Solve the following system of equations, using Cramer'e rule :
`x+y+3z=6`, `x-3y-3z=-4`, `5x-3y+3z=8`

Answer

To solve the system of equations using Cramer's Rule, we follow these steps:

### Given System of Equations:
1. \( x + y + 3z = 6 \) (Equation 1)
2. \( x - 3y - 3z = -4 \) (Equation 2)
3. \( 5x - 3y + 3z = 8 \) (Equation 3)

### Step 1: Formulate the Coefficient Matrix (A) and the Constant Matrix (B)
The coefficient matrix \( A \) and the constant matrix \( B \) can be represented as follows:

\[ A = \begin{bmatrix} 1 & 1 & 3 \\ 1 & -3 & -3 \\ 5 & -3 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 6 \\ -4 \\ 8 \end{bmatrix} \]

### Step 2: Calculate the Determinant of Matrix A (denoted as \( \Delta \))
To find \( \Delta \), we calculate the determinant of matrix \( A \):

\[ \Delta = \begin{vmatrix} 1 & 1 & 3 \\ 1 & -3 & -3 \\ 5 & -3 & 3 \end{vmatrix} \]

Using the determinant formula for a 3x3 matrix:

\[ \Delta = 1 \cdot \begin{vmatrix} -3 & -3 \\ -3 & 3 \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & -3 \\ 5 & 3 \end{vmatrix} + 3 \cdot \begin{vmatrix} 1 & -3 \\ 5 & -3 \end{vmatrix} \]

Calculating the 2x2 determinants:

1. \( \begin{vmatrix} -3 & -3 \\ -3 & 3 \end{vmatrix} = (-3)(3) - (-3)(-3) = -9 - 9 = -18 \)

2. \( \begin{vmatrix} 1 & -3 \\ 5 & 3 \end{vmatrix} = (1)(3) - (-3)(5) = 3 + 15 = 18 \)

3. \( \begin{vmatrix} 1 & -3 \\ 5 & -3 \end{vmatrix} = (1)(-3) - (-3)(5) = -3 + 15 = 12 \)

Substituting back into the determinant calculation:

\[ \Delta = 1(-18) - 1(18) + 3(12) = -18 - 18 + 36 = 0 \]

### Step 3: Analyze the Determinant
Since \( \Delta = 0 \), the system of equations is either inconsistent or has infinitely many solutions. We need to check if the system is consistent.

### Step 4: Check for Consistency
To check for consistency, we can calculate the determinant of the augmented matrix formed by adding the constant matrix \( B \) to \( A \):

\[ \begin{bmatrix} 1 & 1 & 3 & | & 6 \\ 1 & -3 & -3 & | & -4 \\ 5 & -3 & 3 & | & 8 \end{bmatrix} \]

Calculating the determinant of this augmented matrix will help us determine if the system is consistent or inconsistent.

However, from the calculations above, we can conclude that since the determinant of \( A \) is zero, the system is inconsistent.

### Final Conclusion
The system of equations has **no solution**.

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