4. Find the integer solutions of $11 x+15 y=7$
Solution 1: Transform the equation, we get
$$
x=\frac{7-15 y}{11} \text{. }
$$

Since $x$ is an integer, $7-15y$ must be a multiple of 11. By observation, we find that $x_{0}=2, y_{0}=-1$ is a set of integer solutions to this equation.

Therefore, by Theorem 3, the integer solutions to the equation $11 x+15 y=7$ are $\left\{\begin{array}{l}x=2-15 t, \\ y=-1+11 t .\end{array}\right.$ ( $t$ is an integer)
Solution 2: First consider $11 x+15 y=1$.

By observation, it is easy to get
$$
\begin{array}{l}
11 \times(-4)+15 \times(3)=1 . \\
\therefore \quad 11 \times(-4 \times 7)+15 \times(3 \times 7)=7 . \\
\quad x_{0}=-28, y_{0}=21 .
\end{array}
$$

Therefore, $\left\{\begin{array}{l}x=-28-15 t, \\ y=21+11 t .\end{array}\right.$ ( $t$ is an integer)