# How do you write an equation in point slope and slope intercept form given (4,5) & (7,9)?

May 4, 2017

Point-slope form: $y - 5 = \frac{4}{3} \left(x - 4\right)$

Slope intercept form: $y = \frac{4}{3} x - \frac{1}{3}$

#### Explanation:

First find the slope $= m = \frac{\Delta y}{\Delta x} = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$:

$m = \frac{9 - 5}{7 - 4} = \frac{4}{3}$

Point-slope form $y - {y}_{1} = m \left(x - {x}_{1}\right)$:

$y - 5 = \frac{4}{3} \left(x - 4\right)$

To find the slope intercept form $y = m x + b$:

Use the point-slope form and distribute: $y - 5 = \frac{4}{3} x - \frac{4}{3} \cdot \frac{4}{1}$

Simplify:

$y - 5 = \frac{4}{3} x - \frac{16}{3}$

$y = \frac{4}{3} x - \frac{16}{3} + \frac{5}{1}$

$y = \frac{4}{3} x - \frac{16}{3} + \frac{15}{3}$

$y = \frac{4}{3} x - \frac{1}{3}$

Use slope and one point to find the $y$-intercept $b$:

$$y = 4/3x + b

9 = 4/3 * 7/1 + b

9/1 = 28/3 + b


$\frac{27}{3} - \frac{28}{3} = b$

$b = - \frac{1}{3}$

$y = \frac{4}{3} x - \frac{1}{3}$