Editing Equation solving (section)

{{short description|Finding values for variables that make an equation true}}
{{one source|date=December 2009}}
{{redirect|Solution (mathematics)|solutions of constraint satisfaction problems|Constraint satisfaction problem#Resolution|solutions of mathematical optimization problems|Feasible solution}}
{{Image frame|width=220|align=right|caption=The [[quadratic formula]], the symbolic solution of the [[quadratic equation]] {{math|1=''ax''<sup>2</sup> + ''bx'' + ''c'' = 0}}
|content=<math>\overset{}{\underset{}{ x=\frac{-b\pm\sqrt{b^2-4ac} }{2a} } }</math>}}
[[Image:NewtonIteration Ani.gif|alt=Illustration of Newton's method|thumb|An example of using [[Newton–Raphson method]] to solve numerically the equation {{math|1=''f''(''x'') = 0}}]]
In [[mathematics]], to '''solve an equation''' is to find its '''solutions''', which are the values ([[number]]s, [[function (mathematics)|functions]], [[Set (mathematics)|sets]], etc.) that fulfill the condition stated by the [[equation]], consisting generally of two [[expression (mathematics)|expression]]s related by an [[equals sign]]. When seeking a solution, one or more [[variable (mathematics)|variable]]s are designated as ''[[Indeterminate (variable)|unknowns]]''. A solution is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values (one for each unknown) such that, when [[Substitution (algebra)|substituted]] for the unknowns, the equation becomes an [[equality (mathematics)|equality]].
A solution of an equation is often called a '''root''' of the equation, particularly but not only for [[polynomial equation]]s. The set of all solutions of an equation is its [[solution set]].

An equation may be solved either [[Numerical mathematics|numerically]] or symbolically. Solving an equation ''numerically'' means that only numbers are admitted as solutions. Solving an equation ''symbolically'' means that expressions can be used for representing the solutions.

For example, the equation {{math|1=''x'' + ''y'' = 2''x'' – 1}} is solved for the unknown {{mvar|x}} by the expression {{math|1=''x'' = ''y'' + 1}}, because substituting {{math|''y'' + 1}} for {{math|''x''}} in the equation results in {{math|1=(''y'' + 1) + ''y'' = 2(''y'' + 1) – 1}}, a true statement. It is also possible to take the variable {{math|''y''}} to be the unknown, and then the equation is solved by {{math|1=''y'' = ''x'' – 1}}. Or {{math|''x''}} and {{math|''y''}} can both be treated as unknowns, and then there are many solutions to the equation; a symbolic solution is {{math|1=(''x'', ''y'') = (''a'' + 1, ''a'')}}, where the variable {{mvar|a}} may take any value. Instantiating a symbolic solution with specific numbers gives a numerical solution; for example, {{math|1=''a'' = 0}} gives {{math|1=(''x'', ''y'') = (1, 0)}} (that is, {{math|1=''x'' = 1, ''y'' = 0}}), and {{math|1=''a'' = 1}} gives {{math|1=(''x'', ''y'') = (2, 1)}}. 

The distinction between known variables and unknown variables is generally made in the statement of the problem, by phrases such as "an equation ''in'' {{mvar|x}} and {{mvar|y}}", or "solve ''for'' {{math|''x''}} and {{math|''y''}}", which indicate the unknowns, here {{math|''x''}} and {{math|''y''}}.
However, it is common to reserve {{mvar|x}}, {{mvar|y}}, {{mvar|z}}, ... to denote the unknowns, and to use {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, ... to denote the known variables, which are often called [[parameter]]s. This is typically the case when considering [[polynomial equation]]s, such as [[quadratic equation]]s. However, for some problems, all variables may assume either role.

Depending on the context, solving an equation may consist to find either any solution (finding a single solution is enough), all solutions, or a solution that satisfies further properties, such as belonging to a given [[interval (mathematics)|interval]]. When the task is to find the solution that is the ''best'' under some criterion, this is an [[optimization problem]]. Solving an optimization problem is generally not referred to as "equation solving", as, generally, solving methods start from a particular solution for finding a better solution, and repeating the process until finding eventually the best solution.