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# 4.9: Simplify Variable Expressions Involving Integer Subtraction

Difficulty Level: At Grade Created by: CK-12

Credit: sean hobson
Source: https://www.flickr.com/photos/seanhobson/4353671803/

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 L\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.

### Guidance

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

Here are some examples of variable expressions:

•  2xx\begin{align*}-2x-x\end{align*}
• 3x5y\begin{align*}3x-5y\end{align*}
• 10n(8n)\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 10n(8n)\begin{align*}-10n-(-8n)\end{align*}.

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

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

10n(8n)=10n+8n\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.

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

108=2\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 10n\begin{align*}-10n\end{align*} has a negative sign, give the answer a negative sign.

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

The answer is 10n(8n)=2n\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 10n(8n)\begin{align*}-10n-(-8n)\end{align*} when n=5\begin{align*}n=5\end{align*}.

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

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

### Guided Practice

Simplify 33b(18b)+7\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, b\begin{align*}b\end{align*}. Focus on combining the like terms: 33b(18b)\begin{align*}-33b-(-18b)\end{align*}.

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

33b(18b)=33b+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.

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

3318=15\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 33b\begin{align*}-33b\end{align*} has the greater absolute value, your combined term will be negative.

33b(18b)=15b\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 15b\begin{align*}-15b\end{align*} because it does not have a "b\begin{align*}b\end{align*}", so it cannot be combined with 15b\begin{align*}-15b\end{align*}.

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

### Examples

#### Example 1

Simplify 4y6y\begin{align*}-4y-6y\end{align*} and then evaluate for y=2\begin{align*}y=2\end{align*}.

First, notice that both terms have the same variable, y\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 6y\begin{align*}6y\end{align*}. The opposite of 6 is -6. So you have

4y6y=4y+(6y)\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.

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

4+6=10\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.

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

The simplified expression is 4y6y=10y\begin{align*}-4y-6y=-10y\end{align*}.

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

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

#### Example 2

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

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

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

18x(4x)=18x+4x\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.

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

18+4=22\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.

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

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

#### Example 3

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

First, notice that both terms have the same variable, a\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 3a\begin{align*}-3a\end{align*}. The opposite of -3 is 3. So you have

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

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

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

93=6\begin{align*}9-3=6\end{align*}

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

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

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

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

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

Credit: Casey Fleser
Source: https://www.flickr.com/photos/somegeekintn/3439641097

Remember 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 L\begin{align*}L\end{align*}, the number of lawns he ends up mowing.

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

14L1.50L\begin{align*}14L-1.50L\end{align*}

Now Noah can simplify this expression.

Notice that both terms have the same variable, L\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 1.50L\begin{align*}1.50L\end{align*}. The opposite of 1.50 is -1.50. So you have

14L1.50L=14L+(1.50L)\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.

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

141.50=12.50\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 14L\begin{align*}14L\end{align*} has the greater absolute value, the result is positive.

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

The simplified expression is 14L1.50L=12.50L\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 12.50L\begin{align*}12.50L\end{align*} for L=130\begin{align*}L=130\end{align*}.

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

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

### Explore More

Simplify each variable expression.

1. 8m3m\begin{align*}−8m− 3m\end{align*}

2. (7c)(c)\begin{align*}(−7c) − (−c)\end{align*}

3. 19a(4a)\begin{align*}−19a− (−4a)\end{align*}

4. 12a(4a)\begin{align*}12a− (−4a)\end{align*}

5. 6a(4a)\begin{align*}6a− (4a)\end{align*}

6. 19a(24a)\begin{align*}19a− (24a)\end{align*}

7. 13z(4z)\begin{align*}13z− (−4z)\end{align*}

8.  20x(14x)\begin{align*}−20x− (14x)\end{align*}

9.  19a(18a)\begin{align*}−19a− (18a)\end{align*}

10.  56x22x\begin{align*}56x− 22x\end{align*}

11.  34y(6y)\begin{align*}34y− (−6y)\end{align*}

12. 88z(44x)\begin{align*}88z− (−44x)\end{align*}

13. 19a(4a)8x\begin{align*}−19a− (−4a) − 8x\end{align*}

14. 19a(4a)8x2\begin{align*}−19a− (−4a) − 8x− 2\end{align*}

15. 19a(4a)5y2y\begin{align*}−19a− (−4a) − 5y− 2y\end{align*}

1. [1]^ Credit: sean hobson; Source: https://www.flickr.com/photos/seanhobson/4353671803/; License: CC BY-NC 3.0
2. [2]^ Credit: Casey Fleser; Source: https://www.flickr.com/photos/somegeekintn/3439641097; License: CC BY-NC 3.0

## Date Created:

Aug 10, 2015

Aug 26, 2015
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