Step by step
The question
Question: find the slope of the line through (1, −2) and (4, 7).
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: (1, −2)
and (4, 7).
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 = 1 to x = 4 is 3.
Rise: how far we go up.
From y = −2 to y = 7 is 9.
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, 7).
The first point is (1, −2).
Taking away a negative is the same as adding.
So it becomes (7 + 2)/(4 − 1).
7 plus 2 is 9.
4 minus 1 is 3.
The same rise and run as on the grid.
9 divided by 3 is 3.
So m = 3.
A positive slope goes up from left to right.
For every 1 step to the right, it goes up 3.
4 Check
Let's check: take the points the other way round.
−2 − 7 is −9.
1 − 4 is −3.
Minus over minus is plus.
We get 3 again.
The same slope!
5 Pick the choice
Our answer is m = 3.
That is choice C.
6 Why not the others
A is 1/3.
Wrong: run over rise: upside down.
B is 5/3.
Wrong: 7 − 2: lost the minus; 7 − (−2) is 9.
D is 9.
Wrong: 9 is the rise: we still had to divide by the run, 4 − 1.
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.