Consider the system of linear equations
`x_(1)+2x_(2)+x_(3)=3`
`2x_(1)+3x_(2)+x_(3)=3`
`3x_(1)+5x_(2)+2x_(3)=1`
Then, the system has

Options

A. infinite number of solutions
B. exactly 3 solutions
C. a unique solution
D. no solution

Answer

To determine the nature of the solutions for the given system of linear equations, we will follow these steps:

### Step 1: Write the system of equations in matrix form
The given system of equations is:
1. \( x_1 + 2x_2 + x_3 = 3 \)
2. \( 2x_1 + 3x_2 + x_3 = 3 \)
3. \( 3x_1 + 5x_2 + 2x_3 = 1 \)

We can represent this system in the form of a matrix \( A \) and a vector \( B \):
\[ A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & 5 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 3 \\ 3 \\ 1 \end{bmatrix} \]

### Step 2: Calculate the determinant of the coefficient matrix \( A \)
We need to calculate the determinant \( \Delta \) of the matrix \( A \):
\[ \Delta = \begin{vmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & 5 & 2 \end{vmatrix} \]

Using the determinant formula for a 3x3 matrix:
\[ \Delta = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where \( a, b, c \) are the elements of the first row, and \( d, e, f, g, h, i \) are the elements of the remaining rows.

Calculating:
\[ \Delta = 1 \cdot (3 \cdot 2 - 1 \cdot 5) - 2 \cdot (2 \cdot 2 - 1 \cdot 3) + 1 \cdot (2 \cdot 5 - 3 \cdot 3) \]
\[ = 1 \cdot (6 - 5) - 2 \cdot (4 - 3) + 1 \cdot (10 - 9) \]
\[ = 1 \cdot 1 - 2 \cdot 1 + 1 \cdot 1 \]
\[ = 1 - 2 + 1 = 0 \]

### Step 3: Analyze the determinant
Since \( \Delta = 0 \), this indicates that the system of equations may either have infinitely many solutions or no solution.

### Step 4: Calculate the determinants of the augmented matrices
To further analyze the system, we need to calculate the determinants of the matrices formed by replacing the columns of \( A \) with the vector \( B \).

#### Calculate \( \Delta_1 \)
Replace the first column of \( A \) with \( B \):
\[ \Delta_1 = \begin{vmatrix} 3 & 2 & 1 \\ 3 & 3 & 1 \\ 1 & 5 & 2 \end{vmatrix} \]
Calculating \( \Delta_1 \):
\[ = 3(3 \cdot 2 - 1 \cdot 5) - 2(3 \cdot 2 - 1 \cdot 3) + 1(3 \cdot 5 - 3 \cdot 3) \]
\[ = 3(6 - 5) - 2(6 - 3) + 1(15 - 9) \]
\[ = 3 \cdot 1 - 2 \cdot 3 + 1 \cdot 6 \]
\[ = 3 - 6 + 6 = 3 \quad (\Delta_1 \neq 0) \]

### Step 5: Conclusion
Since \( \Delta = 0 \) and \( \Delta_1 \neq 0 \), the system of equations has no solution.

### Final Answer
The system has **no solution**.