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Solve: 8 x + 9 x 5 x = −3 + 15 .

Solution

Start by combining like terms to simplify each side.

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Combine like terms. .
Divide both sides by 12 to isolate x. .
Simplify. .
Check your answer. Let x = 1
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Solve: 7 x + 6 x 4 x = −8 + 26 .

x = 2

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Solve: 11 n 3 n 6 n = 7 17 .

n = −5

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Solve: 11 20 = 17 y 8 y 6 y .

Solution

Simplify each side by combining like terms.

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Simplify each side. .
Divide both sides by 3 to isolate y. .
Simplify. .
Check your answer. Let y = −3
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Notice that the variable ended up on the right side of the equal sign when we solved the equation. You may prefer to take one more step to write the solution with the variable on the left side of the equal sign.

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

c = −3

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Solve: 18 22 = 12 x x 4 x .

x = 4 7

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Solve: −3 ( n 2 ) 6 = 21 .

Solution

Remember—always simplify each side first.

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Distribute. .
Simplify. .
Divide both sides by -3 to isolate n. .
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Check your answer. Let n = −7 .
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Solve: −4 ( n 2 ) 8 = 24 .

n = −6

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Solve: −6 ( n 2 ) 12 = 30 .

n = −5

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Key concepts

  • Division and Multiplication Properties of Equality
    • Division Property of Equality: For all real numbers a, b, c, and c 0 , if a = b , then a c = b c .
    • Multiplication Property of Equality: For all real numbers a, b, c, if a = b , then a c = b c .

Practice makes perfect

Solve Equations Using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation for the variable using the Division Property of Equality and check the solution.

In the following exercises, solve each equation for the variable using the Multiplication Property of Equality and check the solution.

Solve Equations That Need to be Simplified

In the following exercises, solve the equation.

8 a + 3 a 6 a = −17 + 27

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6 y 3 y + 12 y = −43 + 28

y = −1

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−9 x 9 x + 2 x = 50 2

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−5 m + 7 m 8 m = −6 + 36

m = −5

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100 16 = 4 p 10 p p

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−18 7 = 5 t 9 t 6 t

t = 5 2

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5 12 q + 1 2 q = 25 3

q = 24

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0.25 d + 0.10 d = 6 0.75

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0.05 p 0.01 p = 2 + 0.24

p = 56

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Everyday math

Balloons Ramona bought 18 balloons for a party. She wants to make 3 equal bunches. Find the number of balloons in each bunch, b , by solving the equation 3 b = 18 .

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Teaching Connie’s kindergarten class has 24 children. She wants them to get into 4 equal groups. Find the number of children in each group, g , by solving the equation 4 g = 24 .

6 children

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Ticket price Daria paid $36.25 for 5 children’s tickets at the ice skating rink. Find the price of each ticket, p , by solving the equation 5 p = 36.25 .

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Unit price Nishant paid $12.96 for a pack of 12 juice bottles. Find the price of each bottle, b , by solving the equation 12 b = 12.96 .

$1.08

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Fuel economy Tania’s SUV gets half as many miles per gallon (mpg) as her husband’s hybrid car. The SUV gets 18 mpg . Find the miles per gallons, m , of the hybrid car, by solving the equation 1 2 m = 18 .

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Fabric The drill team used 14 yards of fabric to make flags for one-third of the members. Find how much fabric, f , they would need to make flags for the whole team by solving the equation 1 3 f = 14 .

42 yards

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Writing exercises

Frida started to solve the equation −3 x = 36 by adding 3 to both sides. Explain why Frida’s method will result in the correct solution.

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Emiliano thinks x = 40 is the solution to the equation 1 2 x = 80 . Explain why he is wrong.

Answer will vary.

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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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After reviewing this checklist, what will you do to become confident for all objectives?

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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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