The graph of the line y = 3x, passing through (0, 0) and (2, 6) is given below.
- (i) Identify the slope of the line from the graph and explain how you calculated it using the two points on the line.
- (ii) What is the y-intercept of the line? Explain its significance.
- (iii) A water tank is being filled so that the water level y (in cm) after x minutes follows this graph.
(a) How high is the water after 5 minutes?
(b) How many minutes will it take for the water to reach a height of 21 cm? - (iv) Without drawing a new graph, determine whether the point (4, 10) lies on this line. Justify your answer.
- (v) Plot the point where this line intersects the line x = 3. Explain how you found the coordinates.
Step-by-step solution
Idea: For y = mx + d, m is the slope and d the y-intercept. A point lies on the line exactly when its coordinates satisfy y = 3x.
(i) Identify the slope of the line from the graph and explain how you calculated it using the two points on the line.
- From (0, 0) to (2, 6): run = 2 − 0 = 2, rise = 6 − 0 = 6. Slope = riserun = 62 = 3: y goes up 3 for each 1 step right (matching m = 3 in y = 3x).1 mark
(ii) What is the y-intercept of the line? Explain its significance.
- y = 3x is y = 3x + 0, so the y-intercept is 0: the line cuts the y-axis at the origin (0, 0). For the tank in (iii), it means the water level is 0 cm at time 0, so the tank starts empty.1 mark
(iii) A water tank is being filled so that the water level y (in cm) after x minutes follows this graph.
(a) How high is the water after 5 minutes?
(b) How many minutes will it take for the water to reach a height of 21 cm?
- (a) x = 5: y = 3 × 5 = 15 cm.½ mark
- (b) y = 21: 3x = 21 ⇒ x = 7 minutes.½ mark
(iv) Without drawing a new graph, determine whether the point (4, 10) lies on this line. Justify your answer.
- For x = 4 the line gives y = 3 × 4 = 12, not 10. So (4, 10) does not satisfy y = 3x and does not lie on the line (it is 2 units below it).1 mark
(v) Plot the point where this line intersects the line x = 3. Explain how you found the coordinates.
- Every point of x = 3 has x-coordinate 3. On y = 3x, x = 3 gives y = 9. So the lines meet at (3, 9): plot it 3 units right and 9 units up.1 mark
Check: (iii)(b): after 7 minutes, 3 × 7 = 21 cm ✓. (v): (3, 9) satisfies both x = 3 and y = 3x ✓.
Answer to write in the exam
(i)
Slope = y2 − y1x2 − x1 = 6 − 02 − 0 = 62
∴ Slope = 3
(ii)
y = 3x + 0 ⇒ y-intercept = 0
∴ The line passes through (0, 0); at x = 0 the value of y is 0
(iii)
(a) y = 3 × 5 = 15 ⇒ 15 cm
(b) 3x = 21 ⇒ x = 7 ⇒ 7 minutes
(iv)
At x = 4: 3x = 12 ≠ 10
∴ (4, 10) does not lie on y = 3x
(v)
On x = 3, put x = 3 in y = 3x: y = 9
∴ Point of intersection = (3, 9)
Common mistakes that cost marks
- In (i), dividing run by rise and getting 13.
- In (iii)(b), multiplying 21 by 3 instead of dividing.
- In (v), plotting (9, 3) instead of (3, 9).
How this can come in the exam
A candle burns so that its height h cm after t hours is h = 24 − 3t.
(i) What is the height at the start? (ii) What is the slope and what does it mean? (iii) When will the candle burn out? (iv) What is its height after 5 hours?
Show answer
(i) 24 cm (1 mark). (ii) −3: the height falls 3 cm every hour (1 mark). (iii) 24 − 3t = 0 ⇒ t = 8 hours (1 mark). (iv) 24 − 15 = 9 cm (1 mark).Which point lies on the line y = 3x?
- (3, 1)
- (−2, −6)
- (5, 14)
- (1, 4)
Show answer
(B) (−2, −6)
3 × (−2) = −6 ✓.
Try one yourself
For the line y = 4x: find its slope, its y-intercept, and the point where it meets x = 2.5.
Show answer
Slope 4, y-intercept 0, point (2.5, 10).
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