# What is the equation of the line that passes through (-1,2) and is perpendicular to the line that passes through the following points: (5,-3),(7,6) ?

Apr 15, 2016

$y = \frac{- 2 x}{9} + \frac{16}{9}$

#### Explanation:

The general method to solving questions of this nature is this:
1. Find the slope of the line between the last two points
2. From that find slope of final line in answer
3. Deduce equation from the calculated slope and other point

Step 1:
We can find the slope between lines $\left(5 , - 3\right) \mathmr{and} \left(7 , 6\right)$ using the gradient formula:
$m = \frac{y 2 - y 1}{x 2 - x 1}$
$m = \frac{6 - \left(- 3\right)}{7 - 5}$
$m = \frac{9}{2}$

Step 2:
We can find the gradient of the line we're looking for by using this equation:
$m 1 = - \frac{1}{m 2}$ , where m2 would be $\frac{9}{2}$ and m1 would be the gradient we're looking for. Therefore:
$m 1 = - \frac{1}{\frac{9}{2}}$
$m 1 = - \frac{2}{9}$

Step 3:
Knowing the gradient of the line and a point that passes through it, we can use the point gradient formula to get the final answer:
$y - y 1 = m \left(x - x 1\right)$
$y - 2 = - \frac{2}{9} \left(x + 1\right)$
$y - 2 = \frac{- 2 x}{9} - \frac{2}{9}$
$y = \frac{- 2 x}{9} + \frac{16}{9}$