We have seen in this section that systems of linear equations have limited possibilities for solution sets, and we will shortly prove Theorem PSSLS [52] 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.)
Physics
The equation x2 -y2 = 1 has a solution set by itself that has the shape of a hyperbola when plotted. Four of the five different second equations have solution sets that are circles when plotted individually (the last is another hyperbola). Where the hyperbola and circles intersect are the solutions to the system of two equations. As the size and location of the circles vary, the number of intersections varies from four to one (in the order given). The last equation is a hyperbola that"opens" in the other direction. Sketching the relevant equations would be instructive, as was discussed in Example STNE.
need an explanation for this answer? contact us directly to get an explanation for this answer