Can you prove that two lines of equal slope are parallel?
(Hint: Consider the equations of the two lines to be y = mx + d1 and y = mx + d2 )
Step-by-step solution
Idea: Parallel lines are lines in the same plane that never meet. Show that no point can lie on both lines.
- Let the lines be y = mx + d1 and y = mx + d2, with the same slope m and d1 ≠ d2 (different lines).½ mark
- Suppose a point (p, q) lies on both. Then q = mp + d1 and q = mp + d2.½ mark
- Subtracting: 0 = d1 − d2, so d1 = d2. This contradicts d1 ≠ d2.1 mark
- So no point lies on both lines; they never meet and are parallel. (Another view: at every x, the second line is exactly d2 − d1 above the first, a constant gap.)1 mark
Check: y = 2x + 1 and y = 2x − 3: at x = 0 they are 4 apart, at x = 5 they are 11 and 7, still 4 apart ✓.
Answer to write in the exam
Lines: y = mx + d1, y = mx + d2, d1 ≠ d2
Suppose (p, q) is on both: q = mp + d1 and q = mp + d2
⇒ d1 = d2, a contradiction
∴ No common point ⇒ the lines are parallel
Common mistakes that cost marks
- Drawing two examples and calling it a proof.
- Forgetting the case d1 = d2, where the “two” lines are the same line.
How this can come in the exam
Assertion (A): The lines y = 3x + 2 and y = 3x − 5 do not intersect.
Reason (R): Lines with equal slopes and different y-intercepts are parallel.
- Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
- Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
- Assertion (A) is true, but Reason (R) is false.
- Assertion (A) is false, but Reason (R) is true.
Show answer
(A) Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
Both true; R explains A.
Try one yourself
Is the line 6x − 2y = 7 parallel to y = 3x + 1?
Show answer
y = 3x − 72: slope 3, intercept −72 ≠ 1 ⇒ yes, parallel.
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