Use Cramer's rule to solve each system of equations. If D = 0, then use another methodto determine the solution set. See Examples 5–7. 2x - y + 4z = -23x + 2y - z = -3x + 4y + 2z = 17

Verified step by step guidance

1. First, write the system of equations in matrix form: \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the column matrix of variables, and \( B \) is the constant matrix.
2. The coefficient matrix \( A \) is: \[ A = \begin{bmatrix} 2 & -1 & 4 \\ 3 & 2 & -1 \\ 1 & 4 & 2 \end{bmatrix} \]
3. Calculate the determinant \( D \) of matrix \( A \). If \( D \neq 0 \), proceed with Cramer's Rule. If \( D = 0 \), use another method to solve the system.
4. If \( D \neq 0 \), find the determinants \( D_x \), \( D_y \), and \( D_z \) by replacing the respective columns of \( A \) with the constant matrix \( B = \begin{bmatrix} -2 \\ -3 \\ 17 \end{bmatrix} \).
5. Solve for \( x \), \( y \), and \( z \) using Cramer's Rule: \( x = \frac{D_x}{D} \), \( y = \frac{D_y}{D} \), \( z = \frac{D_z}{D} \).

Key Concepts

Cramer's Rule

Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, provided the determinant of the coefficient matrix is non-zero. It utilizes determinants to express the solution of each variable as a ratio of two determinants: the determinant of the coefficient matrix and the determinant of a modified matrix where one column is replaced by the constants from the equations.

Determinants

A determinant is a scalar value that can be computed from the elements of a square matrix and provides important information about the matrix, such as whether it is invertible. For a 3x3 matrix, the determinant can be calculated using a specific formula involving the elements of the matrix, and if the determinant equals zero, it indicates that the system of equations may have either no solution or infinitely many solutions.

Alternative Methods for Solving Systems

When the determinant of the coefficient matrix is zero (D = 0), Cramer's Rule cannot be applied, and alternative methods must be used to find the solution set. These methods include substitution, elimination, or matrix row reduction techniques, which can help determine if the system has no solutions or infinitely many solutions by analyzing the relationships between the equations.