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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

We use Cartesian Coordinates to 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

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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 'Point-Slope 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:

  • point 'A' is (6,4) (at x is 6, y is 4)
  • point 'B' is (2,3) (at x is 2, y is 3)

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

Looking at this diagram ..

Slope m = change in ychange in x = yA − yBxA − xB

In other words, we:

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

Like this:

m = change in ychange in x = 4−36−2 = 14 = 0.25

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

m = change in ychange in x = 3−42−6 = −1−4 = 0.25

Same answer.


Step 2: The 'Point-Slope Formula'

Now put that slope and one point into the 'Point-Slope Formula'

Start with the 'point-slope' formula (x1 and y1 are the coordinates of a point on the line):

y − y1 = m(x − x1)

We can choose any point on the line for x1 and y1, so let's just use point (2,3):

y − 3 = m(x − 2)

We already calculated the slope 'm':

m = change in ychange in x = 4−36−2 = 14

And we have:

y − 3 = 14(x − 2)

That is an answer, but we can simplify it further.

Step 3: Simplify

Multiply 14 and (x−2):y − 3 = x424
Simplify:y = x4 + 52

And we get:

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

Example: 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

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And we get:

y − 6 = −2(x − 1)

Simplify 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:

A vertical line's gradient is undefined (because we cannot divide by 0):

m = yA − yBxA − xB = 4 − 12 − 2 = 30 = undefined

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

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