The following system of equations `5x-7y+3z=3, 5x+y+3z=7 and 5x+3y+2z=5` is

Options

A. Consistent with trivial solution
B. Consistent with a unique non trivial solution
C. Consistent with infinite solutions
D. Inconsistent with no solution

Answer

To determine the nature of the given system of equations, we will analyze the system step by step.

### Given Equations:
1. \( 5x - 7y + 3z = 3 \) (Equation 1)
2. \( 5x + y + 3z = 7 \) (Equation 2)
3. \( 5x + 3y + 2z = 5 \) (Equation 3)

### Step 1: Form the Coefficient Matrix and the Augmented Matrix
The coefficient matrix \( A \) and the augmented matrix \( [A|B] \) can be formed as follows:

**Coefficient Matrix \( A \)**:
\[ A = \begin{bmatrix} 5 & -7 & 3 \\ 5 & 1 & 3 \\ 5 & 3 & 2 \end{bmatrix} \]

**Augmented Matrix \( [A|B] \)**:
\[ [A|B] = \begin{bmatrix} 5 & -7 & 3 & | & 3 \\ 5 & 1 & 3 & | & 7 \\ 5 & 3 & 2 & | & 5 \end{bmatrix} \]

### Step 2: Calculate the Determinant of the Coefficient Matrix
To determine the nature of the solutions, we calculate the determinant \( \Delta \) of the coefficient matrix \( A \).

\[ \Delta = \begin{vmatrix} 5 & -7 & 3 \\ 5 & 1 & 3 \\ 5 & 3 & 2 \end{vmatrix} \]

Calculating the determinant using the formula for a 3x3 matrix:
\[ \Delta = 5 \begin{vmatrix} 1 & 3 \\ 3 & 2 \end{vmatrix} - (-7) \begin{vmatrix} 5 & 3 \\ 5 & 2 \end{vmatrix} + 3 \begin{vmatrix} 5 & 1 \\ 5 & 3 \end{vmatrix} \]

Calculating the 2x2 determinants:
1. \( \begin{vmatrix} 1 & 3 \\ 3 & 2 \end{vmatrix} = (1)(2) - (3)(3) = 2 - 9 = -7 \)

2. \( \begin{vmatrix} 5 & 3 \\ 5 & 2 \end{vmatrix} = (5)(2) - (3)(5) = 10 - 15 = -5 \)

3. \( \begin{vmatrix} 5 & 1 \\ 5 & 3 \end{vmatrix} = (5)(3) - (1)(5) = 15 - 5 = 10 \)

Substituting back into the determinant calculation:
\[ \Delta = 5(-7) + 7(-5) + 3(10) = -35 - 35 + 30 = -40 \]

### Step 3: Analyze the Determinant
Since \( \Delta \neq 0 \) (specifically \( \Delta = -40 \)), it indicates that the system of equations has a unique solution.

### Conclusion
The system of equations is consistent and has a unique solution.

### Final Answer
The system of equations is **consistent with a unique solution**.

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