Classify the following system of equations as consistent or inconsistent . If consistent, solve them :
`3x-2y=4`, `6x-4y=10`

Answer

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

### Step 1: Write the equations in matrix form
The given equations are:
1. \(3x - 2y = 4\)
2. \(6x - 4y = 10\)

We can express these equations in the form of \(Ax = b\), where:
\[ A = \begin{pmatrix} 3 & -2 \\ 6 & -4 \end{pmatrix}, \quad x = \begin{pmatrix} x \\ y \end{pmatrix}, \quad b = \begin{pmatrix} 4 \\ 10 \end{pmatrix} \]

### Step 2: Calculate the determinant of matrix A
To determine if the system is consistent or inconsistent, we need to calculate the determinant of matrix \(A\):
\[ \text{det}(A) = (3)(-4) - (6)(-2) \]
Calculating this gives:
\[ \text{det}(A) = -12 + 12 = 0 \]

### Step 3: Analyze the determinant
Since the determinant of matrix \(A\) is \(0\), we conclude that the system of equations is inconsistent. This means there are no solutions to the system.

### Final Conclusion
The given system of equations is inconsistent.

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