An equation is analogous to a scale into which weights are placed. When equal weights of something (e.g., grain) are placed into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, an equal amount of grain must be removed from the other pan to keep the scale in balance. More generally, an equation remains in balance if the same operation is performed on its both sides.

An identity is an equation that is true for all possible values of the variable(s) it contains. Many identities are known in algebra and calculus. In the process of solving an equation, an identity is often used to simplify an equation, making it more easily solvable.

In algebra, an example of an identity is the difference of two squares:

where θ is limited to between 0 and 45 degrees, one may use the above identity for the product to give:

Two equations or two systems of equations are equivalent, if they have the same set of solutions. The following operations transform an equation or a system of equations into an equivalent one – provided that the operations are meaningful for the expressions they are applied to:

is a univariate algebraic (polynomial) equation with integer coefficients and

since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than individually.

In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.

Equations can be classified according to the types of operations and quantities involved. Important types include: