### Learning Objectives

By the end of this section, you will be able to:

- Solve equations using the Division and Multiplication Properties of Equality
- Solve equations that need to be simplified

Before you get started, take this readiness quiz.

Simplify: $\mathrm{-7}\left(\frac{1}{\mathrm{-7}}\right).$

If you missed this problem, review Example 4.28.

What is the reciprocal of $-\frac{3}{8}?$

If you missed this problem, review Example 4.29.

Evaluate $9x+2$ when $x=\mathrm{-3}.$

If you missed this problem, review Example 3.56.

### Solve Equations Using the Division and Multiplication Properties of Equality

We introduced the Multiplication and Division Properties of Equality in Solve Equations Using Integers; The Division Property of Equality and Solve Equations with Fractions. We modeled how these properties worked using envelopes and counters and then applied them to solving equations (See Solve Equations Using Integers; The Division Property of Equality). We restate them again here as we prepare to use these properties again.

### Division and Multiplication Properties of Equality

Division Property of Equality: For all real numbers $a,b,c,$ and $c\ne 0,$ if $a=b,$ then $\frac{a}{c}=\frac{b}{c}.$

Multiplication Property of Equality: For all real numbers $a,b,c,$ if $a=b,$ then $ac=bc.$

When you divide or multiply both sides of an equation by the same quantity, you still have equality.

Let’s review how these properties of equality can be applied in order to solve equations. Remember, the goal is to ‘undo’ the operation on the variable. In the example below the variable is multiplied by $4,$ so we will divide both sides by $4$ to ‘undo’ the multiplication.

### Example 8.13

Solve: $4x=\mathrm{-28}.$

Solve: $3y=\mathrm{-48}.$

Solve: $4z=\mathrm{-52}.$

In the previous example, to ‘undo’ multiplication, we divided. How do you think we ‘undo’ division?

### Example 8.14

Solve: $\frac{\phantom{\rule{0.4em}{0ex}}a}{\mathrm{-7}}=\mathrm{-42}.$

Solve: $\frac{\phantom{\rule{0.4em}{0ex}}b}{\mathrm{-6}}=\mathrm{-24}.$

Solve: $\frac{\phantom{\rule{0.4em}{0ex}}c}{\mathrm{-8}}=\mathrm{-16}.$

### Example 8.15

Solve: $-r=2.$

Solve: $-k=8.$

Solve: $-g=3.$

### Example 8.16

Solve: $\frac{2}{3}\phantom{\rule{0.1em}{0ex}}x=18.$

Solve: $\frac{2}{5}\phantom{\rule{0.1em}{0ex}}n=14.$

Solve: $\frac{5}{6}\phantom{\rule{0.1em}{0ex}}y=15.$

### Solve Equations That Need to be Simplified

Many equations start out more complicated than the ones we’ve just solved. First, we need to simplify both sides of the equation as much as possible

### Example 8.17

Solve: $8x+9x-5x=\mathrm{-3}+15.$

Solve: $7x+6x-4x=\mathrm{-8}+26.$

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

### Example 8.18

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

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

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

### Example 8.19

Solve: $\mathrm{-3}(n-2)-6=21.$

Solve: $\mathrm{-4}(n-2)-8=24.$

Solve: $\mathrm{-6}(n-2)-12=30.$

### Links To Literacy

*Links to Literacy*activity, "Everybody Wins" will provide you with another view of the topics covered in this section.

### Section 8.2 Exercises

#### 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.

$8x=32$

$\mathrm{-5}c=55$

$\mathrm{-90}=6y$

$\mathrm{-16}p=\mathrm{-64}$

$0.25z=3.25$

$\mathrm{-3}x=0$

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

$\frac{x}{4}=15$

$\mathrm{-20}=\frac{\phantom{\rule{0.4em}{0ex}}q}{\mathrm{-5}}$

$\frac{y}{9}=\mathrm{-6}$

$\frac{m}{\mathrm{-12}}=5$

$\frac{2}{3}\phantom{\rule{0.1em}{0ex}}y=18$

$-\frac{5}{8}\phantom{\rule{0.1em}{0ex}}w=40$

$-\frac{2}{5}=\frac{1}{10}\phantom{\rule{0.1em}{0ex}}a$

**Solve Equations That Need to be Simplified**

In the following exercises, solve the equation.

$8a+3a-6a=\mathrm{-17}+27$

$\mathrm{-9}x-9x+2x=50-2$

$100-16=4p-10p-p$

$\frac{7}{8}\phantom{\rule{0.1em}{0ex}}n-\frac{3}{4}\phantom{\rule{0.1em}{0ex}}n=9+2$

$0.25d+0.10d=6-0.75$

#### 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 $3b=18.$

**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 $4g=24.$

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

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

**Fuel economy** Tania’s SUV gets half as many miles per gallon (mpg) as her husband’s hybrid car. The SUV gets $\text{18 mpg}.$ Find the miles per gallons, $m,$ of the hybrid car, by solving the equation $\frac{1}{2}\phantom{\rule{0.1em}{0ex}}m=18.$

**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 $\frac{1}{3}\phantom{\rule{0.1em}{0ex}}f=14.$

#### Writing Exercises

Frida started to solve the equation $\mathrm{-3}x=36$ by adding $3$ to both sides. Explain why Frida’s method will result in the correct solution.

Emiliano thinks $x=40$ is the solution to the equation $\frac{1}{2}\phantom{\rule{0.1em}{0ex}}x=80.$ Explain why he is wrong.

#### Self Check

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

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