Solve the system of equations using Cramer's rule: \left\{\begin{array}{l} 3x - 5y + 4z = 5 \\ 5x + 2y + z = 0 \\ 2x + 3y - 2z = 3 \end{array} ight.. First, evaluate the main determinant $D$ using the coefficients of the variables. By expanding by minors, we find $D = -37$. Next, evaluate $D_x$ by using the constants to replace the coefficients of $x$; expanding by minors yields $D_x = -74$. Then, evaluate $D_y$ by replacing the coefficients of $y$ with the constants, yielding $D_y = 111$. Finally, evaluate $D_z$ by replacing the coefficients of $z$ with the constants, which yields $D_z = 148$. Find the values of $x$, $y$, and $z$ by substituting into the formulas: $x = \frac{D_x}{D} = \frac{-74}{-37} = 2$, $y = \frac{D_y}{D} = \frac{111}{-37} = -3$, and $z = \frac{D_z}{D} = \frac{148}{-37} = -4$. Write the solution as the ordered triple $(2, -3, -4)$ and check that it is a solution to all three original equations.