Step by step
The question
Question: write the line through (0, −2) and (3, 4) 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, 4).
The first point is (0, −2).
Taking away a negative is the same as adding.
So it becomes (4 + 2)/(3 − 0).
4 plus 2 is 6.
3 minus 0 is 3.
6 divided by 3 is 2.
So m = 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 (0, −2) to (x, y).
The rise is y − (−2): the change in y.
The run is x − 0: the change in x.
Put them in: rise = m × run becomes y − (−2) = 2(x − 0).
Taking away a negative is the same as adding.
Taking away 0 changes nothing.
So it becomes y + 2 = 2x.
This is called point-slope form.
It is made from a point and the slope.
3 Write as y = mx + b
We want y by itself.
The + 2 is in the way.
To remove + 2, we take away 2.
We do it on both sides, so they stay equal.
+ 2 and − 2 make zero.
Cross them out.
So y = 2x − 2.
4 Check
Let's check: put x = 0 into y = 2x − 2.
We get −2, the y of (0, −2).
Now x = 3 into y = 2x − 2.
We get 4, the y of (3, 4).
And the number in front of x is 2: the slope.
Right!
Let's look back.
First: find the slope.
Then: rise = m × run.
Then: write as y = mx + b.
Last: check.