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Slope of a line · 2 marks

1. Consider the point on the line AB whose x-coordinate is 8. What is the y-coordinate of this point? Express this as an ordered pair.
2. What is the x-coordinate of the point on the line AB whose y-coordinate is 6? Express this as an ordered pair.

  1. 1. Consider the point on the line AB whose x-coordinate is 8. What is the y-coordinate of this point? Express this as an ordered pair.
  2. 2. What is the x-coordinate of the point on the line AB whose y-coordinate is 6? Express this as an ordered pair.
xy−2−112345678−11234567890A (2, 0)B (4, 3)
Answer: 1. (8, 9) 2. (6, 6). The line rises 3 for every 2 to the right.

Step-by-step solution

Idea: From A(2, 0) to B(4, 3) the line goes 2 right and 3 up, so its slope is 32. Keep repeating “2 right, 3 up” (or use the equation y = 32(x − 2)) to reach the required point.

xy−2−112345678−11234567890A (2, 0)B (4, 3)(6, 6)(8, 9)

1. Consider the point on the line AB whose x-coordinate is 8. What is the y-coordinate of this point? Express this as an ordered pair.

  1. Slope of AB = 3 − 04 − 2 = 32. From x = 2 to x = 8 the run is 6.½ mark
  2. Rise = 32 × 6 = 9, so y = 0 + 9 = 9. The point is (8, 9). (Stepping “2 right, 3 up”: (4, 3) → (6, 6) → (8, 9).)½ mark
(8, 9)

2. What is the x-coordinate of the point on the line AB whose y-coordinate is 6? Express this as an ordered pair.

  1. From y = 0 to y = 6 the rise is 6. Run = rise ÷ slope = 6 ÷ 32 = 4.½ mark
  2. x = 2 + 4 = 6. The point is (6, 6).½ mark
(6, 6)
1. (8, 9) 2. (6, 6).

Check: The equation of AB is 3x − 2y = 6. (8, 9): 24 − 18 = 6 ✓. (6, 6): 18 − 12 = 6 ✓.

Answer to write in the exam

1.

Slope of AB = 3 − 04 − 2 = 32

Run from A to x = 8: 8 − 2 = 6

Rise = 32 × 6 = 9 ⇒ y = 0 + 9 = 9

∴ Point: (8, 9)

2.

Rise from A to y = 6: 6 − 0 = 6

Run = 6 ÷ 32 = 4 ⇒ x = 2 + 4 = 6

∴ Point: (6, 6)

Common mistakes that cost marks

  • Using the slope upside down (23), which gives (8, 4) and (11, 6).
  • Measuring the run from the origin instead of from A: the line starts at x = 2, not 0.

How this can come in the exam

MCQ (1 mark)

A line passes through (1, 2) and (3, 6). Its point with x = 5 is

  1. (5, 8)
  2. (5, 10)
  3. (5, 12)
  4. (5, 9)
Show answer

(B) (5, 10)
Slope = 42 = 2; from x = 3 to 5 the rise is 4, so y = 10.

Try one yourself

A line passes through (0, 1) and (3, 3). Find its point with x = 9.

Show answer

Slope 23; run 9 ⇒ rise 6 ⇒ (9, 7).

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