1. **State the problem:** Solve the system of equations using Cramer's rule: $$\begin{cases} 15x - 4y - 7z = 14 \\ -4x + 6y - 2z = 34 \\ 4x - 3y - 2z + 12z = 1 \end{cases}$$ 2. **Write the coefficient matrix $A$, variable vector $\mathbf{x}$, and constant vector $\mathbf{b}$:** $$A = \begin{bmatrix} 15 & -4 & -7 \\ -4 & 6 & -2 \\ 4 & -3 & 12 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 14 \\ 34 \\ 1 \end{bmatrix}$$ 3. **Calculate the determinant of $A$, $\det(A)$:** $$\det(A) = 15 \times \begin{vmatrix} 6 & -2 \\ -3 & 12 \end{vmatrix} - (-4) \times \begin{vmatrix} -4 & -2 \\ 4 & 12 \end{vmatrix} + (-7) \times \begin{vmatrix} -4 & 6 \\ 4 & -3 \end{vmatrix}$$ Calculate each minor: $$\begin{aligned} M_{11} &= 6 \times 12 - (-2) \times (-3) = 72 - 6 = 66 \\ M_{12} &= (-4) \times 12 - (-2) \times 4 = -48 + 8 = -40 \\ M_{13} &= (-4) \times (-3) - 6 \times 4 = 12 - 24 = -12 \end{aligned}$$ So, $$\det(A) = 15 \times 66 + 4 \times (-40) - 7 \times (-12) = 990 - 160 + 84 = 914$$ 4. **Calculate determinants for $x$, $y$, and $z$ by replacing respective columns with $\mathbf{b}$:** - For $x$, replace first column: $$A_x = \begin{bmatrix} 14 & -4 & -7 \\ 34 & 6 & -2 \\ 1 & -3 & 12 \end{bmatrix}$$ $$\det(A_x) = 14 \times \begin{vmatrix} 6 & -2 \\ -3 & 12 \end{vmatrix} - (-4) \times \begin{vmatrix} 34 & -2 \\ 1 & 12 \end{vmatrix} + (-7) \times \begin{vmatrix} 34 & 6 \\ 1 & -3 \end{vmatrix}$$ Calculate minors: $$\begin{aligned} M_{11} &= 6 \times 12 - (-2) \times (-3) = 72 - 6 = 66 \\ M_{12} &= 34 \times 12 - (-2) \times 1 = 408 + 2 = 410 \\ M_{13} &= 34 \times (-3) - 6 \times 1 = -102 - 6 = -108 \end{aligned}$$ So, $$\det(A_x) = 14 \times 66 + 4 \times 410 - 7 \times (-108) = 924 + 1640 + 756 = 3310$$ - For $y$, replace second column: $$A_y = \begin{bmatrix} 15 & 14 & -7 \\ -4 & 34 & -2 \\ 4 & 1 & 12 \end{bmatrix}$$ $$\det(A_y) = 15 \times \begin{vmatrix} 34 & -2 \\ 1 & 12 \end{vmatrix} - 14 \times \begin{vmatrix} -4 & -2 \\ 4 & 12 \end{vmatrix} + (-7) \times \begin{vmatrix} -4 & 34 \\ 4 & 1 \end{vmatrix}$$ Calculate minors: $$\begin{aligned} M_{11} &= 34 \times 12 - (-2) \times 1 = 408 + 2 = 410 \\ M_{12} &= (-4) \times 12 - (-2) \times 4 = -48 + 8 = -40 \\ M_{13} &= (-4) \times 1 - 34 \times 4 = -4 - 136 = -140 \end{aligned}$$ So, $$\det(A_y) = 15 \times 410 - 14 \times (-40) - 7 \times (-140) = 6150 + 560 + 980 = 7690$$ - For $z$, replace third column: $$A_z = \begin{bmatrix} 15 & -4 & 14 \\ -4 & 6 & 34 \\ 4 & -3 & 1 \end{bmatrix}$$ $$\det(A_z) = 15 \times \begin{vmatrix} 6 & 34 \\ -3 & 1 \end{vmatrix} - (-4) \times \begin{vmatrix} -4 & 34 \\ 4 & 1 \end{vmatrix} + 14 \times \begin{vmatrix} -4 & 6 \\ 4 & -3 \end{vmatrix}$$ Calculate minors: $$\begin{aligned} M_{11} &= 6 \times 1 - 34 \times (-3) = 6 + 102 = 108 \\ M_{12} &= (-4) \times 1 - 34 \times 4 = -4 - 136 = -140 \\ M_{13} &= (-4) \times (-3) - 6 \times 4 = 12 - 24 = -12 \end{aligned}$$ So, $$\det(A_z) = 15 \times 108 + 4 \times (-140) + 14 \times (-12) = 1620 - 560 - 168 = 892$$ 5. **Calculate the variables using Cramer's rule:** $$x = \frac{\det(A_x)}{\det(A)} = \frac{3310}{914}$$ $$y = \frac{\det(A_y)}{\det(A)} = \frac{7690}{914}$$ $$z = \frac{\det(A_z)}{\det(A)} = \frac{892}{914}$$ 6. **Simplify fractions if possible:** - $x = \frac{3310}{914} = \frac{3310 \div 2}{914 \div 2} = \frac{1655}{457}$ (cannot simplify further) - $y = \frac{7690}{914} = \frac{7690 \div 2}{914 \div 2} = \frac{3845}{457}$ (cannot simplify further) - $z = \frac{892}{914} = \frac{892 \div 2}{914 \div 2} = \frac{446}{457}$ (cannot simplify further) **Final answer:** $$\boxed{\left(x, y, z\right) = \left(\frac{1655}{457}, \frac{3845}{457}, \frac{446}{457}\right)}$$