Solving equation in three variables please help me understand how the following equation with 3 variables and power of 2 is solved and what solution approach is the quickest.  
$$3y^2 - 3 = 0$$
$$4x - 3z^2 = 0$$
$$-6xz+ 6z = 0 $$

A quick solution:
By simplification, rewrite
$$\begin{cases}y^2=1\\(1-x)z=0\\3z^2=4x.\end{cases}$$
Then mentally,
$$y=\pm1,\\x=z=0\lor x=1,z=\pm\frac2{\sqrt3}.$$
