Chapter 4 Algebraic Expressions and Equations

Chapter 4 Algebraic Expressions and Equations 4.1 Evaluating Variable Expressions and Formulas In this chapter we will study the two primary structures in algebra: expressions and equations. These first two sections deal with expressions, which are objects created through order of operations involving one or more variables. Examples of expressions are: A = lw , E = mc 2 , 2! r 2 + 2! rh , x 2 ! 3x + 4

Notice that expressions may be labeled (as the first two are with A and E) or not labeled (as the second two are). In constructing expressions, note that order of operations is used in the expression. Also note that variables are often written together, such as lw or 2xy. When variables are written this way, the implied operation is multiplication. Our first example illustrates how to evaluate these expressions. Example 1

Find the value of each expression when x = 3 and y = !4 . a. b. c. d.

Solution

a.

5xy 2y 2 x 2 ! y2 3xy 2x ! 4y

This expression indicates to multiply 5 by x by y: 5xy = 5 ( 3) ( !4 ) substituting for variables

= !60 multiplying Note how we used parentheses when substituting for variables. This is often a good idea to distinguish negative numbers from subtraction. b.

This expression indicates to square y, then multiply by 2: 2 2y 2 = 2 ( !4 ) substituting for variables

= 2 (16 )

computing the exponent

= 32

multiplying

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c.

d.

This expression indicates to square x, square y, and subtract their results: 2 2 x 2 ! y 2 = ( 3) ! ( !4 ) substituting for variables = 9 ! 16

computing the exponents

= 9 + (!16) = !7

rewriting subtraction as addition adding

In this expression we will compute the numerator and denominators separately, then simplify the resulting fraction: 3( 3) ( !4 ) 3xy = substituting for variables 2x ! 4y 2 ( 3) ! 4 ( !4 ) !36 6 ! (!16) !36 = 6 + 16 !36 = 22 2/ • 18 =! 2/ • 11 18 =! 11 =

computing multiplications rewriting subtraction as addition adding factoring the GCF simplifying fractions

Note that expressions change in value as their variables change in value also. For example, the expression 2xy 2 can be evaluated when x = !3 and y = !2 :

2xy 2 = 2 ( !3) ( !2 )

2

substituting for variables

= 2 ( !3) ( 4 )


= !24

multiplying

However, when x = !2 and y = !3 :

2xy 2 = 2 ( !2 ) ( !3)

2


= 2 ( !2 ) ( 9 )


= !36

multiplying

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Note that the values of this expression are different when the values of the variables change. For this reason, these expressions are often called variable expressions, and are labeled as a variable. Example 2

Find the value for each variable expression when a = !4 and b = 2 . a. b. c. d.

Solution

a.

X = 3ab ! 4b 2 Y = 4a 3 ! 3b 2 6ab V= 2 a ! b2 7ab V= 2a + 4b

Substituting a = !4 and b = 2 , then finding the value of X: X = 3ab ! 4b 2 given expression

= 3( !4 ) ( 2 ) ! 4 ( 2 )

b.

2


= 3( !4 ) ( 2 ) ! 4 ( 4 )


= !24 ! 16 = !24 + (!16) = !40

multiplying converting to addition adding

Substituting a = !4 and b = 2 , then finding the value of Y: Y = 4a 3 ! 3b 2 given expression

= 4 ( !4 ) ! 3( 2 ) 3

2


= 4 ( !64 ) ! 3( 4 )

computing the exponents

= !256 ! 12 = !256 + (!12) = !268

multiplying converting to addition adding

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c.

Substituting a = !4 and b = 2 , then finding the value of V: 6ab V= 2 given expression a ! b2 6 ( !4 ) ( 2 ) = substituting for variables ( !4 )2 ! ( 2 )2

6 ( !4 ) ( 2 ) 16 ! 4 !48 = 16 ! 4 !48 = 12 = !4 =

d.

computing the exponents multiplying subtracting simplifying

Substituting a = !4 and b = 2 , then finding the value of V: 7ab V= given expression 2a + 4b 7 ( !4 ) ( 2 ) = substituting for variables 2 ( !4 ) + 4 ( 2 )

!56 !8 + 8 !56 = 0 which is undefined =

multiplying adding

In this case the expression has no value; the expression itself is undefined. We say that the variables a = !4 and b = 2 are not in the domain of the expression. The domain of an expression is the group of possible replacement values for the expression. For our purposes, the domain will usually consist of numbers which do not result in a zero denominator. We can also find the values of expressions where the variable values are fractions, as the next example illustrates.

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Example 3

Find the value of each expression when s = ! a. b. c. d.

Solution

a.

b.

3 2 and t = . 4 3

6st !8s + 6t 2s 2 ! 3t 2 s2 ! t 2 s2 + t 2 3 2 and t = , then simplifying the resulting expression: 4 3 " 3% " 2% 6st = 6 $ ! ' $ ' substituting for variables # 4 & # 3&

Substituting s = !

" 1% = 6$ ! ' # 2&

multiplying fractions

= !3

multiplying

3 2 and t = , then simplifying the resulting expression: 4 3 " 3% " 2% !8s + 6t = !8 $ ! ' + 6 $ ' substituting for variables # 4& # 3&

Substituting s = !

=6+4 = 10

multiplying fractions adding

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c.

3 2 and t = , then simplifying the resulting expression: 4 3 2 2 " 3% " 2% 2 2 2s ! 3t = 2 $ ! ' ! 3 $ ' substituting for variables # 4& # 3&

Substituting s = !

" 9% " 4% = 2 $ ' ! 3$ ' # 16 & # 9& 9 4 ! 8 3 27 32 = ! 24 24 5 =! 24 =

d.

computing exponents multiplying fractions converting to common denominators adding fractions

3 2 and t = , then simplifying the resulting expression: 4 3 2 2 " 3% " 2% 2 2 $# ! '& ! $# '& s !t 4 3 = substituting for variables 2 2 2 2 s +t " 3% " 2% $# ! '& + $# '& 4 3 9 4 ! = 16 9 computing exponents 9 4 + 16 9 9 4 ! 16 9 • 144 = multiplying by the LCM 9 4 144 + 16 9 9 4 • 144 ! • 144 9 = 16 distributive property 9 4 • 144 + • 144 16 9 81 ! 64 = multiplying fractions 81 + 64 17 = simplifying 145

Substituting s = !

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Formulas for computing various quantities are actually just variable expressions, and can be evaluated accordingly, as the next example illustrates. Example 4

Evaluate the following formulas given the variable values. a. b. c. d.

Solution

a.

b.

P = 2w + 2l ; w = 12, l = 17

A = ! r 2 ; ! = 3.14, r = 6 S = 2! rh ; ! = 3.14, r = 5, h = 12 A = P(1 + r)t ; P = 1250, r = 0.1, t = 6

Substituting the values for the variables: P = 2w + 2l given expression

= 2 (12 ) + 2 (17 )


= 24 + 34 = 58

multiplying adding

Substituting the values for the variables: A = !r2 given expression

= ( 3.14 ) ( 6 )

c.

2


= ( 3.14 ) ( 36 )


= 113.04

multiplying

Substituting the values for the variables: S = 2! rh given expression

= 2 ( 3.14 ) ( 5 ) (12 )

d.


= ( 6.28 ) ( 60 )

multiplying

= 376.8

multiplying

Substituting the values for the variables (and using a calculator): A = P(1 + r)t given expression = 1250(1 + 0.1)6


= 1250 (1.1)

computing the parentheses

6

= 1250 (1.771561)

evaluating the exponent

= 2214.45125

multiplying

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Formulas are used extensively in applications of algebra. The fact that so many other areas utilize formulas is one reason algebra is required for most college majors. Many of these applications of formulas will be explored in the exercises. Terminology expression (or variable expression)

domain (of an expression)

Exercise Set 4.1 Find the value of each expression when x = 5 and y = !3 .

3x + 7y 7x ! 6y 12xy !8xy 5x 2 y !6xy 2 x 2 ! y2 3x 2 ! 2xy 4xy ! 5xy 2 5x 19. 3y + 4xy 4xy 21. 3x + 5y 5xy 2 23. !6x + 10y

5x + 6y 5x ! 8y 15xy !9xy !4x 2 y 15xy 2 y2 ! x 2 5y 2 ! 8xy 9x 2 y ! 8xy 2 !4x 20. 2x + 5y 3x 2 22. 3x ! 5y 4x ! 3y 2 24. !6x ! 10y

1. 3. 5. 7. 9. 11. 13. 15. 17.

2. 4. 6. 8. 10. 12. 14. 16. 18.

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Find the value for each variable expression when a = 4 and b = !3 . 25. X = 4a ! 5b 27. Y = !4a 2 ! 3b 2 29. W = 7a 2b 3 31. W = !4ab + 8a 2 a+b 33. M = 4ab a!b 35. M = 2 4a b 3a + 4b 37. Q = !8ab !3b 2 39. Q = 6a + 2ab

26. Y = !7a ! 8b 28. X = 2a 2 ! 5b 2 30. W = !4a 2b 2 32. W = !3a 2b + 2ab 2 a!b 34. M = 6ab 2a ! 3b 36. M = 5a 2b !6a ! 8b 38. Q = !5ab !5a 2 40. Q = 2ab ! 8b

Find the value of each expression when s = !

1 2 and t = : 2 3

!6st !4s + 6t !3s ! 5t 2s 2 ! 4t 2 3st 2 ! 6s 2t 4s + 5t 52. 3s ! 4t 4s 2 + 9t 2 54. 4s 2 ! 9t 2

5st 2s + 3t 5s ! 4t s 2 ! 2t 2 2st ! 5s 2t s+t 51. s!t s2 ! t 2 53. 2s 2 ! 3t 2

41. 43. 45. 47. 49.

42. 44. 46. 48. 50.

Evaluate the following formulas given the variable values. 55. P = 2w + 2l ; w = 9, l = 15 57. P = 2w + 2l ; w = 4.7, l = 8.6 1 3 59. P = 2w + 2l ; w = 5 , l = 8 2 4 2 61. A = ! r ; ! = 3.14, r = 4 63. A = ! r 2 ; ! = 3.14, r = 1.2 65. S = 2! rh ; ! = 3.14, r = 4, h = 5

56. P = 2w + 2l ; w = 13, l = 19 58. P = 2w + 2l ; w = 5.9, l = 12.4 2 3 60. P = 2w + 2l ; w = 4 , l = 6 3 4 2 62. A = ! r ; ! = 3.14, r = 12 64. A = ! r 2 ; ! = 3.14, r = 1.5 66. S = 2! rh ; ! = 3.14, r = 6, h = 4

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67. 69. 71. 73. 74. 75. 76. 77. 78.

A = lw ; l = 12, w = 7 68. A = lw ; l = 8.5, w = 4.3 70. 1 1 A = lw ; l = 9 , w = 3 72. 2 3 A = P(1 + r)t ; P = 1400, r = 0.08, t = 2

A = P(1 + r)t ; A = P(1 + r)t ; A = P(1 + r)t ; A = P(1 + r)t ; A = P(1 + r)t ;

P = 2000, r = 0.12, t = 3 P = 2500, r = 0.08, t = 4 P = 10000, r = 0.1, t = 5 P = 12500, r = 0.15, t = 10 P = 15000, r = 0.16, t = 20

A = lw ; l = 15, w = 12 A = lw ; l = 12.7, w = 5.2 1 1 A = lw ; l = 6 , w = 4 4 5

(round answer to nearest hundredth) (round answer to nearest hundredth) (round answer to nearest hundredth) (round answer to nearest hundredth)

Answer each of the following application problems. t

79.

80.

81. 82. 83. 84. 85.

r$ ! The formula A = P # 1 + & is used in investment computation. Compute the value of " n% A when P = $8000, r = 0.08, n = 4, and t = 15 . Use a calculator and round your answer to the nearest hundredth. t r$ ! The formula A = P # 1 + & is used in investment computation. Compute the value of " n% A when P = $12000, r = 0.09, n = 4, and t = 20 . Use a calculator and round your answer to the nearest hundredth. The formula C = 23.95d + 0.15(m ! 780) is used to compute the cost of renting a car. Compute the value of C when d = 7 and m = 956. The formula C = 23.95d + 0.15(m ! 780) is used to compute the cost of renting a car. Compute the value of C when d = 14 and m = 1382. The formula B = 29.95 + 0.15m is used to compute the monthly bill for the use of a cellular phone. Compute the value of B when m = 286. The formula B = 29.95 + 0.15m is used to compute the monthly bill for the use of a cellular phone. Compute the value of B when m = 654. 4 The formula V = ! r 3 is used to compute the volume of a sphere. Compute the value 3 of V when ! = 3.1416 and r = 8.4 inches. Use a calculator and round your answer to the nearest tenth.

278

4 3 ! r is used to compute the volume of a sphere. Compute the value 3 of V when ! = 3.1416 and r = 12.86 inches. Use a calculator and round your answer to the nearest hundredth. 5 The formula C = ( F ! 32 ) is used to convert temperature from Fahrenheit to 9 Celsius. Compute the value of C when F = 59°. 5 The formula C = ( F ! 32 ) is used to convert temperature from Fahrenheit to 9 Celsius. Compute the value of C when F = 86°. 5 The formula C = ( F ! 32 ) is used to convert temperature from Fahrenheit to 9 Celsius. Compute the value of C when F = –40°. 5 The formula C = ( F ! 32 ) is used to convert temperature from Fahrenheit to 9 Celsius. Compute the value of C when F = –13°. 9 The formula F = C + 32 is used to convert temperature from Celsius to Fahrenheit. 5 Compute the value of F when C = 30°. 9 The formula F = C + 32 is used to convert temperature from Celsius to Fahrenheit. 5 Compute the value of F when C = 45°. 9 The formula F = C + 32 is used to convert temperature from Celsius to Fahrenheit. 5 Compute the value of F when C = –20°. 9 The formula F = C + 32 is used to convert temperature from Celsius to Fahrenheit. 5 Compute the value of F when C = –40°. The formula P = 120 p 3q 7 is used in statistics to compute probability. Compute the 1 2 value of P when p = and q = . Express your answer as a fraction and as a decimal 3 3 rounded to four decimal places. The formula P = 120 p 3q 7 is used in statistics to compute probability. Compute the 1 1 value of P when p = and q = . Express your answer as a fraction and as a decimal 2 2 rounded to four decimal places.

86. The formula V =

87.

88.

89.

90.

91.

92.

93.

94. 95.

96.

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97. The formula P = 56 p 5 q 3 is used in statistics to compute probability. Compute the 1 1 value of P when p = and q = . Express your answer as a fraction and as a decimal 2 2 rounded to four decimal places. 98. The formula P = 56 p 5 q 3 is used in statistics to compute probability. Compute the 1 2 value of P when p = and q = . Express your answer as a fraction and as a decimal 3 3 rounded to four decimal places. A 99. The formula P = is used in investment computation. Compute the value of (1 + i )n P when A = $20000, i = 0.08, and n = 15. Use a calculator and round your answer to the nearest hundredth. A 100. The formula P = is used in investment computation. Compute the value of (1 + i )n P when A = $40000, i = 0.11, and n = 20. Use a calculator and round your answer to the nearest hundredth. " (1 + i)n ! 1 % 101. The formula F = p $ ' is used in investment computation. Compute the i # & value of F when p = $300, i = 0.08, and n = 20. Use a calculator and round your answer to the nearest hundredth. " (1 + i)n ! 1 % 102. The formula F = p $ ' is used in investment computation. Compute the i # & value of F when p = $400, i = 0.08, and n = 25. Use a calculator and round your answer to the nearest hundredth. " (1 + i)n ! 1 % 103. The formula F = p $ ' is used in investment computation. Compute the i # & value of F when p = $500, i = 0.12, and n = 25. Use a calculator and round your answer to the nearest hundredth. " (1 + i)n ! 1 % 104. The formula F = p $ ' is used in investment computation. Compute the i # & value of F when p = $700, i = 0.11, and n = 30. Use a calculator and round your answer to the nearest hundredth.

280