Step by step
The question
Question: write the line through (−1, 5) and (2, −1) 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 (2, −1).
The first point is (−1, 5).
Taking away a negative is the same as adding.
So it becomes (−1 − 5)/(2 + 1).
−1 minus 5 is −6.
2 plus 1 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 (−1, 5) to (x, y).
The rise is y − 5: the change in y.
The run is x − (−1): the change in x.
Put them in: rise = m × run becomes y − 5 = −2(x − (−1)).
Taking away a negative is the same as adding.
So it becomes y − 5 = −2(x + 1).
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 − 5 = −2x − 2.
We want y by itself.
The − 5 is in the way.
To remove − 5, we add 5.
We do it on both sides, so they stay equal.
− 5 and + 5 make zero.
Cross them out.
−2 plus 5 is 3.
So y = −2x + 3.
4 Check
Let's check: put x = −1 into y = −2x + 3.
We get 5, the y of (−1, 5).
Now x = 2 into y = −2x + 3.
We get −1, the y of (2, −1).
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.