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 has

Options

A. infinite solutions
B. three solutions
C. unique solution
D. no solution

Answer

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

### Given Equations:
1. \( x_1 + 2x_2 + x_3 = 3 \) (Equation 1)
2. \( 2x_1 + 3x_2 + x_3 = 3 \) (Equation 2)
3. \( 3x_1 + 5x_2 + 2x_3 = 1 \) (Equation 3)

### Step 1: Write the equations in matrix form
We can represent the system of equations in matrix form as:
\[ \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & 5 & 2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 1 \end{bmatrix} \]

### Step 2: Form the augmented matrix
The augmented matrix for the system is:
\[ \begin{bmatrix} 1 & 2 & 1 & | & 3 \\ 2 & 3 & 1 & | & 3 \\ 3 & 5 & 2 & | & 1 \end{bmatrix} \]

### Step 3: Perform row operations
We will perform row operations to simplify the augmented matrix.

1. **R2 = R2 - 2R1**:
\[ \begin{bmatrix} 1 & 2 & 1 & | & 3 \\ 0 & -1 & -1 & | & -3 \\ 3 & 5 & 2 & | & 1 \end{bmatrix} \]

2. **R3 = R3 - 3R1**:
\[ \begin{bmatrix} 1 & 2 & 1 & | & 3 \\ 0 & -1 & -1 & | & -3 \\ 0 & -1 & -1 & | & -8 \end{bmatrix} \]

3. **R3 = R3 - R2**:
\[ \begin{bmatrix} 1 & 2 & 1 & | & 3 \\ 0 & -1 & -1 & | & -3 \\ 0 & 0 & 0 & | & -5 \end{bmatrix} \]

### Step 4: Analyze the last row
The last row of the augmented matrix is:
\[ 0x_1 + 0x_2 + 0x_3 = -5 \]
This is a contradiction because it implies that \(0 = -5\), which is not possible.

### Conclusion
Since we have reached a contradiction, the system of equations has no solution.

### Final Answer
The system of linear equations has **no solution**.

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