1. **Stating the problem:** Cramer's Rule is a method to solve a system of linear equations using determinants. 2. **Formula and explanation:** For a system of $n$ equations with $n$ variables, written as $AX = B$, where $A$ is the coefficient matrix, $X$ is the column vector of variables, and $B$ is the constants vector, the solution for each variable $x_i$ is given by: $$x_i = \frac{\det(A_i)}{\det(A)}$$ where $A_i$ is the matrix formed by replacing the $i$-th column of $A$ with the vector $B$. 3. **Important rules:** - The determinant of $A$, $\det(A)$, must not be zero; otherwise, Cramer's Rule cannot be applied. - Calculate determinants carefully using expansion or other methods. 4. **Intermediate work example:** Suppose the system: $$\begin{cases} 2x + 3y = 5 \\ 4x - y = 1 \end{cases}$$ Coefficient matrix: $$A = \begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix}$$ Constants vector: $$B = \begin{bmatrix} 5 \\ 1 \end{bmatrix}$$ Calculate $\det(A)$: $$\det(A) = (2)(-1) - (3)(4) = -2 - 12 = -14$$ Calculate $A_1$ by replacing first column with $B$: $$A_1 = \begin{bmatrix} 5 & 3 \\ 1 & -1 \end{bmatrix}$$ Calculate $\det(A_1)$: $$\det(A_1) = (5)(-1) - (3)(1) = -5 - 3 = -8$$ Calculate $A_2$ by replacing second column with $B$: $$A_2 = \begin{bmatrix} 2 & 5 \\ 4 & 1 \end{bmatrix}$$ Calculate $\det(A_2)$: $$\det(A_2) = (2)(1) - (5)(4) = 2 - 20 = -18$$ 5. **Final solution:** $$x = \frac{\det(A_1)}{\det(A)} = \frac{-8}{-14} = \frac{4}{7}$$ $$y = \frac{\det(A_2)}{\det(A)} = \frac{-18}{-14} = \frac{9}{7}$$ Thus, the solution is $x=\frac{4}{7}$ and $y=\frac{9}{7}$. Cramer's Rule provides a straightforward way to solve linear systems when the determinant of the coefficient matrix is non-zero.