# 4.9: Simplify Variable Expressions Involving Integer Subtraction

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

Noah has started up a lawn mowing business to make some extra money this summer. He charges $14 to mow a lawn. For each lawn he mows, it costs him $1.50 for supplies. If \begin{align*}L\end{align*} stands for the number of lawns Noah mows, how could Noah write and simplify an expression for his profit from mowing lawns this summer? If Noah ends up mowing 130 lawns, how much money does he make this summer?

In this concept, you will learn how to simplify and evaluate variable expressions involving integer subtraction.

### Simplifying Variable Expressions Involving Integer Subtraction

A **variable expression** is a math phrase that has numbers, variables, and operations in it.

Here are some examples of variable expressions:

- \begin{align*}-2x-x\end{align*}
- \begin{align*}3x-5y\end{align*}
- \begin{align*}-10n-(-8n)\end{align*}

In a variable expression, **like terms** are two terms that include the same variable. If a variable expression has like terms, it can be simplified by combining the like terms into one single term.

If two like terms are being subtracted in a variable expression, you can use what you know about integer subtraction to help you to combine the terms.

Here is an example.

Simplify \begin{align*}-10n-(-8n)\end{align*}.

In this variable expression, \begin{align*}-10n\end{align*} and \begin{align*}-8n\end{align*} are like terms because they include the same variable, \begin{align*}n\end{align*}.

First, rewrite this problem as an addition problem. The term being subtracted is \begin{align*}-8n\end{align*} and the opposite of -8 is 8. This means subtracting \begin{align*}-8n\end{align*} is the same as adding \begin{align*}8n\end{align*}.

\begin{align*}-10n-(-8n)=-10n+8n\end{align*}

Now, notice that the two like terms have different signs. So, your next step is to find the absolute values of both integers and subtract.

\begin{align*}|−10|= 10\end{align*} and \begin{align*}|8|= 8\end{align*}

\begin{align*}10-8=2\end{align*}

Then, decide what the sign should be on your final answer. Your answer should take the sign of the original term that had the greater absolute value. Since 10 is greater than 8, and \begin{align*}-10n\end{align*} has a negative sign, give the answer a negative sign.

\begin{align*}-10n+8n=-2n\end{align*}

The answer is \begin{align*}-10n-(-8n)=-2n\end{align*}.

You can **evaluate** variable expressions for specific values. To **evaluate** means to determine what value the expression takes when the variable is equal to a specific number.

Here is an example.

Find \begin{align*}-10n-(-8n)\end{align*} when \begin{align*}n=5\end{align*}.

You could evaluate by substituting \begin{align*}n=5\end{align*} for both \begin{align*}n\end{align*}'s in the expression and simplifying. However, you already know from the previous example that \begin{align*}-10n-(-8n)\end{align*} simplifies to \begin{align*}-2n\end{align*}. This means you can evaluate \begin{align*}-2n\end{align*} for \begin{align*}n=5\end{align*} instead of having to evaluate the original expression.

\begin{align*}-2n=-2(5)=-10\end{align*}

The answer is -10.

### Examples

#### Example 1

Earlier, you were given a problem about Noah and his business mowing lawns.

He charges $14 to mow a lawn, but every lawn he mows costs him $1.50 in supplies. First, he wanted to know an expression that would represent his profit in terms of \begin{align*}L\end{align*}, the number of lawns he ends up mowing.

He gets \begin{align*}14L\end{align*} dollars from mowing \begin{align*}L\end{align*} lawns, but he spends \begin{align*}1.50L\end{align*} dollars in supplies from mowing those same lawns. That means his profit is

\begin{align*}14L-1.50L\end{align*}

Now Noah can simplify this expression.

Notice that both terms have the same variable, \begin{align*}L\end{align*}, so they are like terms. You can rewrite the subtraction problem as an addition problem. The term being subtracted is \begin{align*}1.50L\end{align*}. The opposite of 1.50 is -1.50. So you have

\begin{align*}14L-1.50L=14L+(-1.50L)\end{align*}

Now notice that the two like terms have different signs. Combine by subtracting their absolute values.

\begin{align*}|14|=14\end{align*} and \begin{align*}|-1.50|=1.50\end{align*}

\begin{align*}14-1.50=12.50\end{align*}

The result will have the same sign as the original term with the greater absolute value. Because \begin{align*}14L\end{align*} has the greater absolute value, the result is positive.

\begin{align*}14L+(-1.50L)=12.50L\end{align*}

The simplified expression is \begin{align*}14L-1.50L=12.50L\end{align*}.

Noah also wondered what his profit would be if he ended up mowing 130 lawns during the summer. To figure this out, you can evaluate \begin{align*}12.50L\end{align*} for \begin{align*}L=130\end{align*}.

\begin{align*}12.50L=12.50(130)=1625\end{align*}.

If Noah mowed 130 lawns, his profit would be $1625.

#### Example 2

Simplify \begin{align*}-33b-(-18b)+7\end{align*}.

First, notice that there are three terms. The first two terms are like terms because they have the same variable, \begin{align*}b\end{align*}. Focus on combining the like terms: \begin{align*}-33b-(-18b)\end{align*}.

Next, rewrite this problem as an addition problem. The term being subtracted is \begin{align*}-18b\end{align*} and the opposite of -18 is 18. This means subtracting \begin{align*}-18b\end{align*} is the same as adding 18b.

\begin{align*}-33b-(-18b)=-33b+18b\end{align*}

Now, notice that the two like terms have different signs. Find the absolute values of both integers and subtract.

\begin{align*}|−33|= 33\end{align*} and \begin{align*}|18|= 18\end{align*}

\begin{align*}33-18=15\end{align*}

Next decide what the sign should be on your combined term. Your answer should take the sign of the original term that had the greater absolute value. Since \begin{align*}-33b\end{align*} has the greater absolute value, your combined term will be negative.

\begin{align*}-33b-(-18b)=-15b\end{align*}

Now you can write your final answer. Don't forget to include the +7 from the original expression. 7 is not like terms with \begin{align*}-15b\end{align*} because it does not have a "\begin{align*}b\end{align*}", so it cannot be combined with \begin{align*}-15b\end{align*}.

The answer is \begin{align*}-15b+7\end{align*}.

#### Example 3

Simplify \begin{align*}-4y-6y\end{align*} and then evaluate for \begin{align*}y=2\end{align*}.

First, notice that both terms have the same variable, \begin{align*}y\end{align*}, so they are like terms. You will combine the like terms and then evaluate.

Next, rewrite the subtraction problem as an addition problem. The term being subtracted is \begin{align*}6y\end{align*}. The opposite of 6 is -6. So you have

\begin{align*}-4y-6y=-4y+(-6y)\end{align*}

Now notice that the two like terms have the same sign. Combine by adding their absolute values.

\begin{align*}|-4|=4\end{align*} and \begin{align*}|-6|=6\end{align*}

\begin{align*}4+6=10\end{align*}

The result will have the same sign as the original two terms. Because the original terms were negative, the result is negative.

\begin{align*}-4y+(-6y)=-10y\end{align*}

The simplified expression is \begin{align*}-4y-6y=-10y\end{align*}.

Finally, you can evaluate \begin{align*}-10y\end{align*} for \begin{align*}y=2\end{align*}.

\begin{align*}-10y=-10(2)=-20\end{align*}

The answer is -20.

#### Example 4

Simplify \begin{align*}18x-(-4x)\end{align*}.

First, notice that both terms have the same variable, \begin{align*}x\end{align*}, so they are like terms.

Next, rewrite the subtraction problem as an addition problem. The term being subtracted is \begin{align*}-4x\end{align*}. The opposite of -4 is 4. So you have

\begin{align*}18x-(-4x)=18x+4x\end{align*}

Now notice that the two like terms have the same sign. Combine by adding their absolute values.

\begin{align*}|18|=18\end{align*} and \begin{align*}|4|=4\end{align*}

\begin{align*}18+4=22\end{align*}

The result will have the same sign as the original two terms. Because the original terms were positive, the result is positive.

\begin{align*}18x+4x=22x\end{align*}

The answer is \begin{align*}18x-(-4x)=22x\end{align*}.

#### Example 5

Simplify \begin{align*}-9a-(-3a)\end{align*} and then evaluate for \begin{align*}a=2\end{align*}.

First, notice that both terms have the same variable, \begin{align*}a\end{align*}, so they are like terms. You will combine the like terms and then evaluate.

Next, rewrite the subtraction problem as an addition problem. The term being subtracted is \begin{align*}-3a\end{align*}. The opposite of -3 is 3. So you have

\begin{align*}-9a-(-3a)=-9a+3a\end{align*}

Now notice that the two like terms have different signs. Combine by subtracting their absolute values.

\begin{align*}|-9|=9\end{align*} and \begin{align*}|3|=3\end{align*}

\begin{align*}9-3=6\end{align*}

The result will have the same sign as the original term with the greater absolute value. Because \begin{align*}-9a\end{align*} has the greater absolute value, the result is negative.

\begin{align*}-9a+3a=-6a\end{align*}

The simplified expression is \begin{align*}-9a-(-3a)=-6a\end{align*}.

Finally, you can evaluate \begin{align*}-6a\end{align*} for \begin{align*}a=2\end{align*}.

\begin{align*}-6a=-6(2)=-12.\end{align*}

The answer is -12.

### Review

Simplify each variable expression.

- \begin{align*}-8m-3m\end{align*}
- \begin{align*}(-7c)-(-c)\end{align*}
- \begin{align*}-19a-(-4a)\end{align*}
- \begin{align*}12a-(-4a)\end{align*}
- \begin{align*}6a-(4a)\end{align*}
- \begin{align*}19a-(24a)\end{align*}
- \begin{align*}13z-(-4z)\end{align*}
- \begin{align*}-20x-(14x)\end{align*}
- \begin{align*}-19a-(18a)\end{align*}
- \begin{align*}56x-22x\end{align*}
- \begin{align*}34y-(-6y)\end{align*}
- \begin{align*}88z-(-44x)\end{align*}
- \begin{align*}-19a-(-4a)-8x\end{align*}
- \begin{align*}-19a-(-4a)-8x-2\end{align*}
- \begin{align*}-19a-(-4a)-5y-2y\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 4.9.

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In this concept, you will learn how to simplify and evaluate variable expressions involving integer subtraction.

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