For what value of k will be following pair of linear equations have infinetly many solutions?
$2 x - 3 y = 7$

$(k + 1) x + (1 - 2 k) y = 5 k - 4$

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Solution

Completing the incomplete question as,
For what value of k will be following pair of linear equations have infinetly many solutions?

$2 x - 3 y = 7$

$(k + 1) x + (1 - 2 k) y = 5 k - 4$

Condition required for infinetly many solutions is,

$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$

Given equations are:

$2 x - 3 y = 7$

$(k + 1) x + (1 - 2 k) y = 5 k - 4$

So,

$\frac{2}{k + 1} = \frac{- 3}{1 - 2 k} = \frac{7}{5 k - 4}$

Considering first 2 equations, we get,

$\frac{2}{k + 1} = \frac{- 3}{1 - 2 k}$

$2 (1 - 2 k) = - 3 (k + 1)$

$2 - 4 k = - 3 k - 3$

$\Rightarrow k = 5$

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