# What is the equation of the line passing through (180,3), (2,68)?

Jul 13, 2016

The line is $y = - \frac{65}{178} x + \frac{6117}{89}$

#### Explanation:

The equation for a line takes the form:
$y = m x + b$

Where $m$ is the slope, and $b$ is the y-intercept. All lines (except vertical lines) are described by equations in this form.

To calculate slope, we use the tried-and-true "rise over run" relationship:
$m = \frac{r i s e}{r u n} = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

So for our line we have:
$m = \frac{3 - 68}{180 - 2} = - \frac{65}{178}$

You'll note here that the order of the x and y didn't matter. If we reversed it we'd end up with:
$m = \frac{68 - 3}{2 - 180} = - \frac{65}{178}$

So since we know the slope, all we need to do is plug in the known $\left(x , y\right)$ pair from one of our given points and compute $b$:
$y = - \frac{65}{178} x + b$
$68 = - \frac{65}{178} \cdot 2 + b$
$68 = - \frac{130}{178} + b$
$b = \frac{6117}{89}$

Combining all our results gives us our line:
$y = - \frac{65}{178} x + \frac{6117}{89}$

You can test that this result is correct by plugging in $x = 180$ and observing that the result is $y = 3$.