Did you know that you can evaluate a variable expression for a given value? Have you ever been shopping and had to use math in the store? Take a look at this dilemma.
Hanson went shopping and bought cashews and almonds for his Mom. The cashews were $4.99 per pound, and the almonds were $3.50 per pound. Hanson looked at the prices and wrote down the following variable expression.
\begin{align*}4.99x + 3.50y\end{align*}
If Hanson bought four pounds of cashews and four pounds of almonds, can you figure out how much money he spent? Use the information in this Concept to help you to evaluate this variable expression.
Guidance
Before you can learn how to evaluate a variable expression with a given value, you need to know how to identify a variable expression.
What is a variable expression?
A variable expression is a group of numbers, operations and variables without using an equal sign. A variable is a letter used to represent an unknown quantity. A constant is a number without a variable.
Here is a variable expression.
@$\begin{align*}6a+7\end{align*}@$
In this variable expression, @$\begin{align*}a\end{align*}@$ is the variable and 7 is the constant .
Let’s look at evaluating variable expressions when we have been given a value for the variable.
Evaluate the expression @$\begin{align*}4g + 1.5\end{align*}@$ for @$\begin{align*}g = 8\end{align*}@$ .
Step 1: Substitute 8 for the variable “ @$\begin{align*}g\end{align*}@$ .”
@$\begin{align*}4(8) + 1.5\end{align*}@$
Step 2: Follow the standard order of operations to solve: parentheses, exponents, multiply, divide, add, and then subtract.
@$$\begin{align*}& 4(8) + 1.5 \ (\text{Multiply})\\ & 32 + 1.5 \ (\text{Add})\\ & 33.5\end{align*}@$$
Our answer is 33.5
Evaluate the expression @$\begin{align*}5ab + 2a - 7\end{align*}@$ , when @$\begin{align*}a = 2\end{align*}@$ and @$\begin{align*}b = 4\end{align*}@$ .
Yes, there are. But don’t let that take you off course. If you simply substitute the given values into the expression and use the order of operations, you will end up with the correct answer.
Step 1: Substitute 2 for the variable “ @$\begin{align*}a\end{align*}@$ ” and 4 for the variable “ @$\begin{align*}b\end{align*}@$ .”
@$$\begin{align*}& 5ab + 2a - 7\\ & 5(2)(4) + 2(2) - 7\end{align*}@$$
Step 2: Follow the standard order of operations to solve: parentheses, exponents, multiply, divide, add, and then subtract.
@$$\begin{align*}& 5(2)(4) + 2(2) - 7 \ (\text{Multiply})\\ & 10(4) + 4 - 7\\ & 40 + 4 - 7 \ (\text{Add})\\ & 44 - 7 \ (\text{Subtract})\\ & 37\end{align*}@$$
The answer is 37.
Example A
Evaluate the expression @$\begin{align*}5a + 2b - 17\end{align*}@$ , when @$\begin{align*}a = 3\end{align*}@$ and @$\begin{align*}b = 4\end{align*}@$
Solution: @$\begin{align*}6\end{align*}@$
Example B
Evaluate the expression @$\begin{align*}6xy + 2x - 7\end{align*}@$ , when @$\begin{align*}x=4\end{align*}@$ and @$\begin{align*}y=5\end{align*}@$
Solution: @$\begin{align*}121\end{align*}@$
Example C
Evaluate the expression @$\begin{align*}9x + 18y + 5\end{align*}@$ , when @$\begin{align*}x=-6\end{align*}@$ and @$\begin{align*}y=2\end{align*}@$
Solution: @$\begin{align*}-13\end{align*}@$
Now let's go back to the dilemma from the beginning of the Concept. Here is the variable expression that Hanson wrote.
@$\begin{align*}4.99x + 3.50y\end{align*}@$
We know that he bought 4 pounds of cashews and 4 pounds of almonds. Let's substitute those values into the expression for @$\begin{align*}x\end{align*}@$ and @$\begin{align*}y\end{align*}@$ .
@$\begin{align*}4.99(4) + 3.50(4)\end{align*}@$
Now we can solve for the total cost.
@$\begin{align*}19.96 + 14.00 = $33.96\end{align*}@$
This is our final answer.
Vocabulary
- Variable Expression
- a group of numbers, operations and variables without an equal sign.
- Variable
- a letter used to represent an unknown number
- Constant
- a number in an expression that does not have a variable.
Guided Practice
Here is one for you to try on your own.
Evaluate @$\begin{align*}4x + 3y - 18xy\end{align*}@$ , when @$\begin{align*}x=-3,y=4\end{align*}@$
Solution
First, substitute the given values into the expression for @$\begin{align*}x\end{align*}@$ and @$\begin{align*}y\end{align*}@$ .
@$\begin{align*}4(-3) + 3(4) - 18(-3)(4)\end{align*}@$
Now we can evaluate the expression using the order of operations.
@$\begin{align*}-12 + 12 - -216\end{align*}@$
@$\begin{align*}0 + 216 = 216\end{align*}@$
This is our final answer.
Video Review
Khan Academy Evaluating Variable Expressions
Practice
Directions: Evaluate each variable expression using the given values for each variable.
- @$\begin{align*}6a+7\end{align*}@$ when @$\begin{align*}a\end{align*}@$ is 8
- @$\begin{align*}9x-y\end{align*}@$ when @$\begin{align*}x\end{align*}@$ is 2 and @$\begin{align*}y\end{align*}@$ is 4.
- @$\begin{align*}5a+ b^2\end{align*}@$ when @$\begin{align*}a\end{align*}@$ is 12 and @$\begin{align*}b\end{align*}@$ is 4.
- @$\begin{align*}\frac{8}{x}+2\end{align*}@$ when @$\begin{align*}x\end{align*}@$ is 4
- @$\begin{align*}6x+2.5\end{align*}@$ when @$\begin{align*}x\end{align*}@$ is 2.
- @$\begin{align*}y^2+4\end{align*}@$ when @$\begin{align*}y\end{align*}@$ is 9
- @$\begin{align*}7x+2y\end{align*}@$ when @$\begin{align*}x\end{align*}@$ is 3 and @$\begin{align*}y\end{align*}@$ is 5
- @$\begin{align*}9xy+ x^2\end{align*}@$ when @$\begin{align*}x\end{align*}@$ is 4 and @$\begin{align*}y\end{align*}@$ is 2
- @$\begin{align*}3ab+ b^3\end{align*}@$ when @$\begin{align*}a\end{align*}@$ is 9 and @$\begin{align*}b\end{align*}@$ is 2
- @$\begin{align*}16xy^2+ 14\end{align*}@$ when @$\begin{align*}x\end{align*}@$ is 3 and @$\begin{align*}y\end{align*}@$ is 4
11. @$\begin{align*}6xy + 4x\end{align*}@$ when @$\begin{align*}x\end{align*}@$ is 2 and @$\begin{align*}y\end{align*}@$ is 7
12. @$\begin{align*}16y + 8xy\end{align*}@$ when @$\begin{align*}x\end{align*}@$ is 3 and @$\begin{align*}y\end{align*}@$ is 4
13. @$\begin{align*}3x^2 + 24y\end{align*}@$ when @$\begin{align*}x\end{align*}@$ is 3 and @$\begin{align*}y\end{align*}@$ is 4
14. @$\begin{align*}-xy + 8xy\end{align*}@$ when @$\begin{align*}x\end{align*}@$ is 3 and @$\begin{align*}y\end{align*}@$ is 4
15. @$\begin{align*}22x + 4y - 3xy\end{align*}@$ when @$\begin{align*}x\end{align*}@$ is 3 and @$\begin{align*}y\end{align*}@$ is 4