# How do you write the equations of a line passing through (5,-1) and (8,-3) in standard form?

Jun 13, 2018

$2 x + 3 y = 7$

#### Explanation:

$\text{the equation of a line in "color(blue)"standard form}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{A x + B y = C} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{where A is a positive integer and B, C are integers}$

$\text{to start with obtain the equation in "color(blue)"slope-intercept form}$

•color(white)(x)y=mx+b

$\text{where m is the slope and b the y-intercept}$

$\text{to calculate m use the "color(blue)"gradient formula}$

•color(white)(x)m=(y_2-y_1)/(x_2-x_1)

$\text{let "(x_1,y_1)=(5,-1)" and } \left({x}_{2} , {y}_{2}\right) = \left(8 , - 3\right)$

$m = \frac{- 3 - \left(- 1\right)}{8 - 5} = \frac{- 2}{3} = - \frac{2}{3}$

$y = - \frac{2}{3} x + b \leftarrow \textcolor{b l u e}{\text{is the partial equation}}$

$\text{to find b substitute either of the 2 given points into}$
$\text{the partial equation}$

$\text{using "(8,-3)" then}$

$- 3 = - \frac{16}{3} + b \Rightarrow b = - \frac{9}{3} + \frac{16}{3} = \frac{7}{3}$

$y = - \frac{2}{3} x + \frac{7}{3} \leftarrow \textcolor{red}{\text{in slope-intercept form}}$

$\text{multiply all terms by 3}$

$3 y = - 2 x + 7$

$2 x + 3 y = 7 \leftarrow \textcolor{red}{\text{in standard form}}$