The system of equations $2 x + 6 y + 11 = 0$ , $6 y - 18 z + 1 = 0, 6 x + 20 y - 6 z + 3 = 0$ ,

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is consistent

Has unique solution

is inconsistent

Cannot be determined

answer is C.

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Detailed Solution

$A u g m e n t e d m a t r i x [A D] = [\begin{matrix} 2 & 6 & 0 & - 11 \\ 0 & 6 & - 18 & - 1 \\ 6 & 20 & - 6 & - 3 \end{matrix}] R_{3} \to R_{3} - 3 R_{1} ~ [\begin{matrix} 1 & 6 & 0 & 11 \\ 0 & 6 & - 18 & - 1 \\ 0 & 2 & - 6 & 30 \end{matrix}] R_{3} \to 3 R_{3} - R_{2} ~ [\begin{matrix} 1 & 6 & 0 & 11 \\ 0 & 6 & - 18 & - 1 \\ 0 & 0 & 0 & 91 \end{matrix}] H e r e r a n k A = 2 a n d r a n k [A D] = 3 \Rightarrow r a n k A \neq r a n k [A D] \Rightarrow T h e g i v e n s y s t e m h a s n o s o l u t i o n$