Variables and Expressions

Objective Learn the notion of variable and algebraic expression.

The most important concept in algebra is that of a variable. This is a concept that you may have seen before, but it is important to begin by reviewing and discussing this concept. Remember that a variable is simply a letter or sometimes a symbol used to represent a quantity. For example, suppose b represents the number of boys in the class and g represents the number of girls. A verbal expression like "There are 14 girls in the class." can be written as an algebraic expression or equation, which involves a variable.

g = 14

We can also write a verbal expression from an algebraic one. Because we know that the variable b represents the number of boys in the class, the algebraic equation b = 15 can be translated to the verbal sentence "There are 15 boys in the class."

What is an algebraic expression?

Have a look at these definitions:

• A variable is a letter or symbol used to represent a quantity.

• An algebraic expression consists of one or more numbers and variables along with one or more arithmetic operations.

Some examples of algebraic expressions are shown below.

You might write a table like the following to show the relation between verbal and algebraic expressions.

Verbal Expression	Algebraic Expression
2 more than a number x	x + 2
The square of a number x divided by three times a number y.
The number of eyes in the classroom (Let s represent the number of students in the class.)	2s + 2 or 2(s + 1) (Each student has two eyes and the teacher has two eyes)

Remember the terms that describe expressions like x ⁵. These expressions are called powers. The x is the base of the power and the number 5 is called the exponent. The expression x ⁵ is described verbally as "x raised to the fifth power" or simply "x to the fifth."

Translating between algebraic and verbal expressions.

Example 1

Write an algebraic expression for the sum of 3 and the number x divided by 16.

Solution (3 + x ) ÷ 16 or .

Example 2

Write an algebraic expression for six subtracted from the number y, all raised to the fourth power.

Solution ( y - 6) ⁴

Example 3

Write a verbal expression for 5m - n ² .

Solution

The product of 5 and the number m minus the number n raised to the second power (or n squared)

Example 4

Write a verbal expression for ( x + 2) ^y.

Solution

The sum of the number x and 2, all raised to the exponent y (or the y th power)

There is one final term that you should describe to your students in this lesson. To evaluate an expression means to find its values. Use the following examples to illustrate this definition.

 

Example 5

Evaluate 4 ³ .

Solution

4 ³ = 4 · 4 · 4 = 64

 

Example 6

Evaluate the expression ( x + 3) ² when x = 3.

Solution

Substitute 3 for x in the expression. ( x + 3) ² = (3 + 3) ² = 6 ² = 36

The processes of assigning variables to quantities and of translating between verbal and algebraic expressions are fundamental in algebra.