Level Sets and Roots

We want to solve the system of two equations in two unknowns:

$$\begin{eqnarray*}f(x,y) &=& 0\\ g(x,y) &=& 0 \end{eqnarray*}$$

Taken individually, the set of points that satisfy the first equation is the level set of the function f for the value 0 and the set of points that satisfy the second equation is the level set of g for the value 0. Points that satisfy both equations lie on both the level sets. In other words, the set of solutions to the pair of the equations is the set of points of intersection of the level set of ffor the value 0 and the level set of g for the value 0 . We can use the implicitplot command to plot each of these sets.

    implicitplot({f(x,y)=0,g(x,y)=0},x=-1..3,y=-2..2,
                     scaling=CONSTRAINED,color=red);

We can see from the plot that the level sets intersect at two points, which are located at approximately (.8, .6) and (.8,-.6). Before we go on, note that in this case, since the equations are relatively simple, they can be solved symbolically. The solutions (x,y) are

$$(\frac{-1 + \sqrt{17}}{4}, \frac{\sqrt{-2 + 2\sqrt{17}}}{4})$$

and

$$(\frac{-1 + \sqrt{17}}{4}, -\frac{\sqrt{-2 + 2\sqrt{17}}}{4})$$

which are approximately (.781, .625) and (.781,-625).