# 1.14: Expressions with One or More Variables

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

**Practice**Expressions with One or More Variables

Do you remember Joshua and his summer job from the last Concept?

Well, Joshua loves people and so he is working at the ticket counter. His job is to count the people entering the zoo each day. He does this twice. He counts them in the morning and in the afternoon. Sometimes he has more people come in the morning and sometimes the counts are higher in the afternoon. Joshua loves his job. He loves figuring out how much money the zoo has made from the ticket sales. Joshua has a thing for mental math. While many of his friends think it is too difficult, Joshua enjoys figuring it out in his head.

To enter the zoo for the day, it costs an adult \begin{align*}\$7.00\end{align*} and a child \begin{align*}\$5.00\end{align*}. Joshua has written the following expression to help him to figure out the amount of money that the zoo makes in half a day. He divides his arithmetic up between the morning and the afternoon.

\begin{align*}7x+5y\end{align*}

Here are his counts for Monday.

AM - 65 adults and 75 children

PM - 35 adults and 50 children

Here are his counts for Tuesday.

AM - 70 adults and 85 children

PM - 50 adults and 35 children

Given these counts, how much revenue (money) was collected at the zoo for the entire day on Monday? How much money was collected at the zoo for the entire day on Tuesday? How much money was collected in the two days combined? Joshua can figure this out using his expression.

Can you?

**In this Concept, you will learn how to use a variable expression to solve a real-world problem. Pay close attention. You will need these skills to figure out the zoo revenue for Monday and Tuesday.**

### Guidance

In the last Concept, you learned how to evaluate expressions that had one variable and one operation. In this Concept, you are going to learn how to evaluate expressions that have multiple variables and multiple operations.

**Let’s see what this looks like.**

*Evaluate \begin{align*} 6a+b \end{align*} when \begin{align*}a\end{align*} is 4 and \begin{align*}b\end{align*} is 6.* First, of all, you can see that there are **two variables** in this expression. There are also **two operations** here. The first one is **multiplication**: \begin{align*}6a\end{align*} lets us know that we are going to multiply 6 times the value of \begin{align*}a\end{align*}. The second one is **addition**: the \begin{align*}+ \ b\end{align*} lets us know that we are going to add the value of \begin{align*}b\end{align*}. We have also been given the values of \begin{align*}a\end{align*} and \begin{align*}b\end{align*}. We substitute the given values for each variable into the expression and evaluate it.

\begin{align*}& 6(4)+5\\ & 24+5\\ & 29\end{align*}

**Our answer is 29.** **Notice that we used the order of operations when working through this problem.**

**Order of Operations**

**P - parentheses**

**E - exponents**

**MD - multiplication and division in order from left to right**

**AS - addition and subtraction in order from left to right**

**Whenever we are evaluating expressions with more than one operation in them, always refer back and use the order of operations.**

**Let’s look at another example with multiple variables and expressions.**

*Evaluate \begin{align*} 7b-d \end{align*} when \begin{align*}b\end{align*} is 7 and \begin{align*}d\end{align*} is 11.* First, we substitute the given values in for the variables.

\begin{align*}& 7(7) - 11\\ & 49 - 11\\ & 38\end{align*}

**Our answer is 38.**

**What about when we have a dilemma that is all variables?**

*Evaluate \begin{align*} ab + cd \end{align*} when \begin{align*}a\end{align*} is 4, \begin{align*}b\end{align*} is 3, \begin{align*}c\end{align*} is 10 and \begin{align*}d\end{align*} is 6.*

We work on this one in the same way as the other examples. Begin by substituting the given values in for the variables.

\begin{align*}(4)(3) + (10)(6)\end{align*}

We have two multiplication problems here and one addition. Next, we follow the order of operations to evaluate the expression.

\begin{align*}& 12 + 60\\ & 72\end{align*}

**Our answer is 72.**

Now it is time for you to try a few on your own.

#### Example A

*Evaluate \begin{align*}12x - y \end{align*} when \begin{align*}x\end{align*} is 4 and \begin{align*}y\end{align*} is 9.*

**Solution: 39**

#### Example B

*Evaluate \begin{align*}\frac{12}{a} + 4\end{align*} when \begin{align*}a\end{align*} is 3.*

**Solution: 8**

#### Example C

*Evaluate \begin{align*}5x + 3y \end{align*} when \begin{align*}x\end{align*} is 4 and \begin{align*}y\end{align*} is 8.*

**Solution: 52**

Now back to Joshua and the tickets. Here is the problem once again.

Joshua loves people and so he is working at the ticket counter. His job is to count the people entering the zoo each day. He does this twice. He counts them in the morning and in the afternoon. Sometimes he has more people come in the morning and sometimes the counts are higher in the afternoon. Joshua loves his job. He loves figuring out how much money the zoo has made from the ticket sales. Joshua has a thing for mental math. While many of his friends think it is too difficult, Joshua enjoys figuring it out in his head.

To enter the zoo for the day, it costs an adult \begin{align*}\$7.00\end{align*} and a child \begin{align*}\$5.00\end{align*}. Joshua has written the following expression to help him to figure out the amount of money that the zoo makes in half a day. He divides his arithmetic up between the morning and the afternoon.

\begin{align*}\underline{7x+5y}\end{align*}

Here are his counts for Monday.

AM - 65 adults and 75 children

PM - 35 adults and 50 children

Here are his counts for Tuesday.

AM - 70 adults and 85 children

PM - 50 adults and 35 children

**Begin by underlining all of the important information in the problem, done here already.**

**First, we can start with Monday.** **Our expression remains the same.** **We can use \begin{align*} 7x+5y \end{align*}.**

**For Monday morning, the zoo had 65 adults and 75 children visit. Those are the given values that we can substitute into our expression for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.**

\begin{align*}& 7(65)+5(75)\\
& 455+375\\
& \$830.00\end{align*} **For Monday afternoon, the zoo had 35 adults and 50 children visit. Those are the given values that we can substitute into our expression for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.**

\begin{align*}& 7(35)+5(50) \\ & 245+250\\ & \$495.00\end{align*}

**The total amount of money made on Monday is** \begin{align*}830 + 495 = \$1325.\end{align*} **Next, we can figure out Tuesday.** **For Tuesday morning, the zoo had 70 adults and 85 children visit. Those are the given values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.**

\begin{align*}& 7(70)+5(85)\\ & 490+425\\ & \$915.00\end{align*}

**For Tuesday afternoon, the zoo had 50 adults and 35 children visit. Those are the given values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.**

\begin{align*}& 7(50)+5(35)\\ & 350+175\\ & \$525\end{align*}

**The total amount of money made on Tuesday is** \begin{align*}915 + 525 = \$1440\end{align*}.

**If we wanted to figure out the total amount of revenue for both days combined, we simply add the two totals together.**

\begin{align*} \$1325 + \$1440 = \$2765.00\end{align*}

### Vocabulary

Here are the vocabulary words found in this Concept.

- Evaluate
- to simplify an expression that does not have an equals sign.

- Variable
- a letter, usually lowercase, that is used to represent an unknown quantity.

- Expression
- a number sentence that uses operations but does not have an equals sign

- Variable Expression
- a number sentence that has variables or unknown quantities in it with one or more operations and no equals sign.

- Revenue
- means money

### Guided Practice

Here is one for you to try on your own.

Evaluate \begin{align*} a + ab + cd \end{align*} when \begin{align*}a\end{align*} is 4, \begin{align*}b\end{align*} is 9, \begin{align*}c\end{align*} is 6 and \begin{align*}d\end{align*} is 4.

**Answer**

First, we have to substitute the given values into the expression.

\begin{align*}4 + 4(9) + 6(4)\end{align*}

Now we can evaluate using the order of operations.

\begin{align*}4 + 36 + 24\end{align*}

\begin{align*}64\end{align*}

**This is our answer.**

### Video Review

Here are a few videos for review.

Khan Academy Evaluating an Expression

James Sousa Example of Evaluating an Expression

James Sousa Example of Evaluating an Expression

### Practice

Directions: Evaluate each multi-variable expression when \begin{align*}x = 2 \end{align*} and \begin{align*}y = 3\end{align*}.

1. \begin{align*}2x + y\end{align*}

2. \begin{align*}9x - y \end{align*}

3. \begin{align*}x + y \end{align*}

4. \begin{align*}xy \end{align*}

5. \begin{align*}xy + 3 \end{align*}

6. \begin{align*}9y - 5 \end{align*}

7. \begin{align*}10x - 2y \end{align*}

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

9. \begin{align*}2x + 2y \end{align*}

10. \begin{align*}7x - 3y \end{align*}

11. \begin{align*}3y - 2 \end{align*}

12. \begin{align*}10x - 8 \end{align*}

13. \begin{align*}12x - 3y \end{align*}

14. \begin{align*}9x + 7y \end{align*}

15. \begin{align*}11x - 7y \end{align*}

algebraic

The word*algebraic*indicates that a given expression or equation includes variables.

Algebraic Expression

An expression that has numbers, operations and variables, but no equals sign.Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.Expression

An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.Order of Operations

The order of operations specifies the order in which to perform each of multiple operations in an expression or equation. The order of operations is: P - parentheses, E - exponents, M/D - multiplication and division in order from left to right, A/S - addition and subtraction in order from left to right.Parentheses

Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.revenue

Revenue is money that is earned.substitute

In algebra, to substitute means to replace a variable or term with a specific value.Variable Expression

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

Here you'll learn how to evaluate multi - variable expressions with given values.

## Concept Nodes:

algebraic

The word*algebraic*indicates that a given expression or equation includes variables.

Algebraic Expression

An expression that has numbers, operations and variables, but no equals sign.Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.Expression

An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.Order of Operations

The order of operations specifies the order in which to perform each of multiple operations in an expression or equation. The order of operations is: P - parentheses, E - exponents, M/D - multiplication and division in order from left to right, A/S - addition and subtraction in order from left to right.Parentheses

Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.revenue

Revenue is money that is earned.substitute

In algebra, to substitute means to replace a variable or term with a specific value.Variable Expression

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