Solve the system of equations
$x + 2 y + 3 z = 1 2 x + 3 y + 2 z = 2 3 x + 3 y + 4 z = 1$
Find the value of $14 (x + y + z)$

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Solution

We have
$x + 2 y + 3 z = 1 2 x + 3 y + 2 z = 2 3 x + 3 y + 4 z = 1$
The given system of equations in the matrix form are written as below
$⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 2 & 3 & 2 3 & 3 & 4 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 121 \end{matrix} ⎤ ⎥ ⎦$
$A X = B$
$\Rightarrow X = A^{- 1} B$
Where
$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 2 & 3 & 2 3 & 3 & 4 \end{matrix} ⎤ ⎥ ⎦, X = ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦$ and $B = ⎡ ⎢ ⎣ \begin{matrix} 121 \end{matrix} ⎤ ⎥ ⎦$
$| A | = 1 (12 - 6) - 2 (8 - 6) + 3 (6 - 9)$
$= 6 - 4 - 9 = 7 \neq 0$
Let $C$ be the matrix of cofactors of elements in $| A |$ .
Then
$C = ⎡ ⎢ ⎣ \begin{matrix} C_{11} & C_{12} & C_{13} C_{21} & C_{22} & C_{23} C_{31} & C_{32} & C_{33} \end{matrix} ⎤ ⎥ ⎦$
Here
$C_{11} = ∣ ∣ ∣ \begin{matrix} 3 & 2 3 & 4 \end{matrix} ∣ ∣ ∣ = 6; C_{12} = - ∣ ∣ ∣ \begin{matrix} 2 & 2 3 & 4 \end{matrix} ∣ ∣ ∣ = - 2; C_{13} = ∣ ∣ ∣ \begin{matrix} 2 & 3 3 & 3 \end{matrix} ∣ ∣ ∣ = - 3$
$C_{21} = - ∣ ∣ ∣ \begin{matrix} 2 & 3 3 & 4 \end{matrix} ∣ ∣ ∣ = 1; C_{22} = ∣ ∣ ∣ \begin{matrix} 1 & 3 3 & 4 \end{matrix} ∣ ∣ ∣ = - 5; C_{23} = - ∣ ∣ ∣ \begin{matrix} 1 & 2 3 & 3 \end{matrix} ∣ ∣ ∣ = 3$
$C_{31} = ∣ ∣ ∣ \begin{matrix} 2 & 3 3 & 2 \end{matrix} ∣ ∣ ∣ = - 5; C_{32} = - ∣ ∣ ∣ \begin{matrix} 1 & 3 2 & 2 \end{matrix} ∣ ∣ ∣ = 4; C_{33} = ∣ ∣ ∣ \begin{matrix} 1 & 2 2 & 3 \end{matrix} ∣ ∣ ∣ = - 1$
$∴ C = ⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & - 3 1 & - 5 & 3 - 5 & 4 & - 1 \end{matrix} ⎤ ⎥ ⎦$
$∴ A d j . A = C^{'} = ⎡ ⎢ ⎣ \begin{matrix} 6 & 1 & - 5 - 2 & - 5 & 4 - 3 & 3 & - 1 \end{matrix} ⎤ ⎥ ⎦$
$∴ A^{- 1} = \frac{A d j A}{| A |} = - \frac{1}{7} ⎡ ⎢ ⎣ \begin{matrix} 6 & 1 & - 5 - 2 & - 5 & 4 - 3 & 3 & - 1 \end{matrix} ⎤ ⎥ ⎦$
$= ⎡ ⎢ ⎣ \begin{matrix} 6 / 7 & 1 / 7 & - 5 / 7 - 2 / 7 & - 5 / 7 & 4 / 7 - 3 / 7 & 3 / 7 & - 1 / 7 \end{matrix} ⎤ ⎥ ⎦$
$∴ A^{- 1} B = ⎡ ⎢ ⎣ \begin{matrix} 6 / 7 & 1 / 7 & - 5 / 7 - 2 / 7 & - 5 / 7 & 4 / 7 - 3 / 7 & 3 / 7 & - 1 / 7 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 121 \end{matrix} ⎤ ⎥ ⎦$
$⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} - 3 / 7 8 / 7 - 2 / 7 \end{matrix} ⎤ ⎥ ⎦ (∵ A^{- 1} B = X)$
$∴ x = - 3 / 7, y = 8 / 7, z = - 2 / 7$
$∴ 14 (x + y + z) = 6$

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