chapter 4 algebraic expressions and equations · pdf file269 chapter 4 algebraic expressions...

269

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 = mc2 , 2!r2 + 2!rh , x2 ! 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. 5xy b. 2y

2 c. x

2! y

2

d. 3xy

2x ! 4y

Solution a. 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:

2y2

= 2 !4( )2

substituting for variables

= 2 16( ) computing the exponent

= 32 multiplying

270

c. This expression indicates to square x, square y, and subtract their results:

x2! y

2= 3( )

2! !4( )

2substituting for variables

= 9 !16 computing the exponents

= 9 + (!16) rewriting subtraction as addition

= !7 adding

d. In this expression we will compute the numerator and denominators separately, then simplify the resulting fraction:

3xy

2x ! 4y=

3 3( ) !4( )

2 3( ) ! 4 !4( )substituting for variables

=!36

6 ! (!16)computing multiplications

=!36

6 +16rewriting subtraction as addition

=!36

22adding

= !/2 •18

/2 •11factoring the GCF

= !18

11simplifying fractions

Note that expressions change in value as their variables change in value also. For example, the expression 2xy2 can be evaluated when x = !3 and y = !2 :

2xy2= 2 !3( ) !2( )


= 2 !3( ) 4( ) computing the exponent

= !24 multiplying

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

2xy2= 2 !2( ) !3( )


= 2 !2( ) 9( ) computing the exponent

= !36 multiplying

271

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. X = 3ab ! 4b

2 b. Y = 4a

3! 3b

2

c. V =6ab

a2! b

2

d. V =7ab

2a + 4b

Solution a. Substituting a = !4 andb = 2 , then finding the value of X:

X = 3ab ! 4b2 given expression

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


= 3 !4( ) 2( ) ! 4 4( ) computing the exponent

= !24 !16 multiplying

= !24 + (!16) converting to addition

= !40 adding

b. Substituting a = !4 andb = 2 , then finding the value of Y:

Y = 4a3! 3b2 given expression

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


= 4 !64( ) ! 3 4( ) computing the exponents

= !256 !12 multiplying

= !256 + (!12) converting to addition

= !268 adding

272

c. Substituting a = !4 andb = 2 , then finding the value of V:

V =6ab

a2! b

2given expression

=6 !4( ) 2( )

!4( )2! 2( )


=6 !4( ) 2( )

16 ! 4computing the exponents

=!48

16 ! 4multiplying

=!48

12subtracting

= !4 simplifying

d. Substituting a = !4 andb = 2 , then finding the value of V:

V =7ab

2a + 4bgiven expression

=7 !4( ) 2( )

2 !4( ) + 4 2( )substituting for variables

=!56

!8 + 8multiplying

=!56

0adding

which is undefined

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.

273

Example 3 Find the value of each expression when s = !3

4 and t = 2

3.

a. 6st b. !8s + 6t c. 2s

2! 3t

2

d. s2! t

2

s2+ t

2

Solution a. Substituting s = !3

4 and t = 2

3, then simplifying the resulting expression:

6st = 6 !3

4

"#$

%&'

2

3

"#$

%&'


= 6 !1

2

"#$

%&'

multiplying fractions

= !3 multiplying

b. Substituting s = !3

4 and t = 2


!8s + 6t = !8 !3

4

"#$

%&'+ 6

2

3

"#$

%&'


= 6 + 4 multiplying fractions

= 10 adding

274

c. Substituting s = !3

4 and t = 2


2s2 ! 3t 2= 2 !

3

4

"#$

%&'

2

! 32

3

"#$

%&'

2


= 29

16

"#$

%&'! 3

4

9

"#$

%&'

computing exponents

=9

8!

4

3multiplying fractions

=27

24!

32

24converting to common denominators

= !5

24adding fractions

d. Substituting s = !3

4 and t = 2


s2 ! t 2

s2+ t

2=

!3

4

"#$

%&'

2

!2

3

"#$

%&'

2

!3

4

"#$

%&'

2

+2

3

"#$

%&'


=

9

16!

4

99

16+

4

9

computing exponents

=

9

16!

4

99

16+

4

9

•144

144multiplying by the LCM

=

9

16•144 !

4

9•144

9

16•144 +

4

9•144

distributive property

=81! 64

81+ 64multiplying fractions

=17

145simplifying

275

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. P = 2w + 2l ; w = 12, l = 17 b. A = !r

2 ; ! = 3.14, r = 6 c. S = 2!rh ; ! = 3.14, r = 5, h = 12 d. A = P(1+ r)

t ; P = 1250, r = 0.1, t = 6 Solution a. Substituting the values for the variables:

P = 2w + 2l given expression

= 2 12( ) + 2 17( ) substituting for variables

= 24 + 34 multiplying

= 58 adding

b. Substituting the values for the variables:

A = !r2 given expression

= 3.14( ) 6( )2


= 3.14( ) 36( ) computing the exponent

= 113.04 multiplying

c. Substituting the values for the variables:

S = 2!rh given expression

= 2 3.14( ) 5( ) 12( ) substituting for variables

= 6.28( ) 60( ) multiplying

= 376.8 multiplying

d. Substituting the values for the variables (and using a calculator):

A = P(1+ r)t given expression

= 1250(1+ 0.1)6 substituting for variables

= 1250 1.1( )6

computing the parentheses

= 1250 1.771561( ) evaluating the exponent

= 2214.45125 multiplying

276

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 . 1. 3x + 7y 2. 5x + 6y 3. 7x ! 6y 4. 5x ! 8y 5. 12xy 6. 15xy 7. !8xy 8. !9xy 9. 5x

2y 10. !4x

2y

11. !6xy2 12. 15xy2

13. x2! y

2 14. y2! x

2 15. 3x

2! 2xy 16. 5y

2! 8xy

17. 4xy ! 5xy2 18. 9x

2y ! 8xy

2

19. 5x

3y + 4xy 20. !4x

2x + 5y

21. 4xy

3x + 5y 22. 3x

2

3x ! 5y

23. 5xy2

!6x +10y 24. 4x ! 3y

2

!6x !10y

277

Find the value for each variable expression when a = 4 and b = !3 . 25. X = 4a ! 5b 26. Y = !7a ! 8b 27. Y = !4a

2! 3b

2 28. X = 2a2! 5b

2 29. W = 7a

2b3 30. W = !4a

2b2

31. W = !4ab + 8a2 32. W = !3a

2b + 2ab

2

33. M =a + b

4ab 34. M =

a ! b

6ab

35. M =a ! b

4a2b

36. M =2a ! 3b

5a2b

37. Q =3a + 4b

!8ab 38. Q =

!6a ! 8b

!5ab

39. Q =!3b

2

6a + 2ab 40. Q =

!5a2

2ab ! 8b

Find the value of each expression when s = !1

2 and t = 2

3:

41. 5st 42. !6st 43. 2s + 3t 44. !4s + 6t 45. 5s ! 4t 46. !3s ! 5t 47. s

2! 2t

2 48. 2s2! 4t

2 49. 2st ! 5s

2t 50. 3st

2! 6s

2t

51. s + t

s ! t 52. 4s + 5t

3s ! 4t

53. s2! t

2

2s2! 3t

2 54. 4s

2+ 9t

2

4s2! 9t

2

Evaluate the following formulas given the variable values. 55. P = 2w + 2l ; w = 9, l = 15 56. P = 2w + 2l ; w = 13, l = 19 57. P = 2w + 2l ;w = 4.7, l = 8.6 58. P = 2w + 2l ; w = 5.9, l = 12.4

59. P = 2w + 2l ;w = 51

2, l = 8

3

4 60. P = 2w + 2l ; w = 4

2

3, l = 6

3

4

61. A = !r2 ;! = 3.14, r = 4 62. A = !r

2 ; ! = 3.14, r = 12 63. A = !r

2 ;! = 3.14, r = 1.2 64. A = !r2 ; ! = 3.14, r = 1.5

65. S = 2!rh ;! = 3.14, r = 4, h = 5 66. S = 2!rh ; ! = 3.14, r = 6, h = 4

278

67. A = lw ; l = 12, w = 7 68. A = lw ; l = 15, w = 12 69. A = lw ; l = 8.5, w = 4.3 70. A = lw ; l = 12.7, w = 5.2

71. A = lw ; l = 9 12, w = 3

1

3 72. A = lw ; l = 6 1

4, w = 4

1

5

73. A = P(1+ r)t ; P = 1400, r = 0.08, t = 2

74. A = P(1+ r)t ; P = 2000, r = 0.12, t = 3 (round answer to nearest hundredth)


76. A = P(1+ r)t ; P = 10000, r = 0.1, t = 5



Answer each of the following application problems.

79. The formula A = P 1+r

n

!"#

$%&t

is used in investment computation. Compute the value of

A whenP = $8000, r = 0.08, n = 4, and t = 15 . Use a calculator and round your answer to the nearest hundredth.

80. The formula A = P 1+r

n

!"#

$%&t


A whenP = $12000, r = 0.09, n = 4, and t = 20 . Use a calculator and round your answer to the nearest hundredth. 81. 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. 82. 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. 83. 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. 84. 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.

85. The formula V =4

3!r

3 is used to compute the volume of a sphere. Compute the value

of V when ! = 3.1416 and r = 8.4 inches. Use a calculator and round your answer to the nearest tenth.

279

86. The formula V =4

3!r

3 is used to compute the volume of a sphere. Compute the value

of V when ! = 3.1416 and r = 12.86 inches. Use a calculator and round your answer to the nearest hundredth.

87. The formula C =5

9F ! 32( ) is used to convert temperature from Fahrenheit to

Celsius. Compute the value of C when F = 59°.



Celsius. Compute the value of C when F = 86°.



Celsius. Compute the value of C when F = –40°.



Celsius. Compute the value of C when F = –13°.

91. The formula F =9

5C + 32 is used to convert temperature from Celsius to Fahrenheit.

Compute the value of F when C = 30°.



Compute the value of F when C = 45°.



Compute the value of F when C = –20°.



Compute the value of F when C = –40°. 95. The formula P = 120p

3q7 is used in statistics to compute probability. Compute the

value of P when p = 13

and q = 23

. Express your answer as a fraction and as a decimal

rounded to four decimal places. 96. The formula P = 120p



and q = 12


rounded to four decimal places.

280

97. The formula P = 56p5q3 is used in statistics to compute probability. Compute the


and q = 12


rounded to four decimal places. 98. The formula P = 56p



and q = 23


rounded to four decimal places.

99. The formula P =A

1+ i( )n


P when A = $20000, i = 0.08, and n = 15. Use a calculator and round your answer to the nearest hundredth.

100. The formula P =A

1+ i( )n


P when A = $40000, i = 0.11, and n = 20. Use a calculator and round your answer to the nearest hundredth.

101. The formula F = p(1+ i)

n !1

i

"

#$

%

&' is used in investment computation. Compute the

value of F when p = $300, i = 0.08, and n = 20. Use a calculator and round your answer to the nearest hundredth.


n !1

i

"

#$

%




n !1

i

"

#$

%




n !1

i

"

#$

%



chapter 4 algebraic expressions and equations · pdf file269 chapter 4 algebraic expressions...

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