Equation of a Line from 2 Points
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:
What it Looks Like
With 2 points we can work out the Equation of the Straight Line that goes through them.
Here is a demo of what it looks like. Try dragging the points.
OK, let's discover how to create the equation. First, find the slope ...
Finding Slope (or Gradient) from 2 Points
What is the slope (or gradient) of this line?We know two points:

The slope is the change in height divided by the change in horizontal distance.
Looking at this diagram ...
... the formula is:

So we:
 subtract the Y values,
 subtract the X values
 then divide
Like this:
m = 

= 

= 

= 0.25 
It doesn't matter which point comes first, it still works out the same. Try swapping the points:
m = 

= 

= 

= 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 "pointslope" formula:
y − y_{1} = m(x − x_{1})
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 = 

= 

= 

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 "SlopeIntercept (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
What is the equation of this line?

Start with the "pointslope" formula:
y − y_{1} = m(x − x_{1})
Put in these values:
 x_{1} = 1
 y_{1} = 6
 m = (2−6)/(3−1) = −4/2 = −2
And we get:
y − 6_{} = −2(x − 1_{})
We can change it to "SlopeIntercept (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:
A vertical line's gradient is undefined (because you cannot divide by 0):
But there is still a way of writing the equation: use x= instead of y=, like this: x = 2 