We have a line in $ \Bbb R^3 $ given as intersetion of two planes: $$ \left\{ \begin{aligned} A_1x+B_1y+C_1z + D_1 &=0 \\ A_2x+B_2y+C_2z + D_2 &=0 \\ \end{aligned} \right. $$ How to represent it in parametric form: $$ \left\{ \begin{aligned} x &= x_0 +at\\ y &= y_0 +bt \\ z &= z_0 +ct \\ \end{aligned} \right. $$ ?
example I'm doing is:
$
l:
\left\{
\begin{aligned}
x + y - z + 1 &=0 \\
x - y + z - 1 &=0 \\
\end{aligned}
\right.
$
What you look for is a point on the line $\left(x_0, y_0, z_0\right)$, and a direction vector of the line $\left(a, b, c\right)$.
To find a point on the line, you can for example fix $x$ and find $y,z$ (there are some case where this won't work).
To find a direction vector, note that the vector $\left(A_1,B_1,C_1\right)$ is orthogonal to the first plane therefore to the line. Likewise, $\left(A_2,B_2,C_2\right)$ is orthogonal to the line. If you take their vector product you will get a direction vector.
Another way to find a direction vector, is to find another point on the line and subtract one point from the other.
Row reduce the system of equations. Generically, you will get an equation with two variables and one with three. There is going to be a repeated variable in both, that is your parameter. Write the remaining variables in terms of this common variable and you are done.
In your example:
$ l: \left\{ \begin{aligned} x + y - z + 1 &=0 \\ x - y + z - 1 &=0 \\ \end{aligned} \right. $
Add the first equation to the last to get:
$ l: \left\{ \begin{aligned} x + y - z + 1 &=0 \\ x &=0 \\ \end{aligned} \right. $
We can take $y$ or $z$ as the parameter. For example let's take $z$. Then
$ \begin{aligned} x&=0+0z\\ y&=-1+z\\ z&=0+z \end{aligned} $
The Maple command $$ solve(\{A_1x+B_1y+C_1z+D_1=0,A_2x+B_2y+C_2z+D_2=0\},\{x,y\}) $$ produces the answer $$ \left\{ x={\frac {{\it B_1}\,{\it C_2}\,z-{\it B_2}\,{\it C_1}\,z+{\it B_1 }\,{\it D_2}-{\it D_1}\,{\it B_2}}{{\it A_1}\,{\it B_2}-{\it A_2}\,{\it B_1}} },y=-{\frac {{\it A_1}\,{\it C_2}\,z-{\it A_2}\,{\it C_1}\,z+{\it A_1}\,{ \it D_2}-{\it A_2}\,{\it D_1}}{{\it A_1}\,{\it B_2}-{\it A_2}\,{\it B_1}}} \right\} .$$ The command $$ solve(\{A_1x+B_1y+C_1z+D_1=0,A_2x+B_2y+C_2z+D_2=0\},\{x,y\},parametric=full) $$ investigates all possibilities depending on the coefficients.
