For what value of $k$ will the following pair of linear equations have no solution?
$2 x + 3 y = 1$ and $(3 k - 1) x + (1 - 2 k) y = 2 k + 3$

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Solution

Given equations are,
$2 x + 3 y = 1$ ....(1)
and $(3 k - 1) x + (1 - 2 k) y = 2 k + 3$ .....(2)
The pairs of linear equations have no solution, if
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
$\Rightarrow \frac{2}{3 k - 1} = \frac{3}{1 - 2 k}$
$\Rightarrow 2 (1 - 2 k) = 3 (3 k - 1)$
$\Rightarrow 2 - 4 k = 9 k - 3$
$\Rightarrow 13 k = 5$
$\Rightarrow k = \frac{5}{13}$
Therefore, for $k = \frac{5}{13}$ , the equations have no solution.

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