Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. While in arithmetic only numbers and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra one also uses symbols (such as x and y, or a and b) to denote numbers. These are called variables. This is useful because:

• It allows the generalization of arithmetical equations (and inequalities) to be stated as laws (such as a + b = b + a for all a and b), and thus is the first step to the systematic study of the properties of the real number system.
• It allows reference to numbers which are not known. In the context of a problem, a variable may represent a certain value which is not yet known, but which may be found through the formulation and manipulation of equations.
• It allows the exploration of mathematical relationships between quantities (such as "if you sell x tickets, then your profit will be 3x − 10 dollars").

These three are the main strands of elementary algebra, which should be distinguished from abstract algebra, a more advanced area of study.

In elementary algebra, an "expression" may contain numbers, variables and arithmetical operations. These are usually written (by convention) with 'higher-power' terms on the left (see polynomial); a few examples are:

In more advanced algebra, an expression may also include elementary functions.

An "equation" is the claim that two expressions are equal. Some equations are true for all values of the involved variables (such as a + b = b + a); such equations are called "identities". "Conditional" equations are true for only some values of the involved variables: x − 1 = 4. The values of the variables which make the equation true are called the "solutions" of the equation.