Question

x + y = 1
x + z = − 6
x − y − 2z = 3

Answer 1

These equations can be written as
x+ y + 0z = 1
x + 0y + z = − 6
x − y − 2z = 3


$$\begin{gathered} D=\left|\begin{array}{ccc}1& 1& 0\\ 1& 0& 1\\ 1& -1& -2\end{array}\right| \\ =1(0+1)-1(-2-1)+0(-1-0) \\ =4 \\ {D}_1=\left|\begin{array}{ccc}1& 1& 0\\ -6& 0& 1\\ 3& -1& -2\end{array}\right| \\ =1(0+1)-1(12-3)+0(6-0) \\ =-8 \\ {D}_2=\left|\begin{array}{ccc}1& 1& 0\\ 1& -6& 1\\ 1& 3& -2\end{array}\right| \\ =1(12-3)-1(-2-1)+0(3+6) \\ =12 \\ {D}_3=\left|\begin{array}{ccc}1& 1& 1\\ 1& 0& -6\\ 1& -1& 3\end{array}\right| \\ =1(0-6)-1(3+6)+1(-1-0) \\ =-16 \\ \mathrm{Now}, \\ x=\frac{D_1}{D}=\frac{-8}{4}=-2 \\ y=\frac{D_2}{D}=\frac{12}{4}=3 \\ z=\frac{D_3}{D}=\frac{-16}{4}=-4 \\ \therefore x=-2,y=3\mathrm{and}z=-4 \end{gathered}$$