Example 6 Given three non-negative numbers $a$, $b$, $c$ satisfying $3a + 2b + c = 5$ and $2a + b - 3c = 1$. If $m = 3a + b - 7c$, then the minimum value of $m$ is $\qquad$, and the maximum value of $m$ is $\qquad$
(2000, 14th Junior High School Mathematics Competition in Jiangsu Province)
Given the conditions, we have
$$
a=7 c-3, b=7-11 c \text { . }
$$

Substituting into \( m=3 a+b-7 c \), we get \( m=3 c-2 \).
Since \( a, b, c \) are non-negative,
$$
\therefore\left\{\begin{array}{l}
a=7 c-3 \geqslant 0, \\
b=7-11 c \geqslant 0, \\
c \geqslant 0 .
\end{array} \text { Solving, we get } \frac{3}{7} \leqslant c \leqslant \frac{7}{11} .\right.
$$

Then, \( -\frac{5}{7} \leqslant m \leqslant -\frac{1}{11} \).
Thus, the minimum value of \( m \) is \( -\frac{5}{7} \), and the maximum value is \( -\frac{1}{11} \).