Evaluating Algebraic Expressions
So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as [latex]x+5,\frac{4}{3}\pi {r}^{3}[/latex], or [latex]\sqrt{2{m}^{3}{n}^{2}}[/latex]. In the expression [latex]x+5[/latex], 5 is called a constant because it does not vary and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division. We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.
Example 8: Describing Algebraic Expressions
List the constants and variables for each algebraic expression.- x + 5
- [latex]\frac{4}{3}\pi {r}^{3}[/latex]
- [latex]\sqrt{2{m}^{3}{n}^{2}}[/latex]
Solution
Constants | Variables | |
---|---|---|
1. x + 5 | 5 | x |
2. [latex]\frac{4}{3}\pi {r}^{3}[/latex] | [latex]\frac{4}{3},\pi [/latex] | [latex]r[/latex] |
3. [latex]\sqrt{2{m}^{3}{n}^{2}}[/latex] | 2 | [latex]m,n[/latex] |
Try It 8
List the constants and variables for each algebraic expression.- [latex]2\pi r\left(r+h\right)[/latex]
- 2(L + W)
- [latex]4{y}^{3}+y[/latex]
Example 9: Evaluating an Algebraic Expression at Different Values
Evaluate the expression [latex]2x - 7[/latex] for each value for x.- [latex]x=0[/latex]
- [latex]x=1[/latex]
- [latex]x=\frac{1}{2}[/latex]
- [latex]x=-4[/latex]
Solution
- Substitute 0 for [latex]x[/latex].
[latex]\begin{array}\text{ }2x-7 \hfill& = 2\left(0\right)-7 \\ \hfill& =0-7 \\ \hfill& =-7\end{array}[/latex]
- Substitute 1 for [latex]x[/latex].
[latex]\begin{array}\text{ }2x-7 \hfill& = 2\left(1\right)-7 \\ \hfill& =2-7 \\ \hfill& =-5\end{array}[/latex]
- Substitute [latex]\frac{1}{2}[/latex] for [latex]x[/latex].
[latex]\begin{array}\text{ }2x-7 \hfill& = 2\left(\frac{1}{2}\right)-7 \\ \hfill& =1-7 \\ \hfill& =-6\end{array}[/latex]
- Substitute [latex]-4[/latex] for [latex]x[/latex].
[latex]\begin{array}\text{ }2x-7 \hfill& = 2\left(-4\right)-7 \\ \hfill& =-8-7 \\ \hfill& =-15\end{array}[/latex]
Try It 9
Evaluate the expression [latex]11 - 3y[/latex] for each value for y.a. [latex]y=2[/latex] b. [latex]y=0[/latex] c. [latex]y=\frac{2}{3}[/latex] d. [latex]y=-5[/latex]
SolutionExample 10: Evaluating Algebraic Expressions
Evaluate each expression for the given values.- [latex]x+5[/latex] for [latex]x=-5[/latex]
- [latex]\frac{t}{2t - 1}\\[/latex] for [latex]t=10[/latex]
- [latex]\frac{4}{3}\pi {r}^{3}\\[/latex] for [latex]r=5[/latex]
- [latex]a+ab+b[/latex] for [latex]a=11,b=-8[/latex]
- [latex]\sqrt{2{m}^{3}{n}^{2}}[/latex] for [latex]m=2,n=3[/latex]
Solution
- Substitute [latex]-5[/latex] for [latex]x[/latex].
[latex]\begin{array}\text{ }x+5\hfill&=\left(-5\right)+5 \\ \hfill&=0\end{array}[/latex]
- Substitute 10 for [latex]t[/latex].
[latex]\begin{array}\text{ }\frac{t}{2t-1}\hfill& =\frac{\left(10\right)}{2\left(10\right)-1} \\ \hfill& =\frac{10}{20-1} \\ \hfill& =\frac{10}{19}\end{array}[/latex]
- Substitute 5 for [latex]r[/latex].
[latex]\begin{array}\text{ }\frac{4}{3}\pi r^{3} \hfill& =\frac{4}{3}\pi\left(5\right)^{3} \\ \hfill& =\frac{4}{3}\pi\left(125\right) \\ \hfill& =\frac{500}{3}\pi\end{array}[/latex]
- Substitute 11 for [latex]a[/latex] and –8 for [latex]b[/latex].
[latex]\begin{array}\text{ }a+ab+b \hfill& =\left(11\right)+\left(11\right)\left(-8\right)+\left(-8\right) \\ \hfill& =11-8-8 \\ \hfill& =-85\end{array}[/latex]
- Substitute 2 for [latex]m[/latex] and 3 for [latex]n[/latex].
[latex]\begin{array}\text{ }\sqrt{2m^{3}n^{2}} \hfill& =\sqrt{2\left(2\right)^{3}\left(3\right)^{2}} \\ \hfill& =\sqrt{2\left(8\right)\left(9\right)} \\ \hfill& =\sqrt{144} \\ \hfill& =12\end{array}[/latex]
Try It 10
Evaluate each expression for the given values. a. [latex]\frac{y+3}{y - 3}[/latex] for [latex]y=5[/latex] b. [latex]7 - 2t[/latex] for [latex]t=-2[/latex] c. [latex]\frac{1}{3}\pi {r}^{2}[/latex] for [latex]r=11[/latex] d. [latex]{\left({p}^{2}q\right)}^{3}[/latex] for [latex]p=-2,q=3[/latex] e. [latex]4\left(m-n\right)-5\left(n-m\right)[/latex] for [latex]m=\frac{2}{3},n=\frac{1}{3}[/latex] SolutionLicenses & Attributions
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