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Mathematics LibreTexts

2.2: Evaluate, Simplify, and Translate Expressions (Part 1)

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    4978
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    Skills to Develop

    • Evaluate algebraic expressions
    • Identify terms, coefficients, and like terms
    • Simplify expressions by combining like terms
    • Translate word phrases to algebraic expressions

    Be prepared!

    Before you get started, take this readiness quiz.

    1. Is n ÷ 5 an expression or an equation? If you missed this problem, review Example 2.4.
    2. Simplify 45. If you missed this problem, review Example 2.7.
    3. Simplify 1 + 8 • 9. If you missed this problem, review Example 2.8.

    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 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 2.13:

    Evaluate x + 7 when (a) x = 3 (b) x = 12

    Solution

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

      $$x + 7$$
    Substitute. $$\textcolor{red}{3} + 7$$
    Add. 10

    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.

      $$x + 7$$
    Substitute. $$\textcolor{red}{12} + 7$$
    Add. 19

    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.

     

    exercise 2.25:

    Evaluate: y + 4 when (a) y = 6 (b) y = 15

    exercise 2.26:

    Evaluate: a − 5 when (a) a = 9 (b) a = 17

    Example 2.14:

    Evaluate 9x − 2, when (a) x = 5 (b) 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.

      $$9x - 2$$
    Substitute \(\textcolor{red}{5}\) for x. $$9 \cdot \textcolor{red}{5} - 2$$
    Multiply. $$45 - 2$$
    Subtract. $$43$$

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

      $$9x - 2$$
    Substitute \(\textcolor{red}{1}\) for x. $$9 \cdot \textcolor{red}{1} - 2$$
    Multiply. $$9 - 2$$
    Subtract. $$7$$

    Notice that in part (a) that we wrote 9 • 5 and in part (b) we wrote 9(1). Both the dot and the parentheses tell us to multiply.

    exercise 2.27:

    Evaluate: 8x − 3, when (a) x = 2 (b) x = 1

    exercise 2.28:

    Evaluate: 4y − 4, when (a) y = 3 (b) y = 5

    Example 2.15:

    Evaluate x2 when x = 10.

    Solution

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

      $$x^{2}$$
    Substitute \(\textcolor{red}{10}\) for x. $$\textcolor{red}{10}^{2}$$
    Use the definition of exponent. $$10 \cdot 10$$
    Multiply $$100$$

     When x = 10, the expression x2 has a value of 100.

     

    exercise 2.29:

    Evaluate: x2 when x = 8.

    exercise 2.30:

    Evaluate: x3 when x = 6.

    Example 2.16:

    Evaluate 2x when x = 5.

    Solution

    In this expression, the variable is an exponent.

      $$2^{x}$$
    Substitute \(\textcolor{red}{5}\) for x. $$2^{\textcolor{red}{5}}$$
    Use the definition of exponent. $$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$$
    Multiply $$32$$

    When x = 5, the expression 2x has a value of 32.

    exercise 2.31:

    Evaluate: 2x when x = 6.

    exercise 2.32:

    Evaluate: 3x when x = 4.

    Example 2.17:

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

    Solution

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

      $$3x + 4y − 6$$
    Substitute \(\textcolor{red}{10}\) for x and \(\textcolor{blue}{2}\) for y. $$3(\textcolor{red}{10}) + 4(\textcolor{blue}{2}) − 6$$
    Multiply. $$30 + 8 - 6$$
    Add and subtract left to right. $$32$$

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

    exercise 2.33:

    Evaluate: 2x + 5y − 4 when x = 11 and y = 3

    exercise 2.34:

    Evaluate: 5x − 2y − 9 when x = 7 and y = 8

    Example 2.18:

    Evaluate 2x2 + 3x + 8 when x = 4.

    Solution

    We need to be careful when an expression has a variable with an exponent. In this expression, 2x2 means 2 • x • x and is different from the expression (2x)2, which means 2x • 2x.

      $$2x^{2} + 3x + 8$$
    Substitute \(\textcolor{red}{4}\) for each x. $$2(\textcolor{red}{4})^{2} + 3(\textcolor{red}{4}) + 8$$
    Simplify 42 $$2(16) + 3(4) + 8$$
    Multiply. $$32 + 12 + 8$$
    Add. $$52$$

    exercise 2.35:

    Evaluate: 3x2 + 4x + 1 when x = 3.

    exercise 2.36:

    Evaluate: 6x2 − 4x − 7 when x = 2.

    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, 5x2, 9a, and 13xy.

    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 3x is 3. When we write x, the coefficient is 1, since x = 1 • x. Table 2.36 gives the coefficients for each of the terms in the left column.

    Table 2.36

    Term Coefficient
    7 7
    9a 9
    y 1
    5x2 5

    An algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together. Table 2.37 gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.

    Table 2.37

    Expression Terms
    7 7
    y y
    x + 7 x, 7
    2x + 7y + 4 2x, 7y, 4
    3x2 + 4x2 + 5y + 3 3x2, 4x2, 5y, 3

    Example 2.19:

    Identify each term in the expression 9b + 15x2 + a + 6. Then identify the coefficient of each term.

    Solution

    The expression has four terms. They are 9b, 15x2, a, and 6.

    The coefficient of 9b is 9.

    The coefficient of 15x2 is 15.

    Remember that if no number is written before a variable, the coefficient is 1. So the coefficient of a is 1.

    The coefficient of a constant is the constant, so the coefficient of 6 is 6.

    exercise 2.37:

    Identify all terms in the given expression, and their coefficients: 4x + 3b + 2

    exercise 2.38:

    Identify all terms in the given expression, and their coefficients: 9a + 13a2 + a3

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

    $$5x, 7, n^{2}, 4, 3x, 9n^{2}$$

    Which of these terms are like terms?

    • The terms 7 and 4 are both constant terms.
    • The terms 5x and 3x are both terms with x.
    • The terms n2 and 9n2 both have n2.

    Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms 5x, 7, n2, 4, 3x, 9n2, 7 and 4 are like terms, 5x and 3x are like terms, and n2 and 9n2 are like terms.

    Definition: Like terms

    Terms that are either constants or have the same variables with the same exponents are like terms.

    Example 2.20:

    Identify the like terms: (a) y3, 7x2, 14, 23, 4y3, 9x, 5x2 (b) 4x2 + 2x + 5x2 + 6x + 40x + 8xy

    Solution

    (a) y3, 7x2, 14, 23, 4y3, 9x, 5x2

    Look at the variables and exponents. The expression contains y3, x2, x, and constants. The terms y3 and 4y3 are like terms because they both have y3. The terms 7x2 and 5x2 are like terms because they both have x2. The terms 14 and 23 are like terms because they are both constants. The term 9x does not have any like terms in this list since no other terms have the variable x raised to the power of 1.

    (b) 4x2 + 2x + 5x2 + 6x + 40x + 8xy

    Look at the variables and exponents. The expression contains the terms 4x2, 2x, 5x2, 6x, 40x, and 8xy The terms 4x2 and 5x2 are like terms because they both have x2. The terms 2x, 6x, and 40x are like terms because they all have x. The term 8xy has no like terms in the given expression because no other terms contain the two variables xy.

    exercise 2.39:

    Identify the like terms in the list or the expression: 9, 2x3, y2, 8x3, 15, 9y, 11y2

    exercise 2.40:

    Identify the like terms in the list or the expression: 4x3 + 8x2 + 19 + 3x2 + 24 + 6x3

    Simplify Expressions by Combining Like Terms

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

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

    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 3x + 6x 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.

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

    HOW TO: COMBINE LIKE TERMS

    Step 1. Identify like terms.

    Step 2. Rearrange the expression so like terms are together.

    Step 3. Add the coefficients of the like terms.

    Example 2.21:

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

    Solution
      $$3x + 7 + 4x + 5$$
    Identify the like terms $$\textcolor{red}{3x} + \textcolor{blue}{7} + \textcolor{red}{4x} + \textcolor{blue}{5}$$
    Rearrange the expression, so the like terms are together. $$\textcolor{red}{3x} + \textcolor{red}{4x} + \textcolor{blue}{7} + \textcolor{blue}{5}$$
    Add the coefficients of the like terms. $$\textcolor{red}{7x} + \textcolor{blue}{12}$$
    The original expression is simplified to... $$7x + 12$$

    exercise 2.41:

    Simplify: 7x + 9 + 9x + 8

    exercise 2.42:

    Simplify: 5y + 2 + 8y + 4y + 5

    Example 2.22:

    Simplify the expression: 7x2 + 8x + x2 + 4x.

    Solution
      $$7x^{2} + 8x + x^{2} + 4x$$
    Identify the like terms. $$\textcolor{red}{7x^{2}} + \textcolor{blue}{8x} + \textcolor{red}{x^{2}} + \textcolor{blue}{4x}$$
    Rearrange the expression so like terms are together. $$\textcolor{red}{7x^{2}} + \textcolor{red}{x^{2}} + \textcolor{blue}{8x} + \textcolor{blue}{4x}$$
    Add the coefficients of the like terms. $$\textcolor{red}{8x^{2}} + \textcolor{blue}{12x}$$

    These are not like terms and cannot be combined. So 8x2 + 12x is in simplest form.

    exercise 2.43:

    Simplify: 3x2 + 9x + x2 + 5x

    exercise 2.44:

    Simplify: 11y2 + 8y + y2 + 7y

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