Example 3. As shown in the figure, $A D, B E, C F$ intersect at a point $P$ inside $\triangle A B C$, dividing $\triangle A B C$ into six smaller triangles, with the areas of four of these smaller triangles already given in the figure. Find the area of $\triangle A B C$.
Let the unknown areas of the two smaller triangles be $x$ and $y$, then
$$
\frac{B D}{D C}=\frac{40}{30}=\frac{84+x}{70+y},
$$

i.e., $\frac{84+x}{70+y}=\frac{4}{3}$.
Also, $\frac{A E}{E C}=\frac{70}{y}=\frac{84+x}{40+30}$.

i.e., $\frac{84+x}{70}=\frac{70}{y}$.
Dividing (1) by (2), we get
$$
\frac{70}{70+y}=\frac{4}{3} \cdot \frac{y}{70}, y=35 .
$$

From (2), we get $x=56$.
Therefore, $S_{\triangle A B C}=315$.