The slope of the line through A(−3, −3) and B(0, −5) = □ Answer in simplest form

−2/3. Slope = (−5 − (−3))/(0 − (−3)) = −2/3 = −2/3. Subtract in the same order on top and below.

A line passes through A(3, 2) and B(0, −6). From A to B, y changes by □ and x by □, so the slope = □. Fill in each blank

−8, −3, 8/3. y: −6 − (2) = −8; x: 0 − (3) = −3; slope = −8/−3 = 8/3 (in simplest form).

A line of slope 5 is steeper than a line of slope 2. True or false?

True. Steepness is the slope's size, not its sign: for each 1 across, one line rises or falls 5, the other 2.

Coordinate geometry · Ages 14–15

Slope

Slope = rise ÷ run, both subtracted the same way.

How to do it

Method

Slope = rise ÷ run, both subtracted the same way

A(1, 2), B(5, 8): 8 − 25 − 1 = 32

Another way

Count squares

4 right, 6 up: 64 = 32

In full

A line's slope measures how steep it is: through A(x₁, y₁) and B(x₂, y₂), the slope m = y₂ − y₁x₂ − x₁, the change up over the change across. A(1, 2), B(5, 8): m = 8 − 25 − 1 = 64 = 32. Subtract in the same order on top and below (B minus A both times, or A minus B both times), the same answer; never B minus A on one and A minus B on the other. A line rising to the right has a positive slope, falling to the right a negative one; a horizontal line's slope is 0; a vertical line keeps its x, the bottom is 0, and it has no slope (it is not defined). The bigger a positive slope, the steeper the line. Or count squares on the picture: from one point go right some squares (the run), then up or down some (the rise) to the other; the slope is the rise over the run, negative if it goes down.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
The slope of the line through A(−3, −3) and B(0, −5) =
Answer in simplest form
Show the answer and working
Answer: −23
Slope = −5 − (−3)0 − (−3) = −23 = −23. Subtract in the same order on top and below.
Example 2
A line passes through A(3, 2) and B(0, −6). From A to B, y changes by and x by , so the slope = .
Fill in each blank
Show the answer and working
Answer: −8, −3, 83
y: −6 − (2) = −8; x: 0 − (3) = −3; slope = −8−3 = 83 (in simplest form).
Example 3
A line of slope 5 is steeper than a line of slope 2.
True or false?
Show the answer and working
Answer: True
Steepness is the slope's size, not its sign: for each 1 across, one line rises or falls 5, the other 2.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.The slope of the line through A(1, −1) and B(−3, −3) =
Answer in simplest form
Answer:
2.Read A and B off the picture. The slope of the line AB =
Answer in simplest form
Answer:
3.A line passes through A(−1, 6) and B(−2, 5). From A to B, y changes by and x by , so the slope = .
Fill in each blank
Answer:
4.The line through A(0, −2) and B(4, 2) has slope −11.
True or false?
Answer:
5.A line of slope 3 is steeper than a line of slope −4.
True or false?
Answer:

All the answers

1. 12
2. −1
3. −1, −1, 1
4. False
5. False

Free worksheet: SlopeWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

Start playingPlay and learn in the secondary world, Star Galaxy: animated methods, monster battles, a review of every mistake