Question

Find the value of k for which the following system of equations has infinitely many solutions : x + (k+1)y = 4 and (k+1)x + 9y = 5k+2

Answer 1

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Answer 2

The two equations given are :-

$x + (k + 1)y = 4 \\ \\(k + 1)x + 9y = 5k + 2$

Now,

For infinitely many solutions we have :-

$\frac{a1}{a2} = \frac{b1}{b2} = \frac{c1}{c2}$

Hence,

Accordingly :-

$\frac{1}{k + 1} = \frac{k + 1}{9} = \frac{4}{5k + 2}$

Solving for k

Taking first two cases:-

$\frac{1}{k + 1} = \frac{k + 1}{9} \\ \\ \\ {(k + 1)}^{2}=9 \\ \\ \\k + 1 =\pm \sqrt{9} = \pm 3 \\ \\k = 2\: or \: ( - 4)$

Taking the last case as well

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

Putting down k=2 and k=(-4)

The equation satisfies only when k=2.

Hence,

Value of k is equal to 2