In elementary mathematics, a variable is an alphabetic character representing a number, called the value of the variable, that is arbitrary, not fully specified, or unknown.

Variables are useful for several reasons.

Unknown Values

Variables can represent numbers whose values are not yet known. For example, if the temperature of the current day, $C$, is 20 degrees higher than the temperature of the previous day, $P$, then the problem can be described algebraically as $\displaystyle C=P+20$.

General Formulas

Varying Quantities

Variables may describe mathematical relationships between quantities that vary. For example, the relationship between the circumference, $C$, and diameter, $d$, of a circle is described by $\displaystyle \pi =C/d$.

Variables may also describe general problems without specifying the values of the quantities involved. For example, it can be stated specifically that 5 minutes is equivalent to $\displaystyle 60\times 5=300$ seconds. A more general (algebraic) description may state the number of seconds as $\displaystyle s=60\times m$ , where $m$ is the number of minutes.

Mathematical Properties

Variables may describe some mathematical properties. For example, a basic property of addition is commutativity, which states that the order of numbers being added together does not matter. Commutativity is stated algebraically as $\displaystyle (a+b)=(b+a)$.

Types of Variables

Variables can be used to represent different types of numbers. It is common that many variables appear in the same mathematical formula, and they may play different roles. Some names or qualifiers have been introduced to distinguish them. 

For example, in the general cubic equation $\displaystyle ax^{3}+bx^{2}+cx+d=0$, there are five variables. Four of them ($a$, $b$, $c$, $d$) represent given numbers, which are referred to as the parameters of the equation. The last one, $x$, represents the solution of the equation, which is unknown and must be solved for. To distinguish among the different variables, $x$ is called an unknown, and the variables that are multiplied by $x$ are called coefficients. In this equation, the coefficients are $a$, $b$, and $c$. A number on its own (without an unknown variable) is called a constant; in this case, $d$ represents a constant.

Note that a term of an equation is any value (variable or number) or expression that is separated from another term by a space or a character (such as "$+$"). Therefore, a term may simply be a constant or a variable, or it may include both a coefficient and an unknown variable. In the cubic equation described above, there are four terms: $ax^3$, $bx^2$, $cx$, and $d$. 

Note that unknown variables are often denoted by $x$, $y$, or $z$ and the parameters of equations by $a$, $b$, $c$, or $d$. However, this is not always the case. For example, you might be asked to solve the following equation for $b$:  

$12 - b = 3$ 

In this case, $b$ is an unknown variable, not a parameter of the equation. We can solve this problem and find that $b = 9$.