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Question
x + y + z + w = 2
x − 2y + 2z + 2w = − 6
2x + y − 2z + 2w = − 5
3x − y + 3z − 3w = − 3
Solution
\[D = \begin{vmatrix}1 & 1 & 1 & 1 \\ 1 & - 2 & 2 & 2 \\ 2 & 1 & - 2 & 2 \\ 3 & - 1 & 3 & - 3\end{vmatrix}\]
\[1\begin{vmatrix}- 2 & 2 & 2 \\ 1 & - 2 & 2 \\ - 1 & 3 & - 3\end{vmatrix} - 1\begin{vmatrix}1 & 2 & 2 \\ 2 & - 2 & 2 \\ 3 & 3 & - 3\end{vmatrix} + 1\begin{vmatrix}1 & - 2 & 2 \\ 2 & 1 & 2 \\ 3 & - 1 & - 3\end{vmatrix} - 1\begin{vmatrix}1 & - 2 & 2 \\ 2 & 1 & - 2 \\ 3 & - 1 & 3\end{vmatrix}\]
\[ = 1\left[ - 2\left( 6 - 6 \right) - 2\left( - 3 + 2 \right) + 2\left( 3 - 2 \right) \right] - 1\left[ 1\left( 6 - 6 \right) - 2\left( - 6 - 6 \right) + 2\left( 6 + 6 \right) \right] + 1\left[ 1\left( - 3 + 2 \right) + 2\left( - 6 - 6 \right) + 2\left( - 2 - 3 \right) \right] - 1\left[ 1\left( 3 - 2 \right) + 2\left( 6 + 6 \right) + 2\left( - 2 - 3 \right) \right]\]
\[ = 4 - 48 - 35 - 15\]
\[ = - 94\]
\[ D_1 = \begin{vmatrix}2 & 1 & 1 & 1 \\ - 6 & - 2 & 2 & 2 \\ - 5 & 1 & - 2 & 2 \\ - 3 & - 1 & 3 & - 3\end{vmatrix}\]
\[2\begin{vmatrix}- 2 & 2 & 2 \\ 1 & - 2 & 2 \\ - 1 & 3 & - 3\end{vmatrix} - 1\begin{vmatrix}- 6 & 2 & 2 \\ - 5 & - 2 & 2 \\ - 3 & 3 & - 3\end{vmatrix} + 1\begin{vmatrix}- 6 & - 2 & 2 \\ - 5 & 1 & 2 \\ - 3 & - 1 & - 3\end{vmatrix} - 1\begin{vmatrix}- 6 & - 2 & 2 \\ - 5 & 1 & - 2 \\ - 3 & - 1 & 3\end{vmatrix}\]
\[ = 2\left[ - 2\left( 6 - 6 \right) - 2\left( - 3 + 2 \right) + 2\left( 3 - 2 \right) \right] - 1\left[ - 6\left( 6 - 6 \right) - 2\left( 15 + 6 \right) + 2\left( - 15 - 6 \right) \right] + 1\left[ - 6\left( - 3 + 2 \right) + 2\left( 15 + 6 \right) + 2\left( 5 + 3 \right) \right] - 1\left[ - 6\left( 3 - 2 \right) + 2\left( - 15 - 6 \right) + 2\left( 5 + 3 \right) \right]\]
\[ = 188\]
\[ D_2 = \begin{vmatrix}1 & 2 & 1 & 1 \\ 1 & - 6 & 2 & 2 \\ 2 & - 5 & - 2 & 2 \\ 3 & - 3 & 3 & - 3\end{vmatrix}\]
\[1\begin{vmatrix}- 6 & 2 & 2 \\ - 5 & - 2 & 2 \\ - 3 & 3 & - 3\end{vmatrix} - 2\begin{vmatrix}1 & 2 & 2 \\ 2 & - 2 & 2 \\ 3 & 3 & - 3\end{vmatrix} + 1\begin{vmatrix}1 & - 6 & 2 \\ 2 & - 5 & 2 \\ 3 & - 3 & - 3\end{vmatrix} - 1\begin{vmatrix}1 & - 6 & 2 \\ 2 & - 5 & - 2 \\ 3 & - 3 & 3\end{vmatrix}\]
\[1\left[ - 6\left( 6 - 6 \right) - 2\left( 15 + 6 \right) + 2\left( - 15 - 6 \right) \right] - 2\left[ 1\left( 6 - 6 \right) - 2\left( - 6 - 6 \right) + 2\left( 6 + 6 \right) \right] + 1\left[ 1\left( 15 + 6 \right) + 6\left( - 6 - 6 \right) + 2\left( - 6 + 15 \right) \right] - 1\left[ 1\left( - 15 - 6 \right) - 6\left( 6 + 6 \right) + 2\left( - 6 + 15 \right) \right]\]
\[ = 1\]
\[ D_3 = \begin{vmatrix}1 & 1 & 2 & 1 \\ 1 & - 2 & - 6 & 2 \\ 2 & 1 & - 5 & 2 \\ 3 & - 1 & - 3 & - 3\end{vmatrix}\]
\[1\begin{vmatrix}- 2 & - 6 & 2 \\ 1 & - 5 & 2 \\ - 1 & - 3 & - 3\end{vmatrix} - 1\begin{vmatrix}1 & - 6 & 2 \\ 2 & - 5 & 2 \\ 3 & - 3 & - 3\end{vmatrix} + 2\begin{vmatrix}1 & - 2 & 2 \\ 2 & 1 & 2 \\ 3 & - 1 & - 3\end{vmatrix} - 1\begin{vmatrix}1 & - 2 & - 6 \\ 2 & 1 & - 5 \\ 3 & - 1 & - 3\end{vmatrix}\]
\[ = 1\left[ - 2\left( 15 + 6 \right) + 6\left( - 3 + 2 \right) + 2\left( - 3 - 5 \right) \right] - 1\left[ 1\left( 15 + 6 \right) + 6\left( - 6 - 6 \right) + 2\left( - 6 + 15 \right) \right] + 2\left[ 1\left( - 3 + 2 \right) + 2\left( - 6 - 6 \right) + 2\left( - 2 - 3 \right) \right] - 1\left[ 1\left( - 3 - 5 \right) + 2\left( - 6 + 15 \right) - 6\left( - 2 - 3 \right) \right]\]
\[ = - 141\]
\[ D_4 = \begin{vmatrix}1 & 1 & 1 & 2 \\ 1 & - 2 & 2 & - 6 \\ 2 & 1 & - 2 & - 5 \\ 3 & - 1 & 3 & - 3\end{vmatrix}\]
\[1\begin{vmatrix}- 2 & 2 & - 6 \\ 1 & - 2 & - 5 \\ - 1 & 3 & - 3\end{vmatrix} - 1\begin{vmatrix}1 & 2 & - 6 \\ 2 & - 2 & - 5 \\ 3 & 3 & - 3\end{vmatrix} + 1\begin{vmatrix}1 & - 2 & - 6 \\ 2 & 1 & - 5 \\ 3 & - 1 & - 3\end{vmatrix} - 2\begin{vmatrix}1 & - 2 & 2 \\ 2 & 1 & - 2 \\ 3 & - 1 & - 3\end{vmatrix}\]
\[1\left[ - 2\left( 6 + 15 \right) - 2\left( - 3 - 5 \right) - 6\left( 3 - 2 \right) \right] - 1\left[ 1\left( 6 + 15 \right) - 2\left( - 6 + 15 \right) - 6\left( 6 + 6 \right) \right] + 1\left[ 1\left( - 3 - 5 \right) + 2\left( - 6 + 15 \right) - 6\left( - 2 - 3 \right) \right] - 2\left[ 1\left( - 3 - 2 \right) + 2\left( - 6 + 6 \right) + 2\left( - 2 - 3 \right) \right]\]
\[ = 47\]
Thus,
\[x = \frac{D_1}{D} = \frac{188}{- 94} = - 2\]
\[y = \frac{D_2}{D} = \frac{- 282}{- 94} = 3\]
\[z = \frac{D_3}{D} = \frac{- 141}{- 94} = 1 . 5\]
\[w = \frac{D_4}{D} = \frac{47}{- 94} = - 0 . 5\]
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