Step by step
The question
Question: write the line through (2, 1) and (6, 3) 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 (6, 3).
The first point is (2, 1).
3 minus 1 is 2.
6 minus 2 is 4.
We can make 2/4 simpler: divide top and bottom by 2.
So m = 1/2.
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 (2, 1) to (x, y).
The rise is y − 1: the change in y.
The run is x − 2: the change in x.
Put them in: rise = m × run becomes y − 1 = (1/2) times (x − 2).
This is called point-slope form.
It is made from a point and the slope.
3 Write as y = mx + b
The 1/2 in front of the bracket multiplies every part inside.
So we write 1/2 times each part.
Work out each multiplication.
Now we have y − 1 = (1/2)x − 1.
We want y by itself.
The − 1 is in the way.
To remove − 1, we add 1.
We do it on both sides, so they stay equal.
− 1 and + 1 make zero.
Cross them out.
−1 plus 1 is 0.
So y = (1/2)x.
4 Check
Let's check: put x = 2 into y = (1/2)x.
We get 1, the y of (2, 1).
Now x = 6 into y = (1/2)x.
We get 3, the y of (6, 3).
And the number in front of x is 1/2: the slope.
Right!
Let's look back.
First: find the slope.
Then: rise = m × run.
Then: write as y = mx + b.
Last: check.