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Question
For which value of k will the following pair of linear equations have no solution?
3x + y = 1
(2k – 1)x + (k – 1)y = 2k + 1
Solution
3x + y – 1 = 0
(2k – 1)x + (k – 1)y – (2k + 1) = 0
`a_1/a_2 = 3/(2k-1)`
`b_1/b_2 = 1/(k-1)`
`c_1/c_2 = (-1)/(-2k-1) = 1/(2k+1)`
For no solutions,
`a_1/a_2 = b_1/b_2 ≠ c_1/c_2`
`3/(2k-1) = 1/(k-1) ≠ 1/(2k+1)`
`3/(2k-1) = 1/(k-1)`
3k – 3 = 2k – 1
3k - 2k = -1 + 3
k = 2
Hence, for k = 2, the given equation has no solution.
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