Q.Using Cramer's Rule, solve the system of equations $x+y+z=6,\ 2x-y+z=3,\ x+2y-z=2$.

Cramer's Rule for three equations forms four $3\times3$ determinants — one from the coefficients, and one each with the constants column substituted in for $x$, $y$ and $z$.

Coefficient determinant. $D=\begin{vmatrix} 1 & 1 & 1 \\ 2 & -1 & 1 \\ 1 & 2 & -1 \end{vmatrix} = 1\begin{vmatrix}-1&1\\2&-1\end{vmatrix} - 1\begin{vmatrix}2&1\\1&-1\end{vmatrix} + 1\begin{vmatrix}2&-1\\1&2\end{vmatrix} = 1(-1)-1(-3)+1(5) = -1+3+5=7.$

$D_x$ — replace the $x$-coefficient column $(1,2,1)$ with the constants $(6,3,2)$. $D_x=\begin{vmatrix} 6 & 1 & 1 \\ 3 & -1 & 1 \\ 2 & 2 & -1 \end{vmatrix} = 6(-1)-1(-5)+1(8) = -6+5+8=7.$

$D_y$ — replace the $y$-coefficient column $(1,-1,2)$ with the constants. $D_y=\begin{vmatrix} 1 & 6 & 1 \\ 2 & 3 & 1 \\ 1 & 2 & -1 \end{vmatrix} = 1(-5)-6(-3)+1(1) = -5+18+1=14.$

$D_z$ — replace the $z$-coefficient column $(1,1,-1)$ with the constants. $D_z=\begin{vmatrix} 1 & 1 & 6 \\ 2 & -1 & 3 \\ 1 & 2 & 2 \end{vmatrix} = 1(-8)-1(1)+6(5) = -8-1+30=21.$ …

With three equations, the most common mistake is losing track of which column belongs to which unknown once the constants column starts replacing different coefficient columns in $D_x$, $D_y$ and $D_z$ — it helps to label each determinant clearly and always double-check by substituting the final answer back into t …