How many lines exist that
(i) have a given slope?
(ii) have a given slope and pass through a given point?
- (i) have a given slope?
- (ii) have a given slope and pass through a given point?
Answer: (i) Infinitely many (all parallel to each other). (ii) Exactly one.
Step-by-step solution
Idea: Lines with slope m are y = mx + d; changing d slides the line up or down. Fixing a point fixes d.
(i) have a given slope?
- All lines y = mx + d have slope m; each value of d gives a different line. So there are infinitely many, all parallel.1 mark
Infinitely many
(ii) have a given slope and pass through a given point?
- If the line passes through (p, q): q = mp + d ⇒ d = q − mp, a single value. So there is exactly one such line.1 mark
Exactly one
(i) Infinitely many; (ii) exactly one.
Check: Slope 2 through (1, 5): d = 5 − 2 = 3, so the only line is y = 2x + 3 ✓.
Answer to write in the exam
(i)
Lines with slope m: y = mx + d, d any real number
∴ Infinitely many (parallel) lines
(ii)
Through (p, q): q = mp + d ⇒ d = q − mp (one value)
∴ Exactly one line
Common mistakes that cost marks
- Answering “one” for (i): many different lines can share the same steepness.
- Answering “infinitely many” for (ii): once the point is fixed, the line cannot slide.
How this can come in the exam
MCQ (1 mark)
The line with slope −2 through (0, 5) is
- y = 5x − 2
- y = −2x + 5
- y = 2x + 5
- y = −2x − 5
Show answer
(B) y = −2x + 5
y = mx + d with m = −2, d = 5.
Try one yourself
Find the line with slope 3 passing through (2, 1).
Show answer
1 = 6 + d ⇒ d = −5: y = 3x − 5.
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