Example 9 Rooster one, worth five coins; hen one, worth three coins; three chicks, worth one coin. A hundred coins buy a hundred chickens. Question: How many roosters and hens are there?
Let $x_{1}, x_{2}, x_{3}$ represent the number of roosters, hens, and chicks, respectively. From the conditions, we can derive the following system of indeterminate equations:
$$\left\{\begin{array}{l}
5 x_{1}+3 x_{2}+x_{3} / 3=100, \\
x_{1}+x_{2}+x_{3}=100 .
\end{array}\right.$$

We need to find the non-negative solutions to this system of indeterminate equations. Eliminating $x_{3}$, we get:
$$7 x_{1}+4 x_{2}=100 \text {. }$$

First, we find the non-negative solutions to this indeterminate equation. $x_{1}=0, x_{2}=25$ is a particular solution. From equation (5) and Theorem 4, all non-negative solutions are:
$$\begin{array}{c}
x_{1}=0+4 t, \quad x_{2}=25-7 t \\
0=-[0 / 4] \leqslant t \leqslant[25 / 7]=3
\end{array}$$

That is, 0, 25; 4, 18; 8, 11; 12, 4. Therefore, the possible scenarios for the number of chickens bought are as follows:
\begin{tabular}{|c|cccc|}
\hline$x_{1}$ & 0 & 4 & 8 & 12 \\
\hline$x_{2}$ & 25 & 18 & 11 & 4 \\
\hline$x_{3}$ & 75 & 78 & 81 & 84 \\
\hline
\end{tabular}