We have seen in this section that systems of linear equations have limited possibilities for solution sets, and we will shortly prove Theorem PSSLS that describes these possibilities exactly. This exercise will show that if we relax the requirement that our equations be linear, then the possibilities expand greatly. Consider a system of two equations in the two variables x and y, where the departure from linearity involves simply squaring the variables.
x2 - y2 = 1
x2 + y2 = 4
After solving this system of non-linear equations, replace the second equation in turn by
x2 + 2x + y2 = 3,
x2 + y2 = 1, x2 - 4x + y2 = - 3, - x2 + y2 = 1 and solve each resulting system of two equations in two variables. (This exercise includes suggestions from Don Kreher.)