Worked examples
The question
Question: find the slope of the line through (1, 2) and (3, 6).
(1, 2) and (3, 6)
1 Plot the points
Draw the grid.
The x-axis goes across.
The y-axis goes up and down.
They meet at 0.
That point is called the origin.
Put both points on the grid: (1, 2)
and (3, 6).
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 = 3 is 2.
Rise: how far we go up.
From y = 2 to y = 6 is 4.
3 Use the formula
We call the slope m.
m = rise over run.
m = riserun
The top is the change in y: second y minus first y.
The bottom is the change in x: second x minus first x.
m = y2 − y1x2 − x1
Put the numbers in.
The second point is (3, 6).
The first point is (1, 2).
m = 6 − 23 − 1
6 minus 2 is 4.
3 minus 1 is 2.
The same rise and run as on the grid.
m = 42
4 divided by 2 is 2.
So m = 2.
m = 2
A positive slope goes up from left to right.
For every 1 step to the right, it goes up 2.
4 Check
Let's check: take the points the other way round.
2 − 6 is −4.
1 − 3 is −2.
m = 2 − 61 − 3 = −4−2
Minus over minus is plus.
We get 2 again.
The same slope!
= 2
Let's look back.
First: plot the points.
Then: rise and run.
Then: use the formula.
Last: check.
▶ Watch it solved
The question
Question: find the slope of the line through (−1, 2) and (3, −4).
(−1, 2) and (3, −4)
1 Plot the points
Draw the grid.
The x-axis goes across.
The y-axis goes up and down.
They meet at 0.
That point is called the origin.
Put both points on the grid: (−1, 2)
and (3, −4).
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 = 3 is 4.
Rise: how far we go up or down.
From y = 2 to y = −4 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.
m = riserun
The top is the change in y: second y minus first y.
The bottom is the change in x: second x minus first x.
m = y2 − y1x2 − x1
Put the numbers in.
The second point is (3, −4).
The first point is (−1, 2).
m = −4 − 23 − (−1)
Taking away a negative is the same as adding.
− (−) = +
So it becomes (−4 − 2)/(3 + 1).
m = −4 − 23 + 1
−4 minus 2 is −6.
3 plus 1 is 4.
The same rise and run as on the grid.
m = −64
We can make 6/4 simpler: divide top and bottom by 2.
The minus stays.
So m = −3/2.
m = −32
A negative slope goes down from left to right.
For every 2 steps to the right, it goes down 3.
4 Check
Let's check: take the points the other way round.
2 − (−4) is 6.
−1 − 3 is −4.
m = 2 − (−4)−1 − 3 = 6−4
Plus over minus is minus.
We get −3/2 again.
The same slope!
= −32
Let's look back.
First: plot the points.
Then: rise and run.
Then: use the formula.
Last: check.
▶ Watch it solved
The question
Question: find the slope of the line through (−6, −2) and (−1, −4).
(−6, −2) and (−1, −4)
1 Plot the points
Draw the grid.
The x-axis goes across.
The y-axis goes up and down.
They meet at 0.
That point is called the origin.
Put both points on the grid: (−6, −2)
and (−1, −4).
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 = −6 to x = −1 is 5.
Rise: how far we go up or down.
From y = −2 to y = −4 we go down 2.
Down counts as minus, so the rise is −2.
3 Use the formula
We call the slope m.
m = rise over run.
m = riserun
The top is the change in y: second y minus first y.
The bottom is the change in x: second x minus first x.
m = y2 − y1x2 − x1
Put the numbers in.
The second point is (−1, −4).
The first point is (−6, −2).
m = −4 − (−2)−1 − (−6)
Taking away a negative is the same as adding.
− (−) = +
So it becomes (−4 + 2) over (−1 + 6).
m = −4 + 2−1 + 6
−4 plus 2 is −2.
−1 plus 6 is 5.
The same rise and run as on the grid.
This fraction is as simple as it gets.
So m = −2/5.
m = −25
A negative slope goes down from left to right.
For every 5 steps to the right, it goes down 2.
4 Check
Let's check: take the points the other way round.
−2 − (−4) is 2.
−6 − (−1) is −5.
m = −2 − (−4)−6 − (−1) = 2−5
Plus over minus is minus.
We get −2/5 again.
The same slope!
= −25
Let's look back.
First: plot the points.
Then: rise and run.
Then: use the formula.
Last: check.
▶ Watch it solved
