<s>[INST] 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.  
$$3y^2 - 3 = 0$$
$$4x - 3z^2 = 0$$
$$-6xz+ 6z = 0 $$
 [/INST] $$
\begin{cases}
3y^2-3=0\\
4x-3z^2=0\\
6z-6xz=0
\end{cases}\Longleftrightarrow
\begin{cases}
3y^2-3=0\\
x=\frac{3z^2}{4}\\
6z(1-x)=0
\end{cases}\Longleftrightarrow
\begin{cases}
3y^2-3=0\\
x=\frac{3z^2}{4}\\
6z\left(1-\frac{3z^2}{4}\right)=0
\end{cases}
$$
So, when you solve:
$$6z\left(1-\frac{3z^2}{4}\right)=0$$
We get three solutions for $z$, $z=0$ or $z=\pm\frac{2}{\sqrt{3}}$
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