One-step addition & subtraction equations

Google Classroom

Learn to solve equations like "x + 3 = 9" or "y  - 5 = 8".

Based on our understanding of the balance beam model, we know that we always have to do the same thing to both sides of an equation to keep it true.

But how do we know what to do to both sides of the equation?

Addition and subtraction are inverse operations

Inverse operations are opposite operations that undo or counteract each other.

Here's an example of how subtraction is the inverse operation of addition:

If we start with seven, add three, then subtract three, we get back to seven:

$7 + 3 - 3 = 7$7, plus, 3, minus, 3, equals, 7

Here's an example of how addition is the inverse operation of subtraction:

If we start with five, subtract two, then add two, we get back to five:

$5 - 2 + 2 = 5$5, minus, 2, plus, 2, equals, 5

Solving an addition equation using inverse operations

Let's think about how we can solve for $k$k in the following equation:

$\qquad k + 22 = 29$k, plus, 22, equals, 29

We want to get $k$k by itself on the left hand side of the equation. So, what can we do to undo adding 22?

We can subtract 22 because the inverse operation of addition is subtraction!

Here's how subtracting 22 from each side looks:

$\begin{aligned} k + 22 &= 29 \\\\ k + 22 \blueD{- 22} &= 29 \blueD{- 22}~~~~~~~~~~\small\gray{\text{Subtract 22 from each side.}} \\\\ k &= \greenD{7}~~~~~~~~~~\small\gray{\text{Simplify.}} \end{aligned}$

Let's check our work.

It's always a good idea to check our solution in the original equation to make sure we didn't make any mistakes:

$\qquad$ $\begin{aligned} k +22 &= 29 \\ \greenD{7} +22 &\stackrel{\large?}{=} 29\\ 29 &= 29 \end{aligned}$

Yes, $k = \greenD{7}$k, equals, start color #1fab54, 7, end color #1fab54 is a solution!

Solving a subtraction equation using inverse operations

Now let's try to solve a slightly different type of equation:

$\qquad p - 18 = 3$p, minus, 18, equals, 3

We want to get $p$p by itself on the left hand side of the equation. So, what can we do to cancel out subtracting 18?

We can add 18 because the inverse operation of subtraction is addition!

Here's how adding 18 to each side looks:

$\begin{aligned} p - 18 &= 3 \\\\ p - 18 \blueD{+ 18} &= 3 \blueD{+ 18}~~~~~~~~~~\small\gray{\text{Add 18 to each side.}} \\\\ p &= \greenD{21}~~~~~~~~~~\small\gray{\text{Simplify.}} \end{aligned}$

Let's check our work.

$\qquad$ $\begin{aligned} p - 18 &= 3 \\ \greenD{21} - 18 &\stackrel{\large?}{=} 3\\ 3 &= 3 \end{aligned}$

Yes, $p = \greenD{21}$p, equals, start color #1fab54, 21, end color #1fab54 is a solution!

Summary of how to solve addition and subtraction equations

Cool, so we just solved an addition equation and a subtraction equation. Let's summarize what we did:

Type of equation	Example	First step
Addition equation	$k + 22 = 29$k, plus, 22, equals, 29	Subtract 22 from each side.
Subtraction equation	$p - 18 = 3$p, minus, 18, equals, 3	Add 18 to each side.