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Expressions with One or More Variables

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Joshua loves people and so he is working at the ticket counter at the zoo. 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 \$7.00 and a child \$5.00 . 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.

7x+5y

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 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  6a+b when a is 4 and b is 5. 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 : 6a lets us know that we are going to multiply 6 times the value of a . The second one is addition : the + \ b lets us know that we are going to add the value of b . We have also been given the values of a and b . We substitute the given values for each variable into the expression and evaluate it.

& 6(4)+5\\	 & 24+5\\& 29

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  7b-d when b is 7 and d is 11. First, we substitute the given values in for the variables.

& 7(7) - 11\\& 49 - 11\\& 38

Our answer is 38.

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

Evaluate  ab + cd when a is 4, b is 3, c is 10 and d 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.

(4)(3) + (10)(6)

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

& 12 + 60\\& 72

Our answer is 72.

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

Example A

Evaluate 12x - y when x is 4 and y is 9.

Solution: 39

Example B

Evaluate \frac{12}{a} + 4 when a is 3.

Solution: 8

Example C

Evaluate 5x + 3y when x is 4 and y 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 \$7.00 and a child \$5.00 . 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.

\underline{7x+5y}

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  7x+5y .

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 x and y .

& 7(65)+5(75)\\& 455+375\\& \$830.00

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 x and y .

& 7(35)+5(50) \\& 245+250\\& \$495.00

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

& 7(70)+5(85)\\& 490+425\\& \$915.00

For Tuesday afternoon, the zoo had 50 adults and 35 children visit. Those are the given values for x and y .

& 7(50)+5(35)\\& 350+175\\& \$525

The total amount of money made on Tuesday is 915 + 525 = \$1440 .

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

 \$1325 + \$1440 = \$2765.00

Vocabulary

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 has 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  a + ab + cd when a is 4, b is 9, c is 6 and d is 4.

Answer

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

4 + 4(9) + 6(4)

Now we can evaluate using the order of operations.

4 + 36 + 24

64

This is our answer.

Video Review

Khan Academy Evaluating an Expression

James Sousa Example of Evaluating an Expression

James Sousa Example of Evaluating an Expression

Explore More

Directions: Evaluate each multi-variable expression when x = 2 and y = 3 .

1. 2x + y

2. 9x - y

3. x + y

4. xy

5. xy + 3

6. 9y - 5

7. 10x - 2y

8. 3x + 6y

9. 2x + 2y

10. 7x - 3y

11. 3y - 2

12. 10x - 8

13. 12x - 3y

14. 9x + 7y

15. 11x - 7y

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