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# 7.4: Simplify Sums or Differences of Single Variable Expressions

Difficulty Level: At Grade Created by: CK-12
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Practice Simplify Sums or Differences of Single Variable Expressions
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Estimated7 minsto complete
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Have you ever had to order ice cream for a group of people?

Well, Marc is doing just that. He has taken orders for himself, his sister, his grandparents and two of their friends.

Here is what he was told.

Two vanilla ice cream cones

Three more vanilla ice cream cones

Then another vanilla ice cream cone

We can write Marc's order as a variable expression.

2v+3v+v\begin{align*}2v + 3v + v\end{align*}

This is a way of writing Marc's ice cream order as a variable expression.

Can you simplify this expression?

This Concept will teach you how to simplify single variable expressions. At the end of it, you will be able to simplify Marc's order for him.

### Guidance

You already know that an expression shows how numbers and/or variables are connected by operations, such as addition, subtraction, multiplication, and division.

If an expression has only numbers, you can find its numerical value. However, if an expression includes variables and you do not know the values of those variables, you can simplify the expression. To simplify means to make smaller or simpler. Let's take a look at how to simplify expressions now.

Let's apply this information.

6a+3a\begin{align*}6a+3a\end{align*}

When adding expressions with variables, it is important to remember that only like terms can be combined. For example, 6a\begin{align*}6a\end{align*} and 3a\begin{align*}3a\end{align*} are like terms because both terms include the variable a\begin{align*}a\end{align*}. So, we can combine them.

6a+3a9a\begin{align*}& 6a+3a\\ & 9a\end{align*}

6a+3\begin{align*}6a+3\end{align*}

However, 6a\begin{align*}6a\end{align*} and 3 are not like terms because only one term includes the variable a\begin{align*}a\end{align*}. So, we cannot combine them. The expression 6a+3\begin{align*}6a+3\end{align*} cannot be simplified any further.

Here is another one.

Find the difference 15d2d\begin{align*}15d-2d\end{align*}.

Since 15d\begin{align*}15d\end{align*} and 2d\begin{align*}2d\end{align*} both have the same variable, they are like terms. To find the difference, subtract the numerical parts of the terms the same way you would subtract any whole numbers.

15d2d=13d\begin{align*}15d-2d=13d\end{align*}

The difference is 13d\begin{align*}13d\end{align*}.

We can also see examples that have decimals or even fractions in them. Remember back to your work with rational numbers.

Find the sum 0.4x+1.3x\begin{align*}0.4x+1.3x\end{align*}.

Since 0.4x\begin{align*}0.4x\end{align*} and 1.3x\begin{align*}1.3x\end{align*} both have the same variable, they are like terms. To find the sum, add the numerical parts of the terms the same way you would add any decimals.

0.4x+1.3x=1.7x\begin{align*}0.4x+1.3x=1.7x\end{align*}

The sum is 1.7x\begin{align*}1.7x\end{align*}.

Don’t forget that the word SUM means addition and the word DIFFERENCE means subtraction.

Simplify each sum or difference when possible.

#### Example A

3a+12a\begin{align*}3a+12a\end{align*}

Solution:15a\begin{align*}15a\end{align*}

#### Example B

16x2x\begin{align*}16x-2x\end{align*}

Solution:14x\begin{align*}14x\end{align*}

#### Example C

7y+2x\begin{align*}7y+2x\end{align*}

Solution: Terms are not alike. It is in simplest form.

Here is the original problem once again.

Have you ever had to order ice cream for a group of people?

Well, Marc is doing just that. He has taken orders for himself, his sister, his grandparents and two of their friends.

Here is what he was told.

Two vanilla ice cream cones

Three more vanilla ice cream cones

Then another vanilla ice cream cone

We can write Marc's order as a variable expression.

2v+3v+v\begin{align*}2v + 3v + v\end{align*}

This is a way of writing Marc's ice cream order as a variable expression.

Can you simplify this expression?

If you look at this expression you will see that the variables are all the same. This is because everyone ordered vanilla ice cream cones. Therefore, we can simply add the terms.

2v+3v+v=6v\begin{align*}2v + 3v + v = 6v\end{align*}

### Vocabulary

Here are the vocabulary words in this Concept.

Expression
a number sentence without an equal sign that combines numbers, variables and operations.
Simplify
to make smaller by combining like terms
Sum
Difference
the answer in a subtraction problem.

### Guided Practice

Here is one for you to try on your own.

Simplify each expression.

5a+4a2a+6a\begin{align*}5a + 4a - 2a + 6a\end{align*}

To simplify this expression, we follow the order of operations and combine like terms in order from left to right. Here is what the expression looks like after the first two terms have been combined.

9a2a+6a\begin{align*}9a - 2a + 6a\end{align*}

Now, we can perform the subtraction.

7a+6a\begin{align*}7a + 6a\end{align*}

Finally, we add the last two terms.

13a\begin{align*}13a\end{align*}

### Video Review

Here is a video for review.

### Practice

Directions: Simplify each sum or difference by combining like terms.

1. 6a+7a\begin{align*}6a+7a\end{align*}

2. 7x2x\begin{align*}7x-2x\end{align*}

3. 6y+12y\begin{align*}6y+12y\end{align*}

4. 8a+12a\begin{align*}8a+12a\end{align*}

5. 12y7y\begin{align*}12y-7y\end{align*}

6. 8a+15a\begin{align*}8a+15a\end{align*}

7. 13b9b\begin{align*}13b-9b\end{align*}

8. 22x+19x\begin{align*}22x+19x\end{align*}

9. 45y12y\begin{align*}45y-12y\end{align*}

10. 16a+18a+9a\begin{align*}16a+18a+9a\end{align*}

11. 14x6x+2x\begin{align*}14x - 6x + 2x\end{align*}

12. 21a+14a15a\begin{align*}21a + 14a - 15a\end{align*}

13. 33b+13b+8b\begin{align*}33b + 13b +8b\end{align*}

14. 45x+67x29x\begin{align*}45x+67x-29x\end{align*}

15. 92y+6y54y\begin{align*}92y+6y-54y\end{align*}

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