Find the value of ‘k’ for which the following system of equations represents a pair of coincident lines:
x + 2y = 3; (k − 1)x + (k + 1)y = k + 3.
Answer: k = 3 (the second equation becomes 2x + 4y = 6).
Step-by-step solution
Idea: Coincident lines ⇔ a1a2 = b1b2 = c1c2.
- 1k − 1 = 2k + 1 = 3k + 3.½ mark
- First pair: k + 1 = 2(k − 1) ⇒ k + 1 = 2k − 2 ⇒ k = 3.1 mark
- Check the third ratio: 2k + 1 = 3k + 3 ⇒ 2k + 6 = 3k + 3 ⇒ k = 3 ✓, the same value.1 mark
- So k = 3: the second equation is 2x + 4y = 6, which is 2 × (x + 2y = 3).½ mark
k = 3.
Check: k = 3: 2x + 4y = 6; dividing by 2 gives x + 2y = 3 ✓.
Answer to write in the exam
Coincident ⇒ 1k − 1 = 2k + 1 = 3k + 3
1k − 1 = 2k + 1 ⇒ k + 1 = 2k − 2 ⇒ k = 3
2k + 1 = 3k + 3 ⇒ 2k + 6 = 3k + 3 ⇒ k = 3
∴ k = 3
Common mistakes that cost marks
- Using only one pair of ratios and not checking that the constant ratio also agrees (otherwise the lines could be parallel).
- Cross-multiplying as 1 × (k − 1) = 2 × (k + 1) instead of 1 × (k + 1) = 2 × (k − 1).
How this can come in the exam
MCQ (1 mark)
The lines kx + 2y = 4 and 6x + 4y = 8 coincide when k =
- 2
- 3
- 6
- 12
Show answer
(B) 3
k6 = 24 = 48 = 12 ⇒ k = 3.
Try one yourself
For what k are 2x + 3y = 5 and 4x + ky = 10 coincident?
Show answer
24 = 3k = 510 ⇒ k = 6.
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