Q.Using Cramer's Rule, solve $4x + 3y = 18$ and $2x + 5y = 16$.

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.

Coefficient determinant: $D = \begin{vmatrix} 4 & 3 \\ 2 & 5 \end{vmatrix} = (4)(5)-(3)(2) = 20-6 = 14.$ Since $D\neq 0$, the system has a unique solution.

Replace the $x$-column with the constants $(18,16)$: $D_x = \begin{vmatrix} 18 & 3 \\ 16 & 5 \end{vmatrix} = (18)(5)-(3)(16) = 90-48 = 42.$

Replace the $y$-column with the constants: $D_y = \begin{vmatrix} 4 & 18 \\ 2 & 16 \end{vmatrix} = (4)(16)-(18)(2) = 64-36 = 28.$ …

The most common error is replacing the wrong column when forming D_x or D_y. Keep D's own coefficient columns fixed …