Example 6 Solve the system of equations
$$\left\{\begin{array}{l}
x^{2}+y^{2}+z^{2}=\frac{9}{4} \\
-8 x+6 y-24 z=39
\end{array}\right.$$
From (1), we get $x^{2}+z^{2}=\frac{9}{4}-y^{2}$, which can be transformed into
$$\frac{(8 x)^{2}}{8^{2}\left(\frac{9}{4}-y^{2}\right)}+\frac{(24 z)^{2}}{24^{2}\left(\frac{9}{4}-y^{2}\right)}=1$$

By Theorem 1, we have
$$64\left(\frac{9}{4}-y^{2}\right)+576\left(\frac{9}{4}-y^{2}\right) \geqslant(6 y-39)^{2},$$

Simplifying, we get $(26 y-9)^{2} \leqslant 0$, hence $y=\frac{9}{26}$, and substituting back into the original system of equations yields $x=-\frac{6}{13}, z=-\frac{8}{13}$.