# How do you write the point slope form of the equation given (-5,-1) and (0,-5)?

Jan 3, 2018

$y + 1 = - \frac{4}{5} \left(x + 5\right) \textcolor{w h i t e}{\text{xxx")orcolor(white)("xxx}} y + 5 = - \frac{4}{5} \left(x - 0\right)$

#### Explanation:

Note that the slope between the points $\left(- 5 , - 1\right)$ and $\left(0 , - 5\right)$ is
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{m} = \frac{\Delta y}{\Delta x} = \frac{- 1 - \left(- 5\right)}{- 5 - 0} = \textcolor{g r e e n}{- \frac{4}{5}}$

The general slope-point form for a line with slope $\textcolor{g r e e n}{m}$ through the point $\left(\textcolor{b l u e}{\hat{x}} , \textcolor{red}{\hat{y}}\right)$ is
$\textcolor{w h i t e}{\text{XXX}} y - \textcolor{red}{\hat{y}} = \textcolor{g r e e n}{m} \left(x - \textcolor{b l u e}{\hat{x}}\right)$

We can use either of the given points for our $\left(\textcolor{b l u e}{\hat{x}} , \textcolor{red}{\hat{y}}\right)$.
If, for example, we use $\left(\textcolor{b l u e}{\hat{x}} , \textcolor{red}{\hat{y}}\right) = \left(\textcolor{b l u e}{- 5} , \textcolor{red}{- 1}\right)$
then our slope-point form becomes
$\textcolor{w h i t e}{\text{XXX}} y \textcolor{red}{+ 1} = \textcolor{g r e e n}{- \frac{4}{5}} \left(x \textcolor{b l u e}{+ 5}\right)$
[using $\left(\textcolor{b l u e}{\hat{x}} , \textcolor{red}{\hat{y}}\right) = \left(\textcolor{b l u e}{0} , \textcolor{red}{- 5}\right)$ gives the alternate, but equivalent, version shown in the Answer].

Jan 3, 2018

The point-slope form is $y + 1 = - \frac{4}{5} \left(x + 5\right)$.

#### Explanation:

Slope

You first need to determine the slope from the given points. The formula for determining slope is:

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

where:

$m$ is the slope, $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ are the two points.

Plug the known values into the formula. I'm going to use $\left(- 5 , - 1\right)$ for Point 1 and $\left(0 , - 5\right)$ for Point 2. It doesn't matter which point you make 1 or 2. The result will be the same.

$m = \frac{- 5 - \left(- 1\right)}{0 - \left(- 5\right)}$

Simplify.

$m = \frac{- 5 + 1}{0 + 5}$

Solve.

$m = - \frac{4}{5}$

Point-slope form

$y - {y}_{1} = m \left(x - {x}_{1}\right)$,

where:

$m = - \frac{4}{5}$, $\left({x}_{1} , {y}_{1}\right)$ is one of the points. It doesn't matter which one. I'm going to use $\left(- 5 , - 1\right)$.

Plug in the known values.

$y - \left(- 1\right) = - \frac{4}{5} \left(x - \left(- 5\right)\right)$

$y + 1 = - \frac{4}{5} \left(x + 5\right)$ $\leftarrow$ Point-slope form

You can solve for $y$ to convert it into the slope-intercept form:

$y = m x + b$,

where:

$m$ is the slope, $- \frac{4}{5}$, and $b$ is the y-intercept.

$y + 1 = - \frac{4}{5} \left(x + 5\right)$

Expand the right-hand side.

$y + 1 = - \frac{4}{5} x - 4$

Subtract $- 1$ from both sides.

$y + 1 = - \frac{4}{5} x - 4 - 1$

Simplify.

$y = - \frac{4}{5} x - 5$ $\leftarrow$ Slope-intercept form.