Single-Variable Expressions Study Guide

Introduction to Single-Variable Expressions

"Obvious" is the most dangerous word in mathematics.

—Eric Temple Bell (1883–1960) Mathematician and Science Fiction Author

In this lesson, you'll review the order of operations and learn how to evaluate algebraic expressions.

The addition sentence 3x + 7y, or even just 3x alone is an algebraic expression. An algebraic expression is one or more terms, at least one of which contains a variable, which may or may not contain an operation (such as addition or multiplication).

We have seen how to add, subtract, multiply, and divide terms, but all of our answers have contained variables. To evaluate an expression, we replace variables with real numbers. At first glance, it might seem easy to evaluate an expression once the variables have been replaced with numbers, but we must remember to follow the order of operations, or we will arrive at the wrong answer.

The order of operations is a list that tells us how to go about evaluating an expression. First, handle any operations that are in parentheses, no matter what those operations are. Next, work with the exponents in the expression. After that, you can do multiplication and division. Finally, perform addition and subtraction. Most people use the acronym PEMDAS to help them remember the order of operations:

P  Parentheses       P  Please

E   Exponents         E  Excuse

M  Multiplication      M  My

D   Division             D  Dear

A   Addition            A   Aunt

S   Subtraction       S  Sally

On the left is a list of the order of operations, and on the right is a phrase (Please Excuse My Dear Aunt Sally) that can help you remember the order of operations. The first letter in each word of the phrase is the same as the first letter in each operation.

To evaluate the numerical expression 6(5) + 4, we multiply 6 and 5 first, since multiplication comes before addition in the order of operations: 6(5) = 30. Then, we add: 30 + 4 = 34. If we were to add 5 and 4 first and then multiply by 6, our answer would be 54, which is incorrect. That's why the order of operations is so important.

Example

2(6 + 4) – 4² =

Begin with the operation in parentheses: 6 + 4 = 10. The expression is now 2(10) – 4².

Next, work with the exponents: 4² = 16, and the expression becomes 2(10) – 16.

Multiplication is next: 2(10) = 20, and we are left with 20 – 16.

Finally, subtract: 20 – 16 = 4. The expression 2(6 + 4) – 4² is equal to 4.