Consider the straight line that passes through A (−1, 0), B (1, 2), and C (4, 5). Plot the points and find the slope of the line.
Answer: Slope = CPBP = 5 − 24 − 1 = 1.
Step-by-step solution
Idea: Slope = riserun = y2 − y1x2 − x1 for any two points of the line. Drawing a right-angled step from B across to P and up to C shows the run and the rise.
- Plot A(−1, 0), B(1, 2), C(4, 5) and draw the line through them (see the diagram).1 mark
- Take B and C. Go across from B to P(4, 2), then up to C. Run = BP = 4 − 1 = 3; rise = CP = 5 − 2 = 3.1 mark
- Slope = riserun = CPBP = 5 − 24 − 1 = 33 = 1.½ mark
- Any other pair gives the same: A and B: 2 − 01 − (−1) = 22 = 1.½ mark
The slope of the line is 1.
Check: A and C: 5 − 04 − (−1) = 55 = 1 ✓. Each 1 step right, the line rises 1.
Answer to write in the exam
Plot A(−1, 0), B(1, 2), C(4, 5) and draw line AC
Slope = y2 − y1x2 − x1 (using B and C)
= 5 − 24 − 1 = 33
∴ Slope = 1
Common mistakes that cost marks
- Subtracting in different orders: 5 − 21 − 4 = −1. Use the same point first on top and bottom.
- Using runrise instead of riserun (it happens to give 1 here, but not in general).
- Forgetting that 1 − (−1) = 2, not 0.
How this can come in the exam
MCQ (1 mark)
The slope of the line through (2, 1) and (6, 9) is
- 12
- 2
- 4
- 8
Show answer
(B) 2
9 − 16 − 2 = 84 = 2.
Short answer (2 marks)
Show that (0, 1), (2, 4) and (6, 10) lie on one line by finding slopes.
Show answer
Slope of first two: 32 (1 mark). Slope of last two: 64 = 32. Equal slopes with a common point ⇒ same line (1 mark).Try one yourself
Find the slope of the line through (−2, −1) and (3, 9).
Show answer
9 − (−1)3 − (−2) = 105 = 2.
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