# 7.6: Simplify Variable Expressions Involving Multiple Operations

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

Have you ever calculated train fare? Take a look at this dilemma.

Kara is very excited that she has figured out an expression to figure out the total train fare. She heads downstairs and shares her findings with her Grandpa.

\begin{align*}.85x\end{align*}

“See Grandpa, if I put the number of rides in for \begin{align*}x\end{align*}, then we can figure out the total cost,” Kara explains.

“That is great work Kara, but what about me? That will work great when it is just you and Marc, but I am riding with you today. Seniors ride for .60 per ride.”

Wow! Kara hadn’t even thought of that. Now she has a whole new dilemma.

“I’ve got it,” she said writing some notes on a piece of paper.

Kara began to write an expression and then she thought that she could combine the terms of the expression.

**Do you have an idea what Kara is writing? Think back to the last few Concepts. Think about the expression that she wrote for teen fares and train rides. Now we are going to add to that and simplify. This Concept will teach you all about how this works. Focus on the information in the Concept and you will see this problem again at the end of it.**

### Guidance

Sometimes, you may need to simplify algebraic expressions that involve more than one operation. Use what you know about simplifying sums, differences, products, or quotients of algebraic expressions to help you do this.

When evaluating expressions, it is also important to keep in mind the ** order of operations**. Let's review this order below.

- First, do the computation inside grouping symbols, such as parentheses.
- Second, evaluate any exponents.
- Third, multiply and divide in order from left to right.
- Finally, add and subtract in order from left to right.

**That’s good because the order of operations is always useful in mathematics!! Now let’s look at an example.**

Now let's apply this information.

Simplify this expression \begin{align*}7n+8n \cdot 3\end{align*}

According to the order of operations, you should multiply before you add.

\begin{align*}7n+8n \cdot 3=7n+(8n \cdot 3)\end{align*}.

Separate out the factors and use the commutative property to help you multiply.

\begin{align*}7n+(8n \cdot 3)=7n+(8 \cdot n \cdot 3)+7n+(8 \cdot 3 \cdot n)=7n+(24 \cdot n)=7n+24n\end{align*}

Since \begin{align*}7n\end{align*} and \begin{align*}24n\end{align*} are like terms, add them.

\begin{align*}7n+24n=31n\end{align*}.

**The answer is \begin{align*}31n\end{align*}.**

Simplify this expression \begin{align*}10 p-7p+8p \div 2p\end{align*}.

According to the order of operations, you should divide before you subtract or add.

\begin{align*}10p-7p+8p \div 2p=10p-7p+(8p \div 2p)\end{align*}.

It may help you to rewrite the division as \begin{align*}\frac{8p}{2p}\end{align*} and then separate out the numbers and variables like this.

\begin{align*}10p-7p+ \left(\frac{8p}{2p}\right)=10p-7p+\left(\frac{8 \cdot p}{2 \cdot p}\right)=10p-7p+\left(\frac{8}{2} \cdot \frac{p}{p}\right)=10p-7p+(4 \cdot 1)=10p-7p+4\end{align*}

The order of operations says to add and subtract in order from left to right. So, subtract the like terms \begin{align*}10p\end{align*} and \begin{align*}7p\end{align*} next.

\begin{align*}10p-7p+4=3p+4\end{align*}.

Since \begin{align*}3p\end{align*} and 4 are not like terms, those terms cannot be combined. So, the expression cannot be simplified further.

**The expression, when simplified, is \begin{align*}3p+4\end{align*}. This is our final answer.**

Simplify each expression.

#### Example A

\begin{align*}4a+9a - 7\end{align*}

**Solution:\begin{align*}13a - 7\end{align*}**

#### Example B

\begin{align*}\frac{14x}{2}+9x\end{align*}

**Solution: \begin{align*}16x\end{align*}**

#### Example C

\begin{align*}6b-2b+5b-8\end{align*}

**Solution:\begin{align*}9b-8\end{align*}**

Here is the original problem once again.

Kara is very excited that she has figured out an expression to figure out the total train fare. She heads downstairs and shares her findings with her Grandpa.

\begin{align*}.85x\end{align*}

“See Grandpa, if I put the number of rides in for \begin{align*}x\end{align*}, then we can figure out the total cost,” Kara explains.

“That is great work Kara, but what about me? That will work great when it is just you and Marc, but I am riding with you today. Seniors ride for .60 per ride.”

Wow! Kara hadn’t even thought of that. Now she has a whole new dilemma.

“I’ve got it,” she said writing some notes on a piece of paper.

Kara began to write an expression and then she thought that she could combine the terms of the expression.

**To write an expression that includes Grandpa, Kara can begin with the first expression that she wrote.**

**\begin{align*}.85x\end{align*} this accounts for the teen fare and the number of rides which is unknown so we use \begin{align*}x\end{align*}.**

**Next, we have to include Grandpa. Seniors ride for .60 per ride. The number of rides is still unknown, so we can use \begin{align*}x\end{align*} for that too.**

\begin{align*}.60x\end{align*}

**Because they will be riding together, we can add the two terms.**

\begin{align*}.85x+.60x\end{align*}

**Next, we can simplify the expression.**

**Since both are riding the train together, the number of rides will stay the same. We can add the money amounts and keep the \begin{align*}x\end{align*} the same in the simplified expression.**

\begin{align*}.85 + .60 = 1.45\end{align*}

**Our answer is \begin{align*}1.45x\end{align*}.**

**If Kara multiplies the number of train rides that the three of them take times $1.45 then she will have the total amount of money needed or spent riding the train.**

#### 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.

- Product
- the answer in a multiplication problem.

- Quotient
- the answer in a division problem.

### Guided Practice

Here is one for you to try on your own.

Samera has twice as many pets as Amit has. Kyra has 4 times as many pets as Amit has. Let \begin{align*}a\end{align*} represent the number of pets Amit has.

a. Write an expression to represent the number of pets Samera has.

b. Write an expression to represent the number of pets Kyra has.

c. Write an expression to represent the number of pets Samera and Kyra have all together.

**Answer**

Consider part \begin{align*}a\end{align*} first.

The phrase “twice as many pets as Amit” shows how many pets Samera has. Use a number, an operation sign, or a variable to represent each part of that phrase.

\begin{align*}& \underline{twice} \ as \ many \ pets \ as \ \underline{Amit}\\ & \downarrow \qquad \qquad \qquad \qquad \qquad \downarrow\\ & 2 \times \qquad \qquad \qquad \quad \qquad a\end{align*}

So, the expression \begin{align*}2 \times a\end{align*} or \begin{align*}2a\end{align*} represents the number of pets Samera has.

Consider part \begin{align*}b\end{align*} next.

The phrase “4 times as many pets as Amit” shows how many pets Kyra has. Use a number, an operation sign, or a variable to represent each part of that phrase.

\begin{align*}& \underline{4} \ \underline{times} \ as \ many \ pets \ as \ Amit\\ & \downarrow \quad \downarrow \qquad \qquad \qquad \qquad \downarrow\\ & 4 \quad \times \qquad \qquad \qquad \quad \ \ a\end{align*}

So, the expression \begin{align*}4 \times a\end{align*} or \begin{align*}4a\end{align*} represents the number of pets Kyra has.

Finally, consider part \begin{align*}c\end{align*}.

To find the number of pets Samera and Kyra have “all together,” write an addition expression.

\begin{align*}& (\text{number of pets Samera has}) \ + \ (\text{number of pets Kyra has})\\ & \qquad \qquad \downarrow \qquad \qquad \qquad \qquad \ \ \downarrow \qquad \qquad \quad \quad \ \downarrow\\ & \qquad \qquad 2a \qquad \qquad \qquad \quad \ \ + \qquad \qquad \quad \quad 4a\end{align*}

Simplify the expression.

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

**The number of pets Samera and Kyra have all together can be represented by the expression \begin{align*}6a\end{align*}.**

### Video Review

Here is a video for review.

- This is a James Sousa video about combining like terms to simplify an expression.

### Practice

Directions: Simplify each variable expression involving multiple operations.

1. \begin{align*}6a+4a-2b\end{align*}

2. \begin{align*}16b-4b \cdot 2\end{align*}

3. \begin{align*}22a \div 2+14a\end{align*}

4. \begin{align*}19x-5x \cdot 2\end{align*}

5. \begin{align*}16y-12y \div 2\end{align*}

6. \begin{align*}16a-4a-12b\end{align*}

7. \begin{align*}26a+14a+12b+2b\end{align*}

8. \begin{align*}36a+4a-2b+5b\end{align*}

9. \begin{align*}18a+4a+12y\end{align*}

10. \begin{align*}46a+34a-12b+14b\end{align*}

11. \begin{align*}16y+4y-2x\end{align*}

12. \begin{align*}6x+4x+2x+4y-19z\end{align*}

13. \begin{align*}26y-12y \div 2\end{align*}

14. \begin{align*}36y-12y \div 12\end{align*}

15. \begin{align*}46y+12y \div 2\end{align*}

### Image Attributions

Here you'll learn to simplify variable expressions involving multiple operations.