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Variables (Mathematics Glossary)

What Is a Variable?

A variable is a symbol that stands for a number.

The value of a variable is not fixed, its value can change.

Dictionary Definition

The Merriam-Webster dictionary defines a variable as "a quantity that may assume any one of a set of values; a symbol representing a variable."

Real Examples of Variables

In algebra, some values are fixed, while others are allowed to vary.

The values that are allowed to vary are variables. They are represented by letters and symbols:

a list of letters and symbols that represent variables.

In algebra, values that are not variables are constants. Numbers are always constants. Sometimes letters are used for constants, usually the first letters such as a, b and c, whereas the last letters x, y and z are used for variables.

Understanding Variables

It is easier to understand variables with an example. Let's look at an algebraic equation.

Consider a linear equation:

a linear equation with variables

The y and the x are both variables. We can change their value.

The 2 and the 1 are constants. They both have a fixed value.

y and x can take any value.

However, they are in an equation. This means only certain pairs of y and x are allowed that keep both sides of the equation the same. The left hand side (y) must be the same as the right hand side (2x + 1).

x = 1 ∴ y = 2x + 1 = 2 × 1 + 1 = 3

x = 2 ∴ y = 2x + 1 = 2 × 2 + 1 = 5

x = 3 ∴ y = 2x + 1 = 2 × 3 + 1 = 7

The constants are the same in each case and show which pairs of x and y are allowed by the equation.

Why Are Variables Useful?

There are two uses for variables:

  • A variable can stand in for a number that we don't know yet. This 'unknown' can then be found, like solving a puzzle.

    For example, x stands for a number we don't know yet in the equation below.

    Using algebra, x = 1.

  • A variable can stand in for many different numbers. The value the variable takes varies.

    For example, in a function, a variable is used as an input. By putting different numbers in, the function gives different numbers out.

    The variable x can take a value of 1, 2, 3 etc.

More Real Examples of Variables

The slider below has some more real examples of the uses of variables.

Algebra Lessons
Interactive Test

Here's a second test on variables.
Here's a third test on variables.