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Solve equations with fraction coefficients

In the following exercises, solve the equation by clearing the fractions.

1 4 x 1 2 = 3 4

x = −1

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5 6 y 2 3 = 3 2

y = −1

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1 2 a + 3 8 = 3 4

a = 3 4

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2 = 1 3 x 1 2 x + 2 3 x

x = 4

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2 = 3 5 x 1 3 x + 2 5 x

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1 4 m 4 5 m + 1 2 m = −1

m = 20

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5 6 n 1 4 n 1 2 n = −2

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x + 1 2 = 2 3 x 1 2

x = −3

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1 3 w + 5 4 = w 1 4

w = 9 4

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1 2 x 1 4 = 1 12 x + 1 6

x = 1

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1 2 a 1 4 = 1 6 a + 1 12

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1 3 b + 1 5 = 2 5 b 3 5

b = 12

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1 3 x + 2 5 = 1 5 x 2 5

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1 = 1 6 ( 12 x 6 )

x = 1

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1 4 ( p 7 ) = 1 3 ( p + 5 )

p = −41

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1 5 ( q + 3 ) = 1 2 ( q 3 )

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1 2 ( x + 4 ) = 3 4

x = 5 2

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Solve Equations with Decimal Coefficients

In the following exercises, solve the equation by clearing the decimals.

0.4 x + 0.6 = 0.5 x 1.2

x = 18

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0.7 x + 0.4 = 0.6 x + 2.4

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0.23 x + 1.47 = 0.37 x 1.05

x = 18

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0.48 x + 1.56 = 0.58 x 0.64

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0.9 x 1.25 = 0.75 x + 1.75

x = 20

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1.2 x 0.91 = 0.8 x + 2.29

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0.05 n + 0.10 ( n + 8 ) = 2.15

n = 9

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0.05 n + 0.10 ( n + 7 ) = 3.55

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0.10 d + 0.25 ( d + 5 ) = 4.05

d = 8

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0.10 d + 0.25 ( d + 7 ) = 5.25

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0.05 ( q 5 ) + 0.25 q = 3.05

q = 11

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0.05 ( q 8 ) + 0.25 q = 4.10

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Everyday math

Coins Taylor has $2.00 in dimes and pennies. The number of pennies is 2 more than the number of dimes. Solve the equation 0.10 d + 0.01 ( d + 2 ) = 2 for d , the number of dimes.

d = 18

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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 0.49 s + 0.21 ( s 5 ) = 9.45 for s , to find the number of 49-cent stamps Travis bought.

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Writing exercises

Explain how to find the least common denominator of 3 8 , 1 6 , and 2 3 .

Answers will vary.

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If an equation has several fractions, how does multiplying both sides by the LCD make it easier to solve?

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If an equation has fractions only on one side, why do you have to multiply both sides of the equation by the LCD?

Answers will vary.

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In the equation 0.35 x + 2.1 = 3.85 , what is the LCD? How do you know?

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

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Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

Chapter review exercises

Solve Equations using the Subtraction and Addition Properties of Equality

In the following exercises, determine whether the given number is a solution to the equation.

x + 16 = 31 , x = 15

yes

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In the following exercises, solve the equation using the Subtraction Property of Equality.

In the following exercises, solve the equation using the Addition Property of Equality.

c 3 11 = 9 11

c = 12 11

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In the following exercises, solve the equation.

y + 8 15 = −3

y = 4

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7 x + 10 6 x + 3 = 5

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6 ( n 1 ) 5 n = −14

n = −8

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8 ( 3 p + 5 ) 23 ( p 1 ) = 35

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In the following exercises, translate each English sentence into an algebraic equation and then solve it.

The sum of −6 and m is 25 .

−6 + m = 25; m = 31

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Four less than n is 13 .

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In the following exercises, translate into an algebraic equation and solve.

Rochelle’s daughter is 11 years old. Her son is 3 years younger. How old is her son?

s = 11 − 3; 8 years old

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Tan weighs 146 pounds. Minh weighs 15 pounds more than Tan. How much does Minh weigh?

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Peter paid $9.75 to go to the movies, which was $46.25 less than he paid to go to a concert. How much did he pay for the concert?

c − 46.25 = 9.75; $56.00

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Elissa earned $152.84 this week, which was $21.65 more than she earned last week. How much did she earn last week?

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Solve Equations using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation using the Division Property of Equality.

In the following exercises, solve each equation using the Multiplication Property of Equality.

In the following exercises, solve each equation.

5 r 3 r + 9 r = 35 2

r = 3

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24 x + 8 x 11 x = −7 −14

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Solve Equations with Variables and Constants on Both Sides

In the following exercises, solve the equations with constants on both sides.

3 x + 19 = −47

x = −22

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In the following exercises, solve the equations with variables on both sides.

7 y = 6 y 13

y = −13

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k = −6 k 35

k = −5

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In the following exercises, solve the equations with constants and variables on both sides.

12 x 9 = 3 x + 45

x = 6

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5 n 20 = −7 n 80

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4 u + 16 = −19 u

u = −7

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In the following exercises, solve each linear equation using the general strategy.

6 ( x + 6 ) = 24

x = −2

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( s + 4 ) = 18

s = −22

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23 3 ( y 7 ) = 8

y = 12

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1 3 ( 6 m + 21 ) = m 7

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8 ( r 2 ) = 6 ( r + 10 )

r = 38

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5 + 7 ( 2 5 x ) = 2 ( 9 x + 1 ) ( 13 x 57 )

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4 ( 3.5 y + 0.25 ) = 365

y = 26

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0.25 ( q 8 ) = 0.1 ( q + 7 )

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Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation by clearing the fractions.

2 5 n 1 10 = 7 10

n = 2

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3 4 a 1 3 = 1 2 a + 5 6

a = 14 3

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1 2 ( k + 3 ) = 1 3 ( k + 16 )

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In the following exercises, solve each equation by clearing the decimals.

0.8 x 0.3 = 0.7 x + 0.2

x = 5

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0.36 u + 2.55 = 0.41 u + 6.8

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0.6 p 1.9 = 0.78 p + 1.7

p = −20

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0.10 d + 0.05 ( d 4 ) = 2.05

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Chapter practice test

Determine whether each number is a solution to the equation.
3 x + 5 = 23 .

  1. 6
  2. 23 5

  1. yes
  2. no

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In the following exercises, solve each equation.

−8 x 15 + 9 x 1 = −21

x = −5

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10 y = −5 y + 60

y = 4

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9 m 2 4 m + m = 42 8

m = 6

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( d + 9 ) = 23

d = −32

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1 3 ( 6 m + 21 ) = m 7

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2 ( 6 x + 5 ) 8 = −22

x = −2

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8 ( 3 a + 5 ) 7 ( 4 a 3 ) = 20 3 a

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1 4 p + 1 3 = 1 2

p = 2 3

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0.1 d + 0.25 ( d + 8 ) = 4.1

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Translate and solve: The difference of twice x and 4 is 16 .

2 x − 4 = 16; x = 10

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Samuel paid $25.82 for gas this week, which was $3.47 less than he paid last week. How much did he pay last week?

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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