For the system of linear equations2x−y+3z=53x+2y−z=74x+5y+αz=β,Which of the following is NOT correct?

The system has a unique solution for α ≠ − 5 and β = 8

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For the system of linear equations
$\begin{array}{l} 2 x - y + 3 z = 5 \\ 3 x + 2 y - z = 7 \\ 4 x + 5 y + αz = β, \end{array}$
Which of the following is NOT correct?

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The system has infinitely many solutions for $α = - 5 and β = 9$

The system has a unique solution for $α \neq - 5 and β = 8$

The system is inconsistent for $α = - 5 and β = 8$

The system has infinitely many solutions for $α = - 6 and β = 9$

answer is D.

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Detailed Solution

$Augment matrix [\begin{array}{cccc} 2 & - 1 & 3 & 5 \\ 3 & 2 & - 1 & 7 \\ 4 & 5 & α & β \end{array}] ∼ [\begin{matrix} 2 & - 1 & 3 & 5 \\ 0 & 7 & - 11 & - 1 \\ 0 & 7 & α - 6 & β - 10 \end{matrix}] ∼ [\begin{matrix} 2 & - 1 & 3 & 5 \\ 0 & 7 & - 11 & - 1 \\ 0 & 0 & α + 5 & β - 9 \end{matrix}]$