Convert a line in $ \Bbb R^3 $ given as intersection of two planes to parametric form. 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. 
$
 A: 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.
A: 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} 
$
A: 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.
