Continuous equation of a straight line in the space

If the parametric equations $v_1,v_2$ and $v_3$ are different from $0$, we can isolate the parameter $k$ in all $3$ equations: $$\displaystyle k=\frac{x-a_1}{v_1} \qquad k=\frac{y-a_2}{v_2} \qquad k=\frac{z-a_3}{v_3}$$ In equating the obtained expressions, we have: $$\displaystyle \frac{x-a_1}{v_1} =\frac{y-a_2}{v_2} =\frac{z-a_3}{v_3}$$ which are the continuous equations of the straight line.

Parametric equations of the straight line going through point $A = (-1, 1, 3)$ with $\overrightarrow{v}=(3,-2,1)$ as director vector are: $$\left.\begin{array}{rcl} x &=& -1+3k \\ y&=& 1-2k \\ z&=&3+k\end{array}\right\}$$

Isolating $k$ and by equating: $$\displaystyle \frac{x+1}{3}=\frac{y-1}{-2}=z-3$$ which are the continuous equations of the straight line.