Test the consistency of the following system of equations : (i) 3x-y=2 ,6x-2y=4 (ii) x+5y=1 , 2x+2y=4 (iii) 2x-z=-1, 6x-6y-2z=5 , 3x-y-2z=2

Answer

To test the consistency of the given systems of equations, we will follow these steps for each set of equations:

1. **Write the equations in matrix form.**
2. **Calculate the determinant of the coefficient matrix.**
3. **Determine the consistency based on the value of the determinant.**

Let's solve each part step by step.

### Part (i):
Equations:
1. \( 3x - y = 2 \)
2. \( 6x - 2y = 4 \)

**Step 1: Write in matrix form**
\[ \begin{bmatrix} 3 & -1 \\ 6 & -2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 2 \\ 4 \end{bmatrix} \]

**Step 2: Calculate the determinant of A**
\[ \text{det}(A) = (3)(-2) - (6)(-1) = -6 + 6 = 0 \]

**Step 3: Determine consistency**
Since \(\text{det}(A) = 0\), the system is inconsistent.

### Part (ii):
Equations:
1. \( x + 5y = 1 \)
2. \( 2x + 2y = 4 \)

**Step 1: Write in matrix form**
\[ \begin{bmatrix} 1 & 5 \\ 2 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \end{bmatrix} \]

**Step 2: Calculate the determinant of A**
\[ \text{det}(A) = (1)(2) - (5)(2) = 2 - 10 = -8 \]

**Step 3: Determine consistency**
Since \(\text{det}(A) \neq 0\), the system is consistent.

### Part (iii):
Equations:
1. \( 2x - z = -1 \)
2. \( 6x - 6y - 2z = 5 \)
3. \( 3x - y - 2z = 2 \)

**Step 1: Write in matrix form**
\[ \begin{bmatrix} 2 & 0 & -1 \\ 6 & -6 & -2 \\ 3 & -1 & -2 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -1 \\ 5 \\ 2 \end{bmatrix} \]

**Step 2: Calculate the determinant of A**
\[ \text{det}(A) = 2 \begin{vmatrix} -6 & -2 \\ -1 & -2 \end{vmatrix} - 0 + (-1) \begin{vmatrix} 6 & -2 \\ 3 & -2 \end{vmatrix} \]
Calculating the 2x2 determinants:
\[ \begin{vmatrix} -6 & -2 \\ -1 & -2 \end{vmatrix} = (-6)(-2) - (-2)(-1) = 12 - 2 = 10 \]
\[ \begin{vmatrix} 6 & -2 \\ 3 & -2 \end{vmatrix} = (6)(-2) - (-2)(3) = -12 + 6 = -6 \]
Now substituting back:
\[ \text{det}(A) = 2(10) - 0 - (-1)(-6) = 20 - 6 = 14 \]

**Step 3: Determine consistency**
Since \(\text{det}(A) \neq 0\), the system is consistent.

### Summary of Results:
1. Part (i): Inconsistent
2. Part (ii): Consistent
3. Part (iii): Consistent