Example 1 Given $(5 x+\sqrt{3})-i=3+(\sqrt{2}-y) i, x$ and $y$ are real numbers, find $x$ and $y$.
Solve the system based on the condition of equality of complex numbers:
$$\left\{\begin{array}{l}
5 x+\sqrt{3}=3 \\
-1=\sqrt{2}-y
\end{array}\right.$$

Therefore, $x=\frac{3-\sqrt{3}}{5}, y=1+\sqrt{2}$.