Example 2 Solve the system of equations:
$$
\left\{\begin{array}{l}
x+y+\frac{9}{x}+\frac{4}{y}=10, \\
\left(x^{2}+9\right)\left(y^{2}+4\right)=24 x y .
\end{array}\right.
$$
(1993, Taiyuan Junior High School Mathematics Competition)