# What is the equation of the line passing through (-5,4) and (9,-4)?

Jun 12, 2017

$y = - \frac{4}{7} x + \frac{8}{7}$
or $4 x + 7 y = 8$

#### Explanation:

First up, it's a line, not a curve, so a linear equation. The easiest way to do this (in my view) is using the slope intercept formula which is $y = m x + c$, where $m$ is the slope (the gradient) of the line, and c is the y-intercept.

The first step is the calculate the slope:
If the two points are $\left({x}_{1} , {y}_{1}\right) \text{ and } \left({x}_{2} , {y}_{2}\right)$, then

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

$\implies m = \frac{- 4 - 4}{9 - \left(- 5\right)}$

$\implies m = \frac{- 4 - 4}{9 + 5}$

$\implies m = - \frac{8}{14}$

$\implies m = - \frac{4}{7}$

So we now know a bit of the equation:

$y = - \frac{4}{7} x + c$

To find $c$, substitute in the values for $x$ and $y$ from any one of the two points, so using $\left(- 5 , 4\right)$

$\left(4\right) = - \frac{4}{7} \left(- 5\right) + c$

And solve for c

$\implies 4 = \frac{- 4 \cdot - 5}{7} + c$

$\implies 4 = \frac{20}{7} + c$

$\implies 4 - \frac{20}{7} = c$

$\implies \frac{4 \cdot 7}{7} - \frac{20}{7} = c$

$\implies \frac{28}{7} - \frac{20}{7} = c$

$\implies \frac{8}{7} = c$

Then put in $c$ and you get:

$y = - \frac{4}{7} x + \frac{8}{7}$

If you want, you can rearrange this into the general form:

$\implies y = \frac{1}{7} \left(- 4 x + 8\right)$

$\implies 7 y = - 4 x + 8$

$4 x + 7 y = 8$

And your graph would look like:
graph{4x+7y=8 [-18.58, 21.42, -9.56, 10.44]}

(you can click and drag on the line until you get the points if you want to double check)