Questions: Solve the following system of equations using Cramer's Rule: Type your answer in the box below in the form: (x, y, z) 4x-6y-z=-5 4x-y-2z=-19 -2x-5y+z=26

Transcript text: Solve the following system of equations using Cramer's Rule: Type your answer in the box below in the form: $(x, y, z)$ \[ \begin{array}{l} 4 x-6 y-z=-5 \\ 4 x-y-2 z=-19 \\ -2 x-5 y+z=26 \end{array} \]

Solution

Solution Steps

To solve the system of equations using Cramer's Rule, we first need to find the determinant of the coefficient matrix. Then, we find the determinants of matrices formed by replacing each column of the coefficient matrix with the constants from the right-hand side of the equations. Finally, we calculate the values of $x$, $y$, and $z$ by dividing these determinants by the determinant of the coefficient matrix.

Step 1: Define the System of Equations

We are given the following system of equations: \[ \begin{align_} 4x - 6y - z &= -5 \quad (1) \\ 4x - y - 2z &= -19 \quad (2) \\ -2x - 5y + z &= 26 \quad (3) \end{align_} \]

Step 2: Construct the Coefficient Matrix and Constant Vector

The coefficient matrix $ A $ and the constant vector $ B $ are defined as follows: \[ A = \begin{bmatrix} 4 & -6 & -1 \\ 4 & -1 & -2 \\ -2 & -5 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} -5 \\ -19 \\ 26 \end{bmatrix} \]

Step 3: Calculate the Determinant of the Coefficient Matrix

The determinant of the coefficient matrix $ A $ is calculated as: \[ \text{det}(A) = -22 \]

Step 4: Calculate Determinants for Cramer's Rule

We replace each column of $ A $ with $ B $ to find the determinants needed for Cramer's Rule:

For $ x $: \[ A_x = \begin{bmatrix} -5 & -6 & -1 \\ -19 & -1 & -2 \\ 26 & -5 & 1 \end{bmatrix}, \quad \text{det}(A_x) = 132 \]
For $ y $: \[ A_y = \begin{bmatrix} 4 & -5 & -1 \\ 4 & -19 & -2 \\ -2 & 26 & 1 \end{bmatrix}, \quad \text{det}(A_y) = 66 \]
For $ z $: \[ A_z = \begin{bmatrix} 4 & -6 & -5 \\ 4 & -1 & -19 \\ -2 & -5 & 26 \end{bmatrix}, \quad \text{det}(A_z) = 22 \]

Step 5: Solve for $ x $, $ y $, and $ z $ Using Cramer's Rule

Using Cramer's Rule, we find: \[ x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{132}{-22} = -6 \] \[ y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{66}{-22} = -3 \] \[ z = \frac{\text{det}(A_z)}{\text{det}(A)} = \frac{22}{-22} = -1 \]