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	=	y_A - y_B

	change in x		x_A - x_B

So what we do is

subtract the Y values,
subtract the X values
then divide

Like this:

m =

change in y

change in x

4-3

6-2

= 0.25

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

m =

change in y

change in x

3-4

2-6

-1

-4

= 0.25

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":

Slope m =

change in y

change in x

4-3

6-2

And we have:

y - 3 = (1/4)(x - 2)

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.

Check It!

Let us confirm by testing with the second point (6,4):

y = x/4 + 5/2 = 6/4 + 2.5 = 1.5 + 2.5 = 4

Yes, when x=6 then y=4, so it works!

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 =

y_A - y_B

x_A - x_B

4 - 1

2 - 2

= undefined

But there is still a way of writing the equation ... use "x =" instead of "y=", like this:

x = 2

See It In Action

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