Classify the following system of equations as consistent or inconsistent . If consistent, solve them :
`x+2y=5`, `3x+6y=15`

Answer

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

### Step 1: Write the equations in standard form
The given equations are:
1. \( x + 2y = 5 \) (Equation 1)
2. \( 3x + 6y = 15 \) (Equation 2)

### Step 2: Identify the coefficients and constants
From the equations, we can identify:
- Coefficient matrix \( A \):
\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 6 \end{pmatrix} \]
- Variable matrix \( X \):
\[ X = \begin{pmatrix} x \\ y \end{pmatrix} \]
- Constant matrix \( B \):
\[ B = \begin{pmatrix} 5 \\ 15 \end{pmatrix} \]

### Step 3: Calculate the determinant of matrix \( A \)
The determinant of matrix \( A \) is calculated as follows:
\[ \text{det}(A) = (1)(6) - (3)(2) = 6 - 6 = 0 \]

### Step 4: Classify the system of equations
Since the determinant of matrix \( A \) is 0, the system of equations is classified as **inconsistent**. This means there are no solutions to the system.

### Conclusion
The given system of equations \( x + 2y = 5 \) and \( 3x + 6y = 15 \) is inconsistent, and therefore, there is no solution.

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