# How do you write the equation in slope intercept form given (2, 2), (-1, 4)?

Jul 6, 2016

The slope-intercept form of the equation is $y = - \frac{2}{3} x + \frac{10}{3}$

#### Explanation:

The slope-intercept form of the equation of the line is $y = m x + b$
where $m =$ slope and $b =$ the y intercept

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

${x}_{1} = 2$
${y}_{1} = 2$
${x}_{2} = - 1$
${y}_{2} = 4$

$m = \frac{4 - 2}{- 1 - 2}$

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

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

Now use the point slope formula to solve for the equation of the line.

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

For this situation we are given the slope of $3$ and a point of $\left(2 , 1\right)$

$m = - \frac{2}{3}$
${x}_{1} = 2$
${y}_{1} = 2$

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

$\left(y - 2\right) = - \frac{2}{3} \left(x - 2\right)$

$y - 2 = - \frac{2}{3} x - \frac{4}{3}$

$y \cancel{- 2} \cancel{+ 2} = - \frac{2}{3} x + \frac{4}{3} + 2$

$y = - \frac{2}{3} x + \frac{4}{3} + \frac{6}{3}$

$y = - \frac{2}{3} x + \frac{10}{3}$

Jul 6, 2016

$y = \frac{- 2 x}{3} + \frac{10}{3}$

#### Explanation:

If you have two points on a straight line, there is a lovely formula which allows you to get the equation immediately. It is based on the formula for the slope, so you kill two birds with one stone!

$\frac{y - {y}_{1}}{x - {x}_{1}} = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

$\frac{y - 2}{x - 2} = \frac{4 - 2}{- 1 - 2} = \frac{2}{-} 3 \text{ this value is the slope}$

$\frac{y - 2}{x - 2} = - \frac{2}{3} \text{ cross multiply}$

$3 y - 6 = - 2 x + 4$

$3 y = - 2 x + 10$

$y = \frac{- 2 x}{3} + \frac{10}{3}$