# 4.13: Simplify Variable Expressions Involving Integer Division

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

**Practice**Simplify Variable Expressions Involving Integer Division

Have you ever had to do homework while on vacation?

Molly was working on her homework in the ski lodge. She had one more problem to do before she could go back out for another run on the mountain.

Here is the problem.

\begin{align*}\frac{-12x}{-3}\end{align*}

Molly isn't sure how to work with this problem. She knows that it involves integers, variables and division.

**This Concept will teach you how to help Molly with her dilemma.**

### Guidance

You have used variable expressions with addition, subtraction and multiplication. Now we are going to apply division of integers with variable expressions. **Remember that a** *variable expression***is a math phrase that uses numbers, variables and operations.**

Find the value of this expression \begin{align*}-18x \div (-2)\end{align*}

It may help you to rewrite the problem like this using a fraction bar to divide. Now you can see which values can be divided

\begin{align*}\frac{-18x}{-2}\end{align*}

Then separate out the integers like this.

\begin{align*}\frac{-18x}{-2} = \frac{-18 \cdot x}{-2} = \frac{-18}{-2} \cdot x\end{align*}

Notice that we can divide the integers. The \begin{align*}x\end{align*} remains alone because there isn’t another \begin{align*}x\end{align*}. We separate it out. Then we divide the integer part and add the \begin{align*}x\end{align*} to the answer.

Since \begin{align*}-18 \div (-2) = 18 \div 2 = 9\end{align*}, we know that \begin{align*}\frac{-18}{-2} \cdot x= 9 \cdot x = 9x\end{align*}

**So, the value of the expression is \begin{align*}9x\end{align*}.**

Let’s look at another problem where there is a matching variable too.

\begin{align*}-24y \div 2y\end{align*}

Next, we rewrite the expression using a fraction bar.

\begin{align*}\frac{-24y}{2y}\end{align*}

Now, we can separate the terms.

\begin{align*}&\frac{-24}{2} \cdot \frac{y}{y}\\ &-24 \div 2 = -12\end{align*}

\begin{align*}y \div y =1\end{align*} because the \begin{align*}y\end{align*}’s cancel each other out

\begin{align*}-12(1) = -12\end{align*}

**The value of the expression is -12.**

Now it's time for you to try a few on your own.

#### Example A

\begin{align*}-14a \div -7\end{align*}

**Solution:\begin{align*}2a\end{align*}**

#### Example B

\begin{align*}28ab \div -7b\end{align*}

**Solution:\begin{align*}-4a\end{align*}**

#### Example C

\begin{align*}-6x \div -2\end{align*}

**Solution:\begin{align*}3x\end{align*}**

Here is the original problem once again.

Molly was working on her homework in the ski lodge. She had one more problem to do before she could go back out for another run on the mountain.

Here is the problem.

\begin{align*}\frac{-12x}{-3}\end{align*}

Molly isn't sure how to work with this problem. She knows that it involves integers, variables and division.

Molly is correct.

First, let's complete the division of the numbers. We can follow the rules for dividing integers and divide negative 12 by negative 3. That answer is 4.

Because there isn't another \begin{align*}x\end{align*} to work with, we can simply add that to our quotient.

\begin{align*}4x\end{align*}

**This is the answer.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Quotient
- the answer in a division problem.

- Undefined
- when an integer is divided by 0, the answer of that is undefined.

- Variable Expression
- a math phrase with numbers, operations and variables

### Guided Practice

Here is one for you to try on your own.

\begin{align*}-18ab \div 9b\end{align*}

**Answer**

First, rewrite the expression using a fraction bar.

\begin{align*}\frac{-18ab}{-9b}\end{align*}

Next, separate out the terms.

\begin{align*}&\frac{-18}{-9} \cdot \frac{ab}{b}\\ &-18 \div -9 = 2\end{align*}

\begin{align*}ab \div b= a\end{align*} Notice that the \begin{align*}b\end{align*}’s cancel, but the a doesn’t. It is left as part of the final expression.

**Our final answer is \begin{align*}2a\end{align*}.**

### Video Review

Here is a video for review.

- This is a James Sousa video on dividing integers. It is a supporting video for this Concept

### Practice

Directions: Find each quotient with variable expressions.

1. \begin{align*}36t \div (-9)\end{align*}

2. \begin{align*}-56n \div (-7)\end{align*}

3. \begin{align*}-22n \div -11n\end{align*}

4. \begin{align*}-28n \div 7\end{align*}

5. \begin{align*}18xy \div 2x\end{align*}

6. \begin{align*}72t \div (-9t)\end{align*}

7. \begin{align*}48xy \div (-8y)\end{align*}

8. \begin{align*}54xy \div (-9xy)\end{align*}

9. \begin{align*}16a \div (4a)\end{align*}

10. \begin{align*}-16ab \div (-4b)\end{align*}

11. \begin{align*}-99xy \div (-9x)\end{align*}

12. \begin{align*}121a \div (11b)\end{align*}

13. \begin{align*}-144xy \div (-12)\end{align*}

14. \begin{align*}-169y \div (-13x)\end{align*}

15. \begin{align*}-225xy \div (5z)\end{align*}

Quotient

The quotient is the result after two amounts have been divided.Undefined

An expression in mathematics is undefined when it has no meaning. Division by zero is undefined.Variable Expression

A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.### Image Attributions

Here you'll learn to evaluate and simplify variable expressions involving integer division.

Quotient

The quotient is the result after two amounts have been divided.Undefined

An expression in mathematics is undefined when it has no meaning. Division by zero is undefined.Variable Expression

A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.