How to Find a Linear Equation Using Two Points

By Daphne Sakavellas

Edited by Eman Hamed

Let's say you are given two points: (2, -4) and (3,6) and you need to find the equation of the line they cross through in y=mx+b form, also called slope-intercept form.

(2, -4) & (3, 6)

First, you must find the slope, which is a number that describes both the direction and the steepness of a line.

To do this, you set one of the order pairs as (X1, Y1) and the other as (X2, Y2):

(2, -4) = (X1, Y1)

(3, 6) = (X2, Y2)

Then, plug the values you set into the following equation, which solves for the slope (m), or the change in y/the change in x:

In this example, we would plug in the following:

m = 6-(-4)/3-2

This can be reduced further through simplification.

m = 6+4/1

m = 10/1

m = 10

That means the m of our y=mx+b (A.K.A. the slope) equation is 10. We have y=10x+b so far.

Next, we simply plug in one of the ordered pairs to our equation and solve until b is the last remaining variable.

Point selected: (3, 6)

y = 10x+b

6 = 10(3)+b

6 = 30+b

We then solve for b by rearranging the equation:

6 = 30+b

-24 = b

b = -24

Finally, we have both variables (m and b) needed for our equations:

y = 10x-24

Plug in either of the two points into the equation to check your solution, by plugging in the x-coordinate for x and the y-coordinate for y. You can also graph the equation and check to see if both points are on the line.

Thus, our final equation in this example is y=10x-24.