### Let's Think About It

Dexter is in charge of ticket sales at his town’s water park. He has to report to his boss how many tickets he sells and how much money the water park makes in ticket sales each day - adult tickets ($7), child tickets ($5). Today, Dexter has sold 100 adult and 125 child tickets. Yesterday, he sold 120 adult and 120 child tickets. Can Dexter write an expression to figure out today’s ticket sales, and use this to compare today’s sales to yesterday’s?

In this concept, you will learn how to evaluate expressions that have multiple variables and/or multiple operations.

### Guidance

When evaluating expressions with multiple variables and multiple operations, it is important to remember the order of operations.

**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 you are evaluating an expression with more than one operation in it, always refer back to the order of operations.

Let's look at an example of an expression with multiple variables and operations.

Evaluate \begin{align*} 6a+b \end{align*} when \begin{align*}a\end{align*} is 4 and \begin{align*}b\end{align*} is 5.

First, you can see that there are two variables in this expression, \begin{align*}a\end{align*} and \begin{align*}b\end{align*}. There are also two operations here: multiplication, seen in "6 times the value of " and addition, seen in " ". You are given values for .

First, substitute the given values for each variable into the expression.

\begin{align*}6a+b\\ 6\left(4\right)+5\end{align*}

Then, evaluate the expression according to order of operations.

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

The answer is 29.

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, substitute the given values in for the variables.

\begin{align*}7\left(7\right)-11\end{align*}

Then, evaluate the expression according to order of operations.

\begin{align*} 49 - 11\\ 38\end{align*}

The answer is 38.

Sometimes you may have an expression that is all variables. Evaluate this in the same way.

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.

First, substitute the given values in for the variables.

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

Next, evaluate the expression according to order of operations.

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

The answer is 72.

### Guided Practice

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.

First, substitute the given values into the expression.

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

Next, evaluate according to order of operations.

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

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

The answer is 64.

### Examples

#### Example 1

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

First, substitute 4 for

and .\begin{align*}12x-y & \quad \text{Substitute 4 in place of }x \text{ and 9 for }y\\ 12(4)-(9) & \\\end{align*}

Then, evaluate the expression.

\begin{align*}12(4)-(9) & \quad \text{Multiply }12 \times 4 = 48\\ 48-(9) & \quad \text{Subtract }48 - 9 = 39\\ 39 &\\\end{align*}

The answer is 39.

#### Example 2

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

First, substitute 3 for

.\begin{align*}5x + 3y \end{align*}

Next, evaluate the expression.

\begin{align*}\frac{12}{3}+4 & \quad \text{Divide }12\div 3 = 4\\ 4+4 & \quad \text{Add }4+4=8\\ 8 &\\\end{align*}

The answer is 8.

#### Example 3

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

First, substitute 4 for

and 8 for .\begin{align*}5x+3y & \quad \text{Substitute }4 \text{ for }x \text{ and 8 for }y\\ 5(4)+3(8) &\\\end{align*}

Then, evaluate the expression.

\begin{align*}5(4)+3(8) & \quad \text{Multiply }5 \times 4=20\\ 20+3(8) & \quad \text{Multiply }3 \times 8=24\\ 20+24 & \quad \text{Add }20+24=44\\ 44 &\\\end{align*}

The answer is 44.

### Follow Up

Remember Dexter and his watery tickets?

Dexter needs to write an expression to figure out the total money made from the 100 adult and 125 child tickets he sold today, compared to the 120 adult and 120 child tickets he sold yesterday. The adult tickets cost $7 and the child tickets cost $5.

First, write an expression.

Next, substitute in the given values for today’s ticket sales.

Then, follow order of operations to multiply and then add.

\begin{align*}7(100)+5(125) & \quad \text{Multiply }7 \times 100\\
700+5(125) & \quad \text{Multiply }5 \times 125\\
700+625 & \quad \text{Add }700+625=1325\\
1,325 &\\\end{align*}

The answer is 1,325.

Next, substitute in the given values for yesterday’s ticket sales.

Then, follow order of operations to multiply and then add.

\begin{align*}7(120)+5(120) & \quad \text{Multiply }7 \times 120\\
840+5(120) & \quad \text{Multiply }5 \times 120\\
840+600 & \quad \text{Add }840+600=1440\\
1,440 &\\\end{align*}

The answer is 1,440.

Finally, subtract 1,325 from 1,440 to get the difference.

The answer is 115.

Dexter can report to his boss that today the park sold $1,325 in tickets, which is $115 less than the $1,440 they sold in tickets yesterday.

### Video Review

### Explore More

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*}