Equation of a Line from 2 Points

graph with point (12,5)
Example: The point (12,5) is
12 units along, and 5 units up

Coordinates

We use Cartesian Coordinates to mark a point on a graph by how far along and how far up it is:

With 2 points we can work out the Equation of the Straight Line that goes through them. First find the slope:

Finding Slope (or Gradient) from 2 Points

graph 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 slope is the change in height divided by the change in horizontal distance.

Looking at this diagram ...

graph 2 points

... the formula is:

Slope m  =   change in y   =   yA − yB
change in x xA − xB

So we:

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

Like this:

m =
change in y
change in x
  =
4−3
6−2
  = 
1
4
  = 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.

graph 2 points  

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
  = 
1
4

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!

Another Example

graph 2 points  

What is the equation of this line?

 

Start with the "point-slope" formula:

y − y1 = m(x − x1)

Put in these values:

  • x1 = 1
  • y1 = 6
  • m = (2−6)/(3−1) = −4/2 = −2

And we get:

y − 6 = −2(x − 1)

We can change it to "Slope-Intercept (y = mx + b)" form:

y − 6 = −2x + 2

y = −2x + 8

 

The Big Exception

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

graph vertical line

A vertical line's 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

See It In Action