Example 6 Solve the system of equations
$$
\left\{\begin{array}{l}
x-5 y+18 \sqrt{2 y}=20, \\
6 \sqrt{2 x}-x-5 y=11 .
\end{array}\right.
$$
【Analysis】The basic method for handling irrational equations (systems) is rationalization. First, add and subtract the two equations, then multiply the resulting two equations, and use the difference of squares formula to achieve rationalization. After factoring by grouping, the solution can be completed.
Solving by (2)+(1) and (2)-(1) respectively, we get
$$
\left\{\begin{array}{l}
6 \sqrt{2 x}+18 \sqrt{2 y}=10 y+31, \\
6 \sqrt{2 x}-18 \sqrt{2 y}=2 x-9 .
\end{array}\right.
$$

Multiplying the two equations, we get
$$
\begin{array}{l}
6^{2} \times 2 x-18^{2} \times 2 y \\
=20 x y+62 x-90 y-9 \times 31,
\end{array}
$$

which simplifies to $20 x y-10 x+558 y-279=0$.
Thus, $(10 x+279)(2 y-1)=0$.
Since $x, y \geqslant 0$, we have $10 x+279>0$.
Therefore, $y=\frac{1}{2}$, and subsequently, $x=\frac{9}{2}$.
Upon verification, the original system of equations has a unique solution
$$
(x, y)=\left(\frac{9}{2}, \frac{1}{2}\right) \text {. }
$$