I can understand that the following are not the equations of a line, because both the equations refers to the plane $\pi : x - y + z = 2$:
$ \begin{cases} x - y + z = 2 \\ 3x - 3y + 3z = 6 \end{cases} $
But why the following cartesian equations does not represent a line?
$ \begin{cases} x - y + z = 2 \\ 3x - 3y + 3z = 1 \end{cases} $
We are on $\Bbb R^3$, so a line is describe by two equations
$$ \begin{cases} a_{11}x+a_{12}y+a_{13}z=b_1\\ a_{21}x+a_{22}y+a_{23}z=b_2 \end{cases} $$
such that let $A$ the matrix of coefficients
$$ rk A=rk \begin{pmatrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23} \end{pmatrix}=2. $$
In your case the matrix $A$ has rank one.
Moreover if $rkA=1$ and $rkA|b=1$ then the planes coincide and if $rkA=1$, and $rkA|b=2$ then the planes are parallel.