# How do you write an equation in point-slope form for the given (-3, 5), (-6, 8)?

Jul 19, 2015

Given the points $\left(- 3 , 5\right)$ and $\left(- 6 , 8\right)$

Step 1: Determine the slope
The slope of a line between two points is the change in the y values divided by change in the x values. That is, given two point $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$, the slope is:
$\textcolor{w h i t e}{\text{XXXX}}$$m = \frac{\Delta y}{\Delta x} = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

For the given example this becomes:
$\textcolor{w h i t e}{\text{XXXX}}$$m = \frac{8 - 5}{\left(- 6\right) - \left(- 3\right)}$

$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$$= \frac{3}{- 3} = - 1$

Step 2: Combine the slope with a point on the line for the equation
The point-slope form for a linear equation with
$\textcolor{w h i t e}{\text{XXXX}}$slope $m$
$\textcolor{w h i t e}{\text{XXXX}}$through a point $\left(\hat{x} , \hat{y}\right)$
is
$\textcolor{w h i t e}{\text{XXXX}}$$y - \hat{y} = m \left(x - \hat{x}\right)$

For the given example we could choose either of the given point to use as $\left(\hat{x} , \hat{y}\right)$. For demonstration purposes I have used $\left(\hat{x} , \hat{y}\right) = \left(- 3 , 5\right)$

We have already determined that $m = - 1$

So a point-slope form of the line would be:
$\textcolor{w h i t e}{\text{XXXX}}$$y - 5 = \left(- 1\right) \left(x - \left(- 3\right)\right)$

$\textcolor{w h i t e}{\text{XXXX}}$[You might simplify this as:
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$$y - 5 = - x - 3$
$\textcolor{w h i t e}{\text{XXXX}}$or
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$$y = 2 - x$
$\textcolor{w h i t e}{\text{XXXX}}$but, while both have a simpler appearance,
$\textcolor{w h i t e}{\text{XXXX}}$they are not true "point-slope forms".]