You are told that the line passes through the points $(a_1, b_1)$ and $(a_2, b_2$.
A straight line can be written with parametric equations $x= r_1+ tv_1$ and and $y= r_2+ tv_2$. There are many different parametric equations describing the same line. We can arbitrarily take t= 0 at $(a_1, b_1)$ and t= 1 at $(a_2, b_2)$.
you need four equations to find the values of $r_1$, $v_1$, $r_2$, and $v_2$. Those four equation are:
$a_1= r_1+ 0v_1= r_1$
$b_1= r_2+ 0v_2= r_2$
$a_2= r_1+ 1v_1= r_1+ v_1$ and
$b_2= r_2+ 1v_2= r_2+ v_2$.
From the first two equations, $r_1= a_1$ and $r_2= a_2$. The next two equations give
$a_2= r_1+ v_1= a_1+ v_1$ so $v_1= a_2- a_1$ and $b_2= r_2+ v_2= a_2+ v_2$ so $v_2= b_2- a_2$
The parametric equations for the line are
$x= a_2+ (b_1- a_1)t$
$y= a_2+ (b_2- a_2)t$