Non-linear systems help!

Question

I have a non-linear system of equations, $$\left\{ \begin{array}{rcl} x^2 - xy + 8 = 0 \\ x^2 - 8x + y = 0 \\ \end{array} \right.$$ I have tried equating the expressions (because both equal 0), which tells me: $$x^2 - xy + 8 = x^2 - 8x + y$$ Moving all expressions to the right yields: $$0 = xy - 8x + y - 8$$ Factoring the equation: $$0 = x(y-8) + 1(y-8)$$ $$0 = (x+1)(y-8)$$ $$x=-1$$ $$y=8$$ Problem solved, right? No. When you plug in the values into the equations above, you get a false statement. Allow me to demonstrate: $$x^2 - 8x + y = 0$$ $$(-1)^2 - 8(-1) + (8) = 0$$ $$1 - (-8) + 8 = 0$$ $$1 + 8 + 8 = 0$$ $$17 = 0$$ Can someone please help me solve this system of nonlinear equations? I am stuck.

Answer 1

Hint: You got $x=-1$ OR $y=8$. BTW, cool idea to equate the expressions.