MATHEMATICAL ECONOMICS--EBGN 409

Algebra Review

#### References:

Schaum's Outlines: College Algebra, Spiegel, M. R., 1996. (CA) Ch. 1-7, 9-13, 15, 19, 21.

Algebra is a powerful tool and the foundation of all higher mathematics. Without a strong foundation in algebra, economics can be quite difficult. This quiz is designed to help you review many of the algebraic concepts you will need to do economics.

Fundamental Operations with Numbers: CA Chapter 1

Perform each of the indicated operations.

1. (40+21)(72-38)

(32-15)

(1). 37

(2). 99

(3). 122

(4). 167

2. 4/2 + 6/3 - 2/2 + 3*4

3. 12(-40)(-12)

5(-3)-3(-3)

4. 23

5. 5253

57

6. (23)2

7. 38  - 4224

35       26

8. What is the sum of 1/3 and 1/6

9. What is the sum of -4/15 and -11/24

10. What is the difference of -4/15 and -11/24

(3). 23/120

11.

(3). 6xy<z

12. 4a3b2 + 16ab-4a2

2a2

13. What is the sum of -7/8 and 11/24

14. What is the difference of 1/3 and 1/6

15. What is the difference of 5/7 and -7/11

Evaluate the following expressions, given  x= 2, y = -3, z=1/3, a = 2, b = 1.

16. 4x2

17. (x/y)2-3(b/a)3

Fundamental Operations with Algebraic Expressions: CA Chapter 2

18. Add the algebraic expressions of the following groups:
x2 + y2 - z2 + 2xy - 2yz,  y2 + z2 - x2 + 2yz - 2zx,  z2 + x2 - y2 + 2zx - 2xy,  1 - x2 - y2 - z2

19. Subtract the second expression from the first: a - b + c - d, c - a + d - b

In each of the following find the indicated product of the algebraic expressions.

20. (-2ab3)(4a2b5

21. (x3 - 3xy + y2)(4xy2

Special Products: CA Chapter 3

Find each of the following products.

22. (2x + 3y)(2x - 3y);

23. (x - 1)(x5 + x4 + x3 + x2 + x + 1);

24. (x - 2y + z)2

Factoring: CA Chapter 4; ME 1.3

Factor each expression.

25. 2x2 - 3xy;

26. 9s3t + 15s2t3 - 3s2t2

27. x2 - 9;

28. x2 + 8x + 16;

29. x2 - 6x + 8;

30. 6x2 - xy - 12y2

31. bx - ab + x2 - ax

Fractions: CA Chapter 5

32. Reduction of fractions to lowest terms.

x3y-y3x

x2y-xy2

Multiplication, division and addition of fractions.

33.     x+2xy

3x2___

2y+1

6x

34. 15(x2 +xy-x-y)

10(yx+y2-x-y)

35.

Exponents: CA Chapter 6; ME 1.1

Evaluate each of the following.

36. 5*10-3

37. (5x+3y)

(5x+3y)     with 5x+3y ≠ 0

38. x(1/2)x-1/3

Find an equivalent expression amongst the answers:

39.

40.

41.

42.

43.

Equations: CA Chapter 9

Find the solution(s) of each of the following equations.

44. x/2+x/3=10;

45. ;

Use the axioms of equality to solve equation.

46. 3y - 2(y - 1) = 4(y + 2);

47. 2x-4=36

In each of the following formulas, solve for the indicated letter.

48. s = vot + (1/2)at2, solve for a.

49. 1/p=1/f-1/q, solve for p.

Find all values of x for which

50. x2 = 81;

51. ( x - 1 )2 = 4

Linear Equations in One Unknown: CA Chapter 11; ME 1.4

52. Solve the following equation.

x-2 = x-4

x+2   x+4

53. Solve for x.

ax + a = bx + b

54. A man has \$4000 invested, part at 5% and the remainder at 3% simple interest. The total income per year from these investments is \$168. How much does he have invested at each rate? Solve (4000-x)(0.05) + x(0.03) = 168.

55. A can do a job in 3 days, and B can do the same job in 6 days. How long will it take them if they work together?

Simultaneous Equations in One Unknown: CA Chapter 12; ME 1.6

Solve the following systems.

56. 5x + 2y = 3; 2x + 3y = -1

57. x-2 = y+1 = 2, x+3 - 2y-1 = 1;

3                      4        2

58. 2x - y + z = 3;  x + 3y - 2z = 11;  3x - 2y + 4z = 1

Quadratic Equations in One Unknown: CA Chapter 13; ME 1.4

59. Solve by factoring.

x2 + 5x - 6 = 0

60. Solve by the quadratic formula.

4t2 + 12t + 9 = 0

Solve for x.

61.

62.

Ratio, Proportion and Variation: CA Chapter 15

63. Find the ratio of the following quantity.

4/5 : 8/3

64. In the following proportion determine the value of x.

(3 - x) : (x + 1) = 2 : 1

65. For the following statements write an equation, employing k as the constant of proportionality.

The period of vibration T of a simple pendulum at a given place is proportional to the square root of its length l.

66. The pressure p of a given mass of ideal gas varies inversely as the volume v and directly as the absolute temperature T. To what pressure must 100 cubic feet of helium at 1 atmosphere pressure and 253 degrees temperature be subjected to be compressed to 50 cubic feet when the temperature is 313 degrees?

Inequalities: CA Chapter 19; CH 6.5

67. If a > b and c > d , how is a + c related to b + d .

Find the values of x for which each of the following inequalities holds.

68. 4x + 5 > 2x + 9;

69.

Find the values of x for which

70. x2 - 7x + 12 = 0;

71. x2 - 7x + 12 > 0.

Theory of Equations: CA Chapter 21

72. Is -1 a root of the equation  f(x) = x3 - 7x - 6 = 0 ?

73. Is 2i a root of the equation  f(z) = 2z3 + 3z2 + 8z + 12 = 0 ?

Graphing: CH Chapter 2; ME Chapter 1

Roughly graph the following functions and know their general shape for the quiz.

If you need more practice with graphing you can change parameters on these graphs in the excel file alg-graph.xls

74. y = 3

(5). None of above

75. y=2x + 3

(5). None of above

76. y = x2

(5). None of above

77. y = x3

(4). D

(5). None of above

78. y = x2 + 2x + 1

(5). None of above

79. y = x2 - 2x + 1

(5). None of above

80. y = ex

(5). None of above

81. y = e-x

(5). None of above

82. y = e2x

(5). None of above

83. y = 2ex

(5). None of above

84. y = ln x

(5). None of above

85. y = ln 1/x

86. 7 = xy

(5). None of above