Classify the following system of equations as consistent or inconsistent :
`{:(2x-y=5),(x+y=4):}`

Answer

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

### Step 1: Write the system of equations
The given system of equations is:
1. \( 2x - y = 5 \)
2. \( x + y = 4 \)

### Step 2: Write the equations in matrix form
We can express the system in the form \( AX = B \), where:
- \( A \) is the coefficient matrix,
- \( X \) is the variable matrix,
- \( B \) is the constant matrix.

From the equations, we can identify:
- Coefficient matrix \( A = \begin{pmatrix} 2 & -1 \\ 1 & 1 \end{pmatrix} \)
- Variable matrix \( X = \begin{pmatrix} x \\ y \end{pmatrix} \)
- Constant matrix \( B = \begin{pmatrix} 5 \\ 4 \end{pmatrix} \)

### Step 3: Calculate the determinant of the coefficient matrix \( A \)
The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated as:
\[ \text{det}(A) = ad - bc \]
For our matrix \( A \):
- \( a = 2 \), \( b = -1 \), \( c = 1 \), \( d = 1 \)

Calculating the determinant:
\[ \text{det}(A) = (2)(1) - (-1)(1) = 2 + 1 = 3 \]

### Step 4: Analyze the determinant
Since \( \text{det}(A) = 3 \) and \( 3 \neq 0 \), we conclude that the system of equations is consistent.

### Final Conclusion
The given system of equations is **consistent**.

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