Using the Division and Multiplication Properties of Equality for Multi-Step Equations

Learning Outcomes

Solve a linear equation that requires simplification before using properties of equality
Solve a linear equation that requires a combination of the properties of equality

Many equations start out more complicated than the ones we’ve just solved. Our goal has been to familiarize you with the many ways to apply the addition, subtraction, multiplication, and division properties that are used to solve equations algebraically. Let’s work through an example that will employ the following techniques:

simplify by combining like terms
isolate x by using the division property of equality

Example

Solve: [latex]8x+9x - 5x=-3+15[/latex]

Solution:

First, we need to simplify both sides of the equation as much as possible

Start by combining like terms to simplify each side.

	[latex]8x+9x-5x=-3+15[/latex]
Combine like terms.	[latex]12x=12[/latex]
Divide both sides by 12 to isolate x.	[latex]\Large\frac{12x}{\color{red}{12}}\normalsize =\Large\frac{12}{\color{red}{12}}[/latex]
Simplify.	[latex]x=1[/latex]
Check your answer. Let [latex]x=1[/latex]
[latex]8x+9x-5x=-3+15[/latex]
[latex]8\cdot\color{red}{1}+9\cdot\color{red}{1}-5\cdot\color{red}{1}\stackrel{\text{?}}{=}-3+15[/latex]
[latex]8+9-5\stackrel{\text{?}}{=}-3+15[/latex]
[latex]12=12\quad\checkmark[/latex]

Here is a similar problem for you to try.

Try it

You may not always have the variables on the left side of the equation, so we will show an example with variables on the right side. You will see that the properties used to solve this equation are exactly the same as the previous example.

example

Solve: [latex]11 - 20=17y - 8y - 6y[/latex]

Show Solution

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.

Now you can try solving a similar problem.

try it

In our next example, we have an equation that contains a set of parentheses. We will use the distributive property of multiplication over addition first, simplify, then use the division property to finally solve.

example

Solve: [latex]-3\left(n - 2\right)-6=21[/latex]

Remember—always simplify each side first.

Show Solution

Now you can try a similar problem.

try it

In the following video you will see another example of using the division property of equality to solve an equation as well as another example of how to solve a multi-step equation that includes a set of parentheses.

	[latex]11-20=17y-8y-6y[/latex]
Simplify each side.	[latex]-9=3y[/latex]
Divide both sides by 3 to isolate y.	[latex]\Large\frac{-9}{\color{red}{3}}\normalsize =\Large\frac{3y}{\color{red}{3}}[/latex]
Simplify.	[latex]-3=y[/latex]
Check your answer. Let [latex]y=-3[/latex]
[latex]11-20=17y-8y-6y[/latex]
[latex]11-20\stackrel{\text{?}}{=}17( \color{red}{-3})-8(\color{red}{-3})-6(\color{red}{-3})[/latex]
[latex]11-20\stackrel{\text{?}}{=}-51+24+18[/latex]
[latex]-9=-9\quad\checkmark[/latex]

	[latex]-3(n-2)-6=21[/latex]
Distribute.	[latex]-3n+6-6=21[/latex]
Simplify.	[latex]-3n=21[/latex]
Divide both sides by -3 to isolate n.	[latex]\Large\frac{-3n}{\color{red}{-3}}\normalsize =\Large\frac{21}{\color{red}{-3}}[/latex] [latex]n=-7[/latex]
Check your answer. Let [latex]n=-7[/latex] .
[latex]-3(n-2)-6=21[/latex]
[latex]-3(\color{red}{-7}-2)-6\stackrel{\text{?}}{=}21[/latex]
[latex]-3(-9)-6\stackrel{\text{?}}{=}21[/latex]
[latex]27-6\stackrel{\text{?}}{=}21[/latex]
[latex]21=21\quad\checkmark[/latex]

Module 9: Multi-Step Linear Equations