What is the slope of the line passing through points (2,3) and (4,7)?
- (a)2
- (b)3
- (c)5
- (d)4
Answer
A
Why
Correct — A.
Slope formula: m = (y₂ − y₁) ⁄ (x₂ − x₁)
Rise: 7 − 3 = 4
Run: 4 − 2 = 2
Slope: 4 ÷ 2 = 2 → option (a)
Check: the line y = 2x − 1 gives 3 at x = 2 and 7 at x = 4
Why the others are wrong
- (b)3 — 3 is the x-coordinate of the midpoint (3, 5), not a slope. A slope of 3 from (2, 3) would reach y = 9 at x = 4, not 7.
- (c)5 — 5 is the y-coordinate of the midpoint, (3 + 7) ÷ 2. A slope of 5 from (2, 3) would reach y = 13 at x = 4.
- (d)4 — 4 is the rise alone, 7 − 3. It still has to be divided by the run, 4 − 2 = 2, which gives 2.
Concept
The slope of a line is rise over run: how much y changes for each unit change in x.
Between (x₁, y₁) and (x₂, y₂) it is (y₂ − y₁) ⁄ (x₂ − x₁). Either point can go first, provided the same point goes first in both the top and the bottom.
Key facts
- Slope through two points = (y₂ − y₁) ⁄ (x₂ − x₁).
- Midpoint of (x₁, y₁) and (x₂, y₂) = ((x₁ + x₂)⁄2, (y₁ + y₂)⁄2).
- Two perpendicular lines, neither of them vertical, have slopes whose product is −1.
Study next
Slope of a line perpendicular to a given lineEquation of a line through two points (point-slope form)Perpendicular bisector of a segment
Common traps
- Dividing run by rise: 2 ÷ 4 = 1⁄2.
- Mixing the order of subtraction, (7 − 3) ⁄ (2 − 4) = −2, which flips the sign.
15 Sep 2025, 16:00, Quant Q.23 asks for the slope of a line perpendicular to y = −3x + 7 (keyed 1⁄3).
Quant Q.19 of this paper needs the same step first: the segment from (2, 8) to (6, 4) has slope −1 before its perpendicular bisector, y = x + 2, can be written.
Related PYQs
No directly related past PYQ was found.