Parametric equations for lines and planes

Question

Let me start by explaining that I have a textbook and an answer guide I found online. Our class is going about a thousand miles an hour and for some reason we didn't cover the specifics of solving these types of problems The question is: Write the parametric equations for each of the following lines and planes. a.) $x_2 = 3x_1 -7$ and the solution is supposed to be $(0, -7) + (1,3)t.$

I will omit $b$ and $c$ but now $d$ moves on to planes:

$x_1 - 2x_2 +3x_3 =6 $ and the answer to this is supposed to be $(6,0,0) + (-6,-3,0)t + (-6,0,2)s$

Answer 1

Let $x_1=t$. Thus, $x_2=-7+3t$ and we got $(0,-7)+t(1,3)$.

By the same way we can get the second equation.

Let $x_2=-3t$ and $x_3=2s$.

Thus, $x_1=6-6t-6s$ and we got $(6,0,0)+(-6,-3,0)t+(-6,0,2)s$.