Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - 12.1 Parametric Equations - Exercises - Page 603: 33

$$x= 1+t, \quad y= 1+2t.$$

First, we find the slope $m=\frac{3-1}{ 2-1}=\frac{2}{1}=2$. Now, we have $m=\frac{s}{r}=2$ so take $r=1$ and $s=2$. Then, the parametric equations are $$x=a+rt=1+t, \quad y= b+st=1+2t.$$ That is, $$c(t)=(x(t),y(t))=(1+t,1+2t).$$