Parallel and perpendicular through a point

Text

Given $r:\;y=3x-1$ and the point $P(2,4)$, write the parallel and the perpendicular to $r$ through $P$.

Solution

Parallel: $m=3$, $y-4=3(x-2)\Rightarrow y=3x-2$. Perpendicular: $m'=-\dfrac{1}{3}$, $y-4=-\dfrac13(x-2)\Rightarrow y=-\dfrac13 x+\dfrac{14}{3}$. $\boxed{\,y=3x-2\quad\text{and}\quad y=-\tfrac13 x+\tfrac{14}{3}\,}$