Example 1. Find the integer solutions for $4 x+7 y+14 z=40$.
$$
\begin{array}{l}
\text { Solve } x=\frac{1}{4}(40-7 y-14 z) \\
=10-y-3 z-\frac{1}{4}\left(3 y+2 z\right) . \\
\text { Let } \frac{3 y+2 z}{4}=u \quad(u \in Z) \text {, then } \\
2 z=4 u - 3 y \text {, } \\
z=2 u-y-\frac{y}{2} \text {. } \\
\text { Let } y=2 t \quad(t \in Z) \text {. } \\
\end{array}
$$

Substitute back into the above equations:
$$
\left\{\begin{array}{l}
x=10-7 u+7 t, \\
y=2 t, \\
z=2 u-3 t .
\end{array} \quad(\| 11, u \in Z)\right.
$$