Example 8 Solve the system of equations in the set of real numbers
$$\left\{\begin{array}{l}
x^{2}+y^{2}+z^{2}=\frac{9}{4} \\
-8 x+6 y-24 z=39 .
\end{array}\right.$$
Solution: By the Cauchy-Schwarz inequality, we have $\left(x^{2}+y^{2}+z^{2}\right)\left[(-8)^{2}+6^{2}+(-24)^{2}\right] \geqslant(-8 x+6 y-24 z)^{2}$. Note that the left side of the above equation $=\frac{9}{4} \times(64+36+4 \times 144)=39^{2}$, and $(-8 x+6 y-24 y)^{2}=39^{2}$. By the condition for equality in the Cauchy-Schwarz inequality, we get $\frac{x}{-8}=\frac{y}{6}=\frac{z}{-24}$. Let this ratio be $k$, and by solving the system of equations, we can find $x=-\frac{6}{13}, y=\frac{9}{26}, z=-\frac{18}{13}$.