Question

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

Answer 1

The line is `y = -65/178 x + 6117/89`

Explanation:

The equation for a line takes the form:
`y = mx + 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 = (rise)/(run) = (y_2 - y_1)/(x_2 - x_1)`

So for our line we have:
`m = (3 - 68)/(180 - 2)= -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 = (68-3)/(2-180) = -65/178`

So since we know the slope, all we need to do is plug in the known `(x,y)` pair from one of our given points and compute `b`:
`y = -65/178 x + b`
`68 = -65/178 * 2 + b`
`68 = -130/178 + b`
`b = 6117/89`

Combining all our results gives us our line:
`y = -65/178 x + 6117/89`

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