Example 1 The scoring rules for a football match are: 3 points for a win, 1 point for a draw, and 0 points for a loss. Then, how many different ways can a team score 19 points in 14 matches?
Solution: Let the team win $x$ games, draw $y$ games, and lose $z$ games. According to the problem, we have $\left\{\begin{array}{l}x+y+z=14, \\ 3 x+y=19 .\end{array}\right.$ Eliminating $y$ from these equations, we get $2 x-z=5$. Since $(0,-5)$ is a particular solution, we have $x=n, z=2 n-5, y=14-x-y=19-3 n$. By $x, y, z \geqslant 0$, we get $n=3,4,5,6$. Substituting these values into the general solution, we obtain 4 sets of solutions. Therefore, there are 4 possible scenarios for a team to score 19 points in 14 games.