Step by step
The question
Question: write the line through (−2, −1) and (1, 5) 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 (1, 5).
The first point is (−2, −1).
Taking away a negative is the same as adding.
So it becomes (5 + 1)/(1 + 2).
5 plus 1 is 6.
1 plus 2 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 (−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) = 2(x − (−2)).
Taking away a negative is the same as adding.
So it becomes y + 1 = 2(x + 2).
This is called point-slope form.
It is made from a point and the slope.
3 Write as y = mx + b
The 2 in front of the bracket multiplies every part inside.
So we write 2 times each part.
Work out each multiplication.
Now we have y + 1 = 2x + 4.
We want y by itself.
The + 1 is in the way.
To remove + 1, we take away 1.
We do it on both sides, so they stay equal.
+ 1 and − 1 make zero.
Cross them out.
4 minus 1 is 3.
So y = 2x + 3.
4 Check
Let's check: put x = −2 into y = 2x + 3.
We get −1, the y of (−2, −1).
Now x = 1 into y = 2x + 3.
We get 5, the y of (1, 5).
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.