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.