Step by step
The question
Question: find the slope of the line through (−2, 5) and (4, −1).
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Plot the points
Draw the grid: the x-axis goes across, and the y-axis goes up and down.
They meet at 0, a point called the origin.
Put both points on the grid: (−2, 5)
and (4, −1).
Draw the line through them.
2 Rise and run
We start at the point on the left.
Run: how far we go right.
From x = −2 to x = 4 is 6.
Rise: how far we go up or down.
From y = 5 to y = −1 we go down 6.
Down counts as minus, so the rise is −6.
3 Use the formula
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.
It gives the same rise and run, without counting on the grid.
Put the numbers in.
The second point is (4, −1).
The first point is (−2, 5).
Taking away a negative is the same as adding.
So it becomes (−1 − 5)/(4 + 2).
−1 minus 5 is −6.
4 plus 2 is 6.
The same rise and run as on the grid.
−6 divided by 6 is −1.
So m = −1.
A negative slope goes down from left to right.
For every 1 step to the right, it goes down 1.
4 Check
Let's check: take the points the other way round.
5 − (−1) is 6.
−2 − 4 is −6.
Plus over minus is minus.
We get −1 again.
The same slope!
5 Pick the choice
Our answer is m = −1.
That is choice B.
6 Why not the others
A is −6.
Wrong: −6 is the rise: we still had to divide by the run, 6.
C is 1.
Wrong: lost the minus: the line goes down 6.
D is 2.
Wrong: added the numbers instead of taking them away.
Let's look back.
First: plot the points.
Then: rise and run.
Then: use the formula.
Then: check.
Then: pick the choice.
Last: why not the others.