Apply Cramer's rule to solve the simultaneous equations. `(i)x+2y+3z=2` `x-y+z=3` `5x-11y+z=17`

Answer

To solve the simultaneous equations using Cramer's rule, we follow these steps:

### Given Equations:
1. \( x + 2y + 3z = 2 \) (Equation 1)
2. \( x - y + z = 3 \) (Equation 2)
3. \( 5x - 11y + z = 17 \) (Equation 3)

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

\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & -1 & 1 \\ 5 & -11 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 2 \\ 3 \\ 17 \end{bmatrix} \]

### Step 2: Calculate the Determinant of Matrix A (\( D \))
To apply Cramer's rule, we first need to find the determinant of matrix \( A \).

\[ D = \begin{vmatrix} 1 & 2 & 3 \\ 1 & -1 & 1 \\ 5 & -11 & 1 \end{vmatrix} \]

Calculating the determinant using the formula:

\[ D = 1 \cdot \begin{vmatrix} -1 & 1 \\ -11 & 1 \end{vmatrix} - 2 \cdot \begin{vmatrix} 1 & 1 \\ 5 & 1 \end{vmatrix} + 3 \cdot \begin{vmatrix} 1 & -1 \\ 5 & -11 \end{vmatrix} \]

Calculating the 2x2 determinants:

1. \( \begin{vmatrix} -1 & 1 \\ -11 & 1 \end{vmatrix} = (-1)(1) - (1)(-11) = -1 + 11 = 10 \)

2. \( \begin{vmatrix} 1 & 1 \\ 5 & 1 \end{vmatrix} = (1)(1) - (1)(5) = 1 - 5 = -4 \)

3. \( \begin{vmatrix} 1 & -1 \\ 5 & -11 \end{vmatrix} = (1)(-11) - (-1)(5) = -11 + 5 = -6 \)

Now substituting back into the determinant formula:

\[ D = 1 \cdot 10 - 2 \cdot (-4) + 3 \cdot (-6) = 10 + 8 - 18 = 0 \]

### Step 3: Analyze the Determinant
Since the determinant \( D = 0 \), it indicates that the system of equations does not have a unique solution. This means the equations are either dependent or inconsistent.

### Conclusion
By Cramer's rule, since the determinant of the coefficient matrix is zero, we conclude that the system of equations has no unique solution.

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