System of linear equations

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Question

If the system of equations ${(k + 1)}^{3} x + {(k + 2)}^{3} y = {(k + 3)}^{3}$ , $(k + 1) x + (k + 2) y = k + 3$ , $x + y = 1$ is consistent. Then the value of $k$ is

Moderate

Solution

$|\begin{matrix} {(k + 1)}^{3} & {(k + 2)}^{3} & {(k + 3)}^{3} \\ k + 1 & k + 2 & k + 3 \\ 1 & 1 & 1 \end{matrix}| = 0 \Rightarrow k = - 2$

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