First slide

System of linear equations

Question

The system of equations $- 2 x + y + z = a, x - 2 y + z = b, x + y - 2 z = c$ is consistent if

Moderate

A

$a + b + c = 0$
B

$a + b + c \leq 0$
C

$a + b + c \neq 0$
D

$a + b + c \geq 0$

Solution

$W e h a v e Δ = |\begin{matrix} - 2 & 1 & 1 \\ 1 & - 2 & 1 \\ 1 & 1 & - 2 \end{matrix}| = - 2 (4 - 1) - 1 (- 2 - 1) + 1 (1 + 2) = 0 ∴ F o r c o n s i s t e n c y w e m u s t h a v e Δ_{1} = Δ_{2} = Δ_{3} = 0 \Rightarrow |\begin{matrix} a & 1 & 1 \\ b & - 2 & 1 \\ c & 1 & - 2 \end{matrix}| = 0 \Rightarrow a (4 - 1) - 1 (- 2 b - c) + 1 (b + 2 c) = 0 \Rightarrow 3 (a + b + c) = 0 \Rightarrow a + b + c = 0$

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