Example 3 Find all positive integer pairs $(x, y)$ that satisfy the equation $2 x^{2}+5 y^{2}=11(x y-11)$.
Solving by transposition, factorization, we get $(2 x-y)(x-5 y)=-121$.
Thus $\left\{\begin{array}{l}2 x-y=-121,-11,-1,1,11,121, \\ x-5 y=1,11,121,-121,-11,-1,\end{array}\right.$ where only the system of equations $\left\{\begin{array}{l}2 x-y=1, \\ x-5 y=-121\end{array}\right.$ yields a positive integer solution $(x, y)=(14,27)$.