1. **Problem statement:** Solve the system of equations using Cramer's Rule: $$\begin{cases} 2x + 3y - z = 5 \\ -x + 4y + 2z = 6 \\ 3x - y + z = 4 \end{cases}$$ 2. **Formula and rules:** Cramer's Rule states that for a system $AX = B$, where $A$ is the coefficient matrix, $X$ the variable vector, and $B$ 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 $B$. 3. **Step 1: Write the coefficient matrix $A$ and constant vector $B$:** $$A = \begin{bmatrix} 2 & 3 & -1 \\ -1 & 4 & 2 \\ 3 & -1 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 5 \\ 6 \\ 4 \end{bmatrix}$$ 4. **Step 2: Calculate $\det(A)$:** $$\det(A) = 2 \begin{vmatrix} 4 & 2 \\ -1 & 1 \end{vmatrix} - 3 \begin{vmatrix} -1 & 2 \\ 3 & 1 \end{vmatrix} + (-1) \begin{vmatrix} -1 & 4 \\ 3 & -1 \end{vmatrix}$$ Calculate each minor: $$\begin{aligned} &= 2(4 \times 1 - (-1) \times 2) - 3((-1) \times 1 - 3 \times 2) + (-1)((-1) \times (-1) - 3 \times 4) \\ &= 2(4 + 2) - 3(-1 - 6) - 1(1 - 12) \\ &= 2 \times 6 - 3 \times (-7) - 1 \times (-11) \\ &= 12 + 21 + 11 = 44 \end{aligned}$$ 5. **Step 3: Calculate $\det(A_x)$ by replacing first column with $B$:** $$A_x = \begin{bmatrix} 5 & 3 & -1 \\ 6 & 4 & 2 \\ 4 & -1 & 1 \end{bmatrix}$$ $$\det(A_x) = 5 \begin{vmatrix} 4 & 2 \\ -1 & 1 \end{vmatrix} - 3 \begin{vmatrix} 6 & 2 \\ 4 & 1 \end{vmatrix} + (-1) \begin{vmatrix} 6 & 4 \\ 4 & -1 \end{vmatrix}$$ Calculate each minor: $$\begin{aligned} &= 5(4 \times 1 - (-1) \times 2) - 3(6 \times 1 - 4 \times 2) - 1(6 \times (-1) - 4 \times 4) \\ &= 5(4 + 2) - 3(6 - 8) - 1(-6 - 16) \\ &= 5 \times 6 - 3 \times (-2) - 1 \times (-22) \\ &= 30 + 6 + 22 = 58 \end{aligned}$$ 6. **Step 4: Calculate $\det(A_y)$ by replacing second column with $B$:** $$A_y = \begin{bmatrix} 2 & 5 & -1 \\ -1 & 6 & 2 \\ 3 & 4 & 1 \end{bmatrix}$$ $$\det(A_y) = 2 \begin{vmatrix} 6 & 2 \\ 4 & 1 \end{vmatrix} - 5 \begin{vmatrix} -1 & 2 \\ 3 & 1 \end{vmatrix} + (-1) \begin{vmatrix} -1 & 6 \\ 3 & 4 \end{vmatrix}$$ Calculate each minor: $$\begin{aligned} &= 2(6 \times 1 - 4 \times 2) - 5((-1) \times 1 - 3 \times 2) - 1((-1) \times 4 - 3 \times 6) \\ &= 2(6 - 8) - 5(-1 - 6) - 1(-4 - 18) \\ &= 2 \times (-2) - 5 \times (-7) - 1 \times (-22) \\ &= -4 + 35 + 22 = 53 \end{aligned}$$ 7. **Step 5: Calculate $\det(A_z)$ by replacing third column with $B$:** $$A_z = \begin{bmatrix} 2 & 3 & 5 \\ -1 & 4 & 6 \\ 3 & -1 & 4 \end{bmatrix}$$ $$\det(A_z) = 2 \begin{vmatrix} 4 & 6 \\ -1 & 4 \end{vmatrix} - 3 \begin{vmatrix} -1 & 6 \\ 3 & 4 \end{vmatrix} + 5 \begin{vmatrix} -1 & 4 \\ 3 & -1 \end{vmatrix}$$ Calculate each minor: $$\begin{aligned} &= 2(4 \times 4 - (-1) \times 6) - 3((-1) \times 4 - 3 \times 6) + 5((-1) \times (-1) - 3 \times 4) \\ &= 2(16 + 6) - 3(-4 - 18) + 5(1 - 12) \\ &= 2 \times 22 - 3 \times (-22) + 5 \times (-11) \\ &= 44 + 66 - 55 = 55 \end{aligned}$$ 8. **Step 6: Calculate the variables:** $$x = \frac{\det(A_x)}{\det(A)} = \frac{58}{44} = \frac{29}{22}$$ $$y = \frac{\det(A_y)}{\det(A)} = \frac{53}{44}$$ $$z = \frac{\det(A_z)}{\det(A)} = \frac{55}{44} = \frac{5}{4}$$ **Final answer:** $$\boxed{x = \frac{29}{22}, \quad y = \frac{53}{44}, \quad z = \frac{5}{4}}$$