Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.
-2x - 2y + 3z = 4
5x + 7y - z = 2
2x + 2y - 3z = -4

Verified step by step guidance

1. Write the system of equations in matrix form as \(A\mathbf{x} = \mathbf{b}\), where \(A\) is the coefficient matrix, \(\mathbf{x}\) is the column vector of variables, and \(\mathbf{b}\) is the constants vector. For this system, \(A = \begin{bmatrix} -2 & -2 & 3 \\ 5 & 7 & -1 \\ 2 & 2 & -3 \end{bmatrix}\) and \(\mathbf{b} = \begin{bmatrix} 4 \\ 2 \\ -4 \end{bmatrix}\).
2. Calculate the determinant \(D\) of the coefficient matrix \(A\). This determinant will help determine if Cramer's rule can be applied. Use the formula for the determinant of a 3x3 matrix:
3. \[D = a(ei - fh) - b(di - fg) + c(dh - eg)\]
4. where \(a, b, c, d, e, f, g, h, i\) are the elements of matrix \(A\) arranged as:
5. \[\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\]
6. If \(D \neq 0\), proceed to find \(D_x\), \(D_y\), and \(D_z\) by replacing the respective columns of \(A\) with the vector \(\mathbf{b}\), then calculate each determinant. Finally, solve for each variable using Cramer's rule:
7. \[x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}, \quad z = \frac{D_z}{D}.\]
8. If \(D = 0\), Cramer's rule cannot be used. In that case, use another method such as substitution or elimination to determine the solution set or check for infinite/no solutions.

Key Concepts

Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. It applies when the coefficient matrix has a nonzero determinant (D ≠ 0). Each variable is found by replacing the corresponding column in the coefficient matrix with the constants vector and calculating the determinant ratio.

Determinant of a Matrix

The determinant is a scalar value that can be computed from a square matrix and indicates whether the matrix is invertible. For a system of equations, if the determinant of the coefficient matrix is zero, the system may have infinitely many solutions or no solution, requiring alternative methods.

Alternative Methods for Solving Systems

When the determinant is zero, Cramer's Rule cannot be used. Alternative methods include substitution, elimination, or matrix row reduction (Gaussian elimination) to find if the system has no solution or infinitely many solutions, and to determine the solution set accordingly.