How to Determine the Equation of a Line from 2 Points

In this video, we are going to look at how to find the equation of a line when given two points.

The equation of a line is written in the form $y=mx+b$ , where $m$ is the slope, and $b$ is the y-intercept. When given the points (2,3) and (4,7), let’s see how to find the equation of the line. First, let’s find the slope. The equation for the slope is:
$m=\frac{y_2-y_1}{x_2-x_1}$
$m=\frac{7-3}{4-2}$
$m=\frac{4}{2}$
$m=2$

Now that we have the slope, let’s see how it looks in the equation of a line.
$y=2x+b$

We still don’t know what $b$ is. We do know that the line passes through the two given points. If we pick any one of the two points and plug in their x and y values, then we can solve for $b$ . Let’s pick the first point (2,3), and see what we get for $b$ .
$3=2(2)+b$
$3=4+b$
$-1=b$

Now we know that $b$ is -1 and $m$ is 2. If we go back to our equation of a line and plug these in, then we have:
$y=2x-1$

There is also another way to solve for this as well. The first step is still to find the slope. We already found that the slope is 2 in the last method of solving, so we can just bypass that step. In order to use this other method, we have to use a different form of the equation of a line, called the point-slope form. This is:
$y-y_1=m(x-x_1)$

In this form, $y_1$ and $x_1$ refer to a point, and $m$ is still the slope. If we just plug in the point (2,3) to the equation and solve then we will get:
$y-3=2(x-2)$
$y-3=2x-4$
$y=2x-1$
As you can see, both methods give us the same equation of a line.