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.

Verified step by step guidance

1. Step 1: Write down the coefficient matrix of the system: \[ A = \begin{bmatrix} 4 & -3 & -2 \\ 8 & -6 & -4 \end{bmatrix} \] Note that this is a 2x3 matrix because there are 2 equations and 3 variables (x, y, z).
2. Step 2: Explain why Cramer's Rule cannot be used: Cramer's Rule requires a square coefficient matrix (same number of equations as variables) and a non-zero determinant. Here, the coefficient matrix is not square (2x3), so the determinant is not defined. Therefore, Cramer's Rule cannot be applied.
3. Step 3: Set up the augmented matrix for Gaussian elimination: \[ \left[ \begin{array}{ccc|c} 4 & -3 & -2 & 12 \\ 8 & -6 & -4 & 22 \end{array} \right] \] This matrix includes the coefficients and the constants from the right side of the equations.
4. Step 4: Use row operations to simplify the augmented matrix: - Multiply the first row by 2 and subtract it from the second row to eliminate the x-term in the second row: Row2 = Row2 - 2 * Row1 This will give a new second row.
5. Step 5: Analyze the resulting system: - If the second row becomes a row of zeros with a non-zero constant, the system is inconsistent (no solution). - If the second row becomes all zeros including the constant, the system has infinitely many solutions. - Otherwise, continue with back substitution to find the values of variables.

Key Concepts

Cramer's Rule and Determinants

Cramer's Rule is a method for solving systems of linear equations using determinants. It requires the coefficient matrix to have a nonzero determinant, ensuring the system has a unique solution. If the determinant is zero, Cramer's Rule cannot be applied because the system is either dependent or inconsistent.

Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations by transforming the augmented matrix into row-echelon form. This process uses row operations to simplify the system, making it easier to find solutions or determine if none or infinitely many exist.

System Consistency and Dependency

A system of equations is consistent if it has at least one solution and inconsistent if it has none. Dependency occurs when equations are multiples of each other, leading to infinitely many solutions or no unique solution. Recognizing dependency helps explain why Cramer's Rule fails and guides the use of alternative methods.