# 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:

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.

### What is the equation of this line?

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!

## 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

## See It In Action

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