Solve the following simultaneous equations using Cramer's rule: x + y + z = 4; 2x - 3y + 4z = 33; 3x - 2y - 2z = 2.

Steps to Solve

1. Set up the equations in matrix form

The system of equations can be represented in the matrix form $A\mathbf{x} = \mathbf{b}$, where

$$ A = \begin{bmatrix} 1 & 1 & 1 \ 2 & -3 & 4 \ 3 & -2 & -2 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad \text{and } \mathbf{b} = \begin{bmatrix} 4 \ 33 \ 2 \end{bmatrix}. $$

2. Calculate the determinant of matrix A

We compute the determinant of matrix $A$, denoted as $\Delta$:

$$ \Delta = \begin{vmatrix} 1 & 1 & 1 \ 2 & -3 & 4 \ 3 & -2 & -2 \end{vmatrix}. $$

Using the determinant formula for a 3x3 matrix,

$$ \Delta = 1(-3 \cdot -2 - 4 \cdot -2) - 1(2 \cdot -2 - 4 \cdot 3) + 1(2 \cdot -2 - 3 \cdot 3). $$

3. Calculate the determinants for x, y, and z

Next, we calculate $\Delta_x$, $\Delta_y$, and $\Delta_z$ by replacing the corresponding columns of matrix $A$ with vector $\mathbf{b}$.

• For $\Delta_x$:

$$ \Delta_x = \begin{vmatrix} 4 & 1 & 1 \ 33 & -3 & 4 \ 2 & -2 & -2 \end{vmatrix}. $$

• For $\Delta_y$:

$$ \Delta_y = \begin{vmatrix} 1 & 4 & 1 \ 2 & 33 & 4 \ 3 & 2 & -2 \end{vmatrix}. $$

• For $\Delta_z$:

$$ \Delta_z = \begin{vmatrix} 1 & 1 & 4 \ 2 & -3 & 33 \ 3 & -2 & 2 \end{vmatrix}. $$

4. Solve for x, y, and z

Using Cramer's rule:

$$ x = \frac{\Delta_x}{\Delta}, \quad y = \frac{\Delta_y}{\Delta}, \quad z = \frac{\Delta_z}{\Delta}. $$

5. Final calculations

After calculating the values of $\Delta$, $\Delta_x$, $\Delta_y$, and $\Delta_z$, we substitute them into the equations to find the final values of $x$, $y$, and $z$.