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. x + 2y + 3z = 4 4x + 3y + 2z = 1 -x - 2y - 3z = 0

Verified step by step guidance

1. Identify the system of equations: \(x + 2y + 3z = 4\), \(4x + 3y + 2z = 1\), \(-x - 2y - 3z = 0\).
2. Write the coefficient matrix \(A\), the variable matrix \(X\), and the constant matrix \(B\): \(A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 3 & 2 \\ -1 & -2 & -3 \end{bmatrix}\), \(X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}\), \(B = \begin{bmatrix} 4 \\ 1 \\ 0 \end{bmatrix}\).
3. Calculate the determinant \(D\) of the coefficient matrix \(A\).
4. If \(D \neq 0\), use Cramer's Rule to find \(x\), \(y\), and \(z\) by calculating determinants \(D_x\), \(D_y\), and \(D_z\) where each is obtained by replacing the corresponding column of \(A\) with \(B\).
5. If \(D = 0\), use another method such as substitution or elimination to solve the system.

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. If the determinant of the coefficient matrix (D) is zero, it indicates that the system of equations may have either no solution or infinitely many solutions, necessitating the use of alternative methods to find the solution set.

Alternative Methods for Solving Systems

When Cramer's Rule cannot be applied due to a zero determinant, alternative methods such as substitution, elimination, or matrix row reduction can be used to solve the system. These methods involve manipulating the equations to isolate variables or reduce the system to a simpler form, allowing for the determination of solutions even when the original system does not yield a unique solution.