3.8. Find the equation of the line passing through the point $P(2, -3)$ and the intersection point of the lines $3x + 2y - 4 = 0$ and $x - y + 5 = 0$.
Solution.

Let's use the equation (3.21) of the pencil of lines $3 x+2 y-4+\lambda(x-y+5)=0$. Since the desired line passes through the point $P(2, -3)$, we can find the parameter $\lambda$ by substituting the coordinates of this point into the equation of the pencil of lines:

$$
3 \cdot 2 + 2 \cdot (-3) - 4 + \lambda(2 + 3 + 5) = 0, \lambda = 0.4
$$

The desired equation is obtained by substituting the found $\lambda$ into the equation of the pencil of lines. We have $3 x + 2 y - 4 + 0.4 \cdot (x - y + 5) = 0, 3.4 x + 1.6 y - 2 = 0$.