Simple systems with a second-degree equation

A solution of a system in two unknowns is a pair (x, y) that turns both equations into true equalities. If one equation is of the first degree, we express one unknown through the other and substitute into the second equation, which gives a quadratic equation in one unknown. If the system has the form x + y = s, xy = p, then by the converse of Vieta's theorem x and y are the roots of z² − sz + p = 0. If x² − y² = a and x − y = b, we use x² − y² = (x − y)(x + y) to get x + y = a : b and then solve a linear system. Often there are two solutions; it helps to check each in the equations. Such systems appear in problems about shapes with known sides and area.