If the system of equations 3 x + y + 4 z = 3 2 x + y ? z = ? 3 x + 2 y + z = 4 has no solution, then the value of is equal to : [2026]

Question

If the system of equations

$\begin{matrix} 3 x + y + 4 z = 3 \\ 2 x + α y - z = - 3 \\ x + 2 y + z = 4 \end{matrix}$

has no solution, then the value of $α$ is equal to : [2026]

Smart Achievers · Accepted Answer

(1)

$for no solution Δ = 0$

$| \begin{matrix} 3 & 1 & 4 \\ 2 & α & - 1 \\ 1 & 2 & 1 \end{matrix} |$ $= 0$

$\Rightarrow 3 (α + 2) + 1 (- 1 - 2) + 4 (4 - α) = 0$

$\Rightarrow 19 - α = 0 \Rightarrow α = 19$

$For α = 19$

$Δ_{x} =$ $| \begin{matrix} 3 & 1 & 4 \\ - 3 & 19 & - 1 \\ 4 & 2 & 1 \end{matrix} |$ $= 3 (21) + 1 (- 1) + 4 (- 82)$

$\neq 0$

$∴$ $no solution for α = 19$

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