Line of a proper pencil through a point

Text

In the pencil of lines through $P(2,1)$, i.e. $y-1=m(x-2)$, find the one that also passes through $Q(5,7)$.

Solution

The slope comes from the two points: $m=\frac{7-1}{5-2}=\frac{6}{3}=2.$ Substituting: $y-1=2(x-2)\Rightarrow y=2x-3$. $\boxed{\,y=2x-3\,}$