Editing Equation (section)

==Properties==

Two equations or two systems of equations are ''equivalent'', if they have the same set of solutions. The following operations transform an equation or a system of equations into an equivalent one – provided that the operations are meaningful for the expressions they are applied to:
* [[addition|Adding]] or [[subtraction|subtracting]] the same quantity to both sides of an equation. This shows that every equation is equivalent to an equation in which the right-hand side is zero.
* [[Multiplication|Multiplying]] or [[division (mathematics)|dividing]] both sides of an equation by a non-zero quantity.
* Applying an [[identity (mathematics)|identity]] to transform one side of the equation. For example, [[polynomial expansion|expanding]] a product or [[factorization of polynomials|factoring]] a sum.
* For a system: adding to both sides of an equation the corresponding side of another equation, multiplied by the same quantity.

If some [[function (mathematics)|function]] is applied to both sides of an equation, the resulting equation has the solutions of the initial equation among its solutions, but may have further solutions called [[extraneous solution]]s. For example, the equation <math>x=1</math> has the solution <math>x=1.</math> Raising both sides to the exponent of 2 (which means applying the function <math>f(s)=s^2</math> to both sides of the equation) changes the equation to <math>x^2=1</math>, which not only has the previous solution but also introduces the extraneous solution, <math>x=-1.</math> Moreover, if the function is not defined at some values (such as 1/''x'', which is not defined for ''x'' = 0), solutions existing at those values may be lost. Thus, caution must be exercised when applying such a transformation to an equation.

The above transformations are the basis of most elementary methods for [[equation solving]], as well as some less elementary ones, like [[Gaussian elimination]].