Example 1 Solve the indeterminate equation $143 x-77 y=187$.
Since $(143,77)=11$, and $11 \mid 187$, the original equation has integer solutions, and is equivalent to $13 x-7 y=17$.
Taking modulo 7 on both sides of (1) gives $6 x \equiv 3(\bmod 7) \Rightarrow 2 x \equiv 1(\bmod 7)$,
so $x \equiv 4(\bmod 7)$.
We can take $x=4$, substituting into (1) gives $y=5$.
Thus, we obtain a particular solution of (1) as $x=4, y=5$.
By Theorem 2, all integer solutions of the original equation are $\left\{\begin{array}{l}x=4+7 t, \\ y=5+13 t\end{array} \quad(t \in \mathbf{Z})\right.$.