How do you write in point slope form an equation of the line through each pair of points (2,7) and (-4,1)?

Apr 5, 2015

Two points uniquely identify a line, and the formula we must use is the following: when given two points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$, the line passing through both of them is

$\frac{y - {y}_{2}}{{y}_{1} - {y}_{2}} = \setminus \frac{x - {x}_{2}}{{x}_{1} - {x}_{2}}$

Plugging the values $\left({x}_{1} , {y}_{1}\right) = \left(2 , 7\right)$ and $\left({x}_{2} , {y}_{2}\right) = \left(- 4 , 1\right)$ into that formula, we get

$\setminus \frac{y - 1}{7 - 1} = \setminus \frac{x + 4}{2 + 4} \setminus \iff \setminus \frac{y - 1}{6} = \setminus \frac{x + 4}{6} \setminus \iff y - 1 = \left(x + 4\right)$

Which can be brought into the standard form:

$y - 1 = x + 4 \setminus \iff y = x + 5$

You can easily check that this is the right solution, because for both points $\left(2 , 7\right)$ and $\left(- 4 , 1\right)$ you have that the $y$ coordinate equals the $x$ one plus $5$.