# System of linear equations

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# $\frac{1}{x}+\frac{2}{y}+\frac{1}{z}=2,$ $\frac{3}{x}-\frac{4}{y}-\frac{2}{z}=1,$ and $\frac{2}{x}+\frac{5}{y}-\frac{2}{z}=3,$ Then the value of $y$  is___________.

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Solution

## $D=\left|\begin{array}{ccc}1& 2& 1\\ 3& -4& -2\\ 2& 5& -2\end{array}\right|=45,$               ${D}_{1/x}=3\left|\begin{array}{ccc}1& 2& 1\\ 2& -4& -2\\ 2& 5& -2\end{array}\right|=36,$ ${D}_{1/y}=3\left|\begin{array}{ccc}1& 1& 1\\ 3& 2& -2\\ 2& 3& -2\end{array}\right|=9,$             $\text{\hspace{0.17em}}\text{\hspace{0.17em}}{D}_{1/z}=3\left|\begin{array}{ccc}1& 2& 1\\ 3& -4& 2\\ 2& 5& -3\end{array}\right|=-9$$x=\frac{5}{4},\text{\hspace{0.17em}}\text{\hspace{0.17em}}y=5,\text{\hspace{0.17em}}\text{\hspace{0.17em}}z=-5$Hence value of $y$  is $5.$

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The number of solutions of $2x+y=4,\text{\hspace{0.17em}}x-2y=2,\text{\hspace{0.17em}}3x+5y=6$  is