# How to calculate the Slope Intercept equation from two points

A common problem in linear equations is of how to determine the equation of a line, when given any two points on that line.

Here's a simple step by step guide on how to work out the equation in Slope Intercept Form (y = mx + b)

For example, given the two points (1,5) and (3,1), let's work out the solution. Lets call the (1,5) "Point 1" and (3,1) "Point 2" for reference.

Step 1: Determine the Slope (m)

The Slope of a line is determined by the change in the value of y which corresponds to a given change in the value of x. We can use the two points we have been given to determine the slope:

m=
 Point2.y - Point1.y Point2.x - Point1.x
Subtract the two y values from eachother, and then divide this through by the difference of the x values

Substituting our values for the Point:

m=
 1 - 5 3 - 1
Note: It doesn't matter whether you subtract Point 2 from Point 1 or Point 1 from Point 2, but it is important to keep the ORDER of the subtraction the same on the numerator and denominator
m=
 -4 2
Do the subtraction on numerator and denominator
m=
-2
Our slope, m, is -2, so it slopes from top left to bottom right

Step 2: Determining the Y-Intercept (b)

The Y intercept is the point where the line intercepts the Y Axis (which is where x = 0). So by using this definition, and EITHER of the two points, we can work back from the point

Our slope, m, is -2. This means for every 1 unit of x we increase, y will decrease by 2 y-units (since m is -2), and conversely, for every x-unit we decrease, y will increase by 2 y-units (since -1 x -2 = 2)!

Taking the first point (1,5), x is 1. To get to the Y axis, we need to subtract 1 x-unit, so a decrease of 1 x-unit will require an increase of 2 y-units.

So the Y Intercept is 5 + 2 = 7

In mathematical terms, the general form of working a point back to the Y Axis using the slope of the line is:

b =
Point1.y - (m × Point1.x)

Taking Point 1, (1,5)

b =
5 - (-2 x 1)
=
5 - (-2)
=
5 + 2
=
5 + 2

Note that you could have ALSO used Point2, (3, 1):

b =
1 - (-2 x 3)
=
1 - (-6)
=
1 + 6
=
7

So our final equation for the line through the points (1,5) and (3,1), in Slope-Intercept form, is

y = -2 x + 7

See this example, and try your own problems using our Linear Equation solver with a Graph!