Which of the following is true about the system of equations given by:
$2 x + 4 y + 6 z = 8 x + 2 y + 3 z = 5 x + y + 3 z = 4$

It has unique solution.

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It is inconsistant.

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It has

3

solution.

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cant be determined.

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Solution

The correct option is B It is inconsistant.
The given system is
$⎡ ⎢ ⎣ \begin{matrix} 2 & 4 & 6 1 & 2 & 3 1 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 854 \end{matrix} ⎤ ⎥ ⎦$
$A X = B$
$A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 4 & 6 1 & 2 & 3 1 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦, X = ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ and B = ⎡ ⎢ ⎣ \begin{matrix} 854 \end{matrix} ⎤ ⎥ ⎦$
Clearly, on simplifying, $| A | = 0$
$A_{y} = ∣ ∣ ∣ ∣ \begin{matrix} 3 & 8 & - 1 - 6 & 5 & 2 6 & 4 & 0 \end{matrix} ∣ ∣ ∣ ∣ \neq 0 and A_{x} = 0, A_{z} = 0$
Thus, the system of equation is inconsistent.

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2 x + p y + 6 z = 8, x + 2 y + q z = 5, x + y + 3 z = 4

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2 x + 4 y + 6 z = 8

x + 2 y + 3 z = 5

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\frac{2}{x} - \frac{3}{y} + \frac{3}{z} = 10 \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 10 \frac{3}{x} - \frac{1}{y} + \frac{2}{z} = 13

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