Question

How do you write the equation of a line given (5,1), (8,-2)?

Answer 1

Use the formula for slope to calculate the slope then use the point-slope formula to obtain the equation for the line.

See full explanation below:

Explanation:

First, use the two points to determine the slope of the line:

The slope can be found by using the formula: `m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))`

Where `m` is the slope and (`color(blue)(x_1, y_1)`) and (`color(red)(x_2, y_2)`) are the two points on the line.

Substituting the values from the problem gives:

`m = (color(red)(-2) - color(blue)(1))/(color(red)(8) - color(blue)(5))`

`m = -3/3`

`m = -1`

We can now use the point-slope formula using either point from the problem and the slope we calculated to determine the equation of the line.

The point-slope formula states: `(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))`

Where `color(blue)(m)` is the slope and `color(red)(((x_1, y_1)))` is a point the line passes through.

Substitution gives:

`(y - color(red)(-2)) = color(blue)(-1)(x - color(red)(8))`

`(y + color(red)(2)) = color(blue)(-1)(x - color(red)(8))`

We can solve for `y` to put this equation into the more familiar slope-intercept form:

`y + color(red)(2) = color(blue)(-1)x - (color(blue)(-1) xx color(red)(8))`

`y + color(red)(2) = -x - (-8)`

`y + color(red)(2) = -x + 8`

`y + color(red)(2) - 2 = -x + 8 - 2`

`y + 0 = -x + 6`

`y = -x + 6`