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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.
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
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
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
And we get:
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
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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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The U.S. Census Bureau statistics tell us that there are at least 151,671different last names and 5,163 different first names in common use inthe United States. Some names are more common than others.
There are 48,348 people named John Smith in the United States. There are 1,051 people named James Bond, 111 people named Harry Potter , 471 people named George Bush, and 34 people named Emily Dickinson. However, Johnny Cash (35 people) songs aside there are, statistically speaking, very few boys named Sue.
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