1. The problem is to understand and apply Cramer's Rule to solve a system of linear equations. 2. Cramer's Rule states that for a system of $n$ linear equations with $n$ unknowns, if the determinant of the coefficient matrix $D$ is nonzero, then each variable $x_i$ can be found as $$ x_i = \frac{D_i}{D} $$ where $D_i$ is the determinant of the matrix formed by replacing the $i$-th column of $D$ with the constants vector. 3. For a 2x2 system: $$ a_{11}x + a_{12}y = b_1 $$ $$ a_{21}x + a_{22}y = b_2 $$ The coefficient matrix is $$ D = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = a_{11}a_{22} - a_{12}a_{21} $$ 4. Replace the first column with constants for $x$: $$ D_x = \begin{vmatrix} b_1 & a_{12} \\ b_2 & a_{22} \end{vmatrix} = b_1a_{22} - a_{12}b_2 $$ Similarly, replace the second column for $y$: $$ D_y = \begin{vmatrix} a_{11} & b_1 \\ a_{21} & b_2 \end{vmatrix} = a_{11}b_2 - b_1a_{21} $$ 5. Find $x$ and $y$: $$ x = \frac{D_x}{D}, \quad y = \frac{D_y}{D} $$ 6. If $D \neq 0$, this gives a unique solution. 7. For larger systems (3x3 or more), the process is analogous, using appropriate determinants. This method is a reliable way to solve linear systems when determinants are easy to compute.