But why does ‘m’ represent the slope?
Answer: For any two points (x1, y1), (x2, y2) on y = mx + d: y2 − y1x2 − x1 = (mx2 + d) − (mx1 + d)x2 − x1 = m(x2 − x1)x2 − x1 = m. So the slope is always m, whichever two points we use.
Step-by-step solution
Given: A line in slope-intercept form: y = mx + d
Idea: Slope = riserun. The points on the line obey y = mx + d, so substitute this for each y; the d‘s cancel and m is left.
- Let A(x1, y1) and B(x2, y2) be any two points on y = mx + d with x1 ≠ x2. Since they lie on the line, y1 = mx1 + d and y2 = mx2 + d.1 mark
- Slope of AB = y2 − y1x2 − x1 = (mx2 + d) − (mx1 + d)x2 − x1 = mx2 − mx1x2 − x1.1 mark
- Take m common on top: m(x2 − x1)x2 − x1 = m (cancel x2 − x1, which is not zero).½ mark
- So the slope is m for every pair of points: we can read the slope straight from y = mx + d, and the slope is the same everywhere on the line.½ mark
Because for any two points on y = mx + d, (y₂ − y₁)/(x₂ − x₁) = m(x₂ − x₁)/(x₂ − x₁) = m.
Check: y = 3x + 1: points (0, 1) and (2, 7) give 62 = 3 = m ✓.
Answer to write in the exam
Let (x1, y1), (x2, y2) lie on y = mx + d ⇒ y1 = mx1 + d, y2 = mx2 + d
Slope = y2 − y1x2 − x1 = (mx2 + d) − (mx1 + d)x2 − x1
= m(x2 − x1)x2 − x1
∴ Slope = m
Common mistakes that cost marks
- Forgetting to cancel d: (mx2 + d) − (mx1 + d) = mx2 − mx1, because d − d = 0.
- Not noting that x2 − x1 ≠ 0. You cannot cancel a zero; that is why vertical lines (all x equal) have no slope value.
How this can come in the exam
MCQ (1 mark)
The slope of the line y = −4x + 7 is
- 7
- 4
- −4
- −74
Show answer
(C) −4
In y = mx + d, m = −4.
Try one yourself
Use the points with x = 0 and x = 4 on y = 12x − 3 to confirm its slope.
Show answer
(0, −3) and (4, −1): −1 − (−3)4 − 0 = 24 = 12 = m ✓.
More questions like this
- Consider the line 5x + y = 3. Rewrite it as y = −5x + 3. Can you now find its slope? What does it mean?
- Can you use the points A, B and C to verify that the slope of this line is indeed −5?
- The following diagrams represent ski hills. Rank the hills in order of their steepness, from least to greatest.
- The ramp at a loading dock rises 2.5 metres over a run of 4 metres. Find the slope of the ramp.
- Find the slope of the line l in each of the following diagrams.
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