Line equation - parametric and canonical

Question

Let's say I have a line in R3:

$$ l:\begin{cases} x-3y+3z=0\\ x+2y-2z=2 \end{cases} $$ How to change it to canonical and parametric equation?

Answer 1

To get a parametric equation, add an equation: $$ax+by+cz=t$$ Where $t$ is your parameter and $a$, $b$, $c$ are such that the system of three equations is regular. Here it's enough to let $t=y$. Then solve for $x$, $y$, $z$ and you get them as functions of $t$.

Notice that with $t=x$, the system is not regular, since after removing $x$ from the first two equations, you get a system in $y$ and $z$ with null dterminant.

What do you call a canonical equation of the line ?