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.
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:
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:
The slope is the change in height divided by the change in horizontal distance.
Looking at this diagram ...
... the formula is:
Slope m = change in ychange in x = yA − yBxA − xB
- subtract the Y values,
- subtract the X values
- then divide
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
Step 2: The "Point-Slope Formula"
Now put the 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 = (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 Slope-Intercept (y = mx + b) form.
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!
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
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:
x = 2