# How do you write a general linear equation for the line that crosses (-2,1) and (2,-2)?

Sep 1, 2015

$y = - \frac{3}{4} x - \frac{1}{2}$

#### Explanation:

In order to determine the equation of a line given two points that are on that line, you need to deterime two things

• the line's slope;
• the line's y-intercept.

For a line that passes through two points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$, the slope of the line is defined as

$\textcolor{b l u e}{m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}}$

Use the coordinates of the two points given to you to determine the slope of the line - it doesn't matter which point you take to be $\left({x}_{1} , {y}_{1}\right)$ and which you take $\left({x}_{2} , {y}_{2}\right)$.

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

Now you need to find its $y$-intercept. The slope-intercept form for a line is given by the equation

$\textcolor{b l u e}{y = m x + b} \text{ }$, where

$m$ - the slope of the line;
$b$ - the $y$-intercept.

Pick one of the two points and use its coordinates to replace $x$ and $y$ in the slope-intercept form equation, and use the calculated value of $m$ to get the $y$-intercept

$1 = - \frac{3}{4} \cdot \left(- 2\right) + b$

$1 = \frac{3}{2} + b \implies b = 1 - \frac{3}{2} = - \frac{1}{2}$

The slope-intercept equation of the line is

$y = - \frac{3}{4} x - \frac{1}{2}$