Editing Equation solving (section)

==Solution sets==
[[File:Ellipse in coordinate system with semi-axes labelled.svg|thumb|The solution set of the equation {{math|1={{sfrac|''x''<sup>2</sup>|4}} + ''y''<sup>2</sup> = 1}} forms an [[ellipse]] when interpreted as a set of [[Cartesian coordinate]] pairs.]]
{{Main|Solution set}}
The [[solution set]] of a given set of equations or [[inequality (mathematics)|inequalities]] is the [[set (mathematics)|set]] of all its solutions, a solution being a [[tuple]] of values, one for each [[unknown (mathematics)|unknown]], that satisfies all the equations or inequalities.
If the [[solution set]] is empty, then there are no values of the unknowns that satisfy simultaneously all equations and inequalities.

For a simple example, consider the equation 
:<math>x^2=2.</math>
This equation can be viewed as a [[Diophantine equation]], that is, an equation for which only [[integer]] solutions are sought. In this case, the solution set is the [[empty set]], since 2 is not the [[square (algebra)|square]] of an integer. However, if one searches for [[real number|real]] solutions, there are two solutions, {{math|{{radic|2}}}}  and {{math|–{{radic|2}}}}; in other words, the solution set is {{math|{{mset|{{radic|2}}, −{{radic|2}}}}}}.

When an equation contains several unknowns, and when one has several equations with more unknowns than equations, the solution set is often infinite. In this case, the solutions cannot be listed. For representing them, a [[parametrization (geometry)|parametrization]] is often useful, which consists of expressing the solutions in terms of some of the unknowns or auxiliary variables. This is always possible when all the equations are [[linear equation|linear]].

Such infinite solution sets can naturally be interpreted as [[geometry|geometric]] shapes such as [[line (geometry)|lines]], [[curve (geometry)|curves]] (see picture), [[plane (geometry)|planes]], and more generally [[algebraic variety|algebraic varieties]] or [[manifold]]s. In particular, [[algebraic geometry]] may be viewed as the study of solution sets of [[algebraic equation]]s.