# How do you write an equation of a line with point (5,-3), (2,5)?

May 31, 2018

$\frac{8}{- 3}$ = $m$
$y = \frac{8}{-} 3 x + \frac{49}{-} 3$

#### Explanation:

The goal of this question is to find the slope from two ordered pairs (two points on the graph). To do this, use this equation:

$\frac{{Y}_{2} - {Y}_{1}}{{X}_{2} - {X}_{1}}$ =$m$, the slope

Next, let's label our ordered pairs, or points, as ${X}_{1}$, ${Y}_{1}$, ${X}_{2}$, and ${Y}_{2}$. List your ordered pairs. Recall that an ordered pair is in the form $\left(x , y\right)$.

$\left(5 , - 3\right)$ $\left({X}_{1} , {Y}_{1}\right)$
$\left(2 , 5\right)$ $\left({X}_{2} , {Y}_{2}\right)$

Now, plug this information into your equation:

$\frac{{Y}_{2} - {Y}_{1}}{{X}_{2} - {X}_{1}}$ =$m$

$\frac{5 - - 3}{2 - 5}$ = $m$

Two negatives make a positive, so:
$\frac{5 + 3}{2 - 5}$ = $m$

Simplify.

$\frac{8}{- 3}$ = $m$

The slope, $m$, is $\left(8 , - 3\right)$. If you want to continue to find the whole line in the form of $y = m x + b$, use the point-slope formula as shown below. Recall that $m$ is the slope and the ordered pair you'll be using is the one you labeled as $\left({X}_{1} , {Y}_{1}\right)$.

$\left(y - {y}_{1}\right) = m \left(x - {x}_{1}\right)$

Plug in your information:
$\left(y - - 3\right) = \left(\frac{8}{-} 3\right) \left(x - 5\right)$

Distribute:
$\left(y + 3\right) = \left(\frac{8}{-} 3 x\right) + \left(- \frac{40}{3}\right)$

Subtract 3 from both sides to isolate for y:
$y = \left(\frac{8}{-} 3 x\right) + \left(\frac{49}{-} 3\right)$

Remove parentheses:

$y = \frac{8}{-} 3 x + \frac{49}{-} 3$