Q.Using Cramer's Rule, solve the system of equations $3x + 2y = 12$ and $x + y = 5$.

Cramer's Rule forms three determinants from the coefficients and constants, and each unknown is the ratio of its own determinant to the coefficient determinant.

Write the coefficient determinant: $D = \begin{vmatrix} 3 & 2 \\ 1 & 1 \end{vmatrix} = (3)(1)-(2)(1) = 1.$ Since $D\neq 0$, the system has a unique solution.

Replace the $x$-coefficient column with the constants column $(12,5)$ to get $D_x$: $D_x = \begin{vmatrix} 12 & 2 \\ 5 & 1 \end{vmatrix} = (12)(1)-(2)(5) = 12-10=2.$

Replace the $y$-coefficient column with the constants column to get $D_y$: $D_y = \begin{vmatrix} 3 & 12 \\ 1 & 5 \end{vmatrix} = (3)(5)-(12)(1) = 15-12=3.$ …

The most common error is replacing the wrong column when forming $D_x$ or $D_y$ — students sometimes replace the constants column with a coefficient column instead of the other way round. It helps to always keep $D$'s own coefficient c …