# What is the equation of the line with slope  m= 13/7  that passes through  (7/5,4/7) ?

Dec 17, 2015

$65 x - 35 y = 71$

#### Explanation:

Given a slope $m$ and a point $\left(\overline{x} , \overline{y}\right)$
the "slope-point form" of the linear equation is
$\textcolor{w h i t e}{\text{XXX}} \left(y - \overline{y}\right) = m \left(x - \overline{x}\right)$

Given
$\textcolor{w h i t e}{\text{XXX}} m = \frac{13}{7}$
and
$\textcolor{w h i t e}{\text{XXX}} \left(\overline{x} , \overline{y}\right) = \left(\frac{7}{5} , \frac{4}{7}\right)$

The "slope-point form" would be:
$\textcolor{w h i t e}{\text{XXX}} \left(y - \frac{4}{7}\right) = \frac{13}{7} \left(x - \frac{7}{5}\right)$
and this should be a valid answer to the given question.

However, this is ugly, so let's convert it into standard form:
$\textcolor{w h i t e}{\text{XXX}} A x + B y = C$ with $A , B , C \in \mathbb{Z} , A \ge 0$

Multiply both sides by $7$
$\textcolor{w h i t e}{\text{XXX}} 7 y - 4 = 13 x - \frac{91}{5}$

Multiply both sides by $5$ to clear the remaining fraction
$\textcolor{w h i t e}{\text{XXX}} 35 y - 20 = 65 x - 91$

Subtract $\left(35 y - 91\right)$ from both sides to get the variables on one side and the constant on the other
$\textcolor{w h i t e}{\text{XXX}} 71 = 65 x - 35 y$

Exchange sides:
#color(white)("XXX")65x-35y=71