# How do you find an equation of the straight line passing through the points with coordinates (-1,5) and (4,-2), giving your answer in the form ax+by+c=0?

Nov 2, 2016

Please see the explanation for details regarding how one does the requested process.

#### Explanation:

The definition of the slope, m, of the line between two points, $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ is:

$m = \frac{{y}_{2} - {y}_{2}}{{x}_{2} - {x}_{1}}$

Using the given points to compute m:

$m = \frac{- 2 - 5}{4 - - 1} = - \frac{7}{5}$

The slope-intercept form of the equation of a line is:

$y = m x + b$

Using the slope and the point $\left(- 1 , 5\right)$, allows us to substitute -1 for x, 5 for y, and $- \frac{7}{5}$ for m, so that we may find the value of b:

$5 = - \frac{7}{5} \left(- 1\right) + b$

$5 = \frac{7}{5} + b$

$5 - \frac{7}{5} = b$

$\frac{25}{5} - \frac{7}{5} = b$

b = 18/5

The slope-intercept form of the line that goes through the two given points is:

$y = - \frac{7}{5} x + \frac{18}{5}$

But we want the form, $a x + b y + c = 0$, multiply boths side by 5:

$5 y = - 7 x + 18$

Add $7 x - 18$ to both sides:

$7 x + 5 y - 18 = 0$