Learning Outcomes
- Evaluate an expression for a given value
Evaluate Algebraic Expressions
In the last section, we simplified expressions using the order of operations. In this section, we’ll evaluate expressions—again following the order of operations.
To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.
example
Evaluate [latex]x+7[/latex] when
- [latex]x=3[/latex]
- [latex]x=12[/latex]
Solution:
1. To evaluate, substitute [latex]3[/latex] for [latex]x[/latex] in the expression, and then simplify.
|
[latex]x+7[/latex] |
| Substitute. |
[latex]\color{red}{3}+7[/latex] |
| Add. |
[latex]10[/latex] |
When [latex]x=3[/latex], the expression [latex]x+7[/latex] has a value of [latex]10[/latex].
2. To evaluate, substitute [latex]12[/latex] for [latex]x[/latex] in the expression, and then simplify.
|
[latex]x+7[/latex] |
| Substitute. |
[latex]\color{red}{12}+7[/latex] |
| Add. |
[latex]19[/latex] |
When [latex]x=12[/latex], the expression [latex]x+7[/latex] has a value of [latex]19[/latex].
Notice that we got different results for parts 1 and 2 even though we started with the same expression. This is because the values used for [latex]x[/latex] were different. When we evaluate an expression, the value varies depending on the value used for the variable.
example
Evaluate [latex]9x - 2,[/latex] when
- [latex]x=5[/latex]
- [latex]x=1[/latex]
Show Solution
Solution
Remember [latex]ab[/latex] means [latex]a[/latex] times [latex]b[/latex], so [latex]9x[/latex] means [latex]9[/latex] times [latex]x[/latex].
1. To evaluate the expression when [latex]x=5[/latex], we substitute [latex]5[/latex] for [latex]x[/latex], and then simplify.
|
[latex]9x-2[/latex] |
| Substitute [latex]\color{red}{5}[/latex] for x. |
[latex]9\cdot\color{red}{5}-2[/latex] |
| Multiply. |
[latex]45-2[/latex] |
| Subtract. |
[latex]43[/latex] |
2. To evaluate the expression when [latex]x=1[/latex], we substitute [latex]1[/latex] for [latex]x[/latex], and then simplify.
|
[latex]9x-2[/latex] |
| Substitute [latex]\color{red}{1}[/latex] for x. |
[latex]9(\color{red}{1})-2[/latex] |
| Multiply. |
[latex]9-2[/latex] |
| Subtract. |
[latex]7[/latex] |
Notice that in part 1 that we wrote [latex]9\cdot 5[/latex] and in part 2 we wrote [latex]9\left(1\right)[/latex]. Both the dot and the parentheses tell us to multiply.
example
Evaluate [latex]{x}^{2}[/latex] when [latex]x=10[/latex].
Show Solution
Solution
We substitute [latex]10[/latex] for [latex]x[/latex], and then simplify the expression.
|
[latex]x^2[/latex] |
| Substitute [latex]\color{red}{10}[/latex] for x. |
[latex]{\color{red}{10}}^{2}[/latex] |
| Use the definition of exponent. |
[latex]10\cdot 10[/latex] |
| Multiply. |
[latex]100[/latex] |
When [latex]x=10[/latex], the expression [latex]{x}^{2}[/latex] has a value of [latex]100[/latex].
example
[latex]\text{Evaluate }{2}^{x}\text{ when }x=5[/latex].
Show Solution
Solution
In this expression, the variable is an exponent.
|
[latex]2^x[/latex] |
| Substitute [latex]\color{red}{5}[/latex] for x. |
[latex]{2}^{\color{red}{5}}[/latex] |
| Use the definition of exponent. |
[latex]2\cdot2\cdot2\cdot2\cdot2[/latex] |
| Multiply. |
[latex]32[/latex] |
When [latex]x=5[/latex], the expression [latex]{2}^{x}[/latex] has a value of [latex]32[/latex].
example
[latex]\text{Evaluate }3x+4y - 6\text{ when }x=10\text{ and }y=2[/latex].
Show Solution
Solution
This expression contains two variables, so we must make two substitutions.
|
[latex]3x+4y-6[/latex] |
| Substitute [latex]\color{red}{10}[/latex] for x and [latex]\color{blue}{2}[/latex] for y. |
[latex]3(\color{red}{10})+4(\color{blue}{2})-6[/latex] |
| Multiply. |
[latex]30+8-6[/latex] |
| Add and subtract left to right. |
[latex]32[/latex] |
When [latex]x=10[/latex] and [latex]y=2[/latex], the expression [latex]3x+4y - 6[/latex] has a value of [latex]32[/latex].
example
[latex]\text{Evaluate }2{x}^{2}+3x+8\text{ when }x=4[/latex].
Show Solution
Solution
We need to be careful when an expression has a variable with an exponent. In this expression, [latex]2{x}^{2}[/latex] means [latex]2\cdot x\cdot x[/latex] and is different from the expression [latex]{\left(2x\right)}^{2}[/latex], which means [latex]2x\cdot 2x[/latex].
|
[latex]2x^2+3x+8[/latex] |
| Substitute [latex]\color{red}{4}[/latex] for each x. |
[latex]2{(\color{red}{4})}^{2}+3(\color{red}{4})+8[/latex] |
| Simplify [latex]{4}^{2}[/latex] . |
[latex]2(16)+3(4)+8[/latex] |
| Multiply. |
[latex]32+12+8[/latex] |
| Add. |
[latex]52[/latex] |
In the video below we show more examples of how to substitute a value for variable in an expression, then evaluate the expression.
