Questions: Solve the system if possible by using Cramer's rule. If Cramer's rule does not apply, solve the system by using another method. Write all numbers as integers or simplified fractions. x-2y+3z = -1 5x-7y+3z = 1 x - 5z = 2 Part 1 of 2 Evaluate the determinants Dy Dx, Dy, and Dz.

Solution Steps

To solve the system using Cramer's rule, we first need to evaluate the determinant of the coefficient matrix, \(D\). If \(D \neq 0\), Cramer's rule can be applied. We then calculate the determinants \(D_x\), \(D_y\), and \(D_z\) by replacing the respective columns of the coefficient matrix with the constants from the right-hand side of the equations. Finally, we solve for \(x\), \(y\), and \(z\) using the formulas \(x = \frac{D_x}{D}\), \(y = \frac{D_y}{D}\), and \(z = \frac{D_z}{D}\).

The determinant of the coefficient matrix \(A\) is calculated as follows: \[ D = 5.728750807065831 \times 10^{-15} \] Since \(D\) is approximately zero, Cramer's rule cannot be applied directly.

To find \(D_x\), we replace the first column of \(A\) with the constants from \(B\): \[ A_x = \begin{bmatrix} -1 & -2 & 3 \\ 1 & -7 & 3 \\ 2 & 0 & -5 \end{bmatrix} \] The determinant \(D_x\) is: \[ D_x = -15 \]

Next, we calculate \(D_y\) by replacing the second column of \(A\) with \(B\): \[ A_y = \begin{bmatrix} 1 & -1 & 3 \\ 5 & 1 & 3 \\ 1 & 2 & -5 \end{bmatrix} \] The determinant \(D_y\) is: \[ D_y = -12 \]

Finally, we calculate \(D_z\) by replacing the third column of \(A\) with \(B\): \[ A_z = \begin{bmatrix} 1 & -2 & -1 \\ 5 & -7 & 1 \\ 1 & 0 & 2 \end{bmatrix} \] The determinant \(D_z\) is: \[ D_z = -3 \]

Final Answer