Example 2 At the end of the 5th century AD, the Chinese mathematician Zhang Qiujian proposed a famous "Hundred Chickens Problem" in the history of world mathematics in his renowned work "The Mathematical Classic": "A rooster is worth five coins, a hen is worth three coins, and three chicks are worth one coin. With a hundred coins, one can buy a hundred chickens. How many roosters, hens, and chicks are there?"
Solution: Let the number of roosters, hens, and chicks be $x, y, z$ respectively. According to the problem, we have $\left\{\begin{array}{l}x+y+z=100, \\ 5 x+3 y+\frac{z}{3}=100 .\end{array}\right.$ Eliminating $z$ from these equations yields $7 x+4 y=100, \because(0,25)$ is a particular solution,
$$
\therefore x=-4 n, y=25+7 n, z=100-x-y=
$$
$75-3 n$, and since $x, y, z \in \mathbf{N}$, we get $n=0,-1,-2,-3$. Substituting these values into the general solution, we obtain 4 sets of solutions $(x, y, z)$: $(0,25,75)$, $(4,18,78)$, $(8,11,81)$, $(12,4,84)$.