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The given system of linear equations is:

\[
\begin{cases}
2x + 8y + 5z = -5 \\
x + y + z = -2 \\
x + 2y - z = 2 
\end{cases}
\]

We will solve this system using **Cramer's Rule**. Cramer's Rule is applicable for systems of linear equations where the number of equations equals the number of unknowns, and it requires calculating determinants.

For Cramer's Rule, we need to:

1. Find the determinant \( D \) of the coefficient matrix.
2. Find the determinants \( D_x \), \( D_y \), and \( D_z \) by replacing the respective columns with the constants from the right-hand side of each equation.
3. Calculate \( x = \frac{D_x}{D} \), \( y = \frac{D_y}{D} \), and \( z = \frac{D_z}{D} \).

### Step 1: Set up the coefficient matrix

The coefficient matrix \( A \) for this system is:

\[
A = \begin{pmatrix} 2 & 8 & 5 \\ 1 & 1 & 1 \\ 1 & 2 & -1 \end{pmatrix}
\]

And the constants matrix \( B \) is:

\[
B = \begin{pmatrix} -5 \\ -2 \\ 2 \end{pmatrix}
\]

### Step 2: Calculate \( D \) (Determinant of the coefficient matrix)

\[
D = \begin{vmatrix} 2 & 8 & 5 \\ 1 & 1 & 1 \\ 1 & 2 & -1 \end{vmatrix}
\]

Using the cofactor expansion method:

\[
D = 2 \begin{vmatrix} 1 & 1 \\ 2 & -1 \end{vmatrix} - 8 \begin{vmatrix} 1 & 1 \\ 1 & -1 \end{vmatrix} + 5 \begin{vmatrix} 1 & 1 \\ 1 & 2 \end{vmatrix}
\]

Calculating each 2x2 determinant:

1. \( \begin{vmatrix} 1 & 1 \\ 2 & -1 \end{vmatrix} = (1)(-1) - (1)(2) = -1 - 2 = -3 \)
2. \( \begin{vmatrix} 1 & 1 \\ 1 & -1 \end{vmatrix} = (1)(-1) - (1)(1) = -1 - 1 = -2 \)
3. \( \begin{vmatrix} 1 & 1 \\ 1 & 2 \end{vmatrix} = (1)(2) - (1)(1) = 2 - 1 = 1 \)

So,

\[
D = 2(-3) - 8(-2) + 5(1) = -6 + 16 + 5 = 15
\]

### Step 3: Calculate \( D_x \), \( D_y \), and \( D_z \)

#### \( D_x \) (Replace the first column with \( B \))

\[
D_x = \begin{vmatrix} -5 & 8 & 5 \\ -2 & 1 & 1 \\ 2 & 2 & -1 \end{vmatrix}
\]

Expanding along the first row:

\[
D_x = -5 \begin{vmatrix} 1 & 1 \\ 2 & -1 \end{vmatrix} - 8 \begin{vmatrix} -2 & 1 \\ 2 & -1 \end{vmatrix} + 5 \begin{vmatrix} -2 & 1 \\ 2 & 2 \end{vmatrix}
\]

Using the 2x2 determinants calculated previously and solving:

\[
D_x = -5(-3) - 8(0) + 5(-6) = 15 + 0 - 30 = -15
\]

#### \( D_y \) (Replace the second column with \( B \))

\[
D_y = \begin{vmatrix} 2 & -5 & 5 \\ 1 & -2 & 1 \\ 1 & 2 & -1 \end{vmatrix}
\]

Expanding along the first row:

\[
D_y = 2 \begin{vmatrix} -2 & 1 \\ 2 & -1 \end{vmatrix} - (-5) \begin{vmatrix} 1 & 1 \\ 1 & -1 \end{vmatrix} + 5 \begin{vmatrix} 1 & -2 \\ 1 & 2 \end{vmatrix}
\]

Calculating each of these determinants gives:

\[
D_y = 

Solve the system of equations using Cramer's rule:
2x + 8y + 5z = -5
x + y + z = -2
x + 2y - z = 2

This guide provides a step-by-step solution for solving a system of linear equations using Cramer's Rule. Explore the methods to calculate determinants for 3x3 matrices and apply Cramer's Rule to find the values of x, y, and z for the equations: 2x + 8y + 5z = -5, x + y + z = -2, and x + 2y - z = 2.

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