Equation of a Line from 2 Points

	Coordinates I will be using Cartesian Coordinates, where you mark a point on a graph by how far along and how far up it is.
Example: The point (12,5) is 12 units along, and 5 units up.

Finding Slope (or Gradient) from 2 Points

What is the slope (or gradient) of this line? We know two points: point "A" is (6,4) (x is 6 when y is 4) point "B" is (2,3) (x is 2 when y is 3).

The formula is:   Slope m  =   change in y   =   yA - yB change in x xA - xB

So what we do is

• subtract the Y values,
• subtract the X values
• then divide

Like this:

It doesn't matter which point comes first, it still works out the same. Try swapping the points:

Finding an Equation from 2 Points

Now you know how to find the slope, let us look at finding a whole equation.

What is the equation of this line?

The easiest method is to start with the "point-slope" formula:

y - y₁ = m(x - x₁)

We can choose any point on the line as being point "1", so let us just use point (2,3):

y - 3 = m(x - 2)

Use the formula from above for the slope "m":

And we have:

That is an acceptable answer, but we could simplify it further:

y - 3 = x/4 - 2/4

y = x/4 - ½ + 3

y = x/4 + 5/2

Which is now in the "Slope-Intercept (y = mx + b)" form.

The Big Exception

The previous method works nicely except for one particular case: a vertical line:

In that case the gradient is undefined (because you cannot divide by 0): m = yA - yB xA - xB = 4 - 1 2 - 2 = 3 0 = undefined But there is still a way of writing the equation ... use "x =" instead of "y=", like this: x = 2