# What is the equation of the line passing through (-3,-2 ) and (1, -5)?

Jan 30, 2017

$y = - \frac{3}{4} x - \frac{17}{4}$

#### Explanation:

The equation of a line in $\textcolor{b l u e}{\text{point-slope form}}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y - {y}_{1} = m \left(x - {x}_{1}\right)} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where m represents the slope and $\left({x}_{1} , {y}_{1}\right) \text{ a point on the line}$

To calculate m, use the $\textcolor{b l u e}{\text{gradient formula}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) , \left({x}_{2} , {y}_{2}\right) \text{ are 2 coordinate points}$

The 2 points here are (-3 ,-2) and (1 ,-5)

let $\left({x}_{1} , {y}_{1}\right) = \left(- 3 , - 2\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(1 , - 5\right)$

$\Rightarrow m = \frac{- 5 - \left(- 2\right)}{1 - \left(- 3\right)} = \frac{- 3}{4} = - \frac{3}{4}$

We can use either of the points (-3 ,-2), (1 ,-5) as the point on the line since the line passes through both of them.

$\text{Using " (x_1,y_1)=(1,-5)" and } m = - \frac{3}{4}$

Substitute these values into the equation.

$y - \left(- 5\right) = - \frac{3}{4} \left(x - 1\right)$

$\Rightarrow y + 5 = - \frac{3}{4} \left(x - 1\right) \leftarrow \textcolor{red}{\text{ in point-slope form}}$

Distributing and simplifying gives an alternative version of the equation.

$y + 5 = - \frac{3}{4} x + \frac{3}{4}$

$\Rightarrow y = - \frac{3}{4} x + \frac{3}{4} - 5$

$\Rightarrow y = - \frac{3}{4} x - \frac{17}{4} \leftarrow \textcolor{red}{\text{in slope-intercept form}}$