# How do you find an equation of the line containing the given pair of points (0,0) and (5,7)?

Dec 27, 2016

$7 x - 5 y = 0$

#### Explanation:

For any two points on a straight line the ratio of the difference between the $y$ values and the difference between the $x$ values will be a constant. This constant is called the slope of the line.

For the given points $\left(0 , 0\right)$ and $\left(5 , 7\right)$ this slope will be
$\textcolor{w h i t e}{\text{XXX}} m = \frac{\Delta y}{\Delta x} = \frac{7 - 0}{5 - 0} = \frac{7}{5}$

For any arbitrary point on the line $\left(x , y\right)$ and the point $\left(0 , 0\right)$ on the line this slope must be the same value:
$\textcolor{w h i t e}{\text{XXX}} \frac{y - 0}{x - 0} = \frac{7}{5}$
or
$\textcolor{w h i t e}{\text{XXX}} \frac{y}{x} = \frac{7}{5}$
or
$\textcolor{w h i t e}{\text{XXX}} 5 y = 7 x$

There are various other forms in which this could be written; for example, "standard form" would be:
$\textcolor{w h i t e}{\text{XXX}} 7 x - 5 y = 0$