Questions: Use Cramer's Rule to solve the system: 2x1 + x2 - x3 = 1 3x1 + 2x2 + 2x3 = 13 4x1 - 2x2 + 3x3 = 9

Solution Steps

We start with the following system of linear equations: \[ \begin{cases} 2 x_{1} + x_{2} - x_{3} = 1 \\ 3 x_{1} + 2 x_{2} + 2 x_{3} = 13 \\ 4 x_{1} - 2 x_{2} + 3 x_{3} = 9 \end{cases} \]

The coefficient matrix \( A \) and the constant vector \( b \) are defined as follows: \[ A = \begin{bmatrix} 2 & 1 & -1 \\ 3 & 2 & 2 \\ 4 & -2 & 3 \end{bmatrix}, \quad b = \begin{bmatrix} 1 \\ 13 \\ 9 \end{bmatrix} \]

The determinant of the coefficient matrix \( A \) is calculated as: \[ \text{Det}(A) = 33.00 \]

To apply Cramer's Rule, we need to calculate the determinants of matrices formed by replacing each column of \( A \) with the vector \( b \).

The matrix \( A_1 \) is formed by replacing the first column of \( A \) with \( b \): \[ A_1 = \begin{bmatrix} 1 & 1 & -1 \\ 13 & 2 & 2 \\ 9 & -2 & 3 \end{bmatrix} \] The determinant of \( A_1 \) is: \[ \text{Det}(A_1) = 33.00 \]

The matrix \( A_2 \) is formed by replacing the second column of \( A \) with \( b \): \[ A_2 = \begin{bmatrix} 2 & 1 & -1 \\ 3 & 13 & 2 \\ 4 & 9 & 3 \end{bmatrix} \] The determinant of \( A_2 \) is: \[ \text{Det}(A_2) = 66.00 \]

The matrix \( A_3 \) is formed by replacing the third column of \( A \) with \( b \): \[ A_3 = \begin{bmatrix} 2 & 1 & 1 \\ 3 & 2 & 13 \\ 4 & -2 & 9 \end{bmatrix} \] The determinant of \( A_3 \) is: \[ \text{Det}(A_3) = 66.00 \]

Using Cramer's Rule, we can find the values of \( x_1, x_2, \) and \( x_3 \): \[ x_1 = \frac{\text{Det}(A_1)}{\text{Det}(A)} = \frac{33.00}{33.00} = 1 \] \[ x_2 = \frac{\text{Det}(A_2)}{\text{Det}(A)} = \frac{66.00}{33.00} = 2 \] \[ x_3 = \frac{\text{Det}(A_3)}{\text{Det}(A)} = \frac{66.00}{33.00} = 2 \]

The solution to the system of equations is: \[ \begin{array}{c} x_{1} = 1 \\ x_{2} = 2 \\ x_{3} = 2 \end{array} \]

Final Answer