE EQUATIONS 171 j^ |2a+3&=17, jj |3m + llri = 67, I3a + 26 = 18. ' l5m-3n = 5. Elimination by Comparison 227. If x=S-?j, (1) and also x = 2 -\- y, (2) by axiom 5, the two expressions for x must be equal. .'.S-y = 2 + y. By comparing the values of x in the given equations, (1) and (2), we have eliminated x and obtained an equation involving y alone. This method is called elimination by comparison. 228. 1. Solve the equations 2 .r — 3 y = 10 and 5x -{- 2y = 6. Solution f2a;-3y = 10, (1) l5x + 2?/=6. (2) From (1), . = ^« + 3,. (3^ From (2), ^='-''' (4) 6 Comparing the values of x in (3) and (4), 10 + 3y ^ ()-2y 2 5 Solving, y = ~2. Substituting — 2 for y in either (3) or (4), ^ x = 2. To verify^ substitute 2 for x and — 2 for y in the given equations. Rule. — Find an expression for the value of the same unknown member in each eq^iation, equate the two expressions^ and solve the equation thus formed. 172 SIMULTANEOUS SIMPLE EQUATIONS 4. 5. [7x-Sy = lS. 9. 10. 11. 12. Solve by comparison, and verify results : 3x-2 7j = 10, [x-\-y = 70. (5x-\-y = 22, \x + 5y=14:. (2x-\-3y = 24:, {5x-3y = lS. (3x-\-5y=Uy \2x-3y=3. 3v-^2y = 36, [5v-9y = 23. 2s-\-7t = S, 3s + 9t = 9. 4 ?i + 6 V = 19, 3u-2v = ^. 1 11 v'4- 5 w = 87. ( 4:X— 13y = o, [3x + lly= -17. 4:X + 3y=10, 13. Ll2a;-ll^ = -10. 4:X-l-3y = 27. (lSx~3y = 4:, Elimination by Substitution 229. Given 3x-{-2y = 27, (1) and X — y =^4. (2) On solving (2) for x, its value is found to he x = 4= -\- y. If 4 + 2/ is substituted for x in (1^, 3 x will become 3(4 + y), and the resulting equation 3(44-2/) + 2i/=27 (3) will involve y only, x having been eliminated. Solving (3), 2/ = 3. Substituting 3 for y in (2), x = 7. This method is called elimination by substitution. Rule. — Find an expression for the value of either of the un- known numbers in one of the equations. Substitute this value^for that unknown number in the other equation, and solve the residting equation. SIMULTANEOUS SIMPLE EQUATIONS 173 EXERCISES 230. Solve by substitution, and verify results: ,4:y — x = 14. (x-hy = 10, I 6 ic - 7 V = 34. 3. 4. 3ic-4?/ = 26, .x-Sy = 22. '6y -10 x = U, ,y — X = 3. >4-l=3a', .5x + 9 = 3y, 6. 7. 8. 9. 10. 17=3x-\-z, 7 = 3z-2x. •iy = 10-x, a; = 5. 7 2-3a; = 18, z — 5x = l. \y— (3-loy = -x, 1 3 -f 15 2/ = 4 a;. l-x = 3y, .3(l-x) = A0-y. 231. Three standard methods of elimination have been given. Though each is applicable under all circumstances, in special cases each has its peculiar advantages. The student should endeavor to select the method best adapted or to invent a method of his own. EXERCISES 232. Solve by any method, verifying all results : ^ ra; + 2; = 13, .x — z = 5. 5. 2. 2/ = 10, y = 6. 4. {3x-\- U + 3 (4:X-{-5y = —2, 1 5 a; -f- 4 2/ = 2. (5x-y = 2S, l3a; + 5v = 28. 6. (x + 3 = y-3, \2(x-{-3) = 6-y. (5x-y = 12, \x-{-3y = 12. (^2-x) = 3y, \2(2-x) = 2(y-2). (x + l) + (y-2) = 7, (x + l)-{y-2)=5. 174 SIMULTANEOUS SIMPLE EQUATIONS Eliminate before or after clearing of fractions, as may be more advantageous ; 9. 10. aj + ! = ll, 4^5 3^5 11. 12. X 2y_ o o X "^^ -{-'! = 12. 2 3 x-\-y x — y 2 3 8, 3 4 13. p + i (3a;-2/-l)=i + f(2/-l), i(4a. + 32/) = TV(7 2/ + 24). Equations of the form -4-- = c, though not simple equa- X y tions, may be solved as simple equations for some of their roots by first regarding - and - as the unknown numbers. X y 14. Solve the equations • 4 3 14 X y 5 4 10 50 ix'^ y 3* Solution. (2) — (1), 13 _ 208 y 15* . 1 16 " y 15* Substituting (3) in (1), 4_48_14, X 15 5 * " X 2* From (4) and (3), -1— i (1) (2) (3) (*) SIMULTANEOUS SIMPLE EQUATIONS 175 Solve , and verify results : 5_§=-2 2 3 . = 5, 15. . X y 18. a; 2/ 25 . 1 ^ 5 2_^ h - = 6. [x y .a; ^~ * [i-«=-i, '4,3 9 - + - = o' 16. . X y 19. X y ^ 13 25 3,4 11 x'^y 28' X y 12 3 1_ o r 5 4 o 7r- + - = 3» 17. ^ 2 a; ?/ ~~ 20. , 3a; y # +-=23. ^-f = 2^ .2 a; y y 6a; ^ Literal Simultaneous Equations 1. Solve the equations f«^ + % = ^'*> I ca; -f dy = w. (1) xd, (2) x6, (3) -(4) (1) xc, (2) xa, (7) -(6) Solution ax-^by=m cz-\- dy = 71 ddx + bdy = dwi 6ca; + bdy = bn (ad — bc)x = dm'~bn . _ (Zm — bn ad— be* acz 4- bey = cm acx + ady = an {ad — bc)y = an — cm . « = gro — cm ad — be (1) (2) (3) (4) (6) («) (7) (8) In solving literal simultaneous equations, elimination is usually per- formed most easily by addition or subtraction. (i ^ \ . ^1 176 SIMULTANEOUS SIMPLE EQUATIONS Solve for x and y in exercises 2-15, and test results by assigning suitable values to the other letters : 5. by = 171, ay = c. (ax + \bx- (ax — by = m, [ex — dy = r. (ax = by, \x-{-y = ab. (m{x-\-y) = a, [ n(x — 2/) = 2 a. 7. 9. (a(x-y) =5, [bx — cy = n. iaia-x) =b(y-b), [ ax = by. Q ^x + y=-.b-a, I bx — ay H- 2 a6 = 0. (x-y = a-b, [ ax -f by = a^ — b\ Eliminate before or after clearing of fractions as seems best 10. 11. 12. a b .bx — ay = 0, 1 a 1 b 0, X y 1 1 { x + l ^ a + b-\-l IZ/ \y + l a - 6 + f [x-y = 2b. 1 ^ 1 X— a x + y 14. y y --- = -1, X y b 15. a ix = -1. y x-y X , y a b lb c Problems 234. Solve the following problems and verify each solution: ' 1. The length of a lot is 20 yards more than its width, and its perimeter is 360 yards. Find its dimensions. 2. The best Panama hats bought in Colombia cost $5 each, and the cheapest cost 50 cents each. If 18 hats cost $ 45, how many hats of each kind were bought ? SIMULTANEOUS SIMPLE EQUATIONS 177 ^ S. A grocer sold 2 boxes of raspberries and 3 of cherries to one customer for 54 ^, and 3 boxes of raspberries and 2 of cherries to another for 56 ^. Find the price of each per box. 4. A druggist wishes to put 500 grains of quinine into 3-grain and 2-grain capsules. He fills 220 capsules. How many capsules of each size does he fill ? 5. On the Fourth of July, 850 glasses of soda water were sold at a fountain, some at 5.^ each, the others at 10 ^ each. The receipts were $ 55. How many were sold at each price ? 6. A fruit dealer bought 36 pineapples for $ 2.50. He sold some at 12 ^ each and the rest at 10 ^ each, thereby gaining $ 1.50. How many did he sell at each price ? 7. The receipts from 300 tickets for a musical recital were $ 125. Adults were charged 50^ each and children 25^ each. How many tickets of each kind were sold ? 8. A dealer packed 1800 Christmas wreaths in 6 barrels and 6 cases. Later he packed 2250 wreaths in 9 barrels and 7 cases. Find the capacity of a barrel ; of a case. 9. East African hemp is worth $25 more per ton than Mexican hemp. If 2 tons of African hemp are worth $ 25 less than 2| tons of Mexican hemp, find the value of each per ton. 10. A natural bridge in Utah has a span of 60 feet more than its height. If its height were 200 feet less, it would be ^ of its span. Find the height arid the span of the bridge. 11. A chimney at Bolton, England, is 100 feet lower than one at Glasgow, and \ of the height of the latter is 64 feet more than \ of the height of the former. Find the height of each. 12. An errand boy went to the bank to deposit some bills for his employer. Some of them were l-dollar bills, and the rest 2-dollar bills. The number of bills was 38 and their value was $ 50. Find the number of each. MILKENS IST YR. ALG. 12 178 SIMULTANEOUS SIMPLE EQUATIONS 13. A grocer bought 1416 oranges of two sizes. Of one kind it took 360 oranges to fill a box and of the other 48. If there were 10 boxes in all, find the number of boxes of each kind. j^ 14. The cost of firing 20 shots from a Japanese battleship was $ 4040. The shots from the large cannon cost $ 400 each and every shot from the small cannon cost $ 70. How many- shots of each kind were fired ? 15. In Berlin, Germany, a mason received 80^ more in 5 days than a painter received in 6 days, each working 10 hours a day. The former earned the same in 9 days as the latter did in 12 days. What was the hourly wage of each man ? 16. The champion National League baseball team one year won 62 games more than it lost. The team that came second played 154 games, winning 16 less than the first and losing 18 more than the first. How many games did each team win and how many did each lose ? 17. During a rate war between rival steamship lines the pas- sage for 2 immigrants from Bremen to New York cost $ 7.14 less than the normal rate for 1, and the passage for i immi- grants ^4.76 less than the normal cost for 3. Find the nor- mal rate and the reduced rate. 18. A man noticed that a 15-word message by telegraph cost him 40^ and a 22- word message 54^, between the same two cities. Find the charge for the first 10 words and the charge for each additional word. 19. The United States imported 38 million bunches of bananas one year. The cost, at 30 ^ each for the larger bunches and 20 ^ each for the smaller ones, was 8.55 million dollars How many bunches of each size were imported ? 20. A steam pipe was inclosed in a wooden case. The diam- eter of the pipe was | of the diameter of the case. The radius of the case was 2 inches less than the diameter of the pipe. What was the diameter of each ? SIMULTANEOUS SIMPLE EQUATIONS 179 21. A man invested $4000, a part at 5 % and the rest at 4 % . If the annual income from both investments was $ 175, what was the amount of each investment ? 22. At a factory where 1000 men and women were em- ployed, the average daily wage was $ 2.50 for a man and $ 1.50 for a woman. If labor cost $ 2340 per day, how many men were employed ? how many women ? 23. It required 60 inches of tape to bind the four edges of a card on which a photograph was mounted. The length of the card was 6 inches greater than the width. How many inches long was the card ? how many inches wide ? 24. The German railroads carried 153 million first-class and second-class passengers one year. The number would have been 180 million if 4 times as many had traveled first class. How many traveled first class? second class? 25. In one hour 1375 vehicles passed a merchant's door on Broadway, New York City. The horse-drawn vehicles would have equaled the automobiles in number, had there been 50 more of the former and 25 less of the latter. How many of each passed his door ? [^' 26. Probably the highest dock in the world is on the Victoria Nyanza. Its height above sea level is 50 feet less than 15 times its length, and the sum of its height and length is 3950 feet. Find its height above sea level. 27. The American and British tourists to Japan during a re- cent year numbered 2794. If there had been twice as many Americans and 3 times as many British, the number would have been 6682. How many tourists were there from each country ? 28. The number of students attending the University of Berlin at one time was 9277 more than the number attending at Munich, and ^ the number at Berlin plus ^ the number at Munich equaled 9591. How many attended each university ? 180 SIMULTANEOUS SIMPLE EQUATIONS 29. UDder the present contract, it costs $ 24.15 less a year per lamp to maintain electric lights in a certain city than it did under the previous one, and the expense of 6 lamps then was $46.35 more than that of 7 lamps now. Find the present yearly expense per lamp. 30. On the same day two seats in the New York Stock Exchange sold for $ 192,500. If one of them had sold for $ 1500 less, and the other for $ 1000 more, the prices of the two would have been equal. Find the price of each. 31. A man had 10 fox skins, some of which were silver fox, worth $300 a pelt, and some black fox, worth $750 a pelt. If he had had 3 less of the former and 3 more of the latter, the total value would have been $6150. How many had he of each ? 32. The quantity of peanuts raised in the United States in a year is 135 million pounds more than the quantity of all nuts imported, and ^ of the former equals ^ of the latter. Find the number of pounds of nuts imported and of peanuts raised in the United States. 33. The receipts from a football game were $ 700. Admis- sion tickets to the grounds were sold for 50 ^, and to the grand stand for 25 ^ in addition. If twice as many persons had pur- chased tickets for the grand stand, the receipts would have been $ 800. How many tickets of each kind were sold ? 34 A train of 25 cars loaded with iron ore wa§ run out on a dock and the ore emptied into pockets beneath the tracks. The ore filled 7 pockets and ^ of another. To fill this last pocket, then, required 16 tons less than 2 extra car loads. What was the capacity of a car? of a pocket? 35. If 100 pounds of soft coal in burning can evaporate 50 pounds more water than 6 gallons of oil, and if 60 pounds of coal can evaporate 10 pounds less water than 4 gallons of oil, how many pounds of water can 1 pound of coal evaporate ? 1 gallon of oil ? SIMULTANEOUS SIMPLE EQUATIONS 181 36. A proposed tunnel under Bering Strait would be in three sections, each of which would be J of a mile longer, and two of which together would be 12f miles longer than the Simplon tunnel. Find the length of the proposed tunnel ; of the Simplon tunnel. 37. If 1 is added to the numerator of a certain fraction, the value of the fraction becomes f ; if 2 is added to the denom- inator, the value of the fraction becomes |. What is the fraction ? Suggestion. — Let x = the numerator and y = the denominator. 38. If the numerator of a certain fraction is decreased by 2, the value of the fraction is decreased by ^ ; but if the denomi- nator is increased by 4, the value of the fraction is decreased by |. What is the fraction ? 39. A certain number expressed by two digits is equal to 7 times the sum of its digits ; if 27 is subtracted from the num- ber, the difference will be expressed by reversing the order of the digits. What is the number ? Suggestion. — The sum of x tens and y units is (lOx -f y) uAits ; of y tens and x units, (10 j/ + x) units. 40. Find a number that is 3 greater than 6 times the sum of its two digits, if the units' digit is 2 less than the tens' digit. 41. A crew can row 8 miles downstream and back, or 12 miles downstream and halfway back in 1 J hours. What is their rate of rowing in still water and the velocity of the stream ? 42. A man rows 12 miles downstream and back in 11 hours. The current is such that he can row 8 miles downstream in the same time as 3 miles upstream. What is his rate of row- ing in still water, and what is the velocity of the stream ? 43. A quantity of wheat could be thrashed by two machines in 6 days, but the larger machine worked alone for 8 days and was then replaced by the smaller, which finished in 3 days. How long would it have taken the larger machine to thrash all of the wheat ? the smaller machine ? 182 SIMULTANEOUS SIMPLE EQUATIONS THREE UNKNOWN NUMBERS 235. The student has been solving systems of two independ- ent simultaneous equations involving two unknown numbers. In general, Principle. — Every system of independent simidtaneous simple equations^ involving the same number of unknown numbers as there are equations, can be solved, and is satisfied by one and only one set of values of its unknown numbers. EXERCISES 236. 1. Solve . x-\-2y-{-3z = U, (1) 2x + y-\-2z = 10, (2) 3x-\-Ay-Sz = 2. (3) Solution. — Eliminating z by combining (1) and (3), (1) + (3), 4x+ey = 16. (4) Eliminating z by combining (2) and (3), (2) X 3, 6 a; 4- 3 y + 6 2 = 30 (3) X 2, 6x+ Sy-6z= 4 Adding, 12x + ny =34, (5) Eliminating x by combining (5) and (4), (4) X 3, 12x + lSy = 48 (6) (6)-(5), ly =U;.:y = 2. Substituting the value of ?/ in (4), 4 cc + 12 = 16 ; .-. x = \. Substituting the values of x and y in (1), 1 +4 + 3;5 = 14; .'. z = S. Verification. — Substituting x=l, y = 2, and 2: =3 in the given equations, (1) becomes 1 + 4 4- 9 = 14, or 14 = 14 (2) becomes 2 + 2 + 6 = 10, or 10 = 10 and (3) becomes 3 + 8-9= 2, or 2= 2 that is, the given equations are satisfied for a; = 1, ?/ = 2, and « = 8. SIMULTANEOUS SIMPLE EQUATIONS 183 Solve, and verify all results x-\-3y — z — 10, 2. ^ 2 a; + 5 2/ + 4 2; = 57, 3 a; - 1/ + 2 z = 15. x + y-{-z = 53, x + 2y + 3z = 105, a; + 32/ + 4z = 134. x-y + z = 30, 3y-x-z = 12, 7z — y-j-2x = 141. 'Sx-5y-^2z = 5S, 5. lx-{-y — z = 9, 13x-9y-{-Sz = 71. 6. 7. 8. 9. x-\-Sy-{-4:Z = SSy x-\-y-j-z = 29, 6x-\-Sy + Sz = 156. 3x-2y + z = 2, 2a: + 5y + 2z = 27, x-{-3y + 3z = 25. x + iy + iz = 32, ixA-iy-{-iz = 15, ix + \y-\-iz = 12. ix-iy + iz = 5. 10. There are three numbers such that the sum of ^ of the first, ^ of the second, and \ of the third is 12 ; of J of the first, J of the second, and -^ of the third is 9; and the sum of the numbers is 38. What are the numbers? 11. A, B, and C have certain sums of money. If A gives B $100, they will have the same amount; if A gives C $100, C will have twice as much as A; and if B gives C $100, C will have 4 times as much as B. What sum has each ? 12. A quantity of water sufficient to fill three jars of differ- ent sizes will fill the smallest jar 4 times ; the largest jar twice with 4 gallons to spare ; or the second jar 3 times with 2 gal- lons to spare. What is the capacity of each jar? 13. A contractor used 3 scows to convey sand from his dredge to the dumping ground. He was credited by the in- spector for : April 20, scows a, 6, c, a, &, c, a, and 6, 8 loads, 3230 cu. yd. April 21, scows c, a, 6, c, a, &, and c, 7 loads, 2820 cu. yd. April 22, scows a, &, c, a, 6, c, and a, 7 loads, 2870 cu. yd. Eind the capacity of each scow. GRAPHIC SOLUTIONS SIMPLE EQUATIONS 237. When related quantities in a series are to be compared, as, for instance, the population of a town in successive jenrs, recourse is often had to a method of representing quantities by lines. This is called the graphic method. By this method, quantity is photographed in the process of change. The whole range of the variation of a quantity, pre- sented in this vivid pictorial way, is easily comprehended at a glance ; it stamps itself on the memory. 238. In Fig. 1 is shown the population of a town throughout its variations during the first 13 years of the town's exist- ence. The population at the end of 2 years, for example, is repre- sented by the length of the heavy black line drawn upward from 2, and is 4000 ; the popu- lation at the end of 6 years is 7000; at the end of 10 years, 6300^ approximately; and so Fig. 1. on. 239. Every point of the curved line shown in Fig. 1 exhibits a pair of corresponding values of two related quantities, years and population. For instance, the position of E shows that the population at the end of 4 years was 6000. Such a line is called a graph. 184 ^ / » / s. - u / 3 / S 4 1^ y g ^ ? ^ * i ^ I S i ( 1 '. i 4 > e Ye i s are ' 1 1 1 12 13 1 4 1 5 UrilVERSITY OF GRAPHIC SOLUTIONS 185 Graphs are useful in numberless ways. The statistician uses them to present information in a telling way. The broker or merchant uses them to compare the rise and fall of prices. The physician uses them to record the progress of diseases. The engineer uses them in testing materials and in computing. The scientist uses them in his investigations of the laws of nature. In short, graphs may be used whenever two related quantities are to be compared throughout a series of values. 240. The graph in Fig. 2 represents the rate in gallons per day per person at which water was used in New York City during a certain day of 24 hours. - ~^ -^ 120 i / "~ ^ / s, ^ V — -I y \ K W) y \ ^ .^ 80 70 60 ->i ■& § 1 30 n ba 'afj-on iK id light 1 2 Midnight 4 5 A.M. 12 Koon Fm. 2. 18 21 24 P.M. -^ Midnight Thus, if each horizontal space represents 1 hour (from mid- night) and each vertical space 10 gallons, at midnight water was being used at the rate of about 84 gallons per day per per- son; at 6 A.M., about 91 gallons; at 1 p.m., the 13th hour, about 108 gallons ; etc. 1. What was the approximate consumption of water at 2 A.M. ? at noon ? at 1 : 30 p.m. ? at 3 p.m. ? at 6 p.m. ? 2. What was the maximum rate during the day ? the mini- mum rate ? at what time did each occur ? 3. During what hours was the rate most uniform ? What was the rate at the middle of each hour ? 4. What was the average increase per hour between 6 a.m. and 8 a.m. ? the average decrease between 4 p.m. and 8 p.m. ? 186 GRAPHIC SOLUTIONS 241. The graphs in Fig. 3 present to the eye in a forceful way the remarkable contrasts in the months of July of the years 1901 and 1904 by showing the average daily maximum temperature for ten cities that cover the part of the United States east of the Eocky Mountains. -/-^^ t \ 7S z \± . y\ 1 \ ,^ N u i v / V - VA -\ A- I I At - y avX j S L J t 7k Z a 7 1 ,^ V7 3- sL^ \ f 1 \T 7 ^ I t: ^ i ^ - : ^ ! o 7 ^Z L ;i L A 4 ^ . ^ -I ^ t ' « 7 \s r 7 t ^ V-.Z t X 7 \7 Xl » -\t ' bays of the Mcjnth July 1901 July 1904 12 3 4 5 6 7 9 10 11 12 13 14 15 16 17 IS IS 20 2L22 ^ %k^ 26 27 28 29 30 31 Fig. 3. The vertical spaces for 0° to 76° inclusive are omitted. In the follow- ing * temperature ' means 'average maximum temperature.' 1. In which year was the month of July the hotter ? 2. On how many days of July, 1901, was the temperature helow 90° ? How many days of July, 1904, had a temperature ahove 90° ? 3. What was the highest temperature for July, 1901, and on what day did it occur? for July, 1904? Give the lowest temperature for July of each of these years and the date of its occurrence. 4. Which year had the smaller range of temperature for July? How many degrees hotter was the Fourth of July, 1901, than the Fourth of July, 1904 ? 5. Find the difference between the highest temperature of July, 1901, and the lowest temperature of July, 1904. GRAPHIC SOLUTIONS 187 242. Fig. 4 gives two graphs, — one showing the height of water above zero of the gage in the Cumberland Eiver at Nash- ville, the other showing the same thing for the Arkansas Kiver at Little Rock, from daily observations taken in September of the year 1908. 2 3 i 5 6 7 8 10 n 12 13 U 15 16 17 18 19 ^JU. 22;i3212526:i7ii82930 Fig. 4. In giving heights read the graphs to the nearest tenth of a foot. 1. When was the water in the Cumberland highest ? lowest? What was the maximum height ? the minimum ? the range between them ? 2. How many days later than the Cumberland was the Arkansas at its maximum ? How many days earlier than the Cumberland was the Arkansas at its minimum ? 3. What was the range for the Arkansas during the month ? 4. State the difference between the maximum readings for the two rivers ; between their minimum readings. 5. On what days were the readings higher for the Arkansas than for the Cumberland ? 6. What part of the month shows the greatest and most rapid changes in the height of the Cumberland ? the Arkansas ? 7. Which river had the least variation in height during the last half of the month ? 8. Give the time of the greatest change in a single day for either river. How much was this change ? 188 GRAPHIC SOLUTIONS EXERCISES 243. 1. Letting each horizontal space represent 10 years and each vertical space 1 million of population, locate points from the pairs of corresponding values (years and millions of popula- tion given below) and connect these points with a line, thus constructing a population graph of the United States : 1810, 7.2 ; 1820, 9.6 ; 1830, 12.8 ; 1840, 17.1 ; 1850, 23.2 ; 1860, 31.4; 1870, 38.6; 1880, 50.2; 1890, 62.6; 1900, 76.3. 2. From the graph of exercise 1, tell the period during which the increase in the population was greatest ; least. 3. The average price of tin in cents per pound for the months of a certain year was : Jan., 23.4; Feb., 24.7; Mar., 26.2 ; Apr., 27.3; May, 29.3; June, 29.3; July, 28.3; Aug., 281 ; Sept., 26.6 ; Oct., 25.8 ; Nov., 25.4 ; Dec, 25.3. Draw a graph to show the variation in the price of tin during the year with each horizontal space representing 1 month and each vertical space 1 cent. 4. Construct the graph of exercise 3, letting each horizontal space represent 1 cent and each vertical space 1 month. 244. Let X and y be two algebraic quantities so related that ?/ = 2.T — 3. It is evident that we may give x a series of values, and obtain a corresponding series of values of y ; and that the number of such pairs of values of x and y is unlimited. All of these values are represented in the graph of y = 2x — Z. Just as in the preceding illustrations, so in the graph of y = 2x — ^, Fig. 5, values of x are represented by lines laid off on or parallel to an x-axis, X'X, and values of y by lines laid off on or parallel to a y-axis, Y'Y, usually drawn perpendicular to the Fig. 5. aj-axis. GRAPHIC SOLUTIONS 189 For example, the position of P shows that t/ = 3 when a; = 3 ; the position of Q shows that y = o when a; = 4 ; the position of R shows that ?/ = 7 when x = 6\ etc. Evidently every point of the graph gives a pair of corre- sponding values of x and y. 245. Conversely, to locate any point with reference to two axes for the purpose of representing a pair of corresponding values of x and ?/, the value of x may be laid off on the a;-axis as an x-distance, or abscissa, and that of y on the y-axis as a y-distance, or ordinate. If from each of the points on the axes obtained by these measurements, a line parallel to the other axis is drawn, the intersection of these two lines locates the point. Thus, in Fig. 6, to represent the corresponding values a; = 3, y = 3, a point P may be located by measuring 3 units from to ilf on the jc-axis and 3 units from O to iVon the y-axis, and then drawing a line from M parallel to O T, and one from JV parallel to OX, producing these lines until they intersect. 246. The abscissa and ordinate of a point referred to two perpendicular axes are called the rectangular coordinates, or simply the coordinates, of the point. Thus, in Fig. 5, the coordinates of P are OM{ = NF) and MP{= ON). 247. By universal custom positive values of x are laid off from as a zero-point, or origin, toward the right, and neg- ative values toward the lejl. Also positive values of y are laid off up- ivard and negative values dowmvard. The point A in Fig. may be designated as ' the point (2, 3),' or by the equation A = (2, 3). Similarly, B=(-2, 4), C=(-;!, -1), and i)=(l, -2). 77ie abscissa is always written first. Y ^^H ^ ,^ 2— -3-2-10 12 3 4 -^^-tr Illlllr^IIIII Fig. 0. 190 GRAPHIC SOLUTIONS 248. Plotting points and constructing graphs. EXERCISES Note. — The use of paper ruled in small squares, called coordinate paper, is advised in plotting graphs. Draw two axes at right angles to each other and locate : 1. A = {3, 2). 2. B = (3, - 2). 3. 0=(4, 3). 4. i) = (4, -3). 9. i = (0, 4). 10. M=(0, -5). 11. xY=(3, 0). 12. F =(-6, 0). the 5. ^ = (5,5). 6. F = (-5,5). 7. G=(-2,5). 8. //=(-3, -4). 13. Where do all points having the abscissa lie? ordinate ? 14. What are the coordinates of the origin? 16. Construct the graph of the equation 2 y — x = 2. Solution Solving for y, y=l{x-\-2). Values are now given to x and corresponding values are computed for y by means of this equation. The numbers substituted for x need not be large. Convenient numbers to be substituted for x in this instance are the even integers from — 6 to + 6. When x = — 6, y= — 2. These values locate the point A = ( — 6, —2). "When X = — 4, y = — 1. These values serve to locate 5 = (— 4, —1). Other points may be located in the same way. A record of the work should be kept as follows : y=i(x-\-2) Y - A X ^\ y^ G ■'1 ^ ^ <- ^ E V X' ^ ^ D X <^ > c ^ ^ B y* A Y' . Fig. 7. X y Point -6 -2 A -4 - 1 B -2 C 1 D 2 2 E 4 3 F 6 4 a A line drawn through A, B, C, D, etc., is the graph of 2 ?/ - ^ — 2, GRAPHIC SOLUTIONS 191 Construct the graph of each of the following : 16. y = 3x — 7. 19. 3a;— 2/ = 4. 22. 3a; = 2 y. n. y = 2x + l. 20. 4a; — 2/ = 10. 23. 2x-{-y = l. 18. y=2x-l. 21. x-2y = 2. 24. 2x-\-Sy = 6. 249. It can be proved by the principle of the similarity of triangles that: Principle. — The graph of a simple equation is a straight line. For this reason simple equations are sometimes called linear equations. 250. Since a straight line is determined by two points, to plot the graph of a linear equation, ^j^o^ two points and draw a straight line through them. It is often convenient to plot the points where the graph intersects the axes. To find where it intersects the a;-axis, let 2/ = ; to find where it intersects the 2/-axis, let x = 0. Thus, in y=^(x + 2), when y = 0,x = -2, locating C7, Fig. 7; when X = 0, y = 1 , locating D. Draw a straight line through C and D. If the equation lias no absolute term, x = when y = 0, and this method gives only one point. In any case it is desirable, for the sake of accuracy^ to plot points some distance apart, as A and G, in Fig. 7. EXERCISES 251. Construct the graph of each of the following: 1. y = x-2. . 8. 2x-Sy = 6. 15. 8 a; - 3 2/ = - 6. 2. y = 2-x. 9. 2x-\-3y = 0. 16. -2a; + 2/ =-3. 3. y = 9 — 4:X. 10. a; -42/ = 3. 17. — 3 a; + 4^ = 8. 4. 2/ = 4a; -9. 11. 7 X — y = 14:. 18. 5x-hSy = 7i. 5. 2/ = 10 -2a;. 12. 4-a; = 2y. 19. x-iy = 3. 6. y = 2x-10. 13. 3a; + 42/ = 12. 20. ix-{-ly = 2. 1. y = 2x — 4. 14. 5a;— 2 2/ = 10. 21. .7 a; — .3 2/ = .4. 192 GRAPHIC SOLUTIONS ~~ n r \ / c \ nX] Y D \ < '/ u \ V N / \ H / \ V ■R / \ * / \ r~ / A SM \ / \ / 252. Graphic solution of simultaneous linear equations. 1. Let it be required to solve graphically the equations y^2^x, (1) 2/ = 6-^. (2) As in § 248, construct the graph of each equation, as shown in Fig. 8. 1. When aj = — 1, the value of y in (1) is represented by AB, and in (2) by AC. Therefore, when it; = — 1, the equations are not satisfied by ^^' ■ the same values of y. 2. Compare the values of y when ic = ; when a; = 1 ; 2. 3. For what value of x are the values of y in the two equa- tions equal, or coincident ? 4. What values of x and y will satisfy hoth equations ? The required values of x and y^ then, are represented graphi- cally by the coordinates of P, the intersection of the grajyhs. II. Let the given equations x + y==7, 2x + 2y = lL 5. What happens if we try to eliminate either a; or ?/ ? 6. Since y = 7 — x in both equations, what will be the relative positions of any two points plotted for the same value of a;? the relative posi- tions of the two graphs ? 7. The algebraic analysis shows that the equations are indeterminate. The graphic analysis also shows that the equations are inde- terminate, for their graphs coincide. be \ \ \ 4\ \1^ ">« N ^ V, ><> -p ^^ 2^ ^.IZ ^3s&. ^^^ ^'^^ X' ^1.^:1 _^- X ^^--u - ^: T' ^ Y' 254. equations Fig. 11. MILNE'8 IST tr. alo. 13 1. Solve graphically the Uy-Sx = 6, [2x + Sy = 12. Solution. — On plotting the graphs of both equations, as in § 248, it is found that they intersect at a point P, whose coordinates are 1.8 and 2.8, approximately. Hence, a; = 1.8 and y = 2.8. The coordinates of P are estimated to tlie nearest tenth. 194 GRAPHIC SOLUTIONS Note. — In solving simultaneous equations by the graphic method the same axes must be used for the graphs of both equations. Construct the graphs of each of the following systems of equations. Solve, if possible. If there is no solution, tell why. {x-y = 2/ = l, 9. 3. 4. 5. 6. a; -f 2 2/ = 4. 2x-y=z5, 4 X + 2/ = 16. 3x = y-\-9, 2y = 6x-lS. y = 4:X, x-y=S. I a; =1(^ + 4), [y = 2(x-2). (x-\-y= -3, 9. ^ {x-2y=-12. 10. I •'^ + 2/= 4, [y = 2-x. ^^ |aj = 2(2/ + l), 1 21 = 2(2 a; -h y). 12. (^+2/ = 8, l2x~6y=-9. 13. 14. 15. 2x — 5y = 5, 10y=2x-^l. 'Sy = 2x-7, .2x = 6 + 3y. '3(0^-4) =22/, .6(2/ + 6)= 9a;. 16. |10«' + 2/=14, \.Sx-5y=-2. ^^ ^2x-\-3y = S, [3x-r2y = S. 18. 19. 20. 21. 22. 23. (4.y + 3x = 5, {4:X-3y = 3. x + 3y=-6, 2x-4ry=z -12. 4i»-102/ = 0, 2a; + 2/ = 12. 'x-2y=:2, ^2y-Qx = 3. '3a; + 42/ = 10, . 6aj -f-82/ = 20. fa; + f2/ = 3i, 10a;-22/ = 14. REVIEW 195 REVIEW 255. 1. Distinguish between integral and fractional equa- tions; between dependent and independent equations. 2. What is meant by the root of an equation ? by solving an equation ? 3. Define and illustrate equivalent equations. 4. What is a formula ? Give a simple formula that has been used in solving some problem. 5. Find three values for x and y \n x -\- y = 15. What kind of an equation is this, and why ? 6. Define simultaneous equations ; elimination. State the axiom upon which elimination by addition is based ; elimination by comparison. 7. Outline the method of elimination by addition or subtrac- tion; by substitution. 8. State what is meant by a graph. Of what practical use are graphs ? 9. Define abscissa ; ordinate ; coordinates ; origin. 10. In making graphs, vrhere are positive values of x and y laid off ? negative values ? Interpret the equation ^ = (— 4, 3). 11. What is the abscissa of any point of the y-axis? the ordinate of any point of the ic-axis ? What are the coordi- nates of the origin ? 12. Why are simple equations sometimes called linear equa- tions ? 13. Construct the graph of 2?/ = 3 a; — 4. 14. How many points is it necessary to plot in drawing the graph of a simple equation ? Why ? 15. Tell how to determine where a graph crosses the ic-axis; the y-axis. 196 REVIEW 16. In drawing the graph of the equation 3y = 2x, what is the result, if the only points plotted are those where the graph intersects the axes ? What must be done in a case like this ? 17. Of what does the graphical solution of two simultaneous simple equations consist? 18. Solve graphically and algebraically : 2x-Sy = 10, 5x-{-2y= 6. Compare the results obtained. 19. If a system of two linear equations is indeterminate, how will this fact be shown by the graphs of the equations referred to the same axes ? 20. Draw the graphs of the two equations x + y=6y ic = 13 - ?/, and tell the algebraic meaning of the fact that the two graphs do not intersect. 21. From the following select the integral equations; the fractional equations; the numerical equations; the literal equations ; the indeterminate equations : (1) (3) (6) Sx-\- 5y = 19. ax-{-bx = c. 5 « + 2 = 3 a;— 10. (2) (4) (6) -L+l^^Sl. 2^+?^ = 31. ^ + ^ = a'b^ 2a; 3x 3 9 bx dy 22. Classify the following sets of equations as equivalent equations, dependent equations, independent equations, simul- taneous equations, or inconsistent equations : ^^^ U + 5 = ll. ^^^ \Sx- y = 2. (2) p + .^= 7' (4) I ^- ^ = ^' INVOLUTION 256. The process of finding any required power of an ex- pression is called involution. 257. The following illustrate powers of positive numbers, of negative numbers, of powers, of products, and of quotients, and show that every case of involution is an example of multi- plication of equal factors. POWERS OF A POWERS OF A POWERS OF A POSITIVE NUMBER NEGATIVE NUMBER POWER 2 = 2^ -2=(-2y 4 = 2^ 2 4 = 22 -2 4 = (-2)2 4 1G = (22)2 = 2* 2 8 = 2« -2 -8=(-2y 4 64=(22)» = 2« 2 16 = 2* -2 16 = (-2)* 4 256 = (22)* = 2« POWER OP A PRODUCT (2 . 3)=^ = (2 . 3) X (2 . 3) = 2 . 2 . 3 . 3 = 22 . 32. POWER OP A QUOTIENT 2 2^22 3 ■ 3 ~ 32' (!)■ 258. From these examples and § 78, it is seen that, for in- volution : Law of Signs. — All powers of a positive number are posi- tive; even powers of a negative number are positive, and odd powers are negative. 197 198 INVOLUTION 259. From the examples in § 257 observe that, for involution: Law of Exponents. — Tlie exponent of a power of a number is equal to the exponent of the number multiplied by the exponent of the power to which the number is to be raised. 260. The last two examples in § 257 illustrate the following : Principles. — 1. Any power of a product is equal to the prod- uct of its factors each raised to that power. 2. Any power of the quotient of two numbers is equal to the quotient of the numbers each raised to that power. 261. Axiom 6. — Tlie same powers of equal numbers are equal. Thus, if a; = 3, ic2 = 32, or 9 ; also x^ = 3*, or 81 ; etc. 262. Involution of monomials. EXERCISES 1. "What is the third power of 4 a^6 ? Solution (4 a%) 3 = 4 a35 X 4 a^fe X 4 a% = 64 a^b^ 2. What is the fifth power of - 2 ab^? Solution (-2a62)5= -2ab^x -2ah^ x -2ab'^x -2ab^x -2ab^= - 32 aS^io. To raise an integral term to any power : E,ULE. — liaise the numerical coefficient to the required, power and annex to it each letter with an exponent equal to the product of its exponent by the exponent of the required power. Make the power positive or negative according to the law of signs. Raise to the power indicated : 3. (a6V)l 7. (-4cy)3. 11. (-1)^. 4. (a^b^cy. 8. (-2aV)^ 12. (- 1)^^'. 5. (2a^cy. 9. (abcx)"^. 13. (3 bey. 6. (Ta^m'y. 10. (2 eV)^ 14. (2 aV)«. • INVOLUTION 199 16. What is the square of - ^-^9 V 7 62c j It Solution 6 a3x2 25 cfiv*^ 7 h-^c 7 62c 49 6*c2 To raise a fraction to any power : KuLE. — Raise both numerator and denominator to the re- quired power and prefix the proper sign to the result. Raise to the ■ power indicated : "■ & " i-s- '" {-■>)- "■ (;;)• "■ {-& " (-&T - m- - (-ej- - (f)-- 263. Involution of polynomials. The following are type forms of squares of polynomials : §85, (a + a;)2 = a2 + 2aa; + ar^. § 88, (a-xy==a^-2ax-{-x'. § 91, (a-x + yy = a^-{-x^-i-f-2ax-\-2ay-2xy. EXERCISES 264. Raise to the second power : 1. 2a-^b. 4. Sx-4.f. 7. 2« + 35-4c. 2. 2a-6. 5. 5m^-ll. 8. oa^-l+^w^ 3. x-\-3y. 6. 4rs2 + ^. 9. 3r' + 2s-^t\ Raise to the required power by multiplication : 10. (x + yy. 12. (x + yy. 14. (a; + y)*. 11. (x-yy. 13. (x-yy. 15. (x-yy. 200 INVOLUTION THE BINOMIAL FORMULA 265. By actual multiplication, (a + xy = a» + 3 a^o; + 3 ax^ + a^- (a-xf=a'-3a'x + 3ax''-af. (a-{-xy = a'-j-4:a^x-\-6a^x^ + 4:a3^-^x*. (a — a;)^ = a* — 4 a^x -f- 6 a V — 4 aar^ + x\ From the expansions just given, and as many others as the student may wish to obtain by multiplication, the following observations may be made in regard to any positive integral power of any binomial, the letter a standing for the first term and x for the second : 1. The 7iumber of terms is one greater than the index of the required power. 2. The first term contains a only ; the last term x only ; all other terms contain both a and x. 3. The exponent of a in the first term is the same as the index of the required power and it decreases 1 in each succeeding term; the exponent of x in the second term is 1, and it increases 1 in each succeeding term. 4. In each term the sum of the exponents of a and x is equal to the index of the required power. 5. The coefficient of the first term is 1 ; the coefficient of the second term is the same as the index of the required power. 6. The coefficient of any term may be found by multiplying the coefficient of the preceding term by the exponent of .a in that term, and dividing this product by the number of the term. 1. All the terms are positive, if both terms of the binomial are positive. 8. The terms are alternately positive and negative, if the second term of the binomial is negative. INVOLUTION 201 EXERCISES 1. Write by inspection the fifth power of (6 ~ ?/)• Solution Substituting b for a and y for x and applying the observations of § 265, (2 and 3 for the letters and exponents, 6 and 6 for the coefficients, and 8 for the signs) we have (b-y)^ = b^-bb*y + 10 b^y^ - 10 62y3 ^5 by*- y^ Note. — Observe that the coefficients of the latter half of the expansion are the same as those of the first half written in the reverse order. Expand : 2. {m 4- n)^. 3. {m — nf. 4. (a-cy. 5. (a-i-bf. 6. (b-\-dy. 7. (x-yy. 12. (a; 4- 4)". 8. (c-ny. 13. {x-^5y. 9. (x-ay. 14. (x-2y. 10. (d-yy, 15. (a; + 2y. 11. (b-\-yy. 16. (a -3/. 17. Write th« expansion of (2 c^ — 5)^ Solution § 265, (a - jc)4 = a4 _ 4 ^^y. ^ q ^2^2 _ 4 ^x* 4. 3^, Substituting 2 c^ for a and 6 for x, we have (2 c2 - 5)* = (2 c-^y - 4 (2 c2)85 + 6 (2 c2)-252 -4 (2 c2)58 + 5^ = 16 c8 - 160 c« + 600 c-* - 1000 c^ + 625. 18. Expand (1 -f if^)^, and test the result. Solution § 265, (a + xy = a^ + Sa^x + Sax^-\-3^. Substituting 1 for a and x^ for x, we have (1 + x2)3 = 13 + 3(l)2(a;2) + 3(l)(x2)2 + (a;2)8 = l + 3a;2 + 3a5* + x«. Test. — When a; = 1, (1 + a;2)8 - 8, and 1 + 3 2;2 + 3 x* + x« = 8 ; hence, (1 + x2)« = 1 + 3 a;2 -f 3 x* + x«, and the expansion is correct. Expand, and test results : 19. {x-\-2yy. 23. (l-3a^y. 27. {\-xy. 20. (2x-y)\ 24. (5a^-a6)^. 28. {\-2x)\ 21. {2x-b)\ 25. (l + a262)4. 29. (^-i)^-' 22. (x'-ioy. 26. (2aa;-6)''. 30. ik^-\yr- Expand : 31. (2. + IJ. 34. ("-ij- 37. {t-^'l 32. f^-^Y. 35. (-¥)■ 38. {H- 33. (^-t\\ \y ^1 36. (!-¥)' 39. ( ■ lA' 40. Expand (r- -s-t)\ Solution Since (j — s— ty may be written in the binomial form, (r — s — 0^, we may substitute (r — s) for a and t for a; in § 265, (a - x)3 = a3 - 3 a^x + 3 aa:^ - ^. Then, we have = (r- s)3 - 8(r -s)2« + 3(r - s)t^-t^ ^ |.3 _ 3 ^2s _^ 3 ^s2 _ gs _ 3 ^(r2_ 2 rs + s^) + 3 rf^ -Sst^- t^ = ,.3 _ 3 ,.2s + 3 ys2 _ s3 _ 3 ^2j 4- 6 rsi - 3 s2^ + 3 r^2 _ 3 ^^2 _ ^3. 41. Expand (a + 6 — c — d)^ Suggestion, (a + 6 — c - c?)^ = (a + & - c + d)^, a binomial form, Expand : 42. (a-\-x-yf. 46. (a-^x + 2y. 43. (a-m-n)3. 47. {a-x-2)\ 44. (a-a; + 2/)^ 48. {a + 2h-Zc)\ 45. (a-x-yy. 49. (a + & + a^ + 2/)'- EVOLUTION 267. Just as (§ 132) one of the two equal factors of a number is its second, or square, root ; so one of the three equal factors of a number is its third, or cube, root; one of the four equal factors, the fourth root ; etc. The second root of a number, as a, is indicated by Va ; the third root by \/a ; the fourth root by ■\/a ; the fifth root by Va; etc. The sign ^ is called the root sign, or the radical sign ; the small figure in its opening is called the index of the root. When no index is written, the second, or square, root is meant. 268. The process 2« = 2 • 2 • 2 = 8 illustrates involution. The process v^8 = ^2 • 2 • 2 = 2 illustrates evolution, which will be defined here as the process of finding a root of a num- ber, or as the inverse of involution, 269. You have learned (§ 132) that every number has two square roots, one positive and the other negative. For example, V25 = + 5 or — 5. The roots may be written together, thus : ±5, read ^plus or minus Jive \' or t5, read ^ minus or plus Jive,^ 270. The square root of — 16 is not 4, for 4^ = -f 16 ; nor — 4, for (—4)^ = + 16. No number so far included in our number system can be a square root of — 16 or of any other negative number. It would be inconvenient and confusing to regard Va as a number only when a is positive. In order to preserve the generality of the discussion of number, it is necessary, there- 204 EVOLUTION fore, to admit square roots of negative numbers into our num- ber system. The square roots of — 16 are written V^=36and -V^=n[6. Such numbers are called imaginary numbers and, in contrast, numbers that do not involve a square root of a negative num- ber are called real numbers. 271. Just as every number has two square roots, so every number has three cube roots, four fourth roots, etc. For example, the cube roots of 8 are the roots of the equa- tion x^ = 8, which later will be found to be 2,-1+ \^^^, and - 1 - V^^ The present discussion is concerned only with real roots. 272. Since 2=^ = 8, ^8 = 2. Since {-2y = -S, , ^ir8 = _2. • Since 2^ = 16 and ( - 2)^ = 16, ^16 = ± 2. Since 2^^ = 32, V^=2. Since (-2)«=-32, ■^/'=^ = -2. A root is odd or even according as its index is odd or even. 273. It follows from the preceding illustrations and from the law of signs for involution (§ 258) that, for real roots : Law of Signs. — An odd root of a number has the same sign as the number. An even root of a iiumber may have either sign. 274. A real root of a number, if it has the same sign as the number itself, is called a principal root of the number. The principal square root of 25 is 5, but not — 5. The principal cube root of 8 is 2 ; of - 8 is - 2. 275. Axiom 7. — The same roots of equal numbers are equal. Thus, if « = 16, Vx = 4 ; if a: = 8, v^ = 2 ; etc. EVOLUTION 205 276. Since (2^ = 2^""^ = 2«, the principal cube root of 2« is Hence, for evolution : Law of Exponents. — The exponent of any root of a number is equal to the exponent of the number divided by the index of the root. • 277. Since (5 a)^ = ^^d^ = 25 a^, the principal square root of 25 a" is _ V25^ = V25. -/^=5a. Hence, for principal roots : Principle. — Any root of a product may be obtained by takirig that root of each of the factors and finding the product of the results. 278. Since f - ] = ^ = ^, the principal fourth root of — is \Sj 3* 81' ^ ^ 81 ^81 ^81 ru\ Since 25 = 20 + 5, 252 _ ^20 + 5)2 = 20^ + 2(20 x 6) + 62 = 626. MILNE'S IST YR. A !.0 — 14 2^=120 w= 2 2^ + ^ = 122 210 EVOLUTION EXERCISES 283. 1. Find the square root of 3844. FIRST PROCESS 38'44l60-l-2 Explanation. — Separating the I — — number into periods of two digits ^^^^ each (Prin., § 281), we find that 2 44 the root is composed of two digits, tens and units. Since the largest 2 44 square in 38 is 6, the tens of the root cannot be greater than 6 tens, or 60. Writing 6 tens in the root, squaring, and subtracting from 3844, we have a remainder of 244. Since the square of a number composed of tens and units is equal to {the square of the tens) + {twice the product of the tens and the units) + (the square of the units)., when the square of the tens has been subtracted, the remainder, 244, is twice the product of the tens and the units, plus the square of the units, or only a little more than twice the product of the tens and the units. Therefore, 244 divided by twice the tens is approximately equal to the units. 2x6 tens, or 120, then, is a trial, or partial divisor. On dividing 244 by the trial divisor, the units' figure is found to be 2. Since twice the tens are to be multiplied by the units, and the units also are to be multiplied by the units to obtain the square of the units, in order to abridge the process the tens and units are first added, forming the complete divisor 122, which is then multiplied by the units. Thus, (120 + 2) multiplied by 2 = 244. Therefore, the square root of 3844 is 62. SECOND PROCESS Explanation. — In practice it is usual to place the figures of the same order in the same column, and to disregard the ciphers on the right of the products. Since any number may be regarded as composed of tens and units, the foregoing processes have a general application. Thus, 346 = 34 tens + 6 units ; 2377 = 237 tens + 7 units. t'= 38'44[62 36 2^=120 u= 2 2 44 2t-^u = 122 244 EVOLUTION 211 2. Find the square root of 104976. Solution 10'49'76| 9 Trial divisor = 2 x 30 = 60 Complete divisor = 60 + 2 =62 1 49 1 24 Trial divisor == 2 x 320 = 640 Complete divisor = 640 + 4 = 644 25 7€ 25 76 Rule. — Separate the number into periods of two figures each, beginning at units. Find the greatest square in the left-hand period and write its root for the first figure of the required root. Square this root, subtract ike result from the left-hand period, and annex to the remainder the next period for a new divi- dend. Double the root already found, with a cipher annexed, for a trial divisor, and by it divide the dividend. The quotient, or quotient diminished, will be the second figure of the root. Add to the tried divisor the figure last found, multiply this complete divisor by the figure of the root last found, subtract the product from the dividend, and to the remainder annex the next period for the next dividend. Proceed in this manner until all the periods have been used. The result will be the square root sought. 1. When the number is not a perfect square, annex periods of decimal ciphers and continue the process. 2. Decimals are pointed off from the decimal point tovs^ard the right. 3. The square root of a common fraction may be obtained by finding the square root of both numerator and denominator separately or by reducing the fraction to a decimal and then finding the root. Find the square root of: 3. 529. 6. 57121. 9. 2480.04. 4. 2209. 7. 42025. 10. 10.9561. 5. 4761. 8. 95481. 11. .001225. 212 EVOLUTION Find the square root of : 12. m. 14. if|. 16. ifj. 18. m 13. lif. 15. tV¥^. 17. lU- 19. Iff. Find the square root to two decimal places : 20. |. 22. |. 24. f. 26. J. 21. |. 23. .6. 25. |. 27. yV ROOTS BY FACTORING 284. The method of finding the cube root of polynomials and of arithmetical numbers, analogous to the one just given for square root, is beyond the scope of this text ; but a method of finding the cube root, or any other root, of a number that is a perfect power of the same degree as the index of the required root is here mentioned because of its simplicity. This method consists in factoring, grouping the factors, and taking the required root of each group. Thus, V42875 = ^5 . 5 . 5 X 7 . 7 . 7 = ^63 X 73 = 5 X 7 z also, Va:* + 2 x3 _ 3 a;2 - 4 a; + 4 = V(x - l)\x + 2)2 = (x-l)(x-h2) • = x^^x-2. EXERCISES 285. Find, by the method of factoring : 1. Square root of a« - 12 a^ + 36. 2. Guhevootoix^-15a:P-\-75x-125. 3. Fourth root of a;^ - 8 a.-« + 24 x^ - 32 a; + 16. 4. Fifth root of a^ - 10 ic* + 40 a^ - 80 a^ -f 80 a; - 32. Find the indicated root : 5. ^3375. 7. -v/262144. 9. ^/4084101. 6. ^1296. 8. V759375. 10. ■v/16777216. RADICALS 286. Thus far the exponents used have been positive integers only, and the laws of exponents have been based on this idea ; but since zero, fractional, and negative exponents may occur in algebraic processes, they must follow the same laws as are given for positive integral exponents ; hence, it becomes neces- sary to discover meanings for these new kinds of exponents, because, for example, in aP, a~^, and at, the exponents 0,-2, and •J cannot show how many times a is used as a factor (§9). 287. Meaning of zero and negative exponents. By notation, §§ 9, 10, a^ = 1 . a • a. (1) Dividing both members of this equation by a, the first mem- ber by subtracting exponents (§ 32) and the second by taking out the factor a, we have a} = l'a. (2) Dividing (2) by a, a« = 1. (3) Dividing (3) by a, o"^ = -• (4) Dividing (4) by a, a"^ = — • (5) The meaning of a zero exponent, illustrated in (3), and of a negative exponent, in (4) and (5), may be stated as follows : . Any number with a zero exponent is equal tol. Any number with a negative exponent is equal to the reciprocal of the same number with a numerically equal positive exponent. 213 214 RADICALS 288. The meaniDg of a negative exponent shows that: Principle. — Any factor may he transferred from one term of a fraction to the other without changing the value of the fraction, provided the sign of the exponent is changed. 289. Meaning of a fractional exponent. Just as, § 276, Vo^ = a^-^ = a, so -y/o? = a^^^ = ai. That is. The numerator of a fractional exponent with positive integral terms indicates a power and the denominator a root. Since the operations may be performed in either order : The fractional exponent as a whole indicates a root of a power or a power of a root, EXERCISES 290. Find a simple value for : 1. 5«. 3. 2-\ 5. (-3)«. 7. (a%\y. 2. 4-1 4. 3-1 6. (-6)-l 8. (-J)"'. 9. Which is the greater, (4.)^ or {\Y? (i)-^ or (^)-3? 10. Write 5 aj~y with positive exponents. Solution. — By § 287 , 5 x-^y'^ = by'^— = ^. Write with positive exponents : 11. 2a;-\ 13. a'^^h-K 15. 4^ah-\ 12. 5a-\ 14. x-^y-^. 16. Saa;-^ 3 a^v 17. Write — f- without a denominator. hoir Solution. — By § 288, ^^ = 3 a^ft-^-ay. Write without a denominator ax _ mn by ' a^ "" a~^lf 18. ^. 19. ???. 20. -U- 21. "''" RADICALS 215 22. Find the value of 16^. First Solution. 16^ = ^/W = v 16 • 16 • 16 = v''(2 . 2 . 2 . 2) (2 . 2 . 2 • 2) (2 . 2 . 2 .2 ) = v/(2 . 2 • 2) (2 . 2 . 2) (2 . 2 . 2)(2 . 2 . 2) = 2.2-2 = 8. Second Solution. 16* = (16^)3 = 23 = 8. In numerical exercises it is usually best to find the root first. Simplify, taking only principal roots : 23. Sl 25. 64i 27. 64"^. 24. si 26. 32^ 28. (-8)"! 29. Which is the greater, 27^ or (- 27)"^? (})^ or (i)~^? 30. Express va^6c'~'* with positive fractional exponents. 2 1 Solution. y/a^bc^ = ahh~^ = ^-^ . Express with positive fractional exponents : 31. Va63. 33. (Vxf, 35. (\/^)-l 32. ^xy. 34. (a/?/)*. 36. SVaJ-^"*- Express roots with radical signs and powers with positive exponents : 37. ^\ 39. xK 41. x^y^. 43. a^-^x^. 38. xi 40. ahi 42. ah-y 44. x^^2/^. Multiply : 45. a^ by a-\ 47. a* by a-*. 49. a^ by a*'. 46. a^bya-^ 48. a by a-\ 50. a;i by xK Divide : 51. a«bya«. 53. a^ by a -I 55. a; 2 by jci 52. a^ by a®. 54. ar^ by «" 2. 56. a;""^ by a^-l 216 RADICALS Solve for values of x corresponding to principal roots by applying axioms 6 and 7 (§§ 261, 275), and test each result; 57. x^ = l. . 61. a;~^ = 6. 58. aj^ = 8. 62. a;"t = 144. 59. a;^ = 81. 63. 25a;-^ = l. 60. \x^=12. 64. a;^ + 32 = 0. 291. An indicated root of a number is called a radical; the number whose root is required is called the radicand. \/5a, (a^)^, y/d^ + 2, and {x-\- y)* are radicals whose radicands are, respectively, 5 a, x^, a^ + 2, and x-{-y. 292. The order of a radical is shown by the index of the root or by the denominator of the fractional exponent. y/a + x.and (6 — xy are radicals of the second order. 293. In the discussion and treatment of radicals only pritv- cipal roots will be considered. Thus, Vl6 will be taken to represent only the principal square root of 16, or 4. The other square root will be denoted by — VlG. 294. Graphical representation of a radical of the second order. In geometry it is shown that the hypotenuse of a right triangle is equal to the square root of the sum of the squares of the other two sides; consequently, a radical of the second order may be represented graphically by the hypotenuse of a right triangle whose other two sides are such that the sum of their squares is equal to the radicand. Thus, to represent V5 graphically, since it may be observed that 5 = 22 + 12, draw OA 2 units in length, then draw AB 1 unit in length in a direction perpendicular to OA. Draw OB, completing the right-angled triangle OAB. Then, the length of OB represents V5 in its relation to the unit length. RADICALS 217 EXERCISES 295. Eepresent graphically : 1. V2. 3. Vl3. 5. V34. 7. V|. 2. ViO. 4. Vl7. 6. V25. 8. Vif. 296. A number that is, or may be, expressed as an integer or as a fraction with integral terms, is called a rational num- ber. 3. I, V^, and .333 are rational numbers. 297. A number that cannot be expressed as an integer or as a fraction with integral terms is called an irrational number. v^^ 4^^ 1 _^ y/S^ and V 1 + V3 are irrational numbers. From § 294, it will be observed that the irrational number V5 can be represented graphically by a line of exact length, though it cannot be represented exactly by decimal figures, for Vb = 2.236..., which is an end- less decimal. 298. When the indicated root of a rational number cannot be obtained exactl}'', the expression is called a surd. V2 is a surd, since 2 is rational but has no rational square root. V 1 + V3 is not a surd, because 1 + V3 is not rational. Radicals may be either rational or irrational, but surds are always irrational. Both Vl and \/S are radicals, but only V3 is a surd. 299. A surd may contain a rational factor, that is, a factor whose radicand is a perfect power of a degree corresponding to the order of the surd. The rational factor may be removed and written as the co- efficient of the irrational factor. In \/8 = V'4 X 2 and Vbi= \/27 x 2, the rational factors are Vi and \/27, respectively ; that is, VS = 2 v^ and V54 = 3 y/2. 300. In the following pages it will he assumed that irrational num- bers obey the same law as rational numbers. For proofs of the generality of these laws, the reader is referred to the author's Advanced Algebra. 218 RADICALS REDUCTION OF RADICALS 301. To reduce a radical to its simplest form. As the work progresses the student will discover the mean- ing of simplest form. EXERCISES 302. 1. Eeduce V20a^to its simplest form by writing the rational factor as the coefficient of the irrational factor. PROCESS V20a«= V4a«x5 = V4a«x V5=2aV5 Explanation. — Since the highest factor of 20 a^ that is a perfect square is 4 a^, \/20 a^ is separated into two factors, a rational factor \/4 a^, and an irrational factor VS ; that is, § 277, V'20 a^ = ■\/4 a'^ x \/5. On finding the square root of 4 a^ and prefixing the root to the irrational factor as a coefficient, the result is 2 a^Vb. 2. Reduce V — 864 to its simplest form. PROCESS 3/ KTTi 3/ /x=-4. Squaring, a; = 16. Verification. — Substituting 16 for x in the given equation and (§ 293) considering only the positive value of Vx — 7 and of Vx, the first member becomes Vl6 -7 - Vl6 = V9 - Vl6 = 3- 4 = - 1 ; but the second member of the given equation is 7 ; hence, x = 16 does not satisfy the equation. That is, the equation has no root, or is impossible. General Directions. — Transpose so that the radical term, if there is but one, or the most comjylex radical term, if there is more than one, may constitute one member of the equation. Then raise each member to a power corresponding to the order of that radical and simplify. If the equation is not freed of radicals by the first involution, proceed again as at first. Solve, and verify results, denoting impossible equations: 4. \/^Tl = 3. 13. Va; + 16-Vx = 2. 5. Va; -h 5 = 4. 14. V2^ - V2 a; - 3 = 1. 6. Vx~^ = l. 15. V2a-+-V2a;- 3 = 1. 7. Va; - a- = b. 16. -^Jx^ -{-x + l = 2 — x. 8. v/«^^ = 2. 17. 3V^^^ = 3.r-3. 9. -Vx -a^=:a. 18. V 3 .'c + 7 + V3 a.- = 7. 10. Vi H- 6 = a. 19.- 2 V« + V4a; — 11 = 1. 11. l+\/a; = 6. 20. 5~V.'M^=V«. 12. 2Va = 6-v^. 21. Vx^ -5d; + 7 + 2 = a;. 234 RADICALS Solve, and verify results, denoting impossible equations : 22. 4:-V4:-Sx-]-9x^ = Sx. 23. V3a; — 5-h V3a; + 7 = 6. 24. \/4a;-|-5 — 2Vx— 1 = 9. 25. V2a;-1+ V2aj + 4=5. 26. V5a;- 1-1 = V5a;-f-16. 27. V^-h3V5a;-16-4 = 0. 28. 2a; -^4 0^- Vl6a^-7 = 1. 29. V^x - V^ = V9a;-32. 30. ■\/2(x' + Sx-5) = (a; + 2) V2. 31. V2(a7 + 1)4-V2a;-1= V8a;-f 1. 32. Solve the equation V2 x — V2 x — 7 = Suggestion. — Clear the eq 33. Solve the equation V2a;-7 Suggestion. — Clear the equation of fractions. V3^ + 15 V3^-f-6 V3a;+ 5 V3x + 1 Suggestion. — Some labor may be saved by reducing each fraction to a mixed number and simplifying before clearing of fractions. Thus, 1 + ^Q = 1 + ^ V3a: + 5 V3a: + 1 Canceling, and dividing both members by 5, 2 ^ 1 \/3x + 5 VSx + 1 Solve and verify : ,^ Vs-1 Vs-3 ' ^^ V2¥+9 V2x + 20 Vs + 5 Vs-1' V2a;-7 V2a;-12 35. Vr:^ ^ V^"^^ 37 V^ + 18 ^ 32 ^ -^ Vm V^^^ ' V»H-2 Va;>6 REVIEW 236 REVIEW 335. 1. Distinguish between involution and evolution. 2. Give the law of exponents for involution ; for evolution. 3. For what values of n between 1 and 12 is (— 2)" posi- tive ? negative ? What is the sign for any power of a positive number ? 4. How is a fraction raised to a power ? How is the root of a fraction found ? Raise -f^ to the second power. V^= ? 5. How is the power of a product found? the root of a product ? 6. What operation is indicated by a radical sign? In what other way may this operation be indicated ? Illustrate. 7. How many values has V25 ? What is the principal square root of 25 ? What is the principal cube root of — 8? 8. What is the index of a root? What index is meant when none is expressed ? 9. Distinguish between real and imaginary numbers and illustrate each. 10. What is the sign of an odd root of a number ? of an even root of a number ? 11. What is the meaning oi aP? 12. When a number has a fractional exponent, what does the numerator of the exponent show ? the denominator ? 13. What is the meaning of x-^ ? How may any factor be transferred from one term of a fraction to the other ? Illustrate by writing without a denominator : — - • 14. Expand (x — yf by the binomial formula. How does the number of terms correspond with the exponent of the power ? What is the coefficient of the first and last terms ? of the second term? How are the coefficients of the other terms obtained ? 236 REVIEW 15. Define and illustrate radical, radicand, entire surd, and mixed surd. 16. What is meant by the order of a radical? Illustrate by giving radicals of different orders. 17. How may a radical of the second order be represented graphically ? Illustrate by representing graphically V26. 18. What is a rational number? an irrational number? From the following select the rational numbers : 8; I; V3; axis. At D and at D', where the ordinates are equal to 0, the value oi ac^—6x-\-5 is 0, and the abscissas are x—1 and x=5. Hence, the roots of the given equation are 1 and 5. The curve obtained by plotting the graph of ar^ — 6 ic + 5, or of any quadratic expression of the form ajr^ + 6jr+c, is a parabola. EXERCISES 357. 1. Solve graphically the equation o^ — 8 a; + 14 = 0. Solution. — Since the coefficient of x is — 8, §356, first substitute 4 for X. Points and their coordinates are given in the table : y = a;2 - 8 a: -f 14. X y Points 4 -2 A 3, 5 -1 B, B' 2, 6 2 c, a 1,7 7 D,D' A , Plotting these points and drawing a smooth curve through them, we have the graph of y = x^ — 8 x + 14, which crosses the x-axis approxi- mately at x = 2.6 and x = 5.4. Hence, to the nearest tenth, the roots of x^ — 8 x -f 14 = are 2.6 and 6.4. MILNE's IST YR. ALG. — 17 268 GRAPHIC SOLUTIONS 2. Solve graphically the equation a^ — 8 ic -f 16 = 0. Solution. — Since the coefficient of x is — 8, §356, first substitute 4 for X. Points and their coordinates are given in the table : y = a;2_8x4-16 X y Points 4 A 3,5 1 B, B> 2,6 4 C, C 1,7 9 D, D< 1 ' ' Dl i° / / I C \ if \ A* B ^ / B A 1 Plotting these points and drawing a smooth curve through them, we have the graph oi y = x^ — Sx + 16, which touches the ic-axis at x = 4. This fact is interpreted graphically to mean that the roots of the equa- tion x^ — Sx + 16 = are equal, both being represented by the abscissa of the point of contact. Hence, the roots are 4 and 4. Note. — The student may show that 4 and 4 are the roots by solving the equation algebraically. 3. Solve graphically the equation a.-^ — 8 ic -f- 18 = 0. Solution. — Since the coefficient of a; is — 8, §356, first substitute 4 for X. Points and their coordinates are given in the table : i -dr — [ fb - A y = a;2-8a; + 18 X y Points 4 2 A 3,5 3 B, B' 2,6 6 C, C 1,7 11 D,D' , Plotting these points and drawing a smooth curve through them, we GRAPHIC SOLUTIONS 259 have the graph of y = x^ — S x + 18^ which neither crosses nor touches the X-axis. This fact is interpreted graphically to mean that the roots of the equation x^ — 8 x + 18 = are imaginary. Note. — The student may show that the roots are imaginary by solving the equation algebraically. In eaxjh of the preceding graphs, the point A, whose ordinate is the least algebraically that any point in the graph has, is called the minimum point. When the coefficient of ic^ is -f 1, it is evident that : Principles. — 1. If the minimum point lies below the x-axis, the roots are real and unequal. 2. If the minimum point lies on the x-axis, the roots are real and equal. 3. If the minim^im point lies above the x-a^is, the roots are imaginary. Solve graphically, giving real roots to the nearest tenth : 4. x^-Ax-\-Z==0. 9. x^-2x-2 = 0. ' 5. a?-(Sx + l =0. 10. x^ = &x-^. 6. a^-4a;=-2. 11. a.-^ + 4 a; + 2 = 0. 7. a? + 2{x-\-V)=0. 12. ar'-2a; + 6 = 0. 8. a^-4a; + 6=0. 13. a^-4a;-l = 0. 14. Solve graphically 4 a; — 2 a.-^ + 1 = 0. Suggestion. — On dividing both members of the given equation by — 2, the coefficient of x^, the equation becomes a:2 _ 2 X - i = 0. The roots may be found by plotting the graph of y = x^ — 2 x — ^. Solve graphically, giving real roots to the nearest tenth: 15 2.^2 + 8.^ + 7 = 0. 17. 12a;-4a^-l=0. 16. 2ar^-12a; + 15 = 0. 18. 11 +8a; + 2a,-2 = 0. Note. — Another method of solving quadratic equations graphically is given in § 379. EQUATIONS IN QUADRATIC FORM 358. An equation that contains but two powers of an unknown number or expression, the exponent of one power being twice that of the other, as oa?^" + 6a;" -|- c = 0, in which n represents any number, is in the quadratic form. EXERCISES 359. 1. Given a;^-10i»2 4- 24 = 0, to find the value of a;. Solution a4 _ 10 a;2 + 24 = 0. Factoring, {x'^ — 4) (ic^ _ 6) = 0. Hence, ic^ _ 4 _ q or ic^ - 6 = ; whence, x= ±2 ot: x= ± V6. Each of these values substituted in the given equation is found to verify ; hence, ± 2 and ± V6 are roots of the equation. 2. Given a;^— oj^ — 6 = 0, to find the values of x. First Solution cc^ - x^ - 6 = 0. Let X* = m, then x^ = m^ and the equation becomes w2 - w - 6 = 0. Solving by factoring, m = 3 or — 2 ; that is, jc^ = 3 or - 2. Raising to the fourth power, a; = 81 or 16. 260 EQUATIONS IN QUADRATIC FORM 261 Second Solution Transposing, x^ — x^ = 6. Completing the square, x^ — x^ + (i)2 = -^. Taking the square root, x* — ^ = ± |. .-. a;^ = 3or -2. Raising to the fourth power, x = 81 or 16. Since x = 16 does not satisfy the given equation, 16 is not a root and should be rejected. Solve, and verify each result : 3. x*-13a^ + 36 = 0. 9. x^-x^ = 6, 4. a;^- 18^2 + 32 = 0. lO. a; + 2Va; = 3. 5. a;*-14a;2-f-45 = 0. n. a;^ _ a;^ - 12 = 0. 6. 2a;^-10a^ + 8=0. 12. (a;-3)2-|-2(a;- 3)=3. 7. 3a;*-llar^ + 8 = 0. 13. (x' - ly - ^x" - 1) = 5. 8. a;i-5a;i + 6=0. 14. (a^-4)2-3(a^ -4) = 10. 15. Solve the equation o^ _ 7 a; + Var' - 7 a; + 18 = 24. Solution x^- Adding 18 to both ra( -7x 3mbe + 18 : + Vx2-7a; + 18 = 24. rs, we have -7. -2. 0) x^-Tx ( + Va;'-^ - 7 X + 18 = 42. and m2 for x2 - 7 x + 18. m2 + m - 42 = 0. m = 6 or (2) Put m for Va;2 - 7 X - Then, transposing, Solving by factoring, f 18 (3) (4) That is. Vx2-7x + 18 = 6, (5) or Squaring (5), Solving, Vx2 - 7 X + 18 = - 7. x2-7x + 18=36. X = 9 or - (6) Since, in accordance w ith § 293, the radical in (6) cannot equal a negative number, Vx2 _ 7 x + 18 = — 7 is an impossible equation. Hence, the only roots of (1) are 9 and — 2. 262 EQUATIONS IN QUADRATIC FORM 16. Solve the equation x + 2V« + 3 = 21, and verify results. Solution a^_9x3+8 = 0. Factoring, (a;3-l)(a;3-8)=0. Therefore, a:8-l=0, or a;3 - 8 = 0. 17. Solve a;2 _ 3 a; 4- 2 Va^ - 3 a; + 6 = 18, and verify results. 18. Solve the equation x^ — 9 a^^ + 8 = 0. (1) (2) (3) (4) If the values of x are found by transposing the known terms in (3) and (4) and then taking the cube root of each member, only one value of X will be obtained from each equation. But if the equations are factored, three values of x are obtained for each. Factoring (3) , (a; - 1) (a;2 + a; + 1) = 0, (5) and (4), (ic - 2) (x2 + 2 a; + 4) = 0. (6) Writing each factor equal to zero, and solving, we have : From (5), a^ = 1, |(- 1 + v^ITf), l{-\- yf^l). (7) From (6), a; = 2, - 1 + V^^, - 1 - V^. (8) Note. — Since- the values of x in (7) are obtained by factoring a^ — 1 = 0, they may be regarded as the three cube roots of the number 1. Also, the values of x in (8) may be regarded as the three cube roots of the numbers (^^271). Solve : 19. a;«-28a;3^27=0. 20. a;^-16 = 0. 21. Find the three cube roots of — 1. 22. Find the three cube roots of — 8. 23. Solve the equation a;* + 4 a:^ — 8 a; + 3 = 0. Solution By applying the factor theorem (§146), the factors of the first mem- ber are found to be a; — 1, a; + 3, and a;^ -f 2 x — 1 ; that is, («-l)(a; + 3)(a;2 + 2a;-l) =0. Solving, a; = 1, - 3, - i ± V2. EQUATIONS IN QUADRATIC FORM 263 Solve, and verify each result : 24. x'-{-x'-4.x = -2. 26. a^-8ic2 + 5a;+6 = 0. 25. x*-4:3^+Sx=-S. 27. a;^-6a^ + 27a; = 10. 28. a;* + 6a^4-7a^-6a;-8 = 0. 29. a;*4-2ic3-10a-2-lla; + 30 = 0. MISCELLANEOUS EXERCISES 360. Solve, aud verify each result : 1. 0.^-250:^4-144 = 0. ^ x^-2o:^-3 = 0. 2. a;^-45or' + 500 = 0. e. a'-3a;^=-2. 3. 2o;*-lla^ + 12 = 0. 7. a;^-2a;* = 3. 4. 5a;*-24ar'4-16 = 0. 8. ic^-lOx* -f 9 = 0. 9. (iB2_l)2_4(ar^-l)+3 = 0. 10. (af^_6)2-7(a;2-6)-30 = 0. 11. (3(^-2xy-2{a:^-2x') = S. 12. a;-6a;^ + 8 = 0. 16. a;-5 + 2Va;-5 =8. 13. a;4-20-9V^=0. 17. 2«- 3V2a; + 5= -5. 14. 2a;^-3a;^+l = 0. 18- 2 aj-6V2o;-l = 8. 15. ^t_7^i^io = 0. 1^-. ^-3 = 21-4V^33; 20. iB^-10a^ + 35ar'-50a;+24 = 0. 21. 4:X'-4:X^-7X^ + 4:X + S = 0. 22. 16a;^-8ar^-31a^^-8.^' + 15=0. 23. ic2-a;-Va;--a;-|-4-8 = 0. 24. a;2-5a; + 2Va;--5a;-2 = 10. SIMULTANEOUS QUADRATIC EQUATIONS 361. Two simultaneous quadratic equations involving two unknown numbers generally lead to equations of the fourth degree, and therefore they cannot usually be solved by quad- ratic methods. However, there are some simultaneous equations involving quadratics that may be solved by quadratic methods, as shown in the following cases. 362. When one equation is simple and the other of higher degree. Equations of this class may be solved by finding the value of one unknown number in terms of the other in the simple equation, and then substituting that value in the other equation. 363. 1. Solve the equations \ ^ ' Solution x + y = 1. (1) -jS.-. 3a;2 + ?/2:^43. (2) From (1), y = 7 — a. (3) Substituting in (2), 3a;2.f-(7_«)2 = 43. (4) Solving, a; = 3 or ^. (6) Substituting 3 for x in (3), 2/ = 4. (6) Substituting \ for x in (3), y = ¥- (7) That is, X and y each have two values ' when a; = 3, y = 4, when iK = ^, y = Y • 264 SIxMULTANEOUS QUADRATIC EQUATIONS 265 Solve, and verify results : ^ (x^ + f = 20, ^ (x^ + xy = 12, \x=2y. ' [x — y = 2. (10x-^y = Sxyy fm^-S^i^rzzlS, y-x = 2. [m-2n = L 4 f^ = 6-2/, (3x(y-\-l)=12, [x'-hf = 72. ' [3x = 2y. (xy(x-2y)=10, ^ n [xy = 10. ' 1: xy(x — 2y)=10, ^ (3rs — 10r = s, 2 - s = - r. 364. An equation that is not affected by interchanging the unknown numbers involved is called a symmetrical equation. 2x^ + xy + 2y^ = 4: and x^-\-y^ = 9 are symmetrical equations. 365. When both equations are symmetrical. Though, equations of this class may usually be solved by substitution, as in §§ 362, 363, it is preferable to find values f or a; + y and x — y and then solve these simple equations for X and y. EXERCISES 1. Solve the equations xy = 10. Solution x + y = 7. (1) xy = 10, (2) Squaring ( 1 ) , x^ +2xy -\-y^ = i9. (3) Multiplying (2) by 4, ixy=z 40. (4) Subtracting (4) from (S), x^-2xy -hy^ = 9. (5) Taking the square root, x — y = ±S. (6) From (1) + (6), a; = 5 or 2. From(l)~(6), y = 2or5. 266 SIMULTANEOUS QUADRATIC EQUATIONS r a^ _l_ y2 __ 25 2. Solve the equations \ ' Squaring (2), Subtracting (1) from (3), Subtracting (4) from (1), Taking the square root, From (2) + (6), From (2) - (6), Solution ^2 + 2/2 _ 26. x^-\-2xy-\- y^ = 49. 2xy = 24. ofi-2xy ■\-y^= 1. x-y = ±\. a; = 4 or 3. 4. 5. Solve, and verify each result : 3 1^.^ + 2/^ = 50, x-\-y = ^, ic2 + i/2 = 34. a; -h ?/ = 9, a;3 + 2/' = 243. ic^ 4- 0^ + 2/2 = 49. fa^ + a.2/ + 2/' = 31, iB=^ + 2/2 = 26. 8. 9. ^ 10. 11. 12. y = 3 or 4. a^ + / = 8, I ^2/ = 12, I a; — a;?/ + ?/ = — 5. a^ + 3a;?/ + ?/2 = 31, xy=e>. a^ + / = 100, 196. ra^+/=i U^ + 2/)' = a^ + / = 13, la^ + 2/2^a.'2/ = 19. (1) (2) (3) (4) (5) (6) 367. An equation all of whose terms are of the same degree with respect to the unknown numbers is called a homogeneous equation. x^ — xy = y^ and Sa^ -\-y^ = are homogeneous equations. An equation like a^ — xy-\-y^ = 21i8 said to be homogeneous in the unknown terms. SIMULTANEOUS QUADRATIC EQUATIONS 267 368. When both equations are quadratic and homogeneous in the unknown terms. Substitute vy for a;, solve for y- in each equation, and com- pare the values of if thus found, forming a quadratic in v. EXERCISES {x^-xy-^y^ = 21, 369. 1. Solve the equations 1 „ ^ . . [y^ — 2xy= —16, Solution a;2-a;y + y2 = 21. (1) 2/2 - 2 xy = - 15. (2) Assume x = vy. (3) Substituting in (1), «2y2_ry2 + y2 = 21. (4) Substituting in (2), y2-2ry2=:_15. (5) Solving (4) for y\ t/2- 21 (6) ^ tj2 - t> + 1 Solving (5) for y'^. «2- 15 . (7) ^ -2t;-l Comparing the values of y"^ 15 21 (8) 2 r - 1 «2 _ t> + 1 Clearing, etc., 6t |2-19t? + 12=r0. (9) Factoring, (r- - 3)(5 r - 4) = 0. (10) .-. r = 3or^. (11) Substituting 3 for v in (7) or in (6), y= ± V3 j (12) and since x = vy^ x = ± 3 v3. J Substituting f for t? in (7) or in (6), 2/ = ± 5 | and since x = vy, x = ± iJ When the double sign is used, as in (12) and in (13), it is understood that the roots shall be associated by taking the upper signs together and the lower signs together. (x=SVB; -3V3; 4; -4 [y= V: Solve, and verify results : .xy-f = S. " I ic2 + y2 = 125. Hence, j - _ {y= VS; - V3; 5 ; - 5. ra^ + 3?/2 = 84, r2a^-3a:2/ + 2/ = 100, 268 SIMULTANEOUS QUADRATIC EQUATIONS ^ ^a^-xy-^f = 21, 5. 6. U2 + 22/' = 27. ' x(x-y)=6, .ic2 + 2/2 = 5. xy^Sf = 20y [af-Sxy = -S. ^ (x'-^xy=.12, .xy + 2y^ = 5. ^ (x^-\-2f==U, [xy — y^ = S. ^ (a^-\-xy = 77, I xy — y^ = 12. 370. Special devices. Many systems of simultaneous equations that belong to one or more of the preceding classes, and many that belong to none of them, may be solved readily by special devices. It is impos- sible to lay down any fixed line of procedure, but the object often aimed at is to find values for any two of the expressions, x-^y, x — y, and xy, from which the values of x and y msLj be obtained. Various manipulations are resorted to in attaining this object, according to the forms of the given equations. 371. 1. Solve the equations EXERCISES x^-\-xy = 12, Solution x2 + iC?/ = 12. xy -{-y^ = 4. Adding (1) and (2), x^-\-2xy + y^ = 16. .-. X -\- y = -\- 4: or — 4 Subtracting (2) from (1), x^-y'^ = S. Dividing (5) by (4), ic - y = + 2 or - 2 Combining (4) and (6), (1) (2) (3) (4) (5) (6) a; = 3or— 3; 2/ = lor — 1. Note. — The first value otx — y corresponds only to the first value of x + y, and the second value only to the second value. Consequently, there are only two pairs of values of x and y. Observe that the given equations belong to the class treated in § 368. The special device adopted here, however, gives a much neater and sim- pler solution than the method presented in that case. SIMULTANEOUS QUADRATIC EQUATIONS 269 2. Solve the equations | ^ + 2/' + aJ + ^ = 14, Solution X'2 + y2 + X + y = U. (1) xy = 3. (2) Adding twice the second equation to the first, x^-\-2xy + y2 + x + y = 20. Completing the square, (x + yy + (x + y) -\- (^)2 = 20|. Taking the square root, x-\-y-\-^=±^. .'. x + y = 4oT — 5. (3) Equations (2) and (3) give two pairs of simultaneous equations, \^-^y = ^ and l^ + y=-^ lxy = 3 [xy = S Solving, the corresponding values of x and y are found to be x = 3; 1; K-5+>/l3); i(-5->/l3); . y = l; 3; i(_5-Vl3); i(_6 + VT3). Sjnnmetrical except as to sign. — When one of the equations is symmetrical and the other would be symmetrical if one or more of its signs were changed, or when both equations are of the latter type, the system may be solved by the methods used for symmetrical equations (§ 365). 3. Solve the equations |^ + 2/'-53, [x — y = 5. Solution x^ + y^ = 53. (1) X - y = 6. (2) Squaring (2) , x^-2xy-hy^ = 25. (3) Subtracting (3) from (1), 2xy = 28. (4) Adding (4) and (1), x^ -{- 2xy -\- y^ = S\. (6) Taking the square root, x + y = ±9. (6) From (6) and (2), x = 7 or - 2 ; and y = 2 or - 7. 4. Solve the equations 270 SIMULTANEOUS QUADRATIC EQUATIONS Division of one equation by the other. — The reduction of equa- tions of higher degree to quadratics is often effected by divid- ing one of the given equations by the other, member hy member. x-y=2. Suggestion. — Dividing the first equation by the second, x'^ + xy -{- y"^ = 13. Therefore, solve the system, r x2 + a:?/ -H 2/2 = 13, U - y = 2, instead of the given system. Elimination of similar terms. — It is often advantageous to eliminate similar terms by addition or subtraction, just as in simultaneous simple equations. 5. Solve the equations {xy + x = 3^j Solution • xy-\-x = :35. (1) xy + y = 32. ^ (2) Subtracting (2) from (1), x — y = 3; whence, y = a: -3. * (3) Substituting (3) in (1) , x(ix-S)+x = 36, or x^ — 2x = 36. Solving, X = 7 or - 5. (4) Substituting (4) in (3), y = 4 or - 8. Solve, using the methods illustrated in exercises 1-5 ; verify : ( m^-\- mn = 2, I mw-f 7r = —1. xy + x = Z2, .xy + y = 27. 8. 'a2-f&2 = l30, ^^ .a-6 = 2. 2x^-3^ = 5, SIMULTANEOUS QUADRATIC EQUATIONS 2T1 372. All the solutions in §§ 362-371 are but illustrations of methods that are important because they are often applicable. The student is urged to use his ingenuity in devising other methods or modifications of these whenever the given system does not yield readily to the devices illustrated, or whenever a simpler solution would result. 1. 3. 5. 6. 9. 10. \^ + f = x^— xy = 48, xy — y- = 12. r a + aft + 28 = 0, a^=-12. rr^ + 2/2 = 40, [xy=^12. fa.-»-f2/3 = 28, 1 a; + 2/ = 4. ra^-|-ar^ = 44, \xy + f=-2%. ^ ric2 + 3ic.y-/ = 9, U + 22/ = 4. 11. 12. 13. 14. 15. 16. 17. 18. MISCELLANEOUS EXERCISES 373. Solve, and verify each result : ^ + 2/ = 3, xy = 2. 5a^-42/' = 44, 4iB2-52/2 = 19. l + x = 61 4if2/ = 96-ary, I a; + 2/ = 6. (Qi? — xy = ^f \xy + f=n. (x{x-\-y) = x, ■x^ + xy + f = l^l, .a^-f 2/2 = 106. (1+x^y, l4 + 4ar' = 2/». (x^-y* = 369, |a-2_2/2 = 9. x^ — .Ty = 6, i^r^ + 2/- = 61. jic + 2/ = 25, 19 ^ ^ + xy-\-f=.l% xf = 19. 20. '^ + 3xy = f^% a; + 3 2/ = 5. 272 SIMULTANEOUS QUADRATIC EQUATIONS Problems 374. 1. The sum of two numbers is 12, and their product is 32. What are the numbers ? 2. The sum of two numbers is 17, and the sum of their squares is 157. What are the numbers ? 3. The difference of two numbers is 1, and the difference of their cubes is 91. What are the numbers ? 4. The area of a rectangular mirror is 88 square feet and its perimeter is 38 feet. Find its dimensions. 5. The perimeter of a rectangular ginseng bed is 18 rods and its area is 20J square rods. What are its demensions ? 6. It takes 52 rods of fence to inclose a rectangular garden containing 1 acre. How long and how wide is the garden ? 7. The product of two numbers is 59 greater than their sum, and the sum of their squares is 170. What are the numbers ? 8. If 63 is subtracted from a certain number expressed by two digits, its digits will be transposed ; and if the number is multiplied by the sum of its digits, the product will be 729. What is the number ? 9. The smallest of the printed muslin flags made in this country has an area of 5^ square inches. If the width were \ of an inch less, the length would be twice the width. Find the length and width. 10. A man's ice bill was $ 18. If ice had cost $ 1 less per ton, he would have received f of a ton more for the same money. How many tons did he use and what was the price per ton ? 11. The car of the airship America is 107 feet longer than it is wide. If its length were ^ as great, the area of the floor would be 184 square feet. Find its dimensions. 12. A Chinaman received $ 1.60 for rolling joss sticks. Had he been paid 8 cents more for each lot of ten thousand, he would have had to roll one lot less for the same amount of money. How many such lots did he roll and how much was he paid per lot ? SIMULTANEOUS QUADRATIC EQUATIONS 273 13. The area of a wall painting in a restaurant in Philadelphia is 340 square feet. If its length were 12 feet less and its width 12 feet greater, it would be square. Find its length and width. 14. Before the reduction in letter postage between the United States and Great Britain, it cost 10 ^ to send a certain letter that now would just go for 4^. What is the weight of the letter, if the postage was reduced 3 ^ an ounce ? 15. If each of a farmer's maple trees had yielded 2 pounds more of sugar, he could have made 750 pounds. If he had had 50 trees more, he could have made 600 pounds. Find the number of trees he had and the yield per tree. 16. A woman in Saxony received 1^ an hour more for making chiffon hats than for weaving straw hats. If she re- ceived 21 ^ for her work on the former and 20^ for her work on the latter, working 14 hours in all, how much did she re- ceive per hour for each ? 17. An Illinois farmer raised broom corn and pressed the 6120 pounds of brush into bales. If he had made each bale 20 pounds heavier, he would have had 1 bale less. How many bales did he press and what was the weight of each ? 18. If the length of the platform of an elevator were 9 feet less and its width 9 feet more, its area would be 361 square feet. If its length were 9 feet more and its width 9 feet less, its area would be 37 square feet. Find its dimensions. 19. A man bought 4 more loads of sand than of gravel, pay- ing $.50 less per load for sand than for gravel. The sand cost him $ 9 and the gravel $ 10. What quantities of each did he buy ? What prices did he pay ? 20. The capital stock of a creamery company is $ 8850. If there had been 10 times as many stockholders, each man's share would have been $45 less. How many stockholders were there and what was each man's share ? MILNE'S IST YR. ALG. 18 274 SIMULTANEOUS QUADRATIC EQUATIONS 21. Mr. Fuller paid $2.25 for some Italian olive oil, and $2 for -J gallon less of French olive oil, which cost $.50 more per gallon. How much of each kind did he buy and at what price ? 22. In papering a room, 18 yards of border were required, while 40 yards of paper ^ yard wide were needed to cover the ceiling exactly. Find the length and breadth of the room. 23. The water surface area of a tank at AVashington, D. C, used in testing ship models is 20,210 square feet. If the length were 3 feet greater, it would be 11 times the width. What are the dimensions of the water surface ? 24. If the elm beetle had killed 500 trees less one year in Albany, New York, the total estimated loss would have been $ 10,000 ; if the value of each tree had been ^ as much, the total loss would have been $ 3750. How many elm trees were killed ? 25. The total area of a window screen whose length is 4 inches greater than its width is 10 square feet. The area inside the wooden frame is 8 square feet. Find the width of the frame. 26. A rectangular skating rink together with a platform around it 25 feet wide covered 37,500 square feet of ground. The area of the platform was ^ of the area of the rink. What were the dimensions of the rink? 27. The course for a 36-mile yacht race is the perimeter of a right triangle, one leg of which is 3 miles longer than the other. How long is each side of the course ? 28. At simple interest a sum of money amounted to $2472 in 9 months and to $ 2528 in 16 months. Find the amount of money at interest and the rate. 29. Two men working together can complete a piece of work in 6| days. If it would take one man 3 days longer than the other to do the work alone, in how many days can each man do the work alone ? GRAPHIC SOLUTIONS QUADRATIC EQUATIONS — TWO UNKNOWN NUMBERS 375. 1. Construct the graph of the equation a?-\-y'^ = 25. Solution. — Solving for y, y= ± \/26 — a;-^. Since any value numerically greater than 5 substituted for x will make the value of y imaginary, we substitute only values of x between and in- cluding — 5 and -f 5. The corresponding values of y, or of ± V25— x^, are recorded in the table below. It will be observed that each value substituted for x, except ± 5, gives two values of y, and that values of x numerically equal give the same values of y ; thus, when x = 2, y = ± 4.6, and also when x = — 2, y=± 4.6. X y ±o ± 1 ±4.9 ± 2 ±4.6 ±3 ±4 ±4 ±3 ± 5 ■^ n "~ "" i J^'* A u* 1 ^ * A Si /\ / ^ ' V \ K / jS L 1 i-* r _i- The values given in the table serve to locate twenty points of the graph of a;2 _|_ y2 _ 25. Plotting these points and drawing a smooth curve through them, the graph is apparently a circle. It may be proved by geometry that this graph is a circle whose radius is 5. The graph of any equation of the form jr^ 4-/- = rMs a circle whose radius is r and. whose center is at the origin. 276 2T6 GRAPHIC SOLUTIONS 2. Construct the graph, of the equation y^ = 3 x -\- 9. Solution. — Solving for y, y = ± VS x + 9, It will be observed that any value smaller than — 3 substituted for x will make y imaginary ; consequently, no point of the graph lies to the left of X = — 3. Beginning with x = — S, we substitute values for x and determine the corresponding values of y, as recorded in the table : X y -3 -2 ±1.7 -1 ±2.4 ±3 1 ±3.5 2 ±3.9 3 ±4.2 -— $K'— === ^ - - --^- — ^- $ :iii±giiii: Plotting these points and drawing a smooth curve through them, the graph obtained is apparently a. parabola (§ 356). The graph of any equation of the form y^ = ax -{■ c is a parabola. 3. Construct the graph of the equation 9 a;^ + 25 1/^ = 225. Solution. — Solving for y, y = ± z ^25 — x^. Since any value numerically greater than 5 substituted for x will make the value of y imaginary, no point of the graph lies farther to the right or to the left of the origin than 5 units ; consequently, we substitute for x only values between and including — 5 and + 5. Corresponding values of x and y are given in the table : X y - ±1 ±2 ±3 ±4 ±5 ±3 ±2.9 ±2.7 ±2.4 ± 1.8 p* r-t L^-^''^'- rH ^ if^^^ ^^1 / ?r ? s ^ \ f V 1^ H *-!< ^ ^ r ^ J _ _ _ _ GRAPHIC SOLUTIONS 277 Plotting the twenty points tabulated on the preceding page, and draw- ing a smooth curve through them, we have the graph of Qx^ + 2oy^ = 225, which is called an ellipse. . The graph of any equation of the form b^x^ + a^^ = a^b^ is an ellipse. 4. Construct the graph of the equation 4:X'—9y^ = 36. SoLUTioy Solving f or y, y = ± § Vx"'^ — 9. Since any value numerically less than 3 substituted for x will make the value of y imaginary, no point of the graph lies between x = + 3 and X = — 3 ; consequentlj^, we substitute for x only ± 3 and values nnraer- ically greater than 3. Corresponding values of x and y are given in the table : X y >^ k / Y'^i - > k > ^^ ±3 ±4 ±6 ±6 ±1.8 ±2.7 ± 3.5 T w i J^ * \ i K^' \ L. J Yi Vs 1 _ Plotting these fourteen points, it is found that half of them are on one side of the j/-axis and half on the other side, and since there are no points of the curve between x = + 3 and x = — 3, the graph has two separate branches^ that is, it is discontinuous. Drawing a smooth curve through each group of points, the two branches thus constructed constitute the graph of the equation 4 x^ — 9 y2 = 36, which is an hijperhola. The graph of any equation of the form b'x^ — ary- = arb"^ is an hyperbola. An hyperbola has two branches and is called a discontinuous curve. 278 GRAPHIC SOLUTIONS 6. Construct the graph of the equation xy = 10. Solution. — Substituting values for x and solving for y, the corre- sponding values found are as given in the table : X y 1 X y 1 10 -1 -10 2 .5 -2 -5 3 3^ -3 -^ 4 n -4 -n 5 2 -5 -2 6 1| -6 -It 7* If -7 -H 8 H -8 -H 9 H -9 -H 10 1 - 10 - 1 — ~ P — — — — — \ \ \ V^^ N ^ * /^ V N ^ &-* . ^ ^ L, > K\ A ^ _ _ _ _ ^ _ _ Plotting these points and drawing a smooth curve through each group of points, the two branches of the curve found constitute the graph of the equation xy — 10, which is an hyperbola. The graph of any equation of the form jr/ = Solve graphically, giving roots to the nearest tenth: 2. 3.2 + 3.-2 = 0. 8. 2x'-x = 6. 3. x^~x-6 = 0. 9. 2a^-x — 15 = 0. 4. 3c^-Sx-4: = 0. 10. 3aj2-|-5a;-28 = 0. 5. x'-2x-W = 0: 11. 6x^-7x-20 = 0. 6. x'-\-5x-{-U = 0. 12. Sx^-\-Ux-15 = 0. 7. a;2_7a;-f.l8 = 0. 13. 15 ar^ -j- 2 a; - 20 = 0. RATIO AND PROPORTION RATIO 381. The relation of two numbers that is expressed by the quotient of the first divided by the second is called their ratio. 382. The sign of ratio is a colon (:). A ratio is expressed also in the form of a fraction. The ratio of a to 6 is written a ; 6 or ^ • The colon is sometimes regarded as derived from the sign of division by omitting the hne. 383. To compare two quantities they must be expressed in terms of a common unit. Thus, to indicate the ratio of 20^ to $1, both quantities must be ex- pressed either in cents or in dollars, as 20^ : 100^ or $ ^ : $ 1. There can be no ratio between 2 pounds and 3 feet. The ratio of two quantities is the ratio of their numerical measures. Thus, the ratio of 4 rods to 5 rods is the ratio of 4 to 5. 384. The first term of a ratio is called the antecedent, and the second, the consequent. Both terms form a couplet. The antecedent corresponds to a dividend or numerator ; the consequent, to a divisor or denominator. . In the ratio a : 6, or - , a is the antecedent, h the consequent, and 6 the terms a and h form a couplet. 284 RATIO AND PROPORTION 285 385. The ratio of the reciprocals of two numbers is called the reciprocal, or inverse, ratio of the numbers. It may be expressed by interchanging the terms of the couplet. The inverse ratio of a to & is - : -• Since - — - = -, the inverse ratio a b aba of a to b may be written - or & : a. a Properties of Ratios ■ 386. It is evident from the definition of a ratio that ratios have the same properties as fractions ; that is, they may be reduced to higher or lower ter7ns, added, subtracted, etc. Hence, Principles. — 1. Multiplying or dividing both terms of a ratio by the same number does not change the value of the ratio. 2. Mvltiplying the antecedent or dividing the consequent of a ratio by any number multiplies the ratio by that number. 3. Dividing the antecedent or multiplying the consequent by any number divides the ratio by that number. EXERCISES 387. 1. What is the ratio of 8 m to 4 m ? of 4 m to 8 m ? 2. Express the ratio of 6 : 9 in its lowest terms ; the ratio 12x:16y; amibm-, 20a6: 106c; (m i- n) : (m^ - n^. 3. Which is the greater ratio, 2:3 or 3:4? 4:9 or 2:5? 4. What is the ratio of ^ to J ? | to | ? | to | ? Suggestion. — When fractions have a common denominator, they have the ratio of their numerat6rs. 5. What is the inverse ratio of 3 : 10 ? of 12 : 7 ? Reduce to lowest terms the ratios expressed by : 6. 10:2. 8. 3:27. 10. ^. 12. 75-100. 7. 12:6. 9. 4:40. 11. ^f. 13. 60-120. 14. What is the ratio of 15 days to 30 days ? of 21 days to 1 week ? of 1 rod to 1 mile ? 286 RATIO AND PROPORTION Find the value of each of the following ratios : 15. ixi^x". 17. 21 : 7-1 19. a'b'x' : a^6V. 16. }a6:|ac. 18. .7m: .Sn. 20. (q^ -y^) : (x — yf. 21. If 9 is subtracted from 4 and then from 5, find the ratio of the first remainder to the second. 22. Change each to a ratio whose antecedent shall be 1 : 5:20; 3a;:12a;; f:f; .4:1.2. 23. When the antecedent is 6 a; and the ratio is I, what is the consequent? PROPORTION 388. An equality of ratios is called a proportion. 3 : 10 = 6 : 20 and a:x = b:y are proportions. The double colon (::) is often used instead of the sign of equality. The double colon has been supposed to represent the extremities of the lines that form the si^ of equality. The proportion a:b = c:dj or a:b::c:d, is read, ' the ratio of a to & is equal to the ratio of c to d,' or ^ a is to 6 as c is to d' 389. In a proportion, the first and fourth terms are called the extremes, and the second and third terms, the means. In a :b = c:d, a and d are the extremes, b and c are the means. 390. Since a proportion is an equality of ratios each of which may be expressed as a fraction, a proportion may be expressed as an equation each member of which is a fraction. Hence, it follows that : General Principle. — The changes that may be made in a proportion without destroyiyig the equality of its ratios correspond to the changes that may be made in the members of an equation without destroying their equality and in the terms of a fraction ivithout altering the vahie of the fraction. RATIO AND PROPORTION 287 Properties of Proportions 391. Pkixciple 1. — In any pi'oporticm the product of the ex- tremes is equal to the product of the means. For, given a:b = c:d, a c Clearing of fractions, ad = be. Test the following by principle 1 to find whether they are true proportions: 1. 6:16 = 3:8. 2. || = if- 3. 7:8=10:12 392. In the proportion a:m = m:b, m is called a mean pro- portional between a and b. By Prin. 1, m^ = ab; .'. m = Va6. Hence, a mean proportional between two numbers is equal to the square root of their product. 1. Show that the mean proportional between 3 and 12 is either 6 or — 6. Write both proportions. 2. Find two mean proportionals between 4 and 25. 393. Principle 2. — Either extreme of a proportion is equal to the product of the means divided by the other extreme. Either mean is equal to the product of the extremes divided by the other mean. For, given a:b = c:d. By Prin. 1, ad = he. Solving for a, d, &, and c, in succession, Ax. 4, he ■, he , ad ad d a c h 1. Solve the proportion 3:4: — x: 20, for x. 2. Solve the proportion x:a = 2m:n, for a;. 288 KATIO AND PROPORTION 3. If a : 6 = & : c, the term c is called a third proportional to a and b. Find, a third proportional to 6 and 2. 4. In the proportion a:b = c-.d, the term d is called a fourth proportional to a, 6, and c. Find a fourth proportional to J, ^y and -J. 394. Principle 3. — If the product of two numbers is equal to the product of two other numbers, one pair of them may be made the extremes and the other pair the means of a proportion. For, given ad= be. Dividing by bd, Ax. 4, ^ = ^ ; b d that is, a :b = c:d. By dividing both members of the given equation, or of be = ad, by the proper numbers, various proportions may be obtained ; but in all of them a and d will be the extremes and b and c the means, or vice versa, as illustrated in the proofs of principles 4 and 5. 1. If a men can do a piece of work in x days, and if b men can do the same work in y days, the number of days' work for one man may be expressed by either ax or by. Form a pro- portion between a, b, x, and y. 2. The formula pd = WD (See p. 166) expresses the physical law that, when a lever just balances, the product of the numerical measures of the power and its distance from the fulcrum is equal to the product of the numerical measures of the weight and its distance from the fulcrum. Express this law by means of a proportion. 395. PtiiNCiPLE 4. — If four numbers are in 2woportion, the ratio of the antecedents is equal to the ratio of the consequents; that isj the numbers are in proportion by alternation. For; given a:b = c:d. Then, Prin. 1, ad = bc. Dividing by cd, Ax. 4, ^ = - ; c d that is, a :c=:b :d. RATIO AND PROPORTION 289 396. Principle 5. — If four numbers are in proportion, the ratio of the second to the first is equal to the ratio of the fourth to the third; that is, the numbers are in proportion by inversion. For, given a:bz=c:d. Then, Prin. 1, ad = bc. .'. be = ad. Dividing by ac, Ax. 4, b_d, a c ' that is, b : a = d : c. 397. Principle 6. — If four numbers are in proportion, the sum of the terms of the first ratio is to either term of the first ratio as the sum of the terms of the second ratio is to the corresponding term of the second ratio ; that is, the numbers are in proportion by composition. For, given a:b = c:d, Then, Ax. 1, a c b' ~d a + 1 : =^+l» b d a + 6. c + d b d ' that is, a + b :b = c-\- d:d. Similarly, taking the given proportion by inversion (Prin. 6), and add- ing 1 to both members, we obtain a-\-b :a = c + d:c. 393. Principle 7. — If four numbers are in propoHion, the difference between the terms of the first ratio is to either term of the first ratio as the difference between the terms of the second ratio is to the corresponding term of the second ratio; that is, the numbers are in proportion by division. For, in the proof of Prin. 6, if 1 is subtracted instead of added, the following proportions are obtained : a — b '. h ^= c — d '. d^ ^^^ a — b : a = c ~ d : c. milne's 1st yr. alg. — 19 290 RATIO AND PROPORTION 399. Principle 8. — If four numbers are in proportion, the sum of the terms of the first ratio is to their difference as the sum of the terms of the second ratio is to their difference; that is, the numbers are in proportion by composition and division. For, given a :b = c :d. By Prin. 6, ■ a + b _cA-d b d (1) By Prin. 7, a — b _c — d b d (2) Dividing (1) by (2), Ax. 4, a + h _c + d , a-h c-d' at is, a + b:a—b = ci-d:G- -d. 400. Principle 9. — In a proportion, if both terms of a couplet or both antecedents, or both consequents are multiplied or divided by the same number, the resulting four numbers form a proportion. For, given a:b = C'.d, a _ c b d' mb nd a b n d n Then, §181, Also, Ax. 3, -•— = -'—, or ma : nb = mc : nd. b n d n 401. Principle 10. — In a series of equal ratios, the sum of all the antecedents is to the sum of all the consequents as any antecedent is to its consequent. For, given a:b = c:d = e:f, or ? = J = i = r,theval b d f Then, Ax. 3, a = br, c = dr, e =fr whence. Ax. 1, a-\-c + e={b + d+f)r, " b + d-\-f b d f that is, a + c + e:6 + d+f=a : 6 or c : d or e :/. RATIO AND PROPORTION 291 EXERCISES 402. 1. Find the value of x in the proportion 3 : 5 = a;: 55. Solution. 3 : 5 = x : 66, Prin. 2, a; = ^ = 33. 5 Find the value of x in each of the following proportions : 2. 2 : 3 = 6 : a;. 5. .'c + 2 : a; = 10 : 6. 3. 5: a; = 4: 3. 6. it- : a; - 1 = 15: 12. 4. l:a; = a;:9. 7. a; + 2: a; — 2 = 3:1. 8. Show that a mean proportional between any two num- bers having like signs has the sign ± . 9. Find two mean proportionals between V2 and V8. 10. Find a third proportional to 4 and 6. 11. Find a fourth proportional to 3, 8, and 7^. Test to see whether the following are true, proportions : 12. 5 J : 3 = 4 : 1^. 14. 5 : 7 a^ = 10 : 14 x. 13. 4:13 = 2:6^ 15. 2.4 a: .8 a = 6 a: 2 a. 16. Given a : b =c : d, to prove that 2a + 3c:2a-3c = 864-12d:86-12d Proof. — Given a :b = c: d. By alternation, Prin. 4, a: c = b : d. Expressing as a fractional equation, - = - • c d Multiplying the first mepaber by f and the second by ^, the equal of |, 2a^ 8ft . 3 c 12 d ' that is, 2 a : 3 c = 8 6 : 12 d. By composition and division, Prin. 8, 2a-{-Sc:2a-3c = Sb + 12 d:Sb- 12 d. When a:b = r: d, prove that: 17. d:b = c: a. 18. c:d = -:-. b a 19. ¥ : d^ = if : c^ 20. a' : 6 V = 1 :d?. 21. b ma : - = mc '2* 22. ac:bd=(^: d\ 292 RATIO AND PROPORTION When a:b = c:d, prove that : 23. a -\- b:c -\-d = a — b:c — d. 24. 2a-i- 5b:2a = 2c + 5d:2c. 25. 4a -Sb:4:C-Sd = a:c. 26. a : a -\- b = a -{- c : a -\- b -\-c -\- d. 27. a-\-b:c + d= Va' + 6^: V^T^- Problems 403. 1. The ratio of the rate of a local train in the New York subway to an express train is 1 : 2. If the local train runs 15 miles an hour, find the rate of the express train. 2. The consumption of gas in New York City one year was to that in Chicago as 7 : 4. If 12 billion cubic feet were con- sumed in Chicago, what was the consumption in New York ? 3. Michigan produces yearly 25 % of the iron ore of the United States. The ratio of Michigan's output to Minne- sota's is 5:8. AVhat per cent of the country's output does Minnesota produce ? 4. The United States manufactured 285 million pens one year. The ratio of the steel pens to the whole number of pens was 18 : 19. How many steel pens were manufactured ? 5. A diver descended 210 feet into a lake. The ratio of this distance to the distance that is usually considered the limit for divers is 7 : 5. Find the usual limit for divers. 6. How many pounds of tea are made from 4200 pounds of the green leaf, if the ratio of the weight of the manufactured tea to that of the green leaf is 5 : 21 ? 7. Two machines, one old and one modern, turn out 960 pins per minute. The ratio of the number turned out by the old machine to the number turned out by the modern one is 1 : 15. How many were turned out by each machine ? RATIO AND PROPORTION " 293 8. Find a number that added to each of the numbers 1, 2, 4, and 7 will give four numbers in proportion. 9. The United States published 20,000 newspapers recently. The relation of this number to those published in the whole world was 2 : 5. How many were published in the world ? 10. The ratio of the greatest length of Lake Erie to the greatest length of Lake Michigan is 5 : 6. What' is the length of each, if Lake Michigan is 50 miles longer than Lake Erie ? 11. The ratio of the loss of life in the Lisbon earthquake to that in the Messina earthquake is 12 : 23. If 55,000. more lives were lost in the latter than in the former, find the loss of life in each earthquake. 12. The length of a giant candle was to that of a Christmas candle as 40 : 1. If 8 times the length of the latter was 96 inches less than that of the former, find the length of each. 13. The wool sales for one week in New York amounted to 555,000 pounds. The ratio of the domestic sales to the foreign was 14 : 23. What were the foreign sales ? 14. Out of a lot of shell caps, 100 times the number rejected by the government inspector for imperfections was to the total number as 3 : 11. If 1097 were accepted, how many were rejected ? 15. In one year Egypt and Russia together sent 9\ million pounds of eggs to Paris. If Egypt had sent twice as many, the ratio of this number to those sent by Russia would have been 1 : 18. How many pounds were sent by each country ? 16. The sum of the three dimensions of a block of ice is 77 inches, and the width, 22 inches, is a mean proportional between the other two dimensions. Find the length and thickness. 17. The ratio of the length of a gold nugget to its width was 11 : 6, but if its length had been ^ of an inch more, the ratio would have been 2 : 1. Find its length and width. 294 RATIO AND PROPORTION 18. The area of the right triangle shown in Fig. 1 may be expressed either as |- ah or as \ ch. Form a proportion whose terms shall be a, b, c, h. Fig. 1. Fig. 2. Fig. 3. 19. In Fig. 2, the perpendicular p, which is 20 feet long, is a mean proportional between a and b, the parts of the diame- ter, which is 50 feet long. Find the length of each part. 20. In Fig. 3, the tangent t is a mean proportional between the whole secant c-j- e, and its external part e. Find the length of ^, if e = 9f and c = 50f . 21. The strings of a musical instrument produce sound by vibrating. The relation between the number of vibrations -Yand N' of two strings, different only in their lengths I and V, is expressed by the proportion J^i N' = V:l. A c string and a g string, exactly alike except in length, vibrate 256 and 384 times per second, respectively. If the c string is 42 inches long, find the length of the g string. 22. If L and I are the lengths of two pendulums and Tand t the times they take for an oscillation, then A pendulum that makes one oscillation per second is approxi- mately 39.1 inches long. How often does a pendulum 156.4 inches long oscillate ? 23. Using the proportion of exercise 22, find how many feet long a pendulum would have to be to oscillate once a minute. GENERAL REVIEW 404. 1. Distinguish, between known and unknown numbers. 2. When x, -t-, or both occur in connection with -f, — , or both in an expression, what is the sequence of operations ? Illustrate by finding the value of: 7 — 3x24-6-7-2. 3. Name and illustrate three ways of indicating multiplica- tion; two ways of indicating division. 4r When is ic" — ?/" exactly divisible hy x-{-y? hj x —y? ^ 5. When is a trinomial a perfect square ? When is a frac- tion in its lowest terms ? What are similar fractions ? 6. By what principle may cancellation be used in reducing ^ fractions to lowest terms ? 7. Factor the following by three different methods : (a2_2)2-a2. 8. Define power; root; like terms; transposition; simul- taneous equations ; surd. 9. Express the following without parentheses : 10. What is the sign of any power of a positive number ? of any even power of a negative number ? of any odd power of a negative number ? 11. How may the involution of a trinomial be performed by the use of the binomial formula ? Illustrate by raising x -{-2y— zto the third power. 296 296 GENERAL REVIEW 12. Explain the meaning of a negative integral exponent ; of a positive fractional exponent ; of a zero exponent. 13. Define evolution ; binomial surd ; similar surds ; conju- gate surds ; symmetrical equation. 14. Is -^2 4- V4 a surd ? State reasons for your answer. 15. Represent VlO inches by a line. 16. Why is it specially important to test the values of the unknown number found in the solution of radical equations ? 17. Define coordinate axes; imaginary number; axiom; coefficient; elimination. 18. How is the degree of an equation determined ? What is the degree of a; + 6 = c ? ofaf-hSx = 7? oi5x-^o(^ = ll? 19. What name is given to an equation of the first degree ? of the second degree ? of any higher degree ? 20. What is a pure quadratic equation ? a complete quad- ratic equation ? Illustrate each. 21. What is the root of an equation ? What is the principle relating to the roots of a pure quadratic equation ? Illustrate by solving the following : 7x^-5 = 28. 22. Give two methods of completing the square in the solu- tion of affected quadratic equations. When is it advantageous to use the Hindoo method ? Solve the following equation by each method : 3a^ + 5a; = 22. 23. Outline the method of solving quadratic equations by factoring. Illustrate by solving the following : 2a:2__5a5 = l2. 24. When is an equation in the quadratic form ? Illustrate. GENERAL REVIEW 297 25. What roots should be associated when the roots of a system of equations are given thus : xz= ±2, y = =f3? 26. Explain how, in the solution of problems, negative roots of quadratic equations, while mathematically correct, are often inadmissible. 27. What is the advantage of letting x^ = y in the graphic solution of a quadratic equation of the form ax^ -{• bx -{- c = 0?_ 28. How does the graph of a quadratic equation show the fact, if the roots are real and equal? real and unequal? imaginary ? 29. What is the form of the graph of a simple equation ? of two inconsistent equations ? of two indeterminate equations ? 30. What is the form of the graph of an equation like aa^ + bx + c = 0? like x'-^-y^^r'? 31. What is meant by the minimum point of a graph ? Solve graphically the following equation and indicate the minimum point of the graph : 32. How many roots has a simple equation? a quadratic equation? a system of two independent simultaneous equa- tions, one simple and the other quadratic ? a system of two independent simultaneous quadratic equations ? 33. Define homogeneous equation ; antecedent ; consequent ; inverse ratio ; proportion. 34. Give and illustrate two principles relating to ratios. Upon what do these principles depend ? 35. Find the ratio of (x — yf tox^ — 2xy + y^. 36. In the following proportion indicate the means; the extremes ; the mean proportional ; the third proportional : x:y = y:z. 37. Find a fourth proportional to 3 a, 9 a, and 5 a. 298 GENERAL REVIEW 38. Add X V2/ + y V^ + Va;y, x^y^— ^x^y — V^, ■\/x^y ■ ^xy^ — V^, and y^x — a? V4 y — Vo"^. 39. Simplify a-J&— a— [a-6 — (2a+6) + (2a— 6) — a]-6S. 40. Divide ic* — 2/* by a? — 2/ by inspection. Test. 41. Separate a^ — 1 into six rational factors. a^ — 6^ — c^ — 2 6c 42. Reduce — to its lowest terms. a2_&2_|_c2_|_2ac 43. Simplify 44. Simplify x-\-\ \ — x ^ — 1 1 + (a — 6)(6 — c) (c — 6)(c — a) (c — a)(a — 6) 45. Reduce to the simplest form : Vl; '\/25a''; ^t . ^1 + 2 Solve the following equations for x : 46. 3a;2_2a?=65. 48. V^^^ = V^ - 1. 47 0.4. 1_3^, *^- "^+2- 2 49. a^ + Va:^ + 16 = 14. Solve the following for the letters involved: '2a; + 3 2/ + 2; = 9, - + -=10, 50. ^^ 2/ 5 + ?=10. ^a; 2/ a^-f/ = 25, a;4-2/ = 7. Solve graphically : a;-2/=l, I a^ 4. 2/2 = 16. 51. 54. 52. 53. 55. a; + 2?/ + 32; = 13, I 3a; + 2/ + 2 2; =11. (^ u^ + xy = 24, 2/2 + a;2/ = 12. 2/ = 3. Find the value of a; in the following proportions :^ 56. 72:6 = a;:4J. 67. a; : 7.2 = 3.9 : 117. GENERAL REVIEW 299 58. The railways of the United States use annually 150 million tons of coal. If the amount used in drawing trains is yig- as much as goes up the smokestacks, how much is used to draw trains ? 59. In one year about 30,000 vessels passed a lighthouse in. Massachusetts. The number that used steam was to the num- ber of the remainder as 1 : 5. How many used steam ? 60. Out of 63 bakeries inspected in a certain city, the num- ber of ' absolutely clean ' ones was 3 more than that of the ' fairly clean ' ones, and the number of ' unsanitary ' ones was 2 less than twice that of the 'absolutely clean' ones. Find the number of bakeries in each class. 61. The number of parts in a certain manufacturer's mower is twice that in his horse rake and ^V that in his binder. If the binder has 3500 parts more than the rake, how many parts has each machine ? 62. Two men earned $ 3.50 one day for picking pine needles. They were paid 25 cents per 100 pounds. How many pounds did each pick, if one picked J as many as the other ? 63. One of the largest rugs ever made in this country con- tains 3180 square feet. Its length is 7 feet greater than its width. What are its dimensions ? 64. Alfred the Great measured time by candles lighted in succession. The number used in a day was ^ the number of inches in the length of each candle, and each burned at the rate of 3 inches per hour. How many candles were used per day and how long was each ? 65. A target used in practice by the United States fleet was 1 foot longer than it was wide and 18 feet longer than the square bull's-eye. The area of the target exclusive of the bull's- eye was 411 square feet. Find the dimensions of each. 66. A good operator usually earns f 1.80 a day by binding derby hats. If she bound 1 dozen more hats and received 5^ less per dozen, she would earn 5 ^ less a day. How many hats does she bind a day and how much does she receive per dozen ? 300 GENERAL REVIEW 67. How far down a river whose current runs 3 miles an hour can a steamboat go and return in 8 hours, if its rate of sailing in still water is 12 miles an hour ? 68. A person who can walk n miles an hour has a hours at his disposal. How far may he ride in a coach that travels m miles an hour and return on foot within the allotted time ? 69. The first copy of The Sun was printed on a sheet 5i inches longer than it was wide. If the length lacked 6 inches of being twice the width, find the dimensions of the sheet. 70. The Lusitania is 26 feet less than 6 times as long as the Clermont, and -^^ of the length of the Lusitania is 11 feet more than -J- of the length of the Clermont. Find the length of each. 71. A woman has 13 square feet to add to the area of the rug she is weaving. She therefore increases the length ^ and the width \j which makes the perimeter 4 feet greater. Find the dimensions of the finished rug. 72. The inventor of toothpicks sold 16,250,000 during his first year of business. Had there been 75,000 more toothpicks in each box, the number of boxes sold would have been 15 fewer. How many boxes did he sell ? 73. Two passengers together have 400 pounds of baggage and are charged, for the excess above the weight allowed free, 40 cents and 60 cents, respectively. If the baggage had be- longed to one of them, he would have been charged $ 1.50. How much baggage is one passenger allowed without charge ? 74. A railway train, after traveling 2 hours at its usual rate, was detained 1 hour by an accident. It then proceeded at I of its former rate,- and arrived 7f hours behind time. If the accident had occurred 50 miles farther on, the train would have arrived 6^ hours behind time. What was the whole dis- tance traveled by the train ? GENERAL REVIEW 301 405. This page contains the questions given in the Elemen- tary Algebra examination of the Regents of the University of the State of New York for June, 1909. References show where the text provides instruction neces- sary to answer these questions. 1. Divide 6 aj3 + 11 a^-1 by 3 it- -14-2x2 (§§38, 108). 2. Find the prime factors of 1 - ^ (§§ 134, 155), 9 a*-90 a^ H-189a2 (§§133,142,155), a'-^b' (§136), 4:X* + Sa^y' + 9y' (§154), aa; + 4a-4a;-16 (§ 145). r3aj + 8 = 4i/4-2, 3. Solve i^^.^ = 3 (§§231,232). to 9 Give an axiom justifying each step in the solution (§ 224). 4. Find a number such that if it is added to 1, 4, 9, 16, respectively, the results will form a proportion (§ 403). 5. Solve Vx -f 1 + Va: - 2 = V2 a; 4- 3 (§§ 362, 353). 6. Find the square root of ^'+^~^-f — - — + 1 9 lo o 2o 5 (§ 280). 7. Simplify i/iree of the following: ■\/—125x^, V^-^, ■^1087"* (§ 301), ^. ^^>^ l2a^ + 2a«/-2/ (§§ 133, 144); (d) x3+15a^-a;-15(§§ 145,134); (e) a'-16 + b'+2ab (§151); (/) 0:^ + 4^^ (§§153, 154). 6. Expand by the binomial formula, showing all the steps ; (2a-a;)5[§§ 265,266]. 7. Solve for » : 2^±J _ ^-zl = 7y-4a; + 36 ^^ ^ 23I, 232). 2 8 16 8. One half of A's money is equal to B's, and five eighths of B's is equal to C's; together they have $1450. How much has each? (§§ 47, 75, 125, 205, 215). 9. Simplify \ "^ ^t"" (§ 200). 1 — a l-f-ic 10. A man bought a suit of clothes for $24 and paid for it in two-dollar bills and fifty-cent pieces, giving twice as many coins as bills. How many bills did he give ? (§§ 125, 215, 234). 11. Five years ago the sum of the ages of A and B was 40 years. B is now four times as old as A. What is the present age of each ? (§§ 125, 215; also § 234). FACTORS AND MULTIPLES 407. This chapter gives a brief treatment of highest common factor (§ 183) and lowest common multiple (§ 189) for the benefit of any who may desire a little more work in these topics than their application affords in fractions, the only place in elementary algebra where they are applied. HIGHEST COMMON FACTOR 408. An expression that is a factor of each of two or more expressions is called a common factor of them. 409. The common factor of two or more expressions that has the largest numerical coefficient and is of the highest degree is called their highest common factor. The common factors of 4 a^h^ and 6 cfih are 2, a, 6, a^, 2 a, 2 6, 2 a^, ah, 2 dby a%, and 2 a-h with sign + or - . Of these, 2 a% (or - 2 a%) has the largest numerical coefficient and is of the highest degree, and is therefore the highest common factor. The highest common factor may be positive or negative, but usually only the positive sign is taken. The highest common factor of two or more expressions is equal to the product of all their common prime factors. 410. Expressions that have no common prime factor, except 1, are said to be prime to each other. EXERCISES 411. 1. Find the h. c. f. of 12 a^hh and 32 a^ftV. Solution The arithmetical greatest common divisor or highest common factor of 12 and 32 is 4. The highest common factor of a^V^c and of a^h^ifi is a^h'^c. Hence, h. c. f. = 4 a%'^c. MILNE'S IST YR. ALG. — 20 305 306 FACTORS AND MULTIPLES KuLE. — To the greatest common divisor of the numerical co- efficients annex each common literal factor with the least exponent it has in any of the expressions. Find the highest common factor of : 2. 10 a^f, 10 x'f, and 15 xy*z. 3. 70a%',21a*b\a,ndS5a'b'. 4. 8 mhi% 28 my, and 56 m'nK 5. 4:b^cd, 6 6V, and 24 a6c^ 6. 3(a + 6)2 and 6(a + 6)3. 7. 6(a + 6)2and4(a-}-6)(a-6). 8. 12 (a - xf, 6 (a - x)'', and (a - x)\ 9. 10 (x-yy^ 'dud 15 (z-y)(x-yf. 10. Whatis theh.c.f.of3a^-3a;2/2and6a:3_i2a^2/ + 6a;y29 PROCESS Sa^-Sxy^ =Sx(x + y)(x-y) 6 x" - 12 xry -\- 6 xy^ = 2- 3 x(x-y)(x-y) .-. h.c.f. = 3a;(a; — y) Explanation. — For convenience in selecting the common factors, the expressions are resolved into their simplest factors. Since the only common prime factors are 3, x, and (x — y), the highest common factor sought (§ 409) is their product Sx{x — y). Find the highest common factor of : 11. a^ — x^ 3ind a^ — 2 ax -{- x^. 12. a"^ -b^ and a^ + 2 ab-\-b^' 13. x^-\-y^ and a^ -\- 2 xy -{- y^. 14. a^-2x-15 2iudx^-x-20. 15. a2 + 7a + 12anda2H-5aH-6. ' FACTORS AND MULTIPLES 307 Find the highest common factor of : 16. aj^ + a^^2 + 2/* and a^ + icy 4- 2/^ 17. ic^ + 2/■^ ^ + ^> and x^y + xy"^. 18. a* + tt26* + 68 and 3a2-3a62 + 36*. 19. ax — y -\-xy—a2LYi(i ay? -{-x^y — a — y. 20. a^b — b — ah + c and ab — ac — b-\-c. 21. l-4a^, l-f-2a;, and4a-16aa^. 22. 24a^/ + 8a;y and 8a:3y3_3aj22^^ 23. 6ar^ + a;-2 and 2a^-lla--f-5. 24. 17 aftc'^d^ - 51 a^ftc^d^ and aftc^d*^ - 3 a^b(^d. 25. 38 xyz - 95 a^y^^ ^nd 34 xy^z - 85 ic^y^s 26. 6r^ + 10/s-4 j-^s^ and 2 r' + 2 ?^s - 4 ? V. 27. a;*-ar^-2ar, a;* - 2 a:^ - 3 a^, and a;* - 3 «» - 4 a:^. 28. 7 Z3«3 + 35Z-'^ + 42 /^ and 7 Z^^ -f- 21 Z^^- 28 J^f-UW, 29. a^-f-a2 -524.2 aa;, a;2-a2+62 4_2 6a;, and a?-a^-b''-2ab. 30. a^-6a; + 5anda;«-5a^ + 7a;-3. Suggestion. — Apply the factor theorem to the second expression. 31 a^-4a;H-3anda^ + ar'-37a;H-35. 32. Q-n^andn^-n-e. Suggestion. — Change 9 — n^ to — (n" — 9) = - (n + 3) (n — 3). 33. l-a^anda^-6a,-2-9a; + 14. 34. (9-a^)2anda^H-2ar'-9x-18. Suggestion. (9 - x^)^ - (x^ — 9)2. 35. (4-c2)2andc3 + 9c2 + 26c-i-24. 36. xy - y^, - (f - x^y), and x^y - xy\ 37. 16-s^2s-s^ ands2-4s + 4. 38. 2/* — a;^ a,-^ 4- y*, and 2/2 + 2 2/a; + a;^ 39. {y-xf{n-mf and (a^^i/ - 2/^) {m-n -2mn' + n^) . 308 FACTORS AND MULTIPLES LOWEST COMMON MULTIPLE 412. An expression that exactly contains each of two or more given expressions is called a common multiple of them. 6 ahx is a common multiple of a, 3 &, 2 x, and 6 ahx. These numbers may have other common multiples, as 12 ahx, 6 a^h'^x, 18 a^fex^, etc. 413. The expression having the smallest numerical coef- ficient and of loivest degree that will exactly contain each of two or more given expressions is called their lowest common multiple. 6 abx is the lowest common multiple of a, 3 6, 2 x, and 6 abx. The lowest common multiple may have either sign + or — , though usually only the positive sign is taken. The lowest common multiple of two or more expressions is equal to the product of all their different prime factors, each factor being used the greatest number of times it occurs in any of the expressions. EXERCISES 414. 1. What is the 1. c. m. of 12 x^y^, 6 a^xy^y and 8 axyz^ ? Solution The lowest common multiple of the numerical coefficients is found as in arithmetic. It is 24. The literal factors of the lowest common multiple are each letter with the highest exponent it has in any of the given expressions. They are, therefore, a^, x^, y'^, and z^. The product of the numerical and literal factors, 24: a^x^y^z^, is the lowest common multiple of the given expressions. Find the lowest common multiple of : 2. a^oi^yj a^xy^, and aa^y. 3. 10 a^ftV, 5 ab\ and 25 b^c^d\ 4. 16 a'b% 24 (^de, and 36 a*b'd^e^ 5. 18 a'br', 12 p^r, and 54 a^Vg. FACTORS AND MULTIPLES 309 6. What is the 1. c. m. oi x^ — 2xy-\- y^, y^ — a^, and a^ + y^? PROCESS a:^-2xy-{-y^ ={x--y){x-y) y^ - x^ = - (xr^ -y^) = - (x-\-y){x-y) .-. 1. c. III. = {x- y)\x ■\-y){^- xy + y^) = (x-yy(a^ + f) EuLE. — Factor the expressions into their prime factors. Find the product of all their different prime factors, using each factor the greatest number of times it occurs in any of the given expressions. The factors of the 1. c. m. may often be selected without separating the expressions into their prime factors. Find the lowest common multiple of : 7. Q^ — y^ and o^ -\-2xy -\-y'^. 8. X? — y^ and x^ — 2 xy -{- y^. 9. x^ — y^, x^ -\- 2 xy -\- y^, and x^ — 2xy-\-y^. 10. a'-n^2iudLSa^-^6ahi+San'. 11. ic^-1 anda2i»2 + a'-6V-62. 12. a^ -\-l, ab — b, a^ + a, and 1 — a^. 13. 2 X -\- y, 2 xy — y^, and 4: x^ — y^. 14. 1 -\- X, X — x^, 1 -j- x^, and a^(l — a?). 15. 2a; + 2, 5a;-5,3a;-3, and 0.-2-1. 16. 1662_1, 1262^35^205-5, and 26. 17. l-2x'-\-x*,(l-xy,Rndl + 2x + x'. 18. b''-5b-{-6, b^^7b + 10, and 6^ - 10 6 + 16. 310 FACTORS AND MULTIPLES Find the lowest common multiple of : 19. oc^ -\- 7 X — S, a^ — 1, X -{- x^, smd S aa^ — 6 ax -\- 3 a. 20. x^ — a%a-2 x, a^ + 2 ax, and a^ - 3 a^ic -f 2 ao?. 21. m^ — ar^, m^ + ma;, 7/i^ -|- mx -f- a^, and (m + a;) x^. 22. 2-3a; + a^, a^ + 4a; + 4, a^ + 3a; + 2, and 1 - a^. 23. a^ — 2/^ ic"* + ^y^ -\-y^, ^ + y^, and a^ + a;?/ + y^. 24. a^ + a;^2/ + ^V^ + 2/^ ^^^ x^ — a^2/ + ^2/^ ~ 2/^- 25. a^ _|_ 4 o[ _^ 4^ 0^2 — 4, 4 - a^, and a^ — 16. 26. a^ - (6 + c)2, 62 - (c + af, and c^ - (a + 6)2. 27. 2 (aa^ - ar^)2, 3 x {oj'x - ar^)^ and 6 (aV - a^) . 28. {yz^ — xyzf, y^ {xz^ — x^, and a?z^ + 2 a;2^ + z^. Suggestion. — In solving the following, use the factor theorem. 29. a^-6a.-2 + llaj-6 anda;^-9a;2 + 26a;-24. 30. a^-5a;2_4^_^20anda^4-2a;2-25a;-50. 31. a^-4a^ + 5a;-2anda^-8a^ + 21a;-18. 32. a^ + 5a^4-7a; + 3 and a^ — 7a^ — 5a; + 75. 33. a^4-2a^-4a;-8, a^-a^ -8a; + 12, a^ + 4a;2- 3 a;- 18. GLOSSARY Abscissa. A distance measured along or parallel to the x-axls. Absolute Term. A term that does not contain an unknown number. Absolute Value. The value of a number without regard to its sign. Addends. Numbers to be added. Addition. The process of finding a simple expression for the algebraic sum of two or more numbers. Affected Quadratic. A quadratic equation that contains both the second and first powers of one unknown number. Algebra. That branch of mathematics which treats of general numbers and the nature and use of equations. It is an extension of arithmetic and it uses both figures and letters to express numbers. Algebraic Expression. A number represented by algebraic symbols. Algebraic Numbers. Positive and negative numbers, whether integers or fractions. Algebraic Sum. The result of adding two or more algebraic numbers. Antecedent. The first term of a ratio. Arrangement. When a polynomial is arranged so that in passing from left to right the several powers of some letter are successively higher or loioer^ the polynomial is said to be arranged according to the ascending or descending powers, respectively, of that letter. Axes of Reference. Two straight lines that intersect, usually at right angles, used to locate a point or points in a plane. Axiom. A principle so simple as to be self-evident. Binomial. An algebraic expression of two terms. ^ Binomial Formula. The formula or principle by means of which any indicated power of a binomial may be expanded. Binomial Quadratic Surd. A binomial surd whose surd or surds are of the second order. 311 312 GLOSSARY Binomial Surd. A binomial, one or both of whose terms are surds. Clearing an Equation of Fractions. The process of changing an equa- tion containing fractions to an equation without fractions. Coefficient. When one of the two factors into which a number can be resolved is a known number, it is usually written first and called the coefficient of the other factor. In a broader sense, either one of the two factors into which a number can be resolved may be considered the coefficient of the other. Co-factor. Same as Coefficient. Common Factor. A factor of each of two or more numbers. Common Multiple. An expression that exactly contains each of two or more given expressions. Complete Quadratic. Same as Affected Quadratic. Complex Fraction. A fraction one or both of whose terms contains a fraction. Compound Expression. Same as Polynomial. Conditional Equation. An equation that is true for only certain values of its letters. Conjugate Surds. Two binomial quadratic surds that differ only in the sign of one of the terms. Consequent. The second term of a ratio. Consistent Equations. Same as Simultaneous Equations. Coordinates. See Bectangular Coordinates. Couplet. The two terms of a ratio. Cube. Same as Third Power. Cube Root. One of the three equal factors of a number. Cubic Surd. A surd of the third order. Degree of an Expression. The term of highest degree in any rational integral expression determines the degree of the expression. Degree of a Term. The sum of the exponents of the literal factors of a rational integral term determines the degree of the term. Denominator. The divisor in an algebraic fraction. GLOSSARY 313 Dependent Equations. Two or more equations that express the same relation between the unknown numbers involved are often called depend- ent equations, for each may be derived from any one of the others. Derived Equations. Same as Dependent Equations. Difference. The result of subtracting one number from another. That is, the difference is the algebraic number that added to the subtra- hend gives the minuend. Dissimilar Fractions. Fractions that have different denominators. Dissimilar Terms. Terms that contain different letters or the same letters with different exponents. Dividend. In division, the number that is divided. Division. The process of finding one of two factors when their product and one of the factors is given. Divisor. In division, the number by which the dividend is divided. Elimination. The process of deriving from a system of simultaneous equations another system involving fewer unknown numbers. Entire Surd. A surd that has no rational coefficient except unity. Equation. A statement of the equality of two numbers or expressions. Equation of the First Degree. Same as Simple Equation. Equation of the Second Degree. Same as Quadratic Equation. Equivalent Equations. Two equations that have the same roots, each equation having all the roots of the other. Even Root. A root whose index is an even number. Evolution. The process of finding any required root of a number. Exponent. A small figure or letter placed at the right and a little above a number to indicate how many times the number is to be used as a factor. Extremes. The first and fourth terms of a proportion. Factor. Each of two or more numbers whose product is a given number. Factoring. The process of separating a number into its factors. Formula. An expression of a principle or a rule in symbols. 314 GLOSSARY Fourth Proportional. The fourth number of four different numbers that form a proportion. Fourth Root. One of the four equal factors of a number. Fraction. In algebra, an indicated division. Fractional Equation. An equation that involves an unknown number in any denominator. Fractional Expression. An expression, any term of which is a fraction. Fulcrum. The point or edge upon which a lever rests. General Number. A literal number to which any value may be assigned. Graph. A picture (line or lines) every point of which exhibits a pair of corresponding values of two related quantities. Graph of an Equation. The line or lines containing all the points, and only those, whose coordinates satisfy a given equation. Higher Equation. An equation that contains a higher power of the unknown number than the second. Highest Common Factor. The common factor of two or more expres- sions that has the largest numerical coefficient and is of the highest degree. It is equal to the product of all the common factors of the expressions. Homogeneous Equation. An equation all of whose terms are of the same degree with respect to the unknown numbers. Identical Equation. An equation whose members are identical, or such that they may be reduced to the same form. Identity. Same as Identical Equation. Imaginary Number. A number that involves an indicated even root of a negative number. Incomplete Quadratic. Same as Pure Quadratic. Inconsistent Equations. Two or more equations that are not satisfied in common by any set of values of the unknown numbers. Independent Equations. Two or more equations that express different relations between the unknown numbers involved, and so cannot be re- duced to the same equation. Indeterminate Equation. An equation that is satisfied by an unlimited number of sets of values of its unknown numbers. GLOSSARY 315 Index of a Power. Same as Exponent. Index of a Root. A small figure or letter written in the opening of a radical sign to indicate what root of a number is sought. Integer. Same as miole Number. Integral Equation. An equation that does not involve an unknown number in any denominator. Integral Expression. An expression that contains no fraction. Inverse Ratio. Same as Beciprocal Ttatio. Involution. The process of finding any required power of an expres- sion. Irrational Equation. An equation involving an irrational root of an unknown number. Irrational Expression. An expression that contains an irrational number. Irrational Number. A number that cannot be expressed as an integer or as a fraction with integral tenns. Known Number. A general number or a number whose value is known. Lever. Any sort of a bar resting on a fixed point or edge. Like Terms. Same as Similar Terms. Linear Equation. Same as Simple Equation. Literal Coefficient. A coefficient composed of letters. Literal Equation. An equation one or more of whose known numbers is expressed by letters. Literal Numbers. Letters that are used for numbers. Lowest Common Denominator. The denominator of lowest degree, having the least numerical coefficient, to which two or more fractions can be reduced. It is equal to the lowest common multiple of the given denominators. Lowest Common Multiple. The expression having the smallest nu- merical coefficient and of lowest degree that will exactly contain each of two or more given expressions. Lowest Terms. When the terms of a fraction have no common factor, the fraction is said to be in its lowest terms. 316 GLOSSARY Mean Proportional. A number that serves as both means of a propor- tion. Means. The second and third terms of a proportion. Members of an Equation. In an equation, the number on the left of the sign of equality is called the first member of the equation, and the number on the right is called the second member. Minimum Point of a graph. The point of a graph that has the alge- braically least ordinate. Minuend. In subtraction, the number from which the subtraction is made. Mixed Coefficient. A coefficient composed of both figures and letters. Mixed Expression. An expression some of whose terms are integral and some fractional. Mixed Number. Same as Mixed Expression. Mixed Surd. A surd that has a rational coefficient. Monomial. An algebraic expression of one term only. Multiplicand. In multiplication, the number multiplied. Multiplication. When the multiplier is a positive integer, the process of taking the multiplicand as many times as there are units in the mul- tiplier. In general, the process of finding a number that is obtained from the multiplicand just as the multiplier is obtained from unity. Multiplier. In multiplication, the number by which the multiplicand is multiplied. Negative Number. A number less than zero. Negative Term. A term preceded by — . Numerator. The dividend in an algebraic fraction. Numerical Coefficient. A coefficient composed of figures. Numerical Equation. An equation all of whose known numbers are expressed by figures. Odd Root. A root whose index is odd. Order of a radical or of a surd is indicated by the index of the root or by the denominator of the fractional exponent. Ordinate. A distance measured along or parallel to the y-axis. GLOSSARY 317 Origin. The intersection of the axes of reference. Perfect Square. An expression that may be separated into two equal factors. Polynomial. An algebraic expression of more than one term. Positive Number. A number greater than zero. Positive Term. A term preceded by +, expressed or understood. Power of a Number. The product obtained when the number is used a certain number of times as a factor. Prime Number. A number that has no factors except itself and 1. Prime to Each Other. Expressions that have no common prime factor except 1 are said to be prime to each other. Principal Root. A real root of a number that has the same sign a.s the number itself. Product. The result of multiplying one number by another. Proportion. An equality of ratios. Pure Quadratic. An equation that contains only the second power of the unknown number. Quadratic Equation. An equation that, when simplified, contains the square of the unknown number, but no higher power. Quadratic Form. An expression that contains but two powers of an unknown number or expression, the exponent of one power being twice that of the other. Quadratic Surd. A surd of the second order. Quotient. The result of dividing one number by another. Radical. An indicated root of a number. Radical Equation. Same as Irrational Equation. Radical Sign. Same as Boot Sign. Radicand. A number whose root is required. Ratio. The relation of two numbers that is expressed by the quotient of the first divided by the second. Rational Expression. An expression that contains no irrational number. Rationalization. The process of multiplying an expression containing a surd by any number that will make the product rational. 318 GLOSSARY Rationalizing Factor. The factor by which a surd expression is multi- plied to render the product rational. Rationalizing the Denominator. The process of reducing a fraction having an irrational denominator to an equal' fraction having a rational denominator. Rational Number. A number that is, or may be, expressed as an integer or as a fraction with integral terms. Real Number. A number that does not involve the even root of a negative number. Reciprocal of a number is 1 divided by the number. Reciprocal of a Fraction is the fraction inverted or 1 divided by the fraction. Reciprocal Ratio. The ratio of the reciprocals of two numbers is called the reciprocal ratio of the numbers. Rectangular Coordinates. The abscissa and ordinate of a point referred to two perpendicular axes are called the rectangular coordinates of the point. Reduction. The process of changing the form of an expression with- out changing its value. Remainder in subtraction. Same as Difference. Root of an Equation. Any number that satisfies the equation. Root of a Number. When the factors of a number are all equal, one of the factors is called a root of the number. Root Sign. The symbol y/ written before a number denotes that a root of the number is sought. Satisfied. When an equation is reduced to an identity by the substi- tution of certain known numbers for the unknown numbers, the equation is said to be satisfied. Second Power. When a number is used twice as a factor, the product is called the second power of the number. Second Root. Same as Square Boot Sign of Addition is +, read ^ plus.'' Sign of a Fraction. The sign written before the dividing line of a fraction. GLOSSARY 319 Sign of Continuation is •••, read ' and so on' ov ^ and so on to.' Sign of Deduction Ls . •., read ' therefore ' or ' hence.' Sign of Division is -^, read ' divided by.' Division is also indicated by a fraction, the numerator being the dividend and the denominator the divisor. Sign of Equality is =, read ' is equal to' or <■ equals.'' Sign of Multiplication is x or the dot (•), read 'multiplied by.'' Multiplication is also indicated by the absence of sign. Sign of Ratio is a colon (:), read ' is to.'' Sign of Subtraction is — , read 'wmwms.' Signs of Aggregation. Signs used to group numbers that are to be re- garded as a single number. They are parentheses, () ; brackets, [] ; braces, {} ; the vinculum, ; and the vertical bar, \ . Signs of Direction. Same as Signs of Quality. Signs of Opposition. Same as Signs of Quality. Signs of Quality. The signs + and — when used to denote positive and negative numbers. Similar Fractions. Fractions that have the same denominator. Similar Radicals. Radicals that in their simplest form are of the same order and have the same radicand. Similar Terms. Terms that contain the same letters with the same exponents. Simple Equation. An integral equation that involves only the first power of one unknown number in any term when similar terms have been united. Simple Expression. Same as Monomial. Simplest Form of a Radical. A radical is in its simplest foi-m when the index of the root is as small as possible, apd when the radicand is in- tegral and contains no factor that is a perfect power whose exponent cor- responds with the index of the root. Simultaneous Equations. Two or more equations that are satisfied by the same set or sets of values of the unknown numbers form a system of simultaneous equations. Solving an Equation. Finding the roots of an equation. Square. Same as Second Power. 320 GLOSSARY Square Root. One of the two equal factors of a number. Substitution. When a particular number takes the place of a letter, or general number, the process is called substitution. Subtraction. The process of finding one of two numbers when their sum and the other number are given. Subtraction is the inverse of addition. Subtrahend. In subtraction, the number that is subtracted. Sum. See Algebraic Sum. Surd. The indicated root of a rational number that cannot be ob- tained exactly. Symmetrical Equation. An equation that is not affected by interchang- ing the unknown numbers involved. Term. An algebraic expression whose parts are not separated by the signs + or — . Terms of a Fraction. The numerator and denominator of a fraction. Third Power. When a number is used three times as a factor, the product is called the third power of the number. Third Proportional. The consequent of the second ratio when the means of a proportion are identical. Third Root. Same as Cube Boot. Transposition. The process of removing a term from one member of an equation to the other. Trinomial. An algebraic expression of three terms. Trinomial Square. A trinomial that is a perfect square. Unknown Number. A number whose value is to be found. Unlike Terms. Same as Dissimilar Terms. Whole Number. A unit or an aggregate of units. X-axis. The horizontal axis of reference is usually called the x-axis. Y-axis. The vertical axis of reference is usually called the y-axis. 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