Q.Solve the following system of equations using Cramer's Rule: $x+y+z=6$, $2x-y+z=3$, $x+2y-z=2$.

$D=\begin{vmatrix}1&1&1\\2&-1&1\\1&2&-1\end{vmatrix}=1[(-1)(-1)-(1)(2)]-1[(2)(-1)-(1)(1)]+1[(2)(2)-(-1)(1)]$

$=1(1-2)-1(-2-1)+1(4+1)=-1+3+5=7$

$D_x=\begin{vmatrix}6&1&1\\3&-1&1\\2&2&-1\end{vmatrix}=6[(-1)(-1)-(1)(2)]-1[(3)(-1)-(1)(2)]+1[(3)(2)-(-1)(2)]$

$=6(1-2)-1(-3-2)+1(6+2)=-6+5+8=7$

$D_y=\begin{vmatrix}1&6&1\\2&3&1\\1&2&-1\end{vmatrix}=1[(3)(-1)-(1)(2)]-6[(2)(-1)-(1)(1)]+1[(2)(2)-(3)(1)]$

$=1(-3-2)-6(-2-1)+1(4-3)=-5+18+1=14$