0.5 – Algebraic Expressions Review

Learning Objectives

  • (0.5.1) – Evaluate and simplify algebraic expressions
  • (0.5.2) – Distinguish between expressions and equations

(0.5.1) – Evaluate and Simplify 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]-4x^2y^3[/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.

[latex]\begin{array}\text{ }\left(-3\right)^{5}=\left(-3\right)\cdot\left(-3\right)\cdot\left(-3\right)\cdot\left(-3\right)\cdot\left(-3\right) \end{array}[/latex]
[latex]\begin{array}\text{ } x^{5}=x\cdot x\cdot x\cdot x\cdot x\end{array}[/latex]
[latex]\begin{array}\text{ }\left(2\cdot7\right)^{3}=\left(2\cdot7\right)\cdot\left(2\cdot7\right)\cdot\left(2\cdot7\right) \end{array}[/latex]
[latex]\begin{array}\text{ } \left(yz\right)^{3}=\left(yz\right)\cdot\left(yz\right)\cdot\left(yz\right)\end{array}[/latex]

In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.

Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

Example: Describing Algebraic Expressions

List the constants and variables for each algebraic expression.

  1. [latex]x + 5[/latex]
  2. [latex]\frac{4}{3}\pi {r}^{3}[/latex]
  3. [latex]-4x^2y^3[/latex]

Try It

Example: Evaluating an Algebraic Expression at Different Values

Evaluate the expression [latex]2x - 7[/latex] for each value for x.

  1. [latex]x=0[/latex]
  2. [latex]x=1[/latex]
  3. [latex]x=\frac{1}{2}[/latex]
  4. [latex]x=-4[/latex]

Try It

Example: Evaluating Algebraic Expressions

Evaluate each expression for the given values.

  1. [latex]x+5[/latex] for [latex]x=-5[/latex]
  2. [latex]\frac{t}{2t - 1}[/latex] for [latex]t=10[/latex]
  3. [latex]\frac{4}{3}\pi {r}^{3}[/latex] for [latex]r=5[/latex]
  4. [latex]a+ab+b[/latex] for [latex]a=11,b=-8[/latex]

Try It


In the following video we present more examples of how to evaluate an expression for a given value.

(0.5.2) – Distinguish between expressions and equations

An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation [latex]2x+1=7[/latex] has the unique solution [latex]x=3[/latex] because when we substitute 3 for [latex]x[/latex] in the equation, we obtain the true statement [latex]2\left(3\right)+1=7[/latex].