Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - 12.1 Parametric Equations - Exercises - Page 603: 31

$$x =3+8t, \quad y= 1-3t.$$

First, we find the slope $m=\frac{4-1}{-5-3}=\frac{3}{-8}$. Now, we have $m=\frac{s}{r}=-\frac{3}{8}$ so take $r=8$ and $s=-3$. Then, the parametric equations are $$x=a+rt=3+8t, \quad y= b+st=1-3t.$$ That is, $$c(t)=(x(t),y(t))=(3+8t,1-3t).$$