Find the value of $k$ for which each of the following systems of equations have infinitely many solution:
$2 x + 3 y = k$
$(k - 1) x + (k + 2) y = 3 k$

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Solution

For the system of equations to have infinitely many solutions, the lines must coincide.

$\Rightarrow$ The ratio of corresponding coefficients must be equal.

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

Equation 1st and 3rd equality ;

$\Rightarrow 6 = k - 1$

$\Rightarrow k = 7$

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Find the value of k for which the following pair of linear equations have infinitely many solutions:

$2 x + 3 y = 7, (k - 1) x + (k + 2) y = 3 k .$

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2 x + 3 y = 7, (k - 1) x + (k + 2) y = 3 k

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