I'm reading the explanation to why the intersection between two planes is a line in the textbook. This seems reasonable enough, but I don't understand the last part of the proof. This indicates that my understandingn of parametric equations may be flawed, which might come back to haunt me later.

According to the book we start with the equations of two planes, which we write as augmented matrices, so:

\begin{array}{rrr|r}
 A\_1 & B\_1 & C\_1 & -D\_1 \\
 A\_2 & B\_2 & C\_2 & -D\_2 
 \end{array}

Through gaussian elimination (omitting some steps) this can be reduced to:

\begin{array}{rrr|r}
 0 & 1 & a & b \\
 1 & 0 & f & g 
 \end{array}

Which would mean that in the intersection of the two planes, the following would apply:

$$y+az=b$$
$$x+fz=g$$

The text then goes on to claim that

(1) z can be treated as a paramater (even though it should be a variable, I've never encountered such paramaters, what would that even mean?) and
(2) that because of this, we could write that:

$$x=g-ft$$
$$y=b-at$$
$$z=t$$

It then goes on to clame (3) that because of this, the solution is a three dimensional line.

Can someone explain these three steps in a little more detail? Or maybe point me to a helpful guide somewhere?