# How do you write an equation in standard form given point (-3,-1) and (2,5)?

Mar 31, 2017

$6 x - 5 y = - 13$

#### Explanation:

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

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{A x + B y = C} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where A is a positive integer and B, C are integers.

Begin by expressing the equation in $\textcolor{b l u e}{\text{point-slope form}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y - {y}_{1} = m \left(x - {x}_{1}\right)} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
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 ,-1) and (2 ,5)

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

$\Rightarrow m = \frac{5 - \left(- 1\right)}{2 - \left(- 3\right)} = \frac{6}{5}$

Either of the 2 given points can be used for $\left({x}_{1} , {y}_{1}\right)$

$\text{Using "m=6/5" and } \left({x}_{1} , {y}_{1}\right) = \left(2 , 5\right)$ we can establish the equation.

$\Rightarrow y - 5 = \frac{6}{5} \left(x - 2\right) \leftarrow \textcolor{red}{\text{ in point-slope form}}$

distribute bracket and rearrange into standard form.

$y - 5 = \frac{6}{5} x - \frac{12}{5}$

multiply ALL terms on both sides by 5

$\Rightarrow 5 y - 25 = 6 x - 12$

$\Rightarrow 6 x - 5 y = - 13 \leftarrow \textcolor{red}{\text{ in standard form}}$