# What is the equation of the line passing through (4,8) and (-9,3)?

Nov 30, 2015

point-slope form:
$y - 8 = \frac{5}{13} \left(x - 4\right)$
or
$y - 3 = \frac{5}{13} \left(x + 9\right)$

slope-intercept form:
$y = \frac{5}{13} x + \frac{84}{13}$

standard form:
$- 5 x + 13 y = 84$

#### Explanation:

Method 1:
Use point slope form
which is $y - {y}_{1} = m \left(x - {x}_{1}\right)$
when given a point $\left({x}_{1} , {y}_{1}\right)$ and the slope $m$
'
In this case, we should first find the slope between the two given points.
This is given by the equation:
$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$
when given the points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$
'
For $\left({x}_{1} , {y}_{1}\right) = \left(4 , 8\right)$ and $\left({x}_{2} , {y}_{2}\right) = \left(- 9 , 3\right)$
By plugging what we know into the slope equation, we can get:
$m = \frac{3 - 8}{- 9 - 4} = \frac{- 5}{- 13} = \frac{5}{13}$
'
from here we can plug in either point and get:
$y - 8 = \frac{5}{13} \left(x - 4\right)$
or
$y - 3 = \frac{5}{13} \left(x + 9\right)$

Method 2:
Use slope intercept form
which is $y = m x + b$
when $m$ is the slope and $b$ is the y-intercept
'
We can find the slope between the two given points using the same steps as above
and get $m = \frac{5}{13}$
'
but this time when we plug in, we will still be missing the $b$ or y-intercept
to find the y-intercept, we need to temporarily plug in one of the given points in for $\left(x , y\right)$ and solve for b
'
so
$y = \frac{5}{13} x + b$
if we plug in $\left(x , y\right) = \left(4 , 8\right)$
we would get:
$8 = \frac{5}{13} \left(4\right) + b$
'
solving for $b$ would get us
$8 = \frac{20}{13} + b$
$b = \frac{84}{13} \mathmr{and} 6 \frac{6}{13}$
'
$y = \frac{5}{13} x + \frac{84}{13}$
$a x + b y = c$
to get $13 y = 5 x + 84$
then subtract $5 x$ from both sides
$- 5 x + 13 y = 84$