# How do you write an equation of a line going through (-1,2), (3,-4)?

Dec 23, 2016

$y - 2 = - \frac{3}{2} \left(x + 1\right)$

or

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

#### Explanation:

To find a linear equation for the line going through these two points we can use the point-slope formula.

However, first we need to determine the slope of the line.

The slope can be found by using the formula: color(red)(m = (y_2 - y_1)/(x_2 - x_1)
Where $m$ is the slope and ($\textcolor{red}{{x}_{1} , {y}_{1}}$) and ($\textcolor{red}{{x}_{2} , {y}_{2}}$) are the two points.

Substituting the two points given in the problem we can solve for $m$ as:

$m = \frac{- 4 - 2}{3 - \left(- 1\right)}$

$m = - \frac{6}{4} = \left(\frac{2}{2}\right) \left(- \frac{3}{2}\right) = 1 \left(- \frac{3}{2}\right)$

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

Now that we have the slope we can use the slope-point formula to write the equation for the line.

The point-slope formula states: $\textcolor{red}{\left(y - {y}_{1}\right) = m \left(x - {x}_{1}\right)}$
Where $\textcolor{red}{m}$ is the slope and ($\textcolor{red}{\left({x}_{1} , {y}_{1}\right)}$) is a point the line passes through.

Substituting the slope of $- \frac{3}{2}$ and using the point $\left(- 1 , 2\right)$ we can get the equation of the line as:

$y - 2 = - \frac{3}{2} \left(x - \left(- 1\right)\right)$

$y - 2 = - \frac{3}{2} \left(x + 1\right)$

If we want this in the slope-intercept for we can solve for $y$ as follows:

$y - 2 = - \frac{3}{2} x - \left(\frac{3}{2} \cdot 1\right)$

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

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

$y - 0 = - \frac{3}{2} x - \frac{3}{2} + 2$

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

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

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

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