Example 10 Find all non-negative solutions of $15 x_{1}+10 x_{2}+6 x_{3}=61$.

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From Example 6, we know the general solution formula is
$$x_{1}=5+2 t_{1}+6 t_{2}, \quad x_{2}=-5-3 t_{1}-6 t_{2}, \quad x_{3}=6-5 t_{2}.$$

Therefore, the non-negative solutions for $t_{1}, t_{2}$ are
$$5+2 t_{1}+6 t_{2} \geqslant 0, \quad-5-3 t_{1}-6 t_{2} \geqslant 0, \quad 6-5 t_{2} \geqslant 0$$

From this, we get
$$-5 / 3-2 t_{2} \geqslant t_{1} \geqslant-5 / 2-3 t_{2}, \quad t_{2} \leqslant 6 / 5$$

Thus, we have
$$-5 / 6 \leqslant t_{2} \leqslant 6 / 5$$

Therefore, $t_{2}=0,1$. It is easy to calculate that when $t_{2}=0$, $t_{1}=-2$; when $t_{2}=1$, $t_{1}=-4,-5$. From the general solution formula, we can find all non-negative solutions are
$$1,1,6 ; 3,1,1 ; \quad 1,4,1$$

The same results can also be obtained from the general solution formula derived in Example 5.