What is the gradient of the line through (1, 5) and (3, 1)?

−2. (1 − 5) ÷ (3 − 1) = −4 ÷ 2.

Which point is 3 to the left of the y-axis and 2 above the x-axis?

(−3, 2). Left is negative x; up is positive y.

A line slopes downhill from left to right. What is true of its gradient?

It is negative. y falls as x increases.

Coordinates and gradient · Cambridge IGCSE Mathematics

In plain words

Every point on a grid has an address: how far across, then how far up. The gradient of a line is how steep it is: how many squares it climbs for each square it goes across.

Three things to know

Coordinates are (x, y): along first, then up. Negative numbers go left and down.
Gradient = change in y ÷ change in x. Between (x₁, y₁) and (x₂, y₂) that is (y₂ − y₁) ÷ (x₂ − x₁).
A line going uphill from left to right has a positive gradient; downhill is negative; a horizontal line has gradient 0.

Worked example

Find the gradient of the line through (1, 2) and (4, 11).

Change in y = 11 − 2 = 9.
Change in x = 4 − 1 = 3.
Gradient = 9 ÷ 3 = 3.

Tips and tricks

Subtract in the same order top and bottom. If you start with the second point for y, start with it for x too.
"Along the corridor, then up the stairs": x before y.

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Coordinates and gradient: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

What is the gradient of the line through (0, 0) and (2, 6)?
- 2
- 3 (the answer)
- 6
- 1/3
6 ÷ 2.
What is the gradient of a horizontal line?
- 0 (the answer)
- 1
- −1
- it has none
It doesn't rise at all.
What is the gradient of the line through (1, 5) and (3, 1)?
- 2
- −2 (the answer)
- −1/2
- 4
(1 − 5) ÷ (3 − 1) = −4 ÷ 2.
Which point is 3 to the left of the y-axis and 2 above the x-axis?
- (3, 2)
- (2, −3)
- (−3, 2) (the answer)
- (−2, 3)
Left is negative x; up is positive y.
A line slopes downhill from left to right. What is true of its gradient?
- It is positive.
- It is negative. (the answer)
- It is zero.
- It is 1.
y falls as x increases.

5 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

3 questions, 6 marks. Write your answers on paper, then check them.

Coordinates and gradient

Cambridge IGCSE Mathematics 0580 · 6 marks · papermunch.org

Name ______________________________ Date ______________

Find the gradient of the line through (0, 3) and (2, 7).[2]
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2. 4 ÷ 2.
I got this one
Find the gradient of the line through (−1, 4) and (3, −4).[2]
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−2. −8 ÷ 4.
I got this one
A line has gradient 3 and passes through (0, 1). Find the y-coordinate of the point on the line where x = 4.[2]
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13. It rises 3 for each 1 across: 1 + 12.
I got this one

Answers: Coordinates and gradient

1. 2. 4 ÷ 2.
2. −2. −8 ÷ 4.
3. 13. It rises 3 for each 1 across: 1 + 12.