Express the following equation and solve them by finding the inverse of coefficient matrix using adjoining method.
$4x+2y-z=3$
$x-2y+z=-8$
$2x-y+z=-7$

$$\begin{array}{l} 4x+2y-z=3 \\ x-2y+z=-8 \\ 2x-y+z=-7 \\ \Rightarrow A=\left[ \begin{array}{l} 4\, \, \, \, \, \, \, \, \, \, 2\, \, \, \, \, \, \, \, -1 \\ 1\, \, \, \, \, \, \, \, -2\, \, \, \, \, \, \, \, \, \, \, \, 1 \\ 2\, \, \, \, \, \, \, -1\, \, \, \, \, \, \, \, \, \, \, \, 1 \end{array} \right] ,\, \, \, \, \, X=\left[ \begin{array}{l} x \\ y \\ z \end{array} \right] ,\, \, \, \, B=\left[ \begin{array}{l} 3 \\ -8 \\ -7 \end{array} \right]  \\ \Rightarrow \left| A \right| =4\left( { -1 } \right) -2\left( { -1 } \right) -1\left( 3 \right)  \\ \Rightarrow \left| A \right| =-4+2-3=-5\ne 0 \\ \Rightarrow \, A\, \, is\, \, invertible,\, \, \left( { X={ A^{ -1 } }B } \right)  \\ Then\, \, co\, \, factor\, \, of\, \, \left| A \right| \, \, are \\ \Rightarrow { a_{ 11 } }=-1,\, \, { a_{ 12 } }=1,\, \, \, { a_{ 13 } }=3 \\ \Rightarrow { a_{ 21 } }=-1,\, \, \, { a_{ 22 } }=6,\, \, \, { a_{ 23 } }=8 \\ \Rightarrow { a_{ 31 } }=0,\, \, \, { a_{ 32 } }=-5,\, \, \, { a_{ 33 } }=-10 \\ \Rightarrow adj\, \, A=\left[ \begin{array}{l} -1\, \, \, \, \, \, \, -1\, \, \, \, \, \, \, \, \, \, 0 \\ 1\, \, \, \, \, \, \, \, \, \, \, \, \, 6\, \, \, \, \, \, \, \, -5 \\ 3\, \, \, \, \, \, \, \, \, \, \, \, 8\, \, \, \, \, \, -10 \end{array} \right]  \\ \Rightarrow X={ A^{ -1 } }B \\ \Rightarrow \left[ \begin{array}{l} x \\ y \\ z \end{array} \right] =\frac { { -1 } }{ 5 } \, \, \left[ \begin{array}{l} -1\, \, \, \, \, \, -1\, \, \, \, \, \, \, 0 \\ 1\; \, \, \, \, \, \, \, \, \, 6\, \, \, \, \, \, -5 \\ 3\, \, \, \, \, \, \, \, \, \, 8\, \, \, \, \, \, \, \, 10 \end{array} \right] \, \, \, \left[ \begin{array}{l} 3 \\ -8 \\ -7 \end{array} \right]  \\ \Rightarrow \left[ \begin{array}{l} x \\ y \\ z \end{array} \right] =\frac { { -1 } }{ 5 } \, \, \, \left[ \begin{array}{l} 5 \\ -10 \\ 15 \end{array} \right] \, \, =\left[ \begin{array}{l} -1 \\ 2 \\ 3 \end{array} \right]  \\ \therefore \, \, x=-1,\, \, y=2\, \, and\, \, z=-3\, \, \, Ans. \end{array}$$