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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). When searching 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 problem of solving an equation may be numeric or symbolic. Solving an equation numerically means that only numbers represented literately (not as a combination of 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 is solved for the unknown ''x'' by the solution , because substituting for ''x'' in the equation results in , a true statement. It is also possible to take the variable ''y'' to be the unknown, and then the equation is solved by . Or ''x'' and ''y'' can both be treated as unknowns, and then there are many solutions to the equation. is a symbolic solution. Instantiating a symbolic solution with specific numbers always gives a numerical solution; for example, gives (that is, and ) and gives . 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 is also a parametrization of the solution set with the parameter being . 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''. ==Overview== In one general case, we have a situation such as :''ƒ'' (''x''1,...,''x''''n'') = ''c'', where ''x''1,...,''x''''n'' are the unknowns, and ''c'' is a constant. Its solutions are the members of the inverse image :''ƒ'' −1() = , where ''T''1×···×''T''''n'' is the domain of the function ''ƒ''. Note that the set of solutions can be the empty set (there are no solutions), a singleton (there is exactly one solution), finite, or infinite (there are infinitely many solutions). For example, an equation such as :3''x'' + 2''y'' = 21''z'' with unknowns ''x'', ''y'' and ''z'', can be solved by first modifying the equation in some way while keeping it equivalent, such as subtracting 21''z'' from both sides of the equation to obtain :3''x'' + 2''y'' − 21''z'' = 0 In this particular case there is not just ''one'' solution to this equation, but an infinite set of solutions, which can be written :. One particular solution is ''x'' = 0, ''y'' = 0, ''z'' = 0. Two other solutions are ''x'' = 3, ''y'' = 6, ''z'' = 1, and ''x'' = 8, ''y'' = 9, ''z'' = 2. In fact, this particular set of solutions describes a ''plane'' in three-dimensional space, which passes through the three points with these coordinates. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Equation solving」の詳細全文を読む スポンサード リンク
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