Question

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

Answer 1

`y = -3/4x - 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 `(x_1, y_1)` and `(x_2, y_2)`, the slope of the line is defined as

`color(blue)(m = (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 `(x_1, y_1)` and which you take `(x_2, y_2)`.

`m = (-2 - 1)/(2 - (-2)) = ((-3))/4 = -3/4`

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

`color(blue)(y = mx + b)" "`, 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 = -3/4 * (-2) + b`

`1 = 3/2 + b implies b = 1 - 3/2 = -1/2`

The slope-intercept equation of the line is

`y = -3/4x - 1/2`