# 12.6: Single Variable Subtraction Equations

**At Grade**Created by: CK-12

**Practice**Single Variable Subtraction Equations

Remember Kelly and the photo?

Well, after the students arrived at the amusement park, Kelly and her friends went to take the picture. The photographer provided them with the following information about cost.

For nine people to be in the photo, the students would pay \begin{align*}$22.50\end{align*}.

But, the man also said there was a discount for a group larger than 5, so the total would be \begin{align*}$18.50\end{align*}.

What was the amount of the discount?

**To figure this out you will need to write an equation and then solve it. This Concept will teach you how to accomplish this task.**

### Guidance

In the last Concept, you learned how to solve single-variable addition equations. Now you are going to learn how to solve single-variable subtraction equations.

**How do we do this?**

We can use inverses once again to solve single-variable subtraction equations. The inverse of subtraction is addition, so we can use the inverse operation to help us in solving each problem.

\begin{align*}x-12=40\end{align*}

**If you think this through, it means “Some number minus twelve is equal to 40.”**

**To figure this out, you can use the inverse of subtraction (addition) and add 12 to both sides of the equation. That will help to get the variable alone and solve the problem.**

\begin{align*}&x-12\ = \ \ 40\\ & \underline{\quad + 12 \quad \ +12}\\ & \qquad x \ \ = \ \ 52\end{align*}

**Notice that -12 + 12 is equal to 0. That got the variable alone on the left side of the equals. On the right side, we added 12 and got an answer of 52.**

**To check this answer, we can substitute it back into the original problem and see if we have a true statement.**

\begin{align*}52-12&=40\\ 40&=40\end{align*}

**Our answer is true so our work is accurate.**

Try and solve a few of these on your own. Write your answer \begin{align*}x = \end{align*}.

#### Example A

\begin{align*}x-9=22\end{align*}

**Solution:\begin{align*}x = 31\end{align*}**

#### Example B

\begin{align*}x-3=46\end{align*}

**Solution:\begin{align*}x = 49\end{align*}**

#### Example C

\begin{align*}x-7=23\end{align*}

**Solution:\begin{align*}x = 30\end{align*}**

Here is the original problem once again.

Remember Kelly and the photo?

Well, after the students arrived at the amusement park, Kelly and her friends went to take the picture. The photographer provided them with the following information about cost.

For nine people to be in the photo, the students would pay \begin{align*}$22.50\end{align*}.

But, the man also said there was a discount for a group larger than 5, so the total would be \begin{align*}$18.50\end{align*}.

What was the amount of the discount?

First, we write a single variable subtraction equation using the given information.

\begin{align*}22.50\end{align*} is the starting cost.

\begin{align*}18.50\end{align*} is the final cost.

\begin{align*}x\end{align*} is the unknown discount.

Here is the equation.

\begin{align*}22.50 - x = 18.50\end{align*}

Now we solve it.

\begin{align*}x = $4.00\end{align*}

**The amount of the discount was \begin{align*}$4.00\end{align*}.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Expression
- a combination of variables, numbers and operations without an equal sign.

- Simplify
- to make smaller

- Inverse
- the opposite. An inverse operation is the opposite operation.

- Sum
- the answer to an addition problem

- Difference
- the answer to a subtraction problem

### Guided Practice

Here is one for you to try on your own.

\begin{align*}y-21=59\end{align*}

**Answer**

To solve this problem, we use the inverse operation.

\begin{align*}y - 21 + 21 = 59 + 21\end{align*}

\begin{align*}y = 80\end{align*}

**This is our answer.**

### Video Review

Here are videos for review.

James Sousa, Solve One Step Equations by Adding and Subtracting Whole Numbers

Other Videos:

- http://www.mathplayground.com/howto_solvevariable.html – This is a video on how to solve a variable equation. It goes through each type of equation step by step.

### Practice

Directions: Solve each single-variable subtraction problem using the inverse operation. Write your answer in the form: variable = _____.

1. \begin{align*}y-5=10\end{align*}

2. \begin{align*}x-7=17\end{align*}

3. \begin{align*}a-4=12\end{align*}

4. \begin{align*}z-6=22\end{align*}

5. \begin{align*}y-9=11\end{align*}

6. \begin{align*}b-5=12\end{align*}

7. \begin{align*}x-8=30\end{align*}

8. \begin{align*}y-7=2\end{align*}

9. \begin{align*}x-9=1\end{align*}

10. \begin{align*}x-19=15\end{align*}

11. \begin{align*}x-18=12\end{align*}

12. \begin{align*}x-29=31\end{align*}

13. \begin{align*}x-15=62\end{align*}

14. \begin{align*}x-22=45\end{align*}

15. \begin{align*}x-19=37\end{align*}

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Difference

The result of a subtraction operation is called a difference.Expression

An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.Simplify

To simplify means to rewrite an expression to make it as "simple" as possible. You can simplify by removing parentheses, combining like terms, or reducing fractions.Sum

The sum is the result after two or more amounts have been added together.### Image Attributions

Here you'll learn to solve single variable subtraction equations.