Q.Using Cramer's Rule, solve $x + 2y + 3z = 14$, $\ 2x - y + z = 3$, $\ 3x + y - z = 2$.

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

Coefficient determinant. $D=\begin{vmatrix} 1 & 2 & 3 \\ 2 & -1 & 1 \\ 3 & 1 & -1 \end{vmatrix} = 1\begin{vmatrix}-1&1\\1&-1\end{vmatrix} - 2\begin{vmatrix}2&1\\3&-1\end{vmatrix} + 3\begin{vmatrix}2&-1\\3&1\end{vmatrix} = 1(0) - 2(-5) + 3(5) = 0+10+15 = 25.$

$D_x$ — replace the $x$-column $(1,2,3)$ with the constants $(14,3,2)$. $D_x=\begin{vmatrix} 14 & 2 & 3 \\ 3 & -1 & 1 \\ 2 & 1 & -1 \end{vmatrix} = 14(0) - 2(-5) + 3(5) = 0+10+15 = 25.$

$D_y$ — replace the $y$-column $(2,-1,1)$ with the constants. $D_y=\begin{vmatrix} 1 & 14 & 3 \\ 2 & 3 & 1 \\ 3 & 2 & -1 \end{vmatrix} = 1(-3-2) - 14(-2-3) + 3(4-9) = -5 + 70 - 15 = 50.$ …

With three equations it is easy to lose track of which column belongs to which unknown once the constants start replacing different columns. Label each determinant clearly and always substitute the final answer back …