So I wanted to find the line of intersection for the two planes $x+3y=7$ and $2y+z = 4$.
So for $z = t$ I get:
$x= 7-(3(4-t))/2$
and
$y = (4-t)/2$
But I want to know how I go from this to the vector equation of the line since that one is easier to understand intuitively.
the one on the form $(x,y,z) = (x_0,y_0,z_0) + t (a,b,c)$.
where $(x_0,y_0,z_0)$ is the starting position (vector) and $(a,b,c)$ is a direction vector of the line.
to find out the starting point of the vector, just put any value of t in the equation you found out.
next to find out the direction ratios just compare the coefficient of t in the equation of x and y.(for z the direction ratio will be one)