Equation of a Line from 2 Points
First, let's see it in action. Here are two points (you can drag them) and the equation of the line through them. Explanations follow.
The Points
Example: The point (12,5) is
12 units along, and 5 units up
We use Cartesian Coordinates to mark a point on a graph by how far along and how far up it is:
Steps
There are 3 steps to find the Equation of the Straight Line :
 1. Find the slope of the line
 2. Put the slope and one point into the "PointSlope Formula"
 3. Simplify
Step 1: Find the 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: Slope m = \frac{change in y}{change in x} = \frac{y_{A} − y_{B}}{x_{A} − x_{B}} 
So we:
 subtract the Y values,
 subtract the X values
 then divide
Like this:
m = \frac{change in y}{change in x} = \frac{4−3}{6−2} = \frac{1}{4} = 0.25
It doesn't matter which point comes first, it still works out the same. Try swapping the points:
m = \frac{change in y}{change in x} = \frac{3−4}{2−6} = \frac{−1}{−4} = 0.25
Step 2: The "PointSlope Formula"
Now put the slope and one point into the "PointSlope Formula"
Start with the "pointslope" formula (x_{1} and y_{1} are the coordinates of a point on the line):
y − y_{1} = m(x − x_{1})
We can choose any point on the line for x_{1} and y_{1}, so let's just use point (2,3):
y − 3_{} = m(x − 2_{})
We already calculated the slope "m":
m = \frac{change in y}{change in x} = \frac{4−3}{6−2} = \frac{1}{4}
And we have:
y − 3_{} = (1/4)(x − 2_{})
That is an answer, but we can simplify it further
Step 3: Simplify
Start with:  y − 3_{} = (1/4)(x − 2_{})  
Put the 1/4 on to x and −2:  y − 3_{} = x/4 − 2/4  
Add 3 to both sides:  y = x/4 − ½_{} + 3  
Calculate −½ + 3 = 5/2:  y = x/4 + 5/2 
And we get
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
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_{})
Simplify to SlopeIntercept (y = mx + b) form:
y − 6_{} = −2x + 2
y = −2x + 8
DONE!
The Big Exception
The previous method works nicely except for one particular case: a vertical line:
A vertical line's gradient is undefined (because we cannot divide by 0): m = \frac{y_{A} − y_{B}}{x_{A} − x_{B}} = \frac{4 − 1}{2 − 2} = \frac{3}{0} = undefined But there is still a way of writing the equation: use x= instead of y=, like this: x = 2 