40. Write the general equation of a line passing through a given point $M$. Write the equation of a line passing through two given points $M$ and $N$. The points are given by coordinates:

$$
M\left(x_{1} ; y_{1}\right), N\left(x_{2} ; y_{2}\right)
$$
40. If the point \( M\left(x_{1} ; y_{1}\right) \) belongs to the line \( A x + B y + C = 0 \), then its coordinates satisfy the equation of this line, i.e.,

\[
A x_{1} + B y_{1} + C = 0
\]

Subtracting the second equation from the first, we get:

\( A\left(x - x_{1}\right) + B\left(y - y_{1}\right) = 0 \) - the general equation of the line passing through the given point.

If the line must also pass through the point \( N\left(x_{2} ; y_{2}\right) \), then its coordinates \( x_{2}, y_{2} \) must also satisfy the obtained equation:

\[
A\left(x_{2} - x_{1}\right) + B\left(y_{2} - y_{1}\right) = 0
\]

6

Thus, we have two relations:

\[
\begin{aligned}
A\left(x - x_{1}\right) & = -B\left(y - y_{1}\right) \\
A\left(x_{2} - x_{1}\right) & = -B\left(y_{2} - y_{1}\right)
\end{aligned}
\]

By dividing one by the other term by term, we get:

\[
\frac{x - x_{1}}{x_{2} - x_{1}} = \frac{y - y_{1}}{y_{2} - y_{1}} \quad \text { or } \quad \frac{x - x_{1}}{x_{2} - x_{1}} - \frac{y - y_{1}}{y_{2} - y_{1}} = 0
\]

This equation defines the line passing through the points \( M \) and \( N \).