Find all solutions to this system of equations

$\displaystyle 3x^2y-3y=0$ (i)

$\displaystyle x^2-3x-2y=0$ (ii)

I take

3(ii) – 2(i) = 0

=> $\displaystyle 6x^2y-6y=3x^3-9x$ (2)

=> $\displaystyle 6y(x^2-1)=3x(x^2-3)$ (3)

Therefor, I have these points.

(0,0)

(±√3, 0)

But x=±1 is also a solution. But that doesn't satisfy the right hand side in (3). I could have seen x=±1 and put it in (i) and (ii). But I didn't see that. Do I have a bad, nonsystematic method for solving this system of equation? How do I do it better?