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Simplify Variable Expressions Involving Integer Subtraction

Combine like terms.

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Simplify Variable Expressions Involving Integer Subtraction

Molly is working on her math homework. She is puzzled about a problem. Take a look.

How would you solve this problem?

14x2x\begin{align*}14x - 2x\end{align*} when x=2\begin{align*}x = -2\end{align*}

Do you know how to do this?

This Concept will show you how to evaluate variable expressions involving subtraction. We'll come back to this one at the end of the Concept.

Guidance

Do you know what a variable expression is?

A variable expression is a math phrase that has numbers, variables and operations in it. In a variable expression we can have like and unlike terms. Like terms can be combined or simplified and unlike terms can not be combined or simplified.

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

These two terms are alike because they both have x\begin{align*}x\end{align*}’s with them. In the last lesson, you learned how to add like terms in a variable expression.

The answer is 2x\begin{align*}2x\end{align*}.

You can use what you know about how to subtract integers to help you find the value of expressions with variables, too.

Remember, you can only subtract like terms in an expression that has variables.

Find the difference 10n(8n)\begin{align*}-10n-(-8n)\end{align*}

Since 10n\begin{align*}-10n\end{align*} and 8n\begin{align*}-8n\end{align*} both have the same variable, they are like terms. Use what you know about how to subtract integers to help you.

The term being subtracted is 8n\begin{align*}-8n\end{align*}. The integer -8 is part of that term. The opposite of that integer is 8, so add 8n\begin{align*}8n\end{align*} to 10n\begin{align*}-10n\end{align*}.

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

Both like terms have different signs. So, find the absolute values of both integers. Then subtract the term whose integer has the lesser absolute value from the other term.

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

Now subtract 10n8n=2n\begin{align*}10n-8n=2n\end{align*}.

Since 10>8, and 10n\begin{align*}-10n\end{align*} has a negative sign, give the answer a negative sign.

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

Remember you can only combine terms that are alike.

We can also evaluate this expression when we are given a value. Let's say that n=5\begin{align*}n = 5\end{align*} in the last problem.

First, we simplify the expression, then we evaluate it using the given value.

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

Use what you have learned to simplify each expression and evaluate when necessary.

Example A

4y6y\begin{align*}-4y-6y\end{align*} when y=2\begin{align*}y = 2\end{align*}

Solution:20\begin{align*}-20\end{align*}

Example B

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

Solution:22x\begin{align*}22x\end{align*}

Example C

9a(3a)\begin{align*}-9a-(-3a)\end{align*} then evaluate if a=2\begin{align*}a = 2\end{align*}

Solution:12\begin{align*}-12\end{align*}

Here is the original problem once again.

Molly is working on her math homework. She is puzzled about a problem. Take a look.

How would you solve this problem?

14x2x\begin{align*}14x - 2x\end{align*} when x=2\begin{align*}x = -2\end{align*}

Do you know how to do this?

First, let's combine like terms.

14x2x=12x\begin{align*}14x - 2x = 12x\end{align*}

Now we substitute the given value for x\begin{align*}x\end{align*} into the expression.

12(2)=24\begin{align*}12(-2) = -24\end{align*}

Vocabulary

Difference
the answer in a subtraction problem.
Integer
the set of whole numbers and their opposites.
Variable Expression
a mathematical phrase that uses numbers, variables and operations, without an equal sign.

Guided Practice

Here is one for you to try on your own.

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

To simplify this expression, we combine the like terms.

In this case, 33b\begin{align*}33b\end{align*} and 18b\begin{align*}-18b\end{align*} are like terms.

33b(18b)=51b\begin{align*}33b -(-18b) = 51b\end{align*}

Now we add the final term.

51b+7\begin{align*}51b+7\end{align*}

Because the two terms left are not alike, we can't simplify these terms any further.

Practice

Directions: 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*}

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Color Highlighted Text Notes

Vocabulary Language: English

Difference

The result of a subtraction operation is called a difference.

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...

Variable Expression

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