Example 6 Solve the equation
$$
\sqrt{2-x}+\sqrt{x+3 y-5}+\sqrt{y+2}=\sqrt{12 y-3} .
$$
By the three-variable mean inequality, we have
$$
\begin{array}{l}
\sqrt{2-x}+\sqrt{x+3 y-5}+\sqrt{y+2} \\
\leqslant 3 \sqrt{\frac{2-x+x+3 y-5+y+2}{3}} \\
=\sqrt{12 y-3} .
\end{array}
$$

The equality holds if and only if $2-x=x+3 y-5=y+2 \geqslant 0$.
Solving this, we get $x=-7, y=7$.
Upon verification, these values satisfy the original equation.