# How do you write an equation of a line going through (0,7), (3,5)?

Apr 14, 2017

$2 x + 3 y = 21$

#### Explanation:

A line going through the points $\left(0 , 7\right)$ and $\left(3 , 5\right)$ has a slope of
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{m} = \frac{\Delta y}{\Delta x} = \frac{7 - 5}{0 - 3} = \textcolor{g r e \cap}{- \frac{2}{3}}$

The general slope-point form for a linear equation is
$\textcolor{w h i t e}{\text{XXX}} y - \textcolor{b l u e}{{y}_{1}} = \textcolor{g r e e n}{m} \left(x - \textcolor{red}{{x}_{1}}\right)$
for a line through the point $\left(\textcolor{red}{{x}_{1}} , \textcolor{b l u e}{{y}_{1}}\right)$ and with a slope of $\textcolor{g r e e n}{m}$

We have already determined $\textcolor{g r e e n}{m} = \textcolor{g r e e n}{- \frac{2}{3}}$
and we can arbitrarily select either of the given points for $\left(\textcolor{red}{{x}_{1}} , \textcolor{b l u e}{{y}_{1}}\right)$

For demonstration purposes, I will use $\left(\textcolor{red}{{x}_{1}} , \textcolor{b l u e}{{y}_{1}}\right) = \left(\textcolor{red}{0} , \textcolor{b l u e}{7}\right)$

So our slope-point form becomes
$\textcolor{w h i t e}{\text{XXX}} y - \textcolor{b l u e}{7} = \textcolor{g r e e n}{- \frac{2}{3}} \left(x - \textcolor{red}{0}\right)$

While this is a perfectly valid answer, it is common to convert this into "standard form": $A x + B y = C$, with $A , B , C \in \mathbb{Z} , A \ge 0$

$\textcolor{w h i t e}{\text{XXX}} y - 7 = - \frac{2}{3} \left(x - 0\right)$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow 3 y - 21 = - 2 x$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow 2 x + 3 y = 21$