Example 2 As shown in Figure 2, the areas of $\triangle A_{1} B_{2} B_{3}$, $\triangle P B_{3} A_{2}$, and $\triangle P A_{3} B_{2}$ are $5, 8, 3$ respectively. Find the area of $\triangle A_{1} A_{2} A_{3}$. (Letters have been changed, the same below)
(1994, Japan Mathematical Olympiad Preliminary Problem)
$$
\begin{array}{c}
\frac{\lambda_{1}}{\mu_{1}}=\frac{\lambda_{0}}{v_{1}}=\frac{S_{\triangle A_{1} A_{2} A_{3}}}{S_{\triangle A_{1} B_{2} B_{3}}} \\
=\frac{\lambda_{1}+\mu_{1}+5+8+3}{5} . \\
\text { Also } \frac{8}{\lambda_{1}}=\frac{\mu_{1}}{3}, \\
\therefore \lambda_{1}=12, \mu_{1}=2, \\
S_{\triangle A_{1} A_{2} A_{3}}=30 .
\end{array}
$$