Step by step
The question
Question: write the line through (1, 2) and (3, 8) in point-slope form.
1 Find the slope
We call the slope m.
m = rise over run.
The top is the change in y: second y minus first y.
The bottom is the change in x: second x minus first x.
Put the numbers in.
The second point is (3, 8).
The first point is (1, 2).
8 minus 2 is 6.
3 minus 1 is 2.
6 divided by 2 is 3.
So m = 3.
2 Rise = m × run
m = rise over run.
So the rise is m times the run.
Like 8/2 is 4, so 8 is 4 times 2.
Take any point (x, y) on the line.
We go from (1, 2) to (x, y).
The rise is y − 2: the change in y.
The run is x − 1: the change in x.
Put them in: rise = m × run becomes y − 2 = 3(x − 1).
This is called point-slope form.
It is made from a point and the slope.
3 Write as y = mx + b
The 3 in front of the bracket multiplies every part inside.
So we write 3 times each part.
Work out each multiplication.
Now we have y − 2 = 3x − 3.
We want y by itself.
The − 2 is in the way.
To remove − 2, we add 2.
We do it on both sides, so they stay equal.
− 2 and + 2 make zero.
Cross them out.
−3 plus 2 is −1.
So y = 3x − 1.
4 Check
Let's check: put x = 1 into y = 3x − 1.
We get 2, the y of (1, 2).
Now x = 3 into y = 3x − 1.
We get 8, the y of (3, 8).
And the number in front of x is 3: the slope.
Right!
Let's look back.
First: find the slope.
Then: rise = m × run.
Then: write as y = mx + b.
Last: check.