section_path: Properties
section_name: Properties
paragraph_text: 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:
  
  If some 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 solutions. For example, the equation x = 1 {\displaystyle x=1} has the solution x = 1. {\displaystyle x=1.} Raising both sides to the exponent of 2 (which means applying the function f (s) = s 2 {\displaystyle f(s)=s^{2}} to both sides of the equation) changes the equation to x 2 = 1 {\displaystyle x^{2}=1}, which not only has the previous solution but also introduces the extraneous solution, x = − 1. {\displaystyle x=-1.} 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.