# How do you find a general form equation for the line through the pair of points (1,2) and (5,4)?

Aug 30, 2017

$x - 2 y = - 3$

#### Explanation:

Step 1: Develop the slope-point form for the line
Given two points $\left(\textcolor{red}{{x}_{1}} , \textcolor{b l u e}{{y}_{1}}\right)$ and $\left({x}_{2} , {y}_{2}\right)$
the slope between the points is
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{m} = \frac{{y}_{2} - \textcolor{b l u e}{{y}_{1}}}{{x}_{2} - \textcolor{red}{{x}_{1}}}$
and
the slope point form is
$\textcolor{w h i t e}{\text{XXX}} y - \textcolor{b l u e}{{y}_{1}} = \textcolor{g r e e n}{m} \left(x - \textcolor{red}{{x}_{1}}\right)$

Using $\left(\textcolor{red}{1} , \textcolor{b l u e}{2}\right)$ as $\left(\textcolor{red}{{x}_{1}} , \textcolor{b l u e}{{y}_{1}}\right)$
and $\left(5 , 4\right)$ as $\left({x}_{2} , {y}_{2}\right)$

We have
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{m} = \frac{4 - \textcolor{b l u e}{2}}{5 - \textcolor{red}{1}} = \textcolor{g r e e n}{\frac{1}{2}}$

and the slope-point form is
$\textcolor{w h i t e}{\text{XXX}} y - \textcolor{b l u e}{2} = \textcolor{g r e e n}{\frac{1}{2}} \left(x - \textcolor{red}{1}\right)$

Step 2: convert the slope point form into standard form
Note that the standard form is
$\textcolor{w h i t e}{\text{XXX}} \textcolor{m a \ge n t a}{A} x + \textcolor{b r o w n}{B} x = \textcolor{p u r p \le}{C}$
with integer values for $\textcolor{m a \ge n t a}{A} , \textcolor{b r o w n}{B} , \textcolor{p u r p \le}{C}$ and $\textcolor{m a \ge n t a}{A} \ge 0$

Starting from the slope-point form
$\textcolor{w h i t e}{\text{XXX}} y - 2 = \frac{1}{2} \left(x - 1\right)$
Multiply both sides by $2$
$\textcolor{w h i t e}{\text{XXX}} 2 y - 4 = x - 1$
Subtract $x$ and add $4$ to both sides
$\textcolor{w h i t e}{\text{XXX}} - x + 2 y = 3$
Multiply both sides by $\left(- 1\right)$ (to make the coefficient of $x$ positive
$\textcolor{w h i t e}{\text{XXX}} x - 2 y = - 3$