# 7.4: Simplify Sums or Differences of Single Variable Expressions

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

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.

\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.

\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, \begin{align*}6a\end{align*}**and** \begin{align*}3a\end{align*}**are like terms because both terms include the variable \begin{align*}a\end{align*} a. So, we can combine them.**

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

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

**However, \begin{align*}6a\end{align*} 6a and 3 are**

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

You can use what you know about like terms and what you know about addition and subtraction to help you simplify expressions with variables.

Here is another one.

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

**Since \begin{align*}15d\end{align*} 15d**

*and***\begin{align*}2d\end{align*}**2d 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.

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

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

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

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

**Since \begin{align*}0.4x\end{align*} 0.4x and \begin{align*}1.3x\end{align*}1.3x 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.**

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

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

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

Simplify each sum or difference when possible.

#### Example A

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

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

#### Example B

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

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

#### Example C

\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.

\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.

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

**This is the answer.**

### 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
- the answer in an addition problem.

- Difference
- the answer in a subtraction problem.

### Guided Practice

Here is one for you to try on your own.

Simplify each expression.

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

**Answer**

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.

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

Now, we can perform the subtraction.

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

Finally, we add the last two terms.

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

**This is our answer.**

### Video Review

Here is a video for review.

- This is a James Sousa video on combining like terms.

### Practice

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

### Image Attributions

Here you'll learn to simplify sums or differences of single variable expressions.