Example 2. There are three balls, A, B, and C. The volume of ball A is $\frac{1}{2}$ the sum of the volumes of B and C, and the volume of ball B is $\frac{1}{3}$ the sum of the volumes of A and C. What fraction of the sum of the volumes of A and B is the volume of C?
Let the volumes of spheres $\mathrm{A}$, $\mathrm{B}$, and $\mathrm{C}$ be $\mathrm{V}_{1}$, $\mathrm{V}_{2}$, and $\mathrm{V}_{3}$, respectively. Suppose the volume of sphere $\mathrm{C}$ is $-\frac{1}{x}$ times the sum of the volumes of spheres $\mathrm{A}$ and $\mathrm{B}$. Then,
$$
\left\{\begin{array}{l}
\mathrm{V}_{1}=\frac{1}{2}\left(\mathrm{~V}_{2}+\mathrm{V}_{3}\right), \\
\mathrm{V}_{2}=\frac{1}{3}\left(\mathrm{~V}_{1}+\mathrm{V}_{3}\right), \\
\mathrm{V}_{3}=\frac{1}{\mathrm{x}}\left(\mathrm{V}_{1}+\mathrm{V}_{2}\right) .
\end{array}\right.
$$

Thus, $\left\{\begin{array}{l}3 V_{1}=V_{1}+V_{2}+V_{3}, \\ 4 V_{2}=V_{1}+V_{2}+V_{3}, \\ (x+1) V_{3}=V_{1}+V_{2}+V_{3} .\end{array}\right.$
or $\left\{\begin{array}{l}\frac{V_{1}}{V_{1}+V_{2}+V_{3}}=\frac{1}{3}, \\ \frac{V_{2}}{V_{1}+V_{2}+V_{3}}=\frac{1}{4}, \\ \frac{V_{3}}{V_{1}+V_{2}+V_{3}}=\frac{1}{x+1} .\end{array}\right.$
Adding the three equations, we get
$$
\frac{1}{3}+\frac{1}{4}+\frac{1}{x+1}=1 \Longrightarrow \frac{1}{x}=\frac{5}{7} .
$$