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Chapter 14,092 wordsPublic domain

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This book includes extensive mathematical expressions and equations, which can not always be easily represented in plain text. The reader is encouraged to download the HTML version of the text, which represents the math more clearly.

For the plain text version, the following conventions are used:

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Square, cube, and other roots are shown by raising a quantity to the appropriate fractional power. For example: [4]^(1/2) is "the square root of 4." [x]^(1/n) is "the nth root of x."

Extra parentheses have been added as needed to clarify the correct order of operations.]

A REVIEW OF ALGEBRA

BY ROMEYN HENRY RIVENBURG, A.M.

HEAD OF THE DEPARTMENT OF MATHEMATICS THE PEDDIE INSTITUTE, HIGHTSTOWN, N.J.

AMERICAN BOOK COMPANY NEW YORK CINCINNATI CHICAGO

A REVIEW OF ALGEBRA.

E. P. 6

PREFACE

In most high schools the course in Elementary Algebra is finished by the end of the second year. By the senior year, most students have forgotten many of the principles, and a thorough review is necessary in order to prepare college candidates for the entrance examinations and for effective work in the freshman year in college. Recognizing this need, many schools are devoting at least two periods a week for part of the senior year to a review of algebra.

For such a review the regular textbook is inadequate. From an embarrassment of riches the teacher finds it laborious to select the proper examples, while the student wastes time in searching for scattered assignments. The object of this book is to conserve the time and effort of both teacher and student, by providing a thorough and effective review that can readily be completed, if need be, in two periods a week for a half year.

Each student is expected to use his regular textbook in algebra for reference, as he would use a dictionary,--to recall a definition, a rule, or a process that he has forgotten. He should be encouraged to _think_ his way out wherever possible, however, and to refer to the textbook only when _forced_ to do so as a last resort.

The definitions given in the General Outline should be reviewed as occasion arises for their use. The whole Outline can be profitably employed for rapid class reviews, by covering the part of the Outline that indicates the answer, the method, the example, or the formula, as the case may be.

The whole scheme of the book is ordinarily to have a page of problems represent a day's work. This, of course, does not apply to the Outlines or the few pages of theory, which can be covered more rapidly. By this plan, making only a part of the omissions indicated in the next paragraph, the essentials of the algebra can be readily covered, if need be, in from thirty to thirty-two lessons, thus leaving time for tests, even if only eighteen weeks, of two periods each, are allotted to the course.

If a brief course is desired, the Miscellaneous Examples (pp. 31 to 35, 50 to 52), many of the problems at the end of the book, and the College Entrance Examinations may be omitted without marring the continuity or the comprehensiveness of the review.

ROMEYN H. RIVENBURG.

CONTENTS

PAGES

OUTLINE OF ELEMENTARY AND INTERMEDIATE ALGEBRA 7-13

ORDER OF OPERATIONS, EVALUATION, PARENTHESES 14

SPECIAL RULES OF MULTIPLICATION AND DIVISION 15

CASES IN FACTORING 16, 17

FACTORING 18

HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE 19

FRACTIONS 20

COMPLEX FRACTIONS AND FRACTIONAL EQUATIONS 21, 22

SIMULTANEOUS EQUATIONS AND INVOLUTION 23, 24

SQUARE ROOT 25

THEORY OF EXPONENTS 26-28

RADICALS 29, 30

MISCELLANEOUS EXAMPLES, ALGEBRA TO QUADRATICS 31-35

QUADRATIC EQUATIONS 36, 37

THE THEORY OF QUADRATIC EQUATIONS 38-41

OUTLINE OF SIMULTANEOUS QUADRATICS 42, 43

SIMULTANEOUS QUADRATICS 44

RATIO AND PROPORTION 45, 46

ARITHMETICAL PROGRESSION 47

GEOMETRICAL PROGRESSION 48

THE BINOMIAL THEOREM 49

MISCELLANEOUS EXAMPLES, QUADRATICS AND BEYOND 50-52

PROBLEMS--LINEAR EQUATIONS, SIMULTANEOUS EQUATIONS, QUADRATIC EQUATIONS, SIMULTANEOUS QUADRATICS 53-57

COLLEGE ENTRANCE EXAMINATIONS 58-80

OUTLINE OF ELEMENTARY AND INTERMEDIATE ALGEBRA

~Important Definitions~

Factors; coefficient; exponent; power; base; term; algebraic sum; similar terms; degree; homogeneous expression; linear equation; root of an equation; root of an expression; identity; conditional equation; prime quantity; highest common factor (H. C. F.); lowest common multiple (L. C. M.); involution; evolution; imaginary number; real number; rational; similar radicals; binomial surd; pure quadratic equation; affected quadratic equation; equation in the quadratic form; simultaneous linear equations; simultaneous quadratic equations; discriminant; symmetrical expression; ratio; proportion; fourth proportional; third proportional; mean proportional; arithmetic progression; geometric progression; S [infinity]

~Special Rules for Multiplication and Division~

1. Square of the sum of two quantities. (x + y)^2.

2. Square of the difference of two quantities. (x - y)^2.

3. Product of the sum and difference of two quantities. (s + t)(s - t).

4. Product of two binomials having a common term. (x + r)(x + m).

5. Product of two binomials whose corresponding terms are similar. (3x + 2t)(2x - 5t).

6. Square of a polynomial. (m - n/3 + k)^2.

7. Sum of two cubes. (x^3 + y^3)/(x + y) = x^2 - xy + y^2.

8. Difference of two cubes. (x^3 - y^3)/(x - y) = x^2 + xy + y^2.

9. Sum or difference of two like powers. (x^7 + y^7)/(x + y), (x^5 - y^5)/(x - y), (x^4 - y^4)/(x - y), (x^4 - y^4)/(x + y).

~Cases in Factoring~

1. Common monomial factor. mx + my - mz = m(x + y - z).

2. Trinomial that is a perfect square. x^2 ± 2xy + y^2 = (x ± y)^2.

3. The difference of two squares. (a) Two terms. x^2 - y^2 = (x + y)(x - y). (b) Four terms. x^2 + 2xy + y^2 - m^2 = (x + y + m)(x + y - m). (c) Six terms. x^2 + 2xy + y^2 - p^2 - 2pq - q^2 = [(x + y) + (p + q)][(x + y) - (p + q)]. (d) Incomplete square. x^4 + x^2 y^2 + y^4 = x^4 + 2x^2 y^2 + y^4 - x^2 y^2 = (x^2 + y^2 + xy)(x^2 + y^2 - xy).

4. Trinomial of the form x^2 + bx + c. x^2 - 5x + 6 = (x - 2)(x - 3).

5. Trinomial of the form ax^2 + bx + c. 20x^2 + 7x - 6 = (4x + 3)(5x - 2).

6. Sum or difference of two cubes. See "Special Rules," 7 and 8. two like powers. See "Special Rules," 9.

7. Common polynomial factor. Grouping. t^2 p + t^2 q - 2mp - 2mq = t^2(p + q) - 2m(p + q) = (p + q)(t^2 - 2m).

8. Factor Theorem. x^3 + 17x - 18 = (x - 1)(x^2 + x + 18).

~H. C. F. and L. C. M.~

a^2 + 2a - 3 = (a + 3)(a - 1).

a^2 + 7a + 12 = (a + 3)(a + 4).

a^4 + 27a = a(a + 3)(a^2 - 3a + 9).

H. C. F. = a + 3.

L. C. M. = (a + 3)(a - 1)(a + 4)a(a^2 - 3a + 9).

~Fractions~

Reduction to lowest terms.

Reduction of a mixed number to an improper fraction.

Reduction of an improper fraction to a mixed number.

Addition and subtraction of fractions.

Multiplication and division of fractions.

Law of signs in division, changing signs of factors, etc.

Complex fractions.

~Simultaneous Equations~

Solved by addition or subtraction. substitution. comparison.

Graphical representation.

~Involution~

Law of signs.

Binomial theorem laws.

Expansion of monomials and fractions. binomials. trinomials.

~Evolution~

Law of signs.

Evolution of monomials and fractions.

Square root of algebraic expressions.

Square root of arithmetical numbers.

Optional Cube root of algebraic expressions. Cube root of arithmetical numbers.

~Theory of Exponents~

Proofs: a^m × a^n = a^(m + n); (a^m)/(a^n) = a^(m - n); (a^m)^n = a^(mn); [a^(mn)]^(1/n) = a^m; (a/b)^n = (a^n)/(b^n); (abc)^n = a^n b^n c^n.

Meaning of fractional exponent. zero exponent. negative exponent.

Four rules To multiply quantities having the same base, add exponents. To divide quantities having the same base, subtract exponents. To raise to a power, multiply exponents. To extract a root, divide the exponent of the power by the index of the root.

~Radicals~

Radical in its simplest form.

Transformation of radicals Fraction under the radical sign. Reduction to an entire surd. Changing to surds of different order. Reduction to simplest form.

Addition and subtraction of radicals.

Multiplication and division of radicals a^(1/n) · b^(1/n) = [ab]^(1/n). ([ab]^(1/n))/(a^(1/n)) = b^(1/n).

Rationalization Monomial denominator. Binomial denominator. Trinomial denominator.

Square root of a binomial surd.

Radical equations. _Always_ check results to avoid extraneous roots.

~Quadratic Equations~

Pure. x^2 = a.

Affected. ax^2 + bx + c = 0.

Methods of solving Completing the square. Formula. Developed from ax^2 + bx + c = 0. Factoring.

Equations in the quadratic form.

Properties of quadratics r_1 = -b/2a + ([b^2 - 4ac]^(1/2))/(2a). r_2 = -b/2a - ([b^2 - 4ac]^(1/2))/(2a). Then r_1 + r_2 = -b/a. r_1 · r_2 = c/a. Discriminant, b^2 - 4ac, and its discussion. Nature or character of the roots.

~Simultaneous Quadratics~

CASE I.

One equation linear.

The other quadratic. 3x - y = 12, x^2 - y^2 = 16.

CASE II.

Both equations homogeneous and of the second degree. x^2 - xy + y^2 = 21, y^2 - 2xy = -15.

CASE III.

Any two of the quantities x + y, x^2 + y^2, xy, x^3 + y^3, x^3 - y^3, x - y, x^2 ± xy + y^2, etc., given. x^2 + y^2 = 41, x + y = 9.

CASE IV. Both equations symmetrical or symmetrical except for sign. Usually one equation of high degree, the other of the first degree. x^5 + y^5 = 242, x + y = 2.

CASE V. Special Devices

I. Solve for a compound unknown, like xy, x + y, (1)/(xy), etc., first. x^2y^2 + xy = 6, x + 2y = -5.

II. Divide the equations, member by member. x^4 - y^4 = 20, x^2 - y^2 = 5.

III. Eliminate the quadratic terms. 4x + 3y = 2xy, 7x - 5y = 5xy.

~Ratio and Proportion~

Proportionals mean, third, fourth.

Theorems 1. Product of means equals product of extremes. 2. If the product of two numbers equals the product of two other numbers, either pair, etc. 3. Alternation. 4. Inversion. 5. Composition. 6. Division. 7. Composition and division. 8. In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent, etc.

Special method of proving four quantities in proportion. Let a/b = x, a = bx, etc.

~Progressions~

Development of formulas. { l = ar^(n - 1). { l = a + (n - 1)d. { S = (ar^n - a)/(r - 1). { S = (n/2)(a + l). { S = (rl - a)/(r - 1). { S = (n/2)[2a + (n - 1)d]. { S[infinity] = (a)/(1 - r).

Insertion of means Arithmetical. Geometrical.

~Binomial Theorem~

Review of binomial theorem laws. See Involution.

Expansion of (a + b)^n.

Finding any term by key number method. r^(th) or (r + 1)^(th) term method.

A REVIEW OF ALGEBRA

ORDER OF OPERATIONS, EVALUATION, PARENTHESES

Order of operations: First of all, raising to a power and extracting a root. Next, multiplication and division. Last of all, addition and subtraction.

Find the value of:

1. 5 · 2^2 - 25^(1/2) ÷ 5 + 2^2 · 8 ÷ 4 - 2.

2. (3 × 6 ÷ 9)/2 - 2[100^(1/2)] ÷ 5 + 4 · 2^3 - (14 · 2)/28.

3. 9 · 2 ÷ 6 + 3 - 2 · 4^2 ÷ 8^(1/3) - 4 + (3 · 2^2)/6.

Evaluate:

4. (a^4 - a^3 + b^3)/([a^2 b^2]^(1/2)) + (c[a^(1/2)] + a^3bc)/(abc), if a = 1, b = 2, c = 3.

5. t^(1/3) + [tm]^(1/3) + m^(1/3), if t = 8, m = 27.

6. (2[3 + 2d + a]^(1/2))/(3[a + b - cx - c]^(1/2)) + ((3c - d)x)/(7ad - [abc]^(1/2)), if a = 5, b = 3, c = -1, d = -2, x = 0.

7. a - {5b - [a - (3c - 3b) + 2c - 3(a - 2b - c)]}, if a = -3, b = 4, c = -5. (_Yale._)

Simplify:

8. m - [2m - {3r - (4r - 2m)}].

9. 2a - [5d + {3c - (a + [2d - 3a + 4c])}].

10. 3c^2 + c(2a - [6c - {3a + c - 4a}]).

SPECIAL RULES OF MULTIPLICATION AND DIVISION

Give results by inspection:

1. (g + 1/2 k)^2.

2. (s - (2m)/3)^2.

3. (2v + 3w)(2v - 3w).

4. (x + 3ts)(x - 7ts).

5. (2l + 3g)(4l - 11g).

6. (a - (2b)/3 + c - d)^2.

7. (x^3 + 8m^3)/(x + 2m).

8. (y^3 - 27k^(3m))/(y - 3k^m).

9. (c^5 - d^5)/(c - d).

10. (e^5 + d^5)/(e + d).

11. (x^4 - y^4)/(x - y).

12. (x^4 - y^4)/(x + y).

13. (a - .03)(a - .0007).

14. (g^n - 1/2)(g^n + 3/4).

15. (t^7 - v^(7/2))/(t - v^(1/2)).

16. (k^32 + 1)(k^16 +1)(k^8 + 1)(k^4 +1)(k^2 +1)(k + 1)(k - 1).

17. [(a + b) + (c + d)][(a + b) - (c + d)].

18. (p - q + r - s)(p - q - r + s).

19. (3m - n - l + 2r)(3m + n - l - 2r).

20. (x + 5)(x - 2)(x - 5)(x + 2).

21. (a^2 + b^2 - c - 2d + 3e)^2.

22. (s + t - 2v/5 + 3w/6 + z^2)^2.

23. (x^5 + 32)/(x + 2).

~References:~ The chapter on Special Rules of Multiplication and Division in any algebra. Special Rules of Multiplication and Division in the Outline in the front of the book.

CASES IN FACTORING

The number of terms in an expression usually gives the clue to the possible cases under which it may come. By applying the _test_ for each and eliminating the _possible_ cases one by one, the right case is readily found. Hence, the number of terms in the expression and a ready and accurate knowledge of the Cases in Factoring are the real keys to success in this vitally important part of algebra.

CASE I. A common monomial factor. Applies to any number of terms.

5cx - 5ct + 5cv - 15 c^2 m + 25 c^3 m^2 = 5c(x - t + v - 3cm + 5c^2 m^2).

CASE II. A trinomial that is a perfect square. Three terms.

x^2 ± 2xm + m^2 = (x ± m)^2.

CASE III. The difference of two squares.

_A._ Two terms. x^2 - y^2 = (x + y)(x - y).

_B._ Four terms.

x^2 + 2xy + y^2 - m^2 = (x^2 + 2xy + y^2) - m^2 = (x + y + m)(x + y - m)

_C._ Six terms. x^2 - 2xy + y^2 - m^2 - 2mn - n^2 = (x^2 - 2xy + y^2) - (m^2 + 2mn + n^2) = (x - y)^2 - (m + n)^2 = [(x - y) + (m + n)][(x - y) - (m + n)].

_D._ An incomplete square. Three terms, and 4th powers or multiples of 4.

c^4 + c^2 d^2 + d^4 = c^4 + 2c^2 d^2 + d^4 - c^2 d^2 = (c^2 + d^2)^2 - c^2 d^2 = (c^2 + d^2 + cd)(c^2 + d^2 - cd).

CASE IV. A trinomial of the form x^2 + bx + c. Three terms.

x^2 + x - 30 = (x + 6)(x - 5).

CASE V. A trinomial of the form ax^2 + bx + c. Three terms.

20x^2 + 7x - 6 = (4x + 3)(5x - 2).

CASE VI.

_A._ The sum or difference of two cubes. Two terms.

x^3 + y^3 = (x + y)(x^2 - xy + y^2); x^3 - y^3 = (x - y)(x^2 + xy + y^2).

_B._ The sum or difference of two like powers. Two terms.

x^4 - y^4 = (x - y)(x^3 + x^2y + xy^2 + y^3); x^5 + y^5 = (x + y)(x^4 - x^3y + x^2 y^2 - xy^3 + y^4).

CASE VII. A common polynomial factor. Any _composite_ number of terms.

t^2 p + t^2 q - t^2 r - g^2 p - g^2 q + g^2 r = t^2 (p + q - r) - g^2 (p + q -r) = (p + q - r)(t^2 - g^2) = (p + q - r)(t + g)(t - g).

CASE VIII. The Factor Theorem. Any number of terms.

x^3 + 17x - 18 = (x - 1)(x^2 + x + 18).

FACTORING

Review the _Cases in Factoring_ (see Outline on preceding pages) and write out the prime factors of the following:

1. 8a^(13) + am^(12).

2. x^7 + y^7.

3. 4x^2 + 11x - 3.

4. m^2 + n^2 - (1 + 2mn).

5. -x^2 + 2x - 1 + x^4.

6. x^(16) - y^(16). (Five factors.)

7. (x + 1)^2 - 5x - 29.

8. x^4 + x^2 y^2 + y^4.

9. x^4 - 11x^2 + 1.

10. x^(2m) + 2 + 1/(x^(2m)).

11. x^(6m) + 13x^(3m) + 12.

12. 4a^2 b^2 - (a^2 + b^2 - c^2)^2.

13. (x^2 - x - 6)(x^2 - x - 20).

14. a^4 - 8a - a^3 + 8.

15. p^3 + 7p^2 + 14p + 8.

16. 18a^2 b + 60ab^2 + 50b^3.

17. x^3 - 7x + 6.

18. 24c^2 d^2 - 47cd - 75.

19. (a^2 - b^2)^2 - (a^2 - ab)^2.

20. a^2 x^3 - (8a^2)/(y^3) - x^3 + 8/(y^3).

21. gt - gk + gl^2 + xt - xk + xl^2.

22. (m - n)(2a^2 - 2ab) + (n - m)(2ab - 2b^2).

23. a^2 - x^2 - y^2 + b^2 + 2ab + 2xy.

24. (2c^2 + 3d^2)a + (2a^2 + 3c^2)d.

25. (n(n - 1))/(1 · 2) a^(n - 2) b^2 + (n(n - 1)(n - 2))/(1 · 2 · 3) a^(n - 3) b^3.

26. (x - x^2)^3 + (x^2 - 1)^3 + (1 - x)^3. (_M. I. T._)

27. (27y^3)^2 - 2(27y^3)(8b^3) + (8b^3)^2. (_Princeton._)

28. (a^3 + 8b^3)(a + b) - 6ab(a^2 - 2ab + 4b^2). (_M. I. T._)

Solve by factoring:

29. x^3 = x.

30. z^2 - 4z - 45 = 0.

31. x^3 - x^2 = 4x - 4.

~Reference:~ The chapter on Factoring in any algebra.

HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE

Define H. C. F. and L. C. M.

Find by factoring the H. C. F. and L. C. M.:

1. 3x^2 - 3x, 12x^2 (x^2 - 1), 18x^3 (x^3 - 1).

2. (x^2 - 1)(x^2 + 5x + 6), (x^2 + 3x)(x^2 - x - 6). (_Harvard._)

3. x^2 - y^2, x^2 + y^2, x^3 + y^3, x^6 + y^6, x^6 - y^6. (_College Entrance Board._)

4. x^3 + x^2 - 2, x^3 + 2x^2 - 3. (_Cornell._)

5. x^5 - 2x^4 + x^2, 2x^4 - 4x^3 - 4x + 6. (_Yale._)

6. x^2 + a^2 - b^2 + 2ax, x^2 - a^2 + b^2 + 2bx, x^2 - a^2 - b^2 - 2ab. (_Harvard._)

7. 2x^2 - x - 15, 3x^2 - 11x + 6, 2x^3 - x^2 - 13x - 6. (_College Entrance Board._)

8. (tv - v^2)^3, v^3 - t^2v, t^3 - v^3, v^2 - 2vt + t^2.

Pick out the H. C. F. and the L. C. M. of the following:

9. 8(x^2 + y)^(32) (t^2 + z)^(19) (m - n^3)^(14), 12(x^2 + y)^(23) (t^2 + z)^(41) (m - n^3)^(17), 18(m - n^3)^(11) (x^2 + y)^(39) (t^2 + z)^(37).

10. 17ax^3 (y + z)^(10) (y - x)^(19) (x + z)^(27), 34a^2 x^4 (y + z)^(11) (y - x)^(21) (x + z)^(13), 51a^3 x^5 (y + z)^4 (x + z)^(32) (y - x)^(29).

~Reference:~ The chapter on H. C. F. and L. C. M. in any algebra.

FRACTIONS

Define: fraction, terms of a fraction, reciprocal of a number.

Look up _the law of signs_ as it applies to fractions. Except for this, fractions in algebra are treated exactly the same as they are in arithmetic.

1. Reduce to lowest terms:

(_a_) 32/24;

(_b_) (a^6 - x^6)/(a^4 - x^4);

(_c_) [(a + b)^2 - (c + d)^2]/[(a + c)^2 - (b + d)^2]. (_M. I. T._)

2. Reduce to a mixed expression:

(_a_) 756/11;

(_b_) (a^3 + b^3)/(a - b).

3. Reduce to an improper fraction:

(_a_) 45-1/8;

(_b_) 9-11/12 qt.;

(_c_) a^2 - ab + b^2 - (b^3)/(a + b).

Add:

4. 5/18 + 7/9 + 11/16 + 5/8.

5. 5/(1 + 2x) - (3x)/(1 - 2x) + (4 - 13x)/(4x^2 - 1).

6. 1/[x(x - a)(x - b)] + 1/[a(a - x)(a - b)] + 1/[b(b - x)(b - a)].

Multiply:

7. 72/121 × 55/56 × 77/90.

8. (b - y)/(a^3 + y^3) × (ca + cy)/(b^2 + by) × (b^6 + y^6)/(b^2 + y^2) × b/c.

Divide:

9. (12/25) ÷ (6/50).

10. [1 - (ab)/(a^2 - ab + b^2)] [1 - (ab)/(a^2 + 2ab + b^2)] ÷ (a^3 - b^3)/(a^3 + b^3). (_Yale._)

11. [(x^4 - y^4)/(x^2 - y^2) ÷ (x + y)/(x^2 - xy)] ÷ [(x^2 + y^2)/(x - y) ÷ (x + y)/(xy - y^2)]. (_Sheffield._)

Simplify:

12. [(4y)/x - (15y^2)/(x^2) + 4] ÷ [4 - (16y)/x + (15y^2)/(x^2)] × [3 - (4x + 20y)/(2x + 5y)].

~Reference:~ The chapter on Fractions in any algebra.

COMPLEX FRACTIONS AND FRACTIONAL EQUATIONS

Define a complex fraction.

Simplify:

1. (3/7 + 4/5)/(2 - 3/7 · 4/5).

2. (2 - 3/2 + 2/3)/(5 - 2/3 + 3/2).

3. 2 - 2/(1 - 1/[1 - 1/(1 + 1/2)]).

4. a/(b^2) - a/[b^2 + (cb)/(a - c/b)]. (_Harvard._)

5. If m = 1/(a + 1), n = 2/(a + 2), p = 3/(a + 3), what is the value of m/(1 - m) + n/(1 - n) + p/(1 - p)? (_Univ. of Penn._)

6. Simplify the expression {x + y - 1/[x + y - xy/(x + y)]}(x^3 - y^3)/(x^2 - y^2). (_Cornell._)

7. Simplify [1 - (2xy)/((x + y)^2)]/[1 + (2xy)/((x - y)^2)] ÷ {(1 - y/x)/(1 + y/x)}^2.

8. Solve (7y + 9)/4 - [y - (2y - 1)/9] = 7.

9. Solve 2-1/3 - (2/5)(x^2 + 3) = (10x)/3 + 1 - (2x^2)/5.

10. How much water must be added to 80 pounds of a 5 per cent salt solution to obtain a 4 per cent solution? (_Yale._)

~Reference:~ See Complex Fractions, and the first part of the chapter on Fractional Equations in any algebra.

FRACTIONAL EQUATIONS

1. Solve for each letter in turn 1/b = 1/p + 1/q.

2. Solve and check:

(5x + 2)/3 - (3 - (3x - 1)/2) = (3x + 19)/2 - ((x + 1)/6 + 3).

3. Solve and check:

(1/2)(x - a/3) - (1/3)(x - a/4) + (1/4)(x - a/5) = 0.

4. Solve (after looking up the special _short_ method):

(3x - 1)/30 + (4x - 7)/15 = x/4 - (2x - 3)/(12x - 11) + (7x - 15)/60.

5. Solve by the special _short_ method:

1/(x - 2) - 1/(x - 3) = 1/(x - 4) - 1/(x - 5).

6. At what time between 8 and 9 o'clock are the hands of a watch (_a_) opposite each other? (_b_) at right angles? (_c_) together?

Work out (_a_) and state the equations for (_b_) and (_c_).

7. The formula for converting a temperature of F degrees Fahrenheit into its equivalent temperature of C degrees Centigrade is C = (5/9)(F - 32). Express F in terms of C, and compute F for the values C = 30 and C = 28. (_College Entrance Exam. Board._)

8. What is the price of eggs when 2 less for 24 cents raises the price 2 cents a dozen? (_Yale._)

9. Solve 2/(x - 2) + 2/(4 - x^2) = 5/(x + 2).

~Reference:~ The Chapter on Fractional Equations in any algebra. Note particularly the special _short_ methods, usually given about the middle of the chapter.

SIMULTANEOUS EQUATIONS

NOTE. Up to this point each topic presented has reviewed to some extent the preceding topics. For example, factoring reviews the special rules of multiplication and division; H. C. F. and L. C. M. review factoring; addition and subtraction of fractions and fractional equations review H. C. F. and L. C. M., etc. From this point on, however, the interdependence is not so marked, and miscellaneous examples illustrating the work already covered will be given very frequently in order to keep the whole subject fresh in mind.

1. Solve by three methods--addition and subtraction, substitution, and comparison: { 5x + y = 11, { 3x + 2y = 1.

Solve and check:

2. { 12R_1 - 11R_2 = b + 12c, { R_1 + R_2 = 2b + c.