Equation Solving

Equation Solving

In mathematics, to solve an equation is to find what values (numbers, functions, sets, etc.) fulfill a condition stated in the form of an equation (two expressions related by equality). These expressions contain one or more unknowns, which are free variables for which values are sought that cause the condition to be fulfilled. To be precise, what is sought are often not necessarily actual values, but, more in general, mathematical expressions. A solution of the equation is an assignment of expressions to the unknowns that satisfies the equation; in other words, expressions such that, when they are substituted for the unknowns, the equation becomes an identity.

For example, the equation x + y = 2x – 1 is solved for the unknown x by the solution x = y + 1, since 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, some of which are (x, y) = (1, 0) – that is, x = 1 and y = 0 – and (x, y) = (2, 1), and, in general, (x, y) = (a + 1, a) for all possible values of a.

Depending on the problem, the task may be to find one solution – any solution will do – or all solutions. The set of all solutions is called the solution set. 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.