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The equations $x + y + z = 0, 2 x - y - 3 z = 0, 3 x - 5 y + 4 z = 0$ have

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$| \begin{matrix} 1 & 1 & 1 \\ 2 & - 1 & - 3 \\ 3 & - 5 & 4 \end{matrix} | = - 43 \neq 0$ $∴$ Given equations has unique solution

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