Classify the following system of equations as consistent or inconsistent :
`{:(5x-6y+4z=15),(7x+4y-3z=19),(2x+y+z=46):}`

Answer

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

### Step 1: Write the system of equations in matrix form
The given system of equations is:
1. \( 5x - 6y + 4z = 15 \)
2. \( 7x + 4y - 3z = 19 \)
3. \( 2x + y + z = 46 \)

We can express this in the form \( A \mathbf{x} = \mathbf{b} \), where:
- \( A \) is the coefficient matrix,
- \( \mathbf{x} \) is the column matrix of variables, and
- \( \mathbf{b} \) is the column matrix of constants.

The coefficient matrix \( A \) is:
\[ A = \begin{pmatrix} 5 & -6 & 4 \\ 7 & 4 & -3 \\ 2 & 1 & 1 \end{pmatrix} \]

The variable matrix \( \mathbf{x} \) is:
\[ \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \]

The constant matrix \( \mathbf{b} \) is:
\[ \mathbf{b} = \begin{pmatrix} 15 \\ 19 \\ 46 \end{pmatrix} \]

### 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 \( A \).

The determinant of a 3x3 matrix \( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \) is given by:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix \( A \):
\[ \text{det}(A) = 5 \left((-6)(1) - (4)(-3)\right) - (-6) \left((7)(1) - (4)(2)\right) + 4 \left((7)(1) - (4)(2)\right) \]

Calculating each term:
1. \( -6 \cdot 1 - 4 \cdot -3 = -6 + 12 = 6 \)
2. \( 7 \cdot 1 - 4 \cdot 2 = 7 - 8 = -1 \)

Now substituting back:
\[ \text{det}(A) = 5(6) + 6(-1) + 4(-1) \]
\[ = 30 - 6 - 4 = 20 \]

### Step 3: Determine consistency
Since the determinant of matrix \( A \) is \( 20 \), which is not equal to \( 0 \), the system of equations is consistent.

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

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