# How do you write the equation of a line in slope intercept, point slope and standard form given (5,3) and (2,1)?

Oct 5, 2016

(see below for the three versions)

#### Explanation:

The slope of the line through $\left(5 , 3\right)$ and $\left(2 , 1\right)$ is
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{m} = \frac{\Delta y}{\Delta x} = \frac{3 - 1}{5 - 2} = \textcolor{g r e e n}{\frac{2}{3}}$

Slope-Point Form: color(black)((y-color(blue)(b))=color(green)(m)(x-color(red)(a))
Using $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right) = \left(\textcolor{red}{5} , \textcolor{b l u e}{3}\right)$
$\textcolor{w h i t e}{\text{XXX}} y - \textcolor{b l u e}{3} = \textcolor{g r e e n}{\frac{2}{3}} \left(x - \textcolor{red}{5}\right)$

Slope-Intercept Form: $\textcolor{b l a c k}{y = \textcolor{g r e e n}{m} x + \textcolor{m a \ge n t a}{k}}$
Starting from the slope-point form:
$\textcolor{w h i t e}{\text{XXX}} y - 3 = \frac{2}{3} x - \frac{10}{3}$
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{\frac{2}{3}} x - \frac{10}{3} + 3$
$\textcolor{w h i t e}{\text{XXX")y=color(green)(2/3)x+color(magenta)(} \left(- \frac{1}{3}\right)}$

Standard Form: $\textcolor{b r o w n}{A} x + \textcolor{\mathmr{and} a n \ge}{B} y = \textcolor{c y a n}{C}$
Starting from the slope-intercept form:
$\textcolor{w h i t e}{\text{XXX}} 3 y = 2 x - 1$
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b r o w n}{2} x \textcolor{\mathmr{and} a n \ge}{- 3} y = \textcolor{c y a n}{1}$