Apply Cramer's rule to solve a system of three linear equations, such as \left\{\begin{array}{l} 3x + 8y + 2z = -5 \\ 2x + 5y - 3z = 0 \\ x + 2y - 2z = -1 \end{array} ight.. First, evaluate the main determinant $D$ using the coefficients of the variables. Next, evaluate the determinants $D_x$, $D_y$, and $D_z$ by substituting the system's constants in place of the respective variable coefficients in each case. Then, calculate the values of the variables using the formulas $x = \frac{D_x}{D}$, $y = \frac{D_y}{D}$, and $z = \frac{D_z}{D}$. Finally, state the solution as an ordered triple and verify it by substituting the values back into all three original equations.