7.1.e – Using the Distributive Property When Solving Equations

Learning Outcomes

Use the distributive property to solve equations containing parentheses

The Distributive Property

As we solve linear equations, we often need to do some work to write the linear equations in a form we are familiar with solving. This section will focus on manipulating an equation we are asked to solve in such a way that we can use the skills we learned for solving multi-step equations to ultimately arrive at the solution.

Parentheses can make solving a problem difficult. To get rid of these unwanted parentheses, we use the distributive property. Using this property, we multiply the number in front of the parentheses by each term inside of the parentheses.

The Distributive Property of Multiplication

For all real numbers [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex], [latex]a(b+c)=ab+ac[/latex].

What this means is that when a number multiplies an expression inside parentheses, you can distribute the multiplication to each term of the expression individually. Then, you can follow the steps we have already practiced to isolate the variable and solve the equation.

Simple distribution and two-step equations

Example

Solve for [latex]a[/latex].

[latex]4\left(2a+3\right)=28[/latex]

Show Solution

In our next example, 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 video that follows, we show another example of how to use the distributive property to solve a multi-step linear equation.

Distribution and combining like terms

example

Solve: [latex]3\left(n - 4\right)-2n=-3[/latex]

Show Solution

Now you can try a few problems that involve distribution.

TRY IT

Distribution and simplifying on both sides

The next example has expressions on both sides that need to be simplified.

example

Solve: [latex]2\left(3k - 1\right)-5k=-2 - 7[/latex]

Show Solution

Now, you give it a try!

TRY IT

In the following video, we present another example of how to solve an equation that requires simplifying before using the addition and subtraction properties.

Using the distribution property on both sides of the equation

In the next example, you will see that there are parentheses on both sides of the equal sign, so you will need to use the distributive property twice. Notice that you are going to need to distribute a negative number, so be careful with negative signs!

Example

Solve for [latex]t[/latex].

[latex]2\left(4t-5\right)=-3\left(2t+1\right)[/latex]

Show Solution

Try It

In the following video, we solve another multi-step equation with two sets of parentheses.

	[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]

	[latex]3(n-4)-2n=-3[/latex]
Distribute on the left.	[latex]3n-12-2n=-3[/latex]
Use the Commutative Property to rearrange terms.	[latex]3n-2n-12=-3[/latex]
Combine like terms.	[latex]n-12=-3[/latex]
Isolate n using the Addition Property of Equality.	[latex]n-12\color{red}{+12}=-3\color{red}{+12}[/latex]
Simplify.	[latex]n=9[/latex]
Check.Substitute [latex]n=9[/latex] into the original equation. [latex]3(n-4)-2n=-3[/latex] [latex]3(\color{red}{9}-4)-2\cdot\color{red}{9}=-3[/latex] [latex]3(5)-18=-3[/latex] [latex]15-18=-3[/latex] [latex]-3=-3\quad\checkmark[/latex] The solution checks.

	[latex]2(3k-1)-5k=-2-7[/latex]
Distribute on the left, subtract on the right.	[latex]6k-2-5k=-9[/latex]
Use the Commutative Property of Addition.	[latex]6k-5k-2=-9[/latex]
Combine like terms.	[latex]k-2=-9[/latex]
Undo subtraction by using the Addition Property of Equality.	[latex]k-2\color{red}{+2}=-9\color{red}{+2}[/latex]
Simplify.	[latex]k=-7[/latex]
Check.Let [latex]k=-7[/latex]. [latex]2(3k-1)-5k=-2-7[/latex] [latex]2(3(\color{red}{-7}-1)-5(\color{red}{-7})=-2-7[/latex] [latex]2(-21-1)-5(-7)=-9[/latex] [latex]2(-22)+35=-9[/latex] [latex]-44+35=-9[/latex] [latex]-9=-9\quad\checkmark[/latex]

Module 7: Linear Equations