Example 4. Find the general solution of the equation $12 x+8 y+36 z=100$.

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Solution: For the equation $12 x+8 y+36 z=100$, dividing each term by the greatest common divisor 4 of the coefficients, we get
$$
3 x+2 y+9 z=25 \text {. }
$$

Divide equation (2) into two equations
$$
\begin{array}{l}
3 x+2 y=u \\
u+9 z=25 .
\end{array}
$$

For equation (3), it is not difficult to see that
$$
3 \cdot(u)+2 \cdot(-u)=u \text {. }
$$

Thus, the general solution of equation (3) is
(I) $\left\{\begin{array}{l}x=u+2 t_{1}, \\ y=-u-3 t_{1}\end{array}\right.$.
For equation (4), similarly,
$$
1 \times(-2)+9 \times(3)=25 \text {. }
$$

Thus, the general solution of equation (4) is
(1) $\left\{\begin{array}{l}u=-2+9 t_{2}, \\ z=3-t_{2} .\end{array}\right.$
By eliminating $u$ from (I) and (1), we get
$$
\left\{\begin{array}{l}
x=-2+2 t_{1}+9 t_{2}, \\
y=2-3 t_{1}-9 t_{2}, \\
z=3-t_{2} .
\end{array}\right.
$$

Here, $t_{1}, t_{2}$ can take any integer.