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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

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Solution

## $D=\left|\begin{array}{cccc}1& 1& 1& 1\\ 1& -2& 2& 2\\ 2& 1& -2& 2\\ 3& -1& 3& -3\end{array}\right|\phantom{\rule{0ex}{0ex}}1\left|\begin{array}{ccc}-2& 2& 2\\ 1& -2& 2\\ -1& 3& -3\end{array}\right|-1\left|\begin{array}{ccc}1& 2& 2\\ 2& -2& 2\\ 3& 3& -3\end{array}\right|+1\left|\begin{array}{ccc}1& -2& 2\\ 2& 1& 2\\ 3& -1& -3\end{array}\right|-1\left|\begin{array}{ccc}1& -2& 2\\ 2& 1& -2\\ 3& -1& 3\end{array}\right|\phantom{\rule{0ex}{0ex}}=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]\phantom{\rule{0ex}{0ex}}=4-48-35-15\phantom{\rule{0ex}{0ex}}=-94\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{D}_{1}=\left|\begin{array}{cccc}2& 1& 1& 1\\ -6& -2& 2& 2\\ -5& 1& -2& 2\\ -3& -1& 3& -3\end{array}\right|\phantom{\rule{0ex}{0ex}}2\left|\begin{array}{ccc}-2& 2& 2\\ 1& -2& 2\\ -1& 3& -3\end{array}\right|-1\left|\begin{array}{ccc}-6& 2& 2\\ -5& -2& 2\\ -3& 3& -3\end{array}\right|+1\left|\begin{array}{ccc}-6& -2& 2\\ -5& 1& 2\\ -3& -1& -3\end{array}\right|-1\left|\begin{array}{ccc}-6& -2& 2\\ -5& 1& -2\\ -3& -1& 3\end{array}\right|\phantom{\rule{0ex}{0ex}}=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]\phantom{\rule{0ex}{0ex}}=188\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{D}_{2}=\left|\begin{array}{cccc}1& 2& 1& 1\\ 1& -6& 2& 2\\ 2& -5& -2& 2\\ 3& -3& 3& -3\end{array}\right|\phantom{\rule{0ex}{0ex}}1\left|\begin{array}{ccc}-6& 2& 2\\ -5& -2& 2\\ -3& 3& -3\end{array}\right|-2\left|\begin{array}{ccc}1& 2& 2\\ 2& -2& 2\\ 3& 3& -3\end{array}\right|+1\left|\begin{array}{ccc}1& -6& 2\\ 2& -5& 2\\ 3& -3& -3\end{array}\right|-1\left|\begin{array}{ccc}1& -6& 2\\ 2& -5& -2\\ 3& -3& 3\end{array}\right|\phantom{\rule{0ex}{0ex}}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]\phantom{\rule{0ex}{0ex}}=1\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{D}_{3}=\left|\begin{array}{cccc}1& 1& 2& 1\\ 1& -2& -6& 2\\ 2& 1& -5& 2\\ 3& -1& -3& -3\end{array}\right|\phantom{\rule{0ex}{0ex}}1\left|\begin{array}{ccc}-2& -6& 2\\ 1& -5& 2\\ -1& -3& -3\end{array}\right|-1\left|\begin{array}{ccc}1& -6& 2\\ 2& -5& 2\\ 3& -3& -3\end{array}\right|+2\left|\begin{array}{ccc}1& -2& 2\\ 2& 1& 2\\ 3& -1& -3\end{array}\right|-1\left|\begin{array}{ccc}1& -2& -6\\ 2& 1& -5\\ 3& -1& -3\end{array}\right|\phantom{\rule{0ex}{0ex}}=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]\phantom{\rule{0ex}{0ex}}=-141\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{D}_{4}=\left|\begin{array}{cccc}1& 1& 1& 2\\ 1& -2& 2& -6\\ 2& 1& -2& -5\\ 3& -1& 3& -3\end{array}\right|\phantom{\rule{0ex}{0ex}}1\left|\begin{array}{ccc}-2& 2& -6\\ 1& -2& -5\\ -1& 3& -3\end{array}\right|-1\left|\begin{array}{ccc}1& 2& -6\\ 2& -2& -5\\ 3& 3& -3\end{array}\right|+1\left|\begin{array}{ccc}1& -2& -6\\ 2& 1& -5\\ 3& -1& -3\end{array}\right|-2\left|\begin{array}{ccc}1& -2& 2\\ 2& 1& -2\\ 3& -1& -3\end{array}\right|\phantom{\rule{0ex}{0ex}}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]\phantom{\rule{0ex}{0ex}}=47\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Thus},\phantom{\rule{0ex}{0ex}}x=\frac{{D}_{1}}{D}=\frac{188}{-94}=-2\phantom{\rule{0ex}{0ex}}y=\frac{{D}_{2}}{D}=\frac{-282}{-94}=3\phantom{\rule{0ex}{0ex}}z=\frac{{D}_{3}}{D}=\frac{-141}{-94}=1.5\phantom{\rule{0ex}{0ex}}w=\frac{{D}_{4}}{D}=\frac{47}{-94}=-0.5\phantom{\rule{0ex}{0ex}}$

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