In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equality sign. When seeking a solution, one or more free variables are designated as unknowns. A solution is an assignment of expressions to the unknown variables that makes the equality in the equation true. In other words, a solution is an expression or a collection of expressions (one for each unknown) such that, when substituted for the unknowns, the equation becomes an identity. A solution of an equation is often also called a root of the equation, particularly but not only for algebraic or numerical equations.
A problem of solving an equation may be numeric or symbolic. Solving an equation numerically means that only numbers represented explicitly as numerals (not as an expression involving variables), are admitted as solutions. Solving an equation symbolically means that expressions that may contain known variables or possibly also variables not in the original equation are admitted as solutions.
For example, the equation x + y = 2x – 1 is solved for the unknown x by the solution x = y + 1, because substituting y + 1 for x in the equation results in (y + 1) + y = 2(y + 1) – 1, a true statement. It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1. Or x and y can both be treated as unknowns, and then there are many solutions to the equation. (x, y) = (a + 1, a) is a symbolic solution. Instantiating a symbolic solution with specific numbers always gives a numerical solution; for example, a = 0 gives (x, y) = (1, 0) (that is, x = 1 and y = 0) and a = 1 gives (x, y) = (2, 1). Note that the distinction between known variables and unknown variables is made in the statement of the problem, rather than the equation. However, in some areas of mathematics the convention is to reserve some variables as known and others as unknown. When writing polynomials, the coefficients are usually taken to be known and the indeterminates to be unknown, but depending on the problem, all variables may assume either role.
Depending on the problem, the task may be to find any solution (finding a single solution is enough) or all solutions. The set of all solutions is called the solution set. In the example above, the solution (x, y) = (a + 1, a) is also a parametrization of the solution set with the parameter being a. It is also possible that the task is to find a solution, among possibly many, that is best in some respect; problems of that nature are called optimization problems; solving an optimization problem is generally not referred to as "equation solving".
A wording such as "an equation in x and y", or "solve for x and y", implies that the unknowns are as indicated: in these cases x and y.