Appendix J: Evaluating Algebraic Expressions

OpenStax

Appendix J: Evaluating Algebraic Expressions

OpenStax

Evaluating 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 algebraic expression means to find the value of the expression when the variable is replaced by a given number. 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 1

Evaluate $x+7$ when

$\phantom{\rule{0.2em}{0ex}}x=3$
$\phantom{\rule{0.2em}{0ex}}x=12$

Solution

a. To evaluate, substitute $3$ for $x$ in the expression, and then simplify.


Substitute.
Add.

When $x=3$ , the expression $x+7$ has a value of $10$ .

b. To evaluate, substitute $12$ for $x$ in the expression, and then simplify.


Substitute.
Add.

When $x=12$ , the expression $x+7$ has a value of $19$ .

Notice that we got different results for parts a) and b) even though we started with the same expression. This is because the values used for $x$ were different. When we evaluate an expression, the value varies depending on the value used for the variable.

TRY IT 1.1

Evaluate:

$y+4\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}$

$\phantom{\rule{0.2em}{0ex}}y=6\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}y=15$

Show Answer

10
19

TRY IT 1.2

Evaluate:

$a-5\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}$

$\phantom{\rule{0.2em}{0ex}}a=9\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}a=17$

Show Answer

4
12

EXAMPLE 2

Evaluate $9x-2,\text{when}\phantom{\rule{0.2em}{0ex}}$

$\phantom{\rule{0.2em}{0ex}}x=5\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}x=1$

Solution

Remember $ab$ means $a$ times $b$ , so $9x$ means $9$ times $x$ .

a. To evaluate the expression when $x=5$ , we substitute $5$ for $x$ , and then simplify.



Multiply.
Subtract.

b. To evaluate the expression when $x=1$ , we substitute $1$ for $x$ , and then simplify.



Multiply.
Subtract.

Notice that in part a) that we wrote $9\cdot 5$ and in part b) we wrote $9\left(1\right)$ . Both the dot and the parentheses tell us to multiply.

TRY IT 2.1

Evaluate:

$8x-3,\text{when}\phantom{\rule{0.2em}{0ex}}$

$\phantom{\rule{0.2em}{0ex}}x=2\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}x=1$

Show Answer

13
5

TRY IT 2.2

Evaluate:

$4y-4,\text{when}\phantom{\rule{0.2em}{0ex}}$

$\phantom{\rule{0.2em}{0ex}}y=3\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}y=5$

Show Answer

8
16

EXAMPLE 3

Evaluate ${x}^{2}$ when $x=10$ .

Solution

We substitute $10$ for $x$ , and then simplify the expression.



Use the definition of exponent.
Multiply.

When $x=10$ , the expression ${x}^{2}$ has a value of $100$ .

TRY IT 3.1

Evaluate:

${x}^{2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=8$ .

Show Answer

64

TRY IT 3.2

Evaluate:

${x}^{3}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=6$ .

Show Answer

216

EXAMPLE 4

$\text{Evaluate}\phantom{\rule{0.2em}{0ex}}{2}^{x}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=5$ .

Solution

In this expression, the variable is an exponent.



Use the definition of exponent.
Multiply.

When $x=5$ , the expression ${2}^{x}$ has a value of $32$ .

TRY IT 4.1

Evaluate:

${2}^{x}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=6$ .

Show Answer

64

TRY IT 4.2

Evaluate:

${3}^{x}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=4$ .

Show Answer

81

EXAMPLE 5

$\text{Evaluate}\phantom{\rule{0.2em}{0ex}}3x+4y-6\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=10\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=2$ .

Solution

This expression contains two variables, so we must make two substitutions.



Multiply.
Add and subtract left to right.

When $x=10$ and $y=2$ , the expression $3x+4y-6$ has a value of $32$ .

TRY IT 5.1

Evaluate:

$2x+5y-4\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=11\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=3$

Show Answer

33

TRY IT 5.2

Evaluate:

$5x-2y-9\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=7\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=8$

Show Answer

10

EXAMPLE 6

$\text{Evaluate}\phantom{\rule{0.2em}{0ex}}2{x}^{2}+3x+8\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=4$ .

Solution

We need to be careful when an expression has a variable with an exponent. In this expression, $2{x}^{2}$ means $2\cdot x\cdot x$ and is different from the expression ${\left(2x\right)}^{2}$ , which means $2x\cdot 2x$ .