Consider finding the roots of $(f_1(x,y), f_2(x,y) )=(x^2+4y^2-9,sin(\pi(x+y)))$.
So we want to solve the system $x^2+4y^2-9=0$, $sin(\pi(x+y))=0$. Now, graphically, we have an ellipse for $f_1$ and a sequence of lines (that looks like a shear) for $f_2$, and we consider where they intersect.
My question is, how is this different than solving $x^2+4y^2-9=sin(\pi(x+y))$. If we're finding where they're both zero, then they're both equal to each other, right? Not quite sure why this is confusing me.