# What is the equation of the line between (-17,14) and (19,6)?

Dec 4, 2015

$y = - \frac{2}{9} x + \frac{92}{2}$

#### Explanation:

First, we find the slope $m$ of the line.
The slope of the line is the change in $y$ per unit of change in $x$. Equivalently, this means that a line with slope $\frac{a}{b}$ will rise $a$ units as $x$ increases by $b$ units. Then, we can find the slope from two points with the following formula:

$m = \left(\text{change in "y)/("change in } x\right) = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

In this case, that gives us

$m = \frac{6 - 14}{19 - \left(- 17\right)} = - \frac{8}{36} = - \frac{2}{9}$

Now, we can write the equation using the point-slope form of a line.
$y - {y}_{1} = m \left(x - {x}_{1}\right)$

Picking either of the points will work, so let's use $\left(19 , 6\right)$ (as an exercise, verify that this gives the same result if you use the other point). This gives us the equation

$y - 6 = - \frac{2}{9} \left(x - 19\right)$

If we wish to put that into the more common slope-intercept form, we can just multiply it out and solve for $y$.

$y - 6 = - \frac{2}{9} x + \frac{38}{9}$

$y = - \frac{2}{9} x + \frac{92}{2}$