Example 1.30. A line is given by the canonical equations

$$
\frac{x-2}{3}=\frac{y+1}{5}=\frac{z-3}{-1} \text {. }
$$

Form the general equations of this line.
Solution. The canonical equations of the line can be written as a system of two independent equations:

$$
\begin{aligned}
& \frac{x-2}{3}=\frac{y+1}{5} \\
& \frac{x-2}{3}=\frac{z-3}{-1}, \text { from which } \quad \begin{array}{r}
5 x-3 y-13=0 \\
x+3 z-11=0
\end{array}
\end{aligned}
$$

We have obtained the general equations of the line, which is now defined by the intersection of two planes, one of which $5 x-3 y-13=0$ is parallel to the $O z$ axis, and the other $x+3 z-11=0$ is parallel to the $O y$ axis.