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. Identify the system of equations: $-2x-2y+3z=4$, $5x+7y-z=2$, $2x+2y-3z=-4$.
2. Write the coefficient matrix $A$, the variable matrix $X$, and the constant matrix $B$: $A=\left[\begin{array}{ccc}-2& -2& 3\\ 5& 7& -1\\ 2& 2& -3\end{array}\right]$, $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$, $B=\left[\begin{array}{c}4\\ 2\\ -4\end{array}\right]$.
3. Calculate the determinant $D$ of the coefficient matrix $A$.
4. If $D\ne 0$, use Cramer's Rule to find $x$, $y$, and $z$ by calculating the determinants $D_x$, $D_y$, and $D_z$ of matrices formed by replacing the respective columns of $A$ with $B$.
5. If $D=0$, use another method such as substitution or elimination to determine the solution set.

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 modified matrix (where one column is replaced by the constants) over the determinant of the coefficient matrix. This method is particularly useful for small systems, typically 2x2 or 3x3.

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix and provides important information about the matrix. It indicates whether a system of equations has a unique solution (non-zero determinant), no solution, or infinitely many solutions (zero determinant). In the context of Cramer's Rule, if the determinant of the coefficient matrix (D) is zero, it suggests that the system is either dependent or inconsistent, necessitating alternative methods for finding solutions.

Alternative Methods for Solving Systems

When the determinant of the coefficient matrix is zero, indicating that the system of equations does not have a unique solution, alternative methods such as substitution, elimination, or matrix row reduction can be employed. These methods help to analyze the relationships between the equations, allowing for the identification of either a single solution, infinitely many solutions, or confirming that no solution exists. Understanding these methods is crucial for effectively addressing systems of equations that Cramer's Rule cannot solve.