The gist of my problem is this: I was wondering if by solving two of the 3 parametric equations for the square root of $t$, which would look like $$ \begin{split} \left(\frac{x - x_1}{a}\right)^{\frac{1}{2}} &= t^{\frac{1}{2}}\text{ and}\\ \left(\frac{y - y_1}{b}\right)^{\frac{1}{2}} &= t^{\frac{1}{2}}, \end{split} $$ and by manipulating the third equation to look like $$ \frac{z - z_1}{c} = t^{\frac{1}{2}}\cdot t^{\frac{1}{2}} , $$ could I substitute the first two equations for $t^{\frac{1}{2}}$ into the third to derive a single equation for a 3d line?
Side note, I apologize for not using the special math notation, as I have yet to become familiar enough with it.