If the system of linear equations kx + (k + 2)y = 12 and (k + 2)x + 9y = 2(k + 5) has infinite solutions, then the value of k is

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Solution

The correct option is A 4
For a system of linear equations $a_{1} x + b_{1} y = c_{1} a n d a_{2} x + b_{2} y = c_{2}$ to have infinitely many solution,
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
∴ For the given system of linear equations, the condition for having infinitely many solutions is
$\frac{k}{k + 2} = \frac{k + 2}{9} = \frac{12}{2 (k + 5)}$
$\Rightarrow \frac{k}{k + 2} = \frac{k + 2}{9}$
$\Rightarrow 9 k = k^{2} + 4 k + 4$
$\Rightarrow k^{2} - 5 k + 4 = 0$
$\Rightarrow k^{2} - 4 k - k + 4 = 0$
$\Rightarrow k (k - 4) - 1 (k - 4) = 0$
$\Rightarrow (k - 1) (k - 4) = 0$
$\Rightarrow k = 1 o r k = 4 . . . . . (i)$

Also, $\frac{k}{k + 2} = \frac{12}{2 (k + 5)}$
$\Rightarrow k^{2} + 5 k = 6 k + 12$
$\Rightarrow k^{2} - k - 12 = 0$
$\Rightarrow k^{2} - 4 k + 3 k - 12 = 0$
$\Rightarrow k (k - 4) + 3 (k - 4) = 0$
$\Rightarrow (k - 4) (k + 3) = 0$
$\Rightarrow k = 4 o r k = - 3 . . . . . . (i i)$
So, from (i) and (ii), it is observed that for k = 4, the condition is satisfied.
Hence, the correct answer is option (1).

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