First slide

System of linear equations

Question

The system of linear equations $3 x - 2 y - k z = 10, 2 x - 4 y - 2 z = 6, x + 2 y - z = 5 m$ is inconsistent if

Moderate

A

$k \neq 3, m \in R$
B

$k = 3, m \neq \frac{4}{5}$
C

$k \neq 3, m \neq \frac{4}{5}$
D

$k = 3, m = \frac{4}{5}$

Solution

$Δ = |\begin{matrix} 3 & - 2 & - k \\ 1 & - 2 & - 1 \\ 1 & 2 & - 1 \end{matrix}| = 0 \Rightarrow k = 3$
$Δ_{x} = |\begin{matrix} 10 & - 2 & - 3 \\ 3 & - 2 & - 1 \\ 5 m & 2 & - 1 \end{matrix}| = 0, Δ_{y} = |\begin{matrix} 3 & 10 & - 3 \\ 1 & 3 & - 1 \\ 1 & 5 m & - 1 \end{matrix}| = 4 (4 - 5 m), Δ_{z} = |\begin{matrix} 3 & - 2 & 10 \\ 1 & - 2 & 3 \\ 1 & 2 & 5 m \end{matrix}| = 4 (4 - 5 m)$
$H e n c e, k = 3 a n d m \neq \frac{4}{5}$

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