# 7.2: Simplifying Expressions

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

## Introduction

*Fares for Seniors and Teens*

Kara is very excited that she has figured out an expression to calculate 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 says, writing some notes on a piece of paper.

Kara begins to write an expression and then she wonders if she could combine the terms of the expression.

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

*What You Will Learn*

In this lesson you will learn how to do the following:

- Simplify sums or differences of single-variable expressions.
- Simplify products or quotients of single-variable expressions.
- Simplify variable expressions involving multiple operations.
- Write and simplify variable expressions describing real-world situations.

*Teaching Time*

I. **Simplify Sums or Differences of Single-Variable Expressions**

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.

Example

\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 describe a specific number of "a's". In other words, 6 "a's" plus 3 more "a's" is 9 "a's". So we can combine them.**

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

Example

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

**However, \begin{align*}6a\end{align*} and 3 are** *not***like terms because only one term includes the variable \begin{align*}a\end{align*}. In other words, 6 "a's" plus 3 is just 6 "a's" plus 3, the 6 and 3 describe numbers of different things, and so cannot be combined. The expression \begin{align*}6a+3\end{align*} 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.

Example

Find the difference between \begin{align*}15d and 2d\end{align*}.

**Since \begin{align*}15d\end{align*}** *and***\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. 15 "d's" minus 2 "d's" equals 13 "d's".**

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

**The difference is \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.

Example

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

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

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

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

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

**7D. Lesson Exercises**

**Simplify each sum or difference when possible.**

- \begin{align*}3a+12a\end{align*}
- \begin{align*}16x-2x\end{align*}
- \begin{align*}7y+2x\end{align*}

*Take a few minutes to check your work with a partner. Did you catch the tricky part of number 3?*

II. **Simplify Products or Quotients of Single-Variable Expressions**

Recall that when you add and subtract terms in an expression, you can only combine like terms.

**However, you can multiply or divide terms whether they are like terms or not.**

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*}. We can multiply them to simplify an expression like this.

\begin{align*}6a \times 3a= 18 \times a \times a=18a^2\end{align*}.

However, even though \begin{align*}6a\end{align*} and 3 are *not* like terms, we can still multiply them, like this.

\begin{align*}6a \times 3=18a\end{align*}.

The Commutative and Associative Properties of Multiplication may help you understand how to multiply expressions with variables. Remember, the *Commutative property***states that factors can be multiplied in any order.** The *Associative property***states that the grouping of factors does not matter.**

Example

\begin{align*}6a(3a)\end{align*}

We can take these two terms and multiply them together.

**First, we multiply the number parts.**

\begin{align*}6 \times 3 = 18\end{align*}

**Next, we multiply the variables.**

\begin{align*}a \cdot a= a^2\end{align*}

**Our answer is \begin{align*}18a^2\end{align*}.**

Example

\begin{align*}5x(8y)\end{align*}

Even though these two terms are different, we can still multiply them together.

**First, we multiply the number parts.**

\begin{align*}5 \times 8 = 40\end{align*}

**Next, we multiply the variables.**

\begin{align*}x \cdot y=xy\end{align*}

**Our answer is \begin{align*}40xy\end{align*}.**

Example

Find the product: \begin{align*}4z \times \frac{1}{2}\end{align*}.

**\begin{align*}4z\end{align*} and \begin{align*}\frac{1}{2}\end{align*} are not like terms, however, you can multiply terms even if they are not like terms.**

Use the commutative and associative properties to rearrange the factors to make it easier to see how they can be multiplied.

According to the commutative property, the order of the factors does not matter.

So, \begin{align*}4z \times \frac{1}{2}=\frac{1}{2}\times 4z\end{align*}.

According to the associative property, the grouping of the factors does not matter. Group the factors so that the numbers are multiplied first.

So, \begin{align*}\frac{1}{2} \times 4z=\frac{1}{2} \times 4 \times z=\left(\frac{1}{2} \times 4\right) \times z\end{align*}.

**Now, multiply.**

\begin{align*}\left(\frac{1}{2} \times 4\right) \times z=\left(\frac{1}{2} \times \frac{4}{1}\right) \times z=\frac{4}{2}\times z=2 \times z=2z.\end{align*}

**The product is \begin{align*}2z\end{align*}.**

*Remember that the word PRODUCT means multiplication and the word QUOTIENT means division.*

Example

Find the quotient: \begin{align*}42c \div 7\end{align*}.

**It may help you to rewrite the problem like this \begin{align*}\frac{42c}{7}\end{align*}. Then separate out the numbers and variables like this.**

\begin{align*}\frac{42c}{7}=\frac{42 \cdot c}{7}=\frac{42}{7} \cdot c\end{align*}

**Now, divide 42 by 7 to find the quotient.**

\begin{align*}\frac{42}{7} \cdot c=6 \cdot c=6c\end{align*}

**The quotient is \begin{align*}6c\end{align*}.**

Example

Find the quotient \begin{align*}50 g \div 10 g\end{align*}.

**It may help you to rewrite the problem like this \begin{align*}\frac{50g}{10g}\end{align*}. Then separate out the numbers and variables like this.**

\begin{align*}\frac{50 g}{10 g}=\frac{50 \cdot g}{10 \cdot g}=\frac{50}{10} \cdot \frac{g}{g}\end{align*}

**Now, divide 50 by 10 and divide \begin{align*}g\end{align*} by \begin{align*}g\end{align*} to find the quotient. Since any number over itself is equal to 1, you know that \begin{align*}\frac{g}{g}=1\end{align*}.**

\begin{align*}\frac{50}{10} \cdot \frac{g}{g}=5x1=5\end{align*}

**The quotient is 5.**

**7E. Lesson Exercises**

**Find each product or quotient.**

- \begin{align*}6a(9a)\end{align*}
- \begin{align*}\frac{15b}{5b}\end{align*}
- \begin{align*}\frac{20c}{4}\end{align*}

*Take a few minutes to check your work with a friend. Then continue to the next section.*

III. **Simplify Variable Expressions Involving Multiple Operations**

**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.**

Example

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*}.**

Example

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.**

**7F. Lesson Exercises**

**Simplify each expression.**

- \begin{align*}4a+9a \cdot 7\end{align*}
- \begin{align*}14x \div 2+9x\end{align*}
- \begin{align*}6b-2b+5b-8\end{align*}

*Take a few minutes to check your work with a friend. Are your answers correct? Fix any errors and then continue.*

IV. **Write and Simplify Variable Expressions Describing Real-World Situations**

A real-world situation can sometimes be represented with an algebraic expression. Key words can help you translate a situation into an algebraic expression.

Example

Samera has twice as many pets as Amit has. Kyra has 4 times as many pets as Amit has. Let "a" 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.

**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*}.**

**Now let’s go back to the problem in the introduction and work on figuring it out.**

## Real Life Example Completed

*Fares for Seniors and Teens*

**Here is the original problem once again. Reread it and underline any important information.**

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

\begin{align*}\underline{.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 says, writing some notes on a piece of paper.

Kara begins to write an expression and then she wonders if 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*} accounts for the teen fare and the unknown number of rides, \begin{align*}x\end{align*}.**

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

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

**Because both expressions use the same variable, 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 by $1.45, then she will have the total amount of money needed to ride the train.**

## Vocabulary

Here are the vocabulary words that are found in this lesson.

- 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 to an addition problem.

- Difference
- the answer to a subtraction problem.

- Product
- the answer to a multiplication problem.

- Quotient
- the answer to a division problem.

- Commutative Property of Multiplication
- states that the product is not affected by the order in which you multiply factors.

- Associative Property of Multiplication
- states that the product is not affected by the groupings of the numbers when multiplying.

## Technology Integration

Khan Academy Variable Expressions

James Sousa, Combining Like Terms

James Sousa, Example of Combining Like Terms

James Sousa, Another Example of Combining Like Terms

James Sousa, Simplifying a Basic Variable Expression by Multiplying

Other Videos:

- http://www.mathplayground.com/mv_simplifying_combining_like_terms.html – This is a Brightstorm video on simplifying and combining like terms.

## Time to 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*}6a+18a\end{align*}

Directions: Simplify each product or quotient.

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

12. \begin{align*}9x(2)\end{align*}

13. \begin{align*}14y(2y)\end{align*}

14. \begin{align*}16a(a)\end{align*}

15. \begin{align*}22x(2x)\end{align*}

16. \begin{align*}18b(2)\end{align*}

17. \begin{align*}\frac{21a}{7}\end{align*}

18. \begin{align*}\frac{22b}{2b}\end{align*}

19. \begin{align*}\frac{25x}{x}\end{align*}

20. \begin{align*}\frac{45a}{9a}\end{align*}

Directions: Simplify each variable expression involving multiple operations.

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

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

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

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

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