Example 9 (1986 National High School Mathematics League Test Question) If $a$, $b, c \in \mathbf{R}$, and $a, b, c$ satisfy the system of equations
$$\left\{\begin{array}{l}
a^{2}-b c-8 a+7=0 \\
b^{2}+c^{2}+b c-6 a+6=0
\end{array}\right.$$

Try to find the range of values for $a$.
Solve: From (1) + (2), we get $b^{2}+c^{2}=-a^{2}+14 a-13$. From (2) - (1), we get $b+c= \pm(a-1)$, so by the corollary, $2(-$ $\left.a^{2}+14 a-13\right) \geqslant[ \pm(a-1)]^{2}$, solving this yields $1 \leqslant a \leqslant 9$, hence the range of values for $a$ is $1 \leqslant a \leqslant 9$.