For what value of k does the pair of equations 5x+2y=2k and 2(k+1)x+ky=3k+4 have an infinite number of solutions?

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Solution

Consider the problem

Given,
$\begin{matrix} 5 x + 2 y = 2 k . . . . (1) 2 (k + 1) x + k y = (3 k + 4) . . . . (2) \end{matrix}$

Here,
$\begin{matrix} a_{1} = 5, b_{1} = 2, c_{1} = 2 k a_{2} = 2 (k + 1), b_{2} = k, c_{2} = 3 k + 4 \end{matrix}$

$\begin{matrix} \Rightarrow \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} ∴ \frac{5}{2 (k + 1)} = \frac{2}{k} = \frac{2 k}{3 k + 4} ∴ \frac{5}{2 (k + 1)} = \frac{2}{k} ∴ 5 k = 2 \times 2 (k + 1) ∴ 5 k = 4 k + 4 ∴ 5 k - 4 k = 4 ∴ k = 4 \end{matrix}$

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5 x + 2 y = 2 k and 2 (k + 1) x + ky = (3 k + 4)

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5 x + 2 y = 2 k, 2 (k + 1) x + k y = (3 k + 4) .

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