Example 7 Given
$$
\begin{array}{l}
\frac{x^{2}}{2^{2}-1^{2}}+\frac{y^{2}}{2^{2}-3^{2}}+\frac{z^{2}}{2^{2}-5^{2}}+\frac{w^{2}}{2^{2}-7^{2}}=1, \\
\frac{x^{2}}{4^{2}-1^{2}}+\frac{y^{2}}{4^{2}-3^{2}}+\frac{z^{2}}{4^{2}-5^{2}}+\frac{w^{2}}{4^{2}-7^{2}}=1, \\
\frac{x^{2}}{6^{2}-1^{2}}+\frac{y^{2}}{6^{2}-3^{2}}+\frac{z^{2}}{6^{2}-5^{2}}+\frac{w^{2}}{6^{2}-7^{2}}=1, \\
\frac{x^{2}}{8^{2}-1^{2}}+\frac{y^{2}}{8^{2}-3^{2}}+\frac{z^{2}}{8^{2}-5^{2}}+\frac{w^{2}}{8^{2}-7^{2}}=1 .
\end{array}
$$

Find the value of $x^{2}+y^{2}+z^{2}+w^{2}$.
Analysis: If we consider the four known equations as a system of four equations in $x, y, z, w$, solving this system to find the values of $x, y, z, w$ would be quite difficult. However, if we view these four equations as a whole and temporarily treat $x, y, z, w$ as known numbers, then these four equations can be transformed into a single equation in $t$ with roots $2^{2}, 4^{2}, 6^{2}, 8^{2}$:
$$
\frac{x^{2}}{t-1^{2}}+\frac{y^{2}}{t-3^{2}}+\frac{z^{2}}{t-5^{2}}+\frac{w^{2}}{t-7^{2}}=1.
$$

At the same time, according to the problem's requirements, we only need to find the value of $x^{2}+y^{2}+z^{2}+w^{2}$, so there is no need to find the values of $x, y, z, w$ individually, but only to consider their overall sum $x^{2}+y^{2}+z^{2}+w^{2}$.
Rearranging equation (1) yields:
$$
\begin{array}{l}
t^{4}-\left(x^{2}+y^{2}+z^{2}+w^{2}+84\right) t^{3}+a_{2} t^{2} \\
+a_{3} t+a_{4}=0.
\end{array}
$$

Since $2^{2}, 4^{2}, 6^{2}, 8^{2}$ are the roots of equation (1), we have:
$$
\left(t-2^{2}\right)\left(t-4^{2}\right)\left(t-6^{2}\right)\left(t-8^{2}\right)=0.
$$

Rearranging gives:
$$
\begin{array}{l}
t^{4}-\left(2^{2}+4^{2}+6^{2}+8^{2}\right) t^{3}+a_{2} t^{2}+a_{3} t \\
+a_{4}=0.
\end{array}
$$

Comparing the coefficients of $t^{3}$ in equations (2) and (3) yields:
$$
x^{2}+y^{2}+z^{2}+w^{2}+84=2^{2}+4^{2}+6^{2}+8^{2}.
$$

Solving for $x^{2}+y^{2}+z^{2}+w^{2}$ gives $x^{2}+y^{2}+z^{2}+w^{2}=36$.