Given the real function [tex]f(x,y) = x^2 - 4xy + 3y^2[/tex], the equation f(x,y) = 0 shows in a graph as 2 straight lines, y=x and y=x/3. For pairs (x,y) between the lines, f(x,y) < 0; for (x,y) outside the lines, f(x,y) > 0. It is easy to prove the above, by substituting y=mx in the equation f(x,y)=0 and finding the values of m (giving either 1 or 1/3). All the behavior above follows by playing with m. The question is, why straight lines? What's the theory for choosing y=mx as the appropriate substitution?