Example 1 Given two lines $A_{1} x+B_{1} y=1$ and $A_{2} x+B_{2} y=1$ intersect at point $P(-7,9)$, find the equation of the line passing through points $P_{1}\left(A_{1}, B_{1}\right), P_{2}\left(A_{2}, B_{2}\right)$.
This indicates that points $P_{1}\left(A_{1}, B_{1}\right)$ and $P_{2}\left(A_{2}, B_{2}\right)$ both satisfy the equation $-7 x+9 y=1$, and the equation of a line passing through two points is unique. Therefore, the required line equation is $-7 x+9 y=1$.
Solution 2 From (1) and (2), we get $k=\frac{B_{2}-B_{1}}{A_{2}-A_{1}}=\frac{7}{9}$.
Using the point-slope form, we have $y-B_{1}=\frac{7}{9}\left(x-A_{1}\right),-7 x+9 y=-7 A_{1}+9 B_{1}=1$.