7. Systems of Equations & Matrices

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. $\begin{vmatrix}-3 & 4 & -5 \\5 & -2 & 0 \\8 & -1 & 3\end{vmatrix}$

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. $\begin{vmatrix}1 & 5 & 6 \\1 & 4 & 5 \\1 & 9 & 10\end{vmatrix}$

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. $\begin{vmatrix}0.5 & 7 & 5 \\0.5 & 3 & 9 \\0.5 & 1 & 3\end{vmatrix}$

Evaluate each determinant.

In Exercises 37–44, use Cramer's Rule to solve each system. $\begin{cases}x + y + z = 0 \\2x - y + z = -1 \\-x + 3y - z = -8\end{cases}$

Evaluate each determinant.

In Exercises 37–44, use Cramer's Rule to solve each system. $\begin{cases}4x - 5y - 6z = -1 \\x - 2y - 5z = -12 \\2x - y = 7\end{cases}$

Evaluate each determinant.

In Exercises 37–44, use Cramer's Rule to solve each system. $\begin{cases}x + y + z = 4 \\x - 2y + z = 7 \\x + 3y + 2z = 4\end{cases}$

In Exercises 37–44, use Cramer's Rule to solve each system. $\begin{cases}x + 2z = 4 \\2y - z = 5 \\2x + 3y = 13\end{cases}$

In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussian elimination to solve the system.

In Exercises 46–51, evaluate each determinant.

In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussian elimination to solve the system.

Use the determinant theorems to evaluate each determinant.

Solve each equation.