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

Answer

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

1. **Write the equations in standard form**:
The given equations are:
\[ x + 2y = 2 \quad (1) \]
\[ 2x + 3y = 3 \quad (2) \]

2. **Identify the coefficients and constants**:
From the equations, we can identify the coefficients of \(x\) and \(y\) and the constants:
- For equation (1): Coefficients are \(a_1 = 1\), \(b_1 = 2\)
- For equation (2): Coefficients are \(a_2 = 2\), \(b_2 = 3\)

3. **Form the matrix \(A\)**:
The coefficient matrix \(A\) is:
\[ A = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix} \]

4. **Calculate the determinant of matrix \(A\)**:
The determinant of matrix \(A\) is calculated as follows:
\[ \text{det}(A) = (1)(3) - (2)(2) = 3 - 4 = -1 \]

5. **Analyze the determinant**:
Since \(\text{det}(A) = -1\) which is not equal to 0, we conclude that the system of equations is consistent.

6. **Final conclusion**:
Therefore, the given system of equations is consistent.