5.1 Evaluating Algebraic Expressions

Evaluating Algebraic Expressions

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Evaluate Algebraic Expressions

Earlier, 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 $x plus 7$ when

ⓐ $x equals 3$
ⓑ $x equals 12$

Show solution

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


Substitute.
Add.

When $x equals 3,$ the expression $x plus 7$ has a value of $10 .$

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


Substitute.
Add.

When $x equals 12,$ the expression $x plus 7$ has a value of $19 .$

Notice that we got different results for parts ⓐ and ⓑ 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.

Example 2

Evaluate $9 x minus 2, when$

ⓐ $x equals 5$
ⓑ $x equals 1$

Show solution

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

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



Multiply.
Subtract.

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



Multiply.
Subtract.

Notice that in part ⓐ that we wrote $9 times 5$ and in part ⓑ we wrote $9 open parenthesis 1 close parenthesis.$ Both the dot and the parentheses tell us to multiply.

Example 5

$Evaluate 3 x + 4 y - 6 when x = 10 and y = 2 .$

Show solution

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



Multiply.
Add and subtract left to right.

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

Example 6

$Evaluate 2 x^{2} + 3 x + 8 when x = 4 .$

Show solution

We need to be careful when an expression has a variable with an exponent. In this expression, $2 x squared$ means $2 \cdot x \cdot x$ and is different from the expression $open parenthesis 2 x close parenthesis squared,$ which means $2 x times 2 x.$



Simplify $4 squared$ .
Multiply.
Add.

Identify Terms, Coefficients, and Like Terms

Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more variables. Some examples of terms are $7, y, 5 x squared, 9 a, and 13 x y.$

The constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number in front of the variable. The coefficient of the term $3 x$ is $3 .$ When we write $x,$ the coefficient is $1,$ since $x = 1 \cdot x .$ Table 5.1 gives the coefficients for each of the terms in the left column.

**Table 5.1**
Term	Coefficient
$9 a$	$9$
$y$	$1$
$5 x squared$	$5$

An algebraic expression may consist of one or more terms added or subtracted. Here, we will only work with terms that are added together. Table 5.2 gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.

**Table 5.2**
Expression	Terms
$7$	$7$
$y$	$y$
$x plus 7$	$x, 7$
$2 x + 7 y + 4$	$2 x, 7 y, 4$
$3 x^{2} + 4 x^{2} + 5 y + 3$	$3 x squared, 4 x squared, 5 y, 3$

Some terms share common traits. Look at the following terms. Which ones seem to have traits in common?

$5 x, 7, n squared, 4, 3 x, 9 n squared$

Which of these terms are like terms?

The terms $7$ and $4$ are both constant terms.
The terms $5 x$ and $3 x$ are both terms with $x .$
The terms $n^{2}$ and $9 n squared$ both have $n^{2} .$

Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms $5 x, 7, n squared, 4, 3 x, 9 n squared,$

$7 and 4 are like terms.$

$5 x and 3 x are like terms.$

$n squared and 9 n squared are like terms.$

Example 8

Identify the like terms:

ⓐ $y cubed, 7 x squared, 14, 23, 4 y cubed, 9 x, 5 x squared$
ⓑ $4 x^{2} + 2 x + 5 x^{2} + 6 x + 40 x + 8 x y$

Show solution

ⓐ $y cubed, 7 x squared, 14, 23, 4 y cubed, 9 x, 5 x squared$

Look at the variables and exponents. The expression contains $y^{3}, x^{2}, x,$ and constants.

The terms $y^{3}$ and $4 y cubed$ are like terms because they both have $y^{3} .$

The terms $7 x squared$ and $5 x squared$ are like terms because they both have $x^{2} .$

The terms $14$ and $23$ are like terms because they are both constants.

The term $9 x$ does not have any like terms in this list since no other terms have the variable $x$ raised to the power of $1 .$

ⓑ $4 x^{2} + 2 x + 5 x^{2} + 6 x + 40 x + 8 x y$

Look at the variables and exponents. The expression contains the terms $4 x squared, 2 x, 5 x squared, 6 x, 40 x, and 8 x y$

The terms $4 x squared$ and $5 x squared$ are like terms because they both have $x^{2} .$

The terms $2 x, 6 x, and 40 x$ are like terms because they all have $x .$

The term $8 x y$ has no like terms in the given expression because no other terms contain the two variables $x y .$

Simplify Expressions by Combining Like Terms

We can simplify an expression by combining the like terms. What do you think $3 x plus 6 x$ would simplify to? If you thought $9 x,$ you would be right!

We can see why this works by writing both terms as addition problems.

The image shows the expression 3 x plus 6 x. The 3 x represents x plus x plus x. The 6 x represents x plus x plus x plus x plus x plus x. The expression 3 x plus 6 x becomes x plus x plus x plus x plus x plus x plus x plus x plus x. This simplifies to a total of 9 x's or the term 9 x.

Add the coefficients and keep the same variable. It doesn’t matter what $x$ is. If you have $3$ of something and add $6$ more of the same thing, the result is $9$ of them. For example, $3$ oranges plus $6$ oranges is $9$ oranges. We will discuss the mathematical properties behind this later.

The expression $3 x plus 6 x$ has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The Commutative Property of Addition says that we can change the order of addends without changing the sum. So we could rearrange the following expression before combining like terms.

Two rows show 3x + 4y - 2x + 6y rearranged as 3x - 2x + 4y + 6y. Turquoise crossing arrows move the terms 4y and -2x so the x terms and y terms are together.

Now it is easier to see the like terms to be combined.

Example 9

Simplify the expression: $3 x + 7 + 4 x + 5 .$

Show solution


Identify the like terms.
Rearrange the expression, so the like terms are together.
Add the coefficients of the like terms.
The original expression is simplified to...

When any of the terms have negative coefficients, the procedure is the same, except that you have to subtract instead of adding to combine like terms.

Example 10

Simplify the expression: $7 x^{2} + 8 x - x^{2} - 4 x .$

Show solution


Identify the like terms.
Rearrange the expression so like terms are together.
Add the coefficients of the like terms.

These are not like terms and cannot be combined. So $6 x squared plus 4 x$ is in simplest form.

Practice Makes Perfect

Evaluate Algebraic Expressions

In the following exercises, evaluate the expression for the given value.

$7 x + 8 when x = 2$

Show solution

$9 x + 7 when x = 3$

$5 x - 4 when x = 6$

Show solution

$8 x - 6 when x = 7$

$x squared when x equals 12$

Show solution

144

$x cubed when x equals 5$

$x^{5} when x = 2$

Show solution

$x^{4} when x = 3$

$3^{x} when x = 3$

Show solution

$4^{x} when x = 2$

$x^{2} + 3 x - 7 when x = 4$

Show solution

$x^{2} + 5 x - 8 when x = 6$

$2 x + 4 y - 5 when x = 7, y = 8$

Show solution

$6 x + 3 y - 9 when x = 6, y = 9$

${(x - y)}^{2} when x = 10, y = 7$

Show solution

${(x + y)}^{2} when x = 6, y = 9$

Show solution

225

$a^{2} + b^{2} when a = 3, b = 8$

Show solution

$r^{2} - s^{2} when r = 12, s = 5$

$2 l + 2 w when l = 15, w = 12$

Show solution

$2 l + 2 w when l = 18, w = 14$

Identify Terms, Coefficients, and Like Terms

In the following exercises, list the terms in the given expression.

$15 x^{2} + 6 x + 2$

Show solution

15x², 6x, 2

$11 x^{2} + 8 x + 5$

$10 y^{3} + y + 2$

Show solution

10y³, y, 2

$9 y^{3} + y + 5$

In the following exercises, identify the coefficient of the given term.

$8 a$

Show solution

$13 m$

$5 r squared$

Show solution

$6 x cubed$

In the following exercises, identify all sets of like terms.

$x cubed, 8 x, 14, 8 y, 5, 8 x cubed$

Show solution

x³ and 8x³; 14 and 5

$6 z, 3 w squared, 1, 6 z squared, 4 z, w squared$

$9 a, a squared, 16 a b, 16 b squared, 4 a b, 9 b squared$

Show solution

16ab and 4ab; 16b² and 9b²

$3, 25 r squared, 10 s, 10 r, 4 r squared, 3 s$

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the given expression by combining like terms.

$10 x plus 3 x$

Show solution

13x

$15 x plus 4 x$

$17 a plus 9 a$

Show solution

26a

$18 z plus 9 z$

$4 c + 2 c + c$

Show solution

$6 y + 4 y + y$

$9 x + 3 x + 8$

Show solution

12x + 8

$8 a + 5 a + 9$

$7 u + 2 + 3 u + 1$

Show solution

10u + 3

$8 d + 6 + 2 d + 5$

$7 p + 6 + 5 p + 4$

Show solution

12p + 10

$8 x + 7 + 4 x - 5$

$10 a + 7 + 5 a - 2 + 7 a - 4$

Show solution

22a + 1

$7 c + 4 + 6 c - 3 + 9 c - 1$

$3 x^{2} + 12 x + 11 + 14 x^{2} + 8 x + 5$

Show solution

17x² + 20x + 16

$5 b^{2} + 9 b + 10 + 2 b^{2} + 3 b - 4$

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