# How do you write an equation in slope intercept form given that the line passes through the points (-2, 2) and (0,5)?

Jun 3, 2015

The answer is $y = \frac{3}{2} x + 5$

Slope intercept form means we're looking for an equation that looks like $y = m x + b$. We need to use the given two points to find the slope, $m$, and the y-intercept, $b$.

To find $m$, we need to find the change in y between two points over the change in x (or "rise over run"). To do this, we use the equation: $m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$.

Using the points $\left(- 2 , 2\right)$ and $\left(0 , 5\right)$ as $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$, we get $m = \frac{5 - 2}{0 - - 2} = \frac{3}{2}$

To find $b$, we look back at the equation $y = m x + b$. We now know $m$ and can use the $x$ and $y$ from either of our points. I'll use $\left(- 2 , 2\right)$, but as you'll see in a moment, we already know $b$ from our other point.

Using $m = \frac{3}{2}$, $x = - 2$, and $y = 2$:
$y = m x + b$ becomes $2 = \frac{3}{2} \cdot - 2 + b$.

Now we solve for $b$:
$2 = \frac{3}{2} \cdot - 2 + b$

Simplify:
$2 = - 3 + b$

Add $3$ to both sides:
$5 = b$

We now know both $m$ and $b$, so our slope-intercept equation becomes:
$y = \frac{3}{2} x + 5$

That $5$ sure looks familiar... the point $\left(0 , 5\right)$ actually gives us $b$ for free since the y-intercept is the point where the line crosses the y-axis, which is true when $x = 0$. This short cut can help you save time, but you should also make sure you know how to find $b$ when you don't get so lucky!