Using the Subtraction and Addition Properties for Multi-Step Equations

Learning Outcomes

Solve a linear equation that needs to be simplified before using the subtraction and addition properties of equality
Check your solution to a linear equation to verify its accuracy

In the examples up to this point, we have been able to isolate the variable with just one operation. Many of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality. You should always simplify as much as possible before trying to isolate the variable.

Example

Solve:

[latex]3x - 7 - 2x - 4=1[/latex]

Solution:
The left side of the equation has an expression that we should simplify before trying to isolate the variable.

	[latex]3x-7-2x-4=1[/latex]
Rearrange the terms, using the Commutative Property of Addition.	[latex]3x-2x-7-4=1[/latex]
Combine like terms.	[latex]x-11=1[/latex]
Add [latex]11[/latex] to both sides to isolate [latex]x[/latex] .	[latex]x-11\color{red}{+11}=1\color{red}{+11}[/latex]
Simplify.	[latex]x=12[/latex]
Check. Substitute [latex]x=12[/latex] into the original equation. [latex]3x-7-2x-4=1[/latex] [latex]3(\color{red}{12})-7-2(\color{red}{12})-4=1[/latex] [latex]36-7-24-4=1[/latex] [latex]29-24-4=1[/latex] [latex]5-4=1[/latex] [latex]1=1\quad\checkmark[/latex] The solution checks.

Now you can try solving a couple equations where you should simplify first.

TRY IT

The last few examples involved simplifying using addition and subtraction. Let’s look at an example where we need to distribute first in order to simplify the equation as much as possible.

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

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.

	[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 9: Multi-Step Linear Equations