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2.2 Solve Equations using the Division and Multiplication Properties of Equality

Isolating a variable by dividing out its coefficient or multiplying by a reciprocal, including equations that need simplifying first.

Elementary Algebra 2e, 2.2 Solve Equations using the Division and Multiplication Properties of Equality

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81 exercises

2.2 Exercise 1

Before you get started, take this readiness quiz.

Simplify: −7(1−7).−7 \left(\frac{1}{−7}\right) .
If you missed this problem, review the reference.

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2.2 Exercise 2

Evaluate 9x+29 x + 2 when x=−3x = −3.
If you missed this problem, review the reference.

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2.2 Exercise 4

Solve: 3y=−41.3 y = −41 .

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2.2 Exercise 5

Solve: 4z=−55.4 z = −55 .

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2.2 Exercise 7

Solve: a−7=−42.\frac{a}{−7} = −42 .

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2.2 Exercise 8

Solve: b−6=−24.\frac{b}{−6} = −24 .

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2.2 Exercise 10

Solve: −k=8.− k = 8 .

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2.2 Exercise 11

Solve: −g=3.− g = 3 .

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2.2 Exercise 13

Solve: 25n=14.\frac{2}{5} n = 14 .

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2.2 Exercise 14

Solve: 56y=15.\frac{5}{6} y = 15 .

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2.2 Exercise 16

Solve: 925=−45z.\frac{9}{25} = - \frac{4}{5} z .

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2.2 Exercise 17

Solve: 56=−83r.\frac{5}{6} = - \frac{8}{3} r .

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2.2 Exercise 19

Solve: 18−27=15c−9c−3c.18 - 27 = 15 c - 9 c - 3 c .

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2.2 Exercise 20

Solve:18−22=12x−x−4x.18 - 22 = 12 x - x - 4 x .

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2.2 Exercise 22

Solve: −4(q−2)−8=24.−4 \left(q - 2\right) - 8 = 24 .

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2.2 Exercise 23

Solve: −6(r−2)−12=30.−6 \left(r - 2\right) - 12 = 30 .

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2.2 Exercise 25

Translate and solve: The number 132 is the product of −12 and y.

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2.2 Exercise 26

Translate and solve: The number 117 is the product of −13 and z.

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2.2 Exercise 28

Translate and solve: nn divided by 7 is equal to −21−21.

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2.2 Exercise 29

Translate and solve: nn divided by 8 is equal to −56−56.

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2.2 Exercise 31

Translate and solve: The quotient of qq and −8−8 is 72.

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2.2 Exercise 32

Translate and solve: The quotient of pp and −9−9 is 81.

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2.2 Exercise 34

Translate and solve: Two-fifths of ff is 16.

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2.2 Exercise 35

Translate and solve: Three-fourths of ff is 21.

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2.2 Exercise 37

Translate and solve: The sum of five-eighths and x is one-fourth.

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2.2 Exercise 38

Translate and solve: The sum of three-fourths and x is five-sixths.

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2.2 Exercise 39

Denae bought 6 pounds of grapes for $10.74. What was the cost of one pound of grapes?

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2.2 Exercise 40

Translate and solve:

Arianna bought a 24-pack of water bottles for $9.36. What was the cost of one water bottle?

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2.2 Exercise 41

Translate and solve:

At JB’s Bowling Alley, 6 people can play on one lane for $34.98. What is the cost for each person?

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2.2 Exercise 42

Andreas bought a used car for $12,000. Because the car was 4-years old, its price was 34\frac{3}{4} of the original price, when the car was new. What was the original price of the car?

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2.2 Exercise 43

Translate and solve:

The annual property tax on the Mehta’s house is $1,800, calculated as 151,000\frac{15}{1,000} of the assessed value of the house. What is the assessed value of the Mehta’s house?

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2.2 Exercise 44

Translate and solve:

Stella planted 14 flats of flowers in 23\frac{2}{3} of her garden. How many flats of flowers would she need to fill the whole garden?

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2.2 Exercise 45

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

8x=568 x = 56

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2.2 Exercise 47

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

−5c=55−5 c = 55

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2.2 Exercise 49

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

−809=15y−809 = 15 y

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2.2 Exercise 51

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

−37p=−541−37 p = −541

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2.2 Exercise 53

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

0.25z=3.250.25 z = 3.25

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2.2 Exercise 55

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

−13x=0−13 x = 0

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2.2 Exercise 57

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

x4=35\frac{x}{4} = 35

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2.2 Exercise 59

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

−20=q−5−20 = \frac{q}{−5}

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2.2 Exercise 61

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

y9=−16\frac{y}{9} = −16

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2.2 Exercise 63

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

m−12=45\frac{m}{−12} = 45

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2.2 Exercise 65

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

−y=6− y = 6

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2.2 Exercise 67

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

−v=−72− v = −72

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2.2 Exercise 69

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

23y=48\frac{2}{3} y = 48

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2.2 Exercise 71

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

−58w=40- \frac{5}{8} w = 40

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2.2 Exercise 73

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

−25=110a- \frac{2}{5} = \frac{1}{10} a

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2.2 Exercise 75

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

−710x=−143- \frac{7}{10} x = - \frac{14}{3}

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2.2 Exercise 77

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

712=−34p\frac{7}{12} = - \frac{3}{4} p

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2.2 Exercise 79

Solve each equation using the Division and Multiplication Properties of Equality and check the solution.

−518=−109u- \frac{5}{18} = - \frac{10}{9} u

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2.2 Exercise 81

Solve each equation requiring simplification.

100−16=4p−10p−p100 - 16 = 4 p - 10 p - p

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2.2 Exercise 83

Solve each equation requiring simplification.

78n−34n=9+2\frac{7}{8} n - \frac{3}{4} n = 9 + 2

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2.2 Exercise 85

Solve each equation requiring simplification.

0.25d+0.10d=6−0.750.25 d + 0.10 d = 6 - 0.75

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2.2 Exercise 87

Solve each equation requiring simplification.

−10(q−4)−57=93−10 \left(q - 4\right) - 57 = 93

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2.2 Exercise 89

Solve each equation requiring simplification.

−10(x+4)−19=85−10 \left(x + 4\right) - 19 = 85

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2.2 Exercise 91

Solve each equation.

910x=90\frac{9}{10} x = 90

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2.2 Exercise 93

Solve each equation.

y+46=55y + 46 = 55

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2.2 Exercise 95

Solve each equation.

w−2=99\frac{w}{−2} = 99

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2.2 Exercise 97

Solve each equation.

27=6a27 = 6 a

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2.2 Exercise 99

Solve each equation.

−x=2− x = 2

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2.2 Exercise 101

Solve each equation.

m−41=−14m - 41 = −14

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2.2 Exercise 103

Solve each equation.

63.90=0.03p63.90 = 0.03 p

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2.2 Exercise 105

Solve each equation.

84=−12z84 = −12 z

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2.2 Exercise 107

Solve each equation.

c−0.3c=35.70c - 0.3 c = 35.70

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2.2 Exercise 109

Solve each equation.

−x=−8− x = −8

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2.2 Exercise 111

Translate to an equation and then solve.

133 is the product of −19−19 and n.

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2.2 Exercise 113

Translate to an equation and then solve.

−152−152 is the product of 8 and q.

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2.2 Exercise 115

Translate to an equation and then solve.

r divided by 12 is equal to −48−48.

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2.2 Exercise 117

Translate to an equation and then solve.

j divided by −20−20 is equal to −80−80.

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2.2 Exercise 119

Translate to an equation and then solve.

The quotient of bb and −6−6 is 18.

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2.2 Exercise 121

Translate to an equation and then solve.

The quotient kk and 22 is −66−66.

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2.2 Exercise 123

Translate to an equation and then solve.

Three-tenths of x is 15.

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2.2 Exercise 125

Translate to an equation and then solve.

Five-halves of v is 50.

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2.2 Exercise 127

Translate to an equation and then solve.

The sum of two-fifths and f is one-half.

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2.2 Exercise 129

Translate to an equation and then solve.

The difference of q and one-eighth is three-fourths.

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2.2 Exercise 131

Translate into an equation and solve.

Balloons Ramona bought 18 balloons for a party. She wants to make 3 equal bunches. How many balloons did she use in each bunch?

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2.2 Exercise 133

Translate into an equation and solve.

Shopping Serena paid $12.96 for a pack of 12 pairs of sport socks. What was the price of pair of sport socks?

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2.2 Exercise 135

Translate into an equation and solve.

MPG John’s SUV gets 18 miles per gallon (mpg). This is half as many mpg as his wife’s hybrid car. How many miles per gallon does the hybrid car get?

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2.2 Exercise 137

Translate into an equation and solve.

Real estate Bea earned $11,700 commission for selling a house, calculated as 6100\frac{6}{100} of the selling price. What was the selling price of the house?

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2.2 Exercise 139

Stamps Travis bought $9.45 worth of 49-cent stamps and 21-cent stamps. The number of 21-cent stamps was 5 less than the number of 49-cent stamps. Solve the equation 0.49s+0.21​(s−5)​​=9.450.49 s + 0.21 ​ \left(s - 5\right) ​ ​ = 9.45 for s, to find the number of 49-cent stamps Travis bought.

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2.2 Exercise 141

Emiliano thinks x=40x = 40 is the solution to the equation 12x=80\frac{1}{2} x = 80. Explain why he is wrong.

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Adapted from Elementary Algebra 2e © 2020 Rice University. Licensed under CC BY 4.0. Source: Elementary Algebra 2e. Imported at revision 77a933c.