# How do you solve y-y_1=m(x-x_1) for m?

Feb 12, 2015

This equation is the point slope form for a straight line.

$y - {y}_{1} = m \left(x - {x}_{1}\right)$

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

The point slope equation is used to determine the equation for a straight line, given the slope $\left(m\right)$ and one point on the line, $\left({x}_{1} , {y}_{1}\right)$. Suppose you have been given a slope of $m = 5$, and a point of ${x}_{1} = 6$ and a point of ${y}_{1} = 2$.

$y - {y}_{1} = m \left(x - {x}_{1}\right)$

Plug in given values.
$y - 2 = 5 \left(x - 6\right)$

Distribute the 5.
$y - 2 = 5 x - 30$

Add 2 to both sides of the equation.
$y = 5 x - 28$

In order to find the slope of a line, using two points on the line, you use the equation $m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$. Notice it is not identical to the first equation you gave, which is because we need two points to determine the slope. Suppose the line goes through points $\left({x}_{1} , {y}_{1}\right) = \left(8 , 10\right)$ and $\left({x}_{2} , {y}_{2}\right) = \left(4 , 2\right)$.

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{10 - 2}{8 - 4} = \frac{8}{4} = 2$

To find the equation of this line using the equation $y = m x + b$, you need to find the y-intercept, $b$.

$b = y - m x$

Plug in the slope and the x and y values for one of the two points. I will use the point $\left(8 , 10\right)$.

$b = 10 - \left(2 \cdot 8\right) = 10 - 16 = - 6$

I get the same answer if I use the other point, $\left(4 , 2\right)$.

$b = 2 - \left(2 \cdot 4\right) = 2 - 8 = - 6$

So the equation of this line is $y = 2 x - 6$.