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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

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#### 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

k = 2

Hence, for k = 2, the given equation has no solution.

Concept: Algebraic Methods of Solving a Pair of Linear Equations - Cross - Multiplication Method

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