For what value of k the following pair of linear equations has no solution 3x y 11 and 6x 2y 18

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

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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Last updated at Oct. 27, 2020 by Teachoo

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Question 3 For what value of k, the pair of linear equations 3x + y = 3 and 6x + ky = 8 does not have a solution.3x + y − 3 = 0 Comparing with a1x + b1y + c1 = 0 ∴ a1 = 3 , b1 = 1 , c1 = −3 6x + ky − 8 = 0 Comparing with a2x + b2y + c2 = 0 ∴ a2 = 6 , b2 = k, c2 = −8 Therefore, a1 = 3 , b1 = 1 , c1 = −3 & a2 = 6 , b2 = k, c2 = −8 𝒂𝟏/𝒂𝟐 𝑎1/𝑎2 = 3/6 𝑎1/𝑎2 = 1/2 𝒃𝟏/𝒃𝟐 𝑏1/𝑏2 = 1/𝑘 𝒄𝟏/𝒄𝟐 𝑐1/𝑐2 = (−3)/(−8) 𝑐1/𝑐2 = 3/8 Since the equations don’t have a solution 𝑎1/𝑎2 = 𝑏1/𝑏2 ≠ 𝑐1/𝑐2 1/2=1/𝑘≠3/8 Thus, 1/2=1/𝑘 k = 2