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

Question

2646 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-01-09T19:40:12

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:

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:

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

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

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

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

$y + 0 = -x + 6$

$y = -x + 6$