Editing Equation solving (section)

==Overview==
One general form of an equation is
:<math>f\left(x_1,\dots,x_n\right)=c,</math>
where {{mvar|f}} is a [[function (mathematics)|function]], {{math|''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>}} are the unknowns, and {{math|''c''}} is a constant. Its solutions are the elements of the [[inverse image]] ([[Fiber (mathematics)|fiber]])
:<math>f^{-1}(c)=\bigl\{(a_1,\dots,a_n)\in D\mid f\left(a_1,\dots,a_n\right)=c\bigr\},</math>
where {{math|''D''}} is the [[Domain of a function|domain]] of the function {{mvar|f}}. The set of solutions can be the [[empty set]] (there are no solutions), a [[singleton (mathematics)|singleton]] (there is exactly one solution), finite, or infinite (there are infinitely many solutions).

For example, an equation such as
:<math>3x+2y=21z,</math>
with unknowns {{math|''x'', ''y''}} and {{math|''z''}}, can be put in the above form by subtracting {{math|21''z''}} from both sides of the equation, to obtain
:<math>3x+2y-21z=0</math>

In this particular case there is not just ''one'' solution, but an infinite set of solutions, which can be written using [[set builder notation]] as
:<math>\bigl\{(x,y,z)\mid 3x+2y-21z=0\bigr\}.</math>

One particular solution is {{math|1=''x'' = 0, ''y'' = 0, ''z'' = 0}}. Two other solutions are {{math|1=''x'' = 3, ''y'' = 6, ''z'' = 1}}, and {{math|1=''x'' = 8, ''y'' = 9, ''z'' = 2}}. There is a unique [[plane (geometry)|plane]] in [[three-dimensional space]] which passes through the three points with these [[coordinates]], and this plane is the set of all points whose coordinates are solutions of the equation.