Classify the following system of equations as consistent or inconsistent :
`{:(x+3y=5),(2x+6y=8):}`

Answer

To classify the given system of equations as consistent or inconsistent, we will follow these steps:

### Step 1: Write down the equations
The given system of equations is:
1. \( x + 3y = 5 \) (Equation 1)
2. \( 2x + 6y = 8 \) (Equation 2)

### Step 2: Form the coefficient matrix and the constant matrix
The coefficient matrix \( A \) and the constant matrix \( B \) can be formed as follows:
- Coefficient matrix \( A \):
\[ A = \begin{pmatrix} 1 & 3 \\ 2 & 6 \end{pmatrix} \]

- Constant matrix \( B \):
\[ B = \begin{pmatrix} 5 \\ 8 \end{pmatrix} \]

### Step 3: Calculate the determinant of the coefficient matrix
To determine if the system is consistent or inconsistent, we need to calculate the determinant of matrix \( A \):
\[ \text{det}(A) = (1)(6) - (3)(2) = 6 - 6 = 0 \]

### Step 4: Analyze the determinant
Since the determinant of the coefficient matrix \( A \) is \( 0 \), we conclude that the system of equations is inconsistent.

### Conclusion
The system of equations is inconsistent, meaning there is no solution to the given equations.

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