Let `2x+ay+6z=8, x+2y+bz=5 and x+y+3z=4` be three equations. If these 3 equations are consistent, then

Options

A. `b=3, a ne 2`
B. `a=2, b ne 3`
C. `a ne 2, b ne 3`
D. `a ne 2, b = 4`

Answer

To determine the conditions under which the given system of equations is consistent, we will analyze the equations step by step.

### Given Equations:
1. \( 2x + ay + 6z = 8 \) (Equation 1)
2. \( x + 2y + bz = 5 \) (Equation 2)
3. \( x + y + 3z = 4 \) (Equation 3)

### Step 1: Write the Augmented Matrix
We can express the system of equations in the form of an augmented matrix:

\[ \begin{bmatrix} 2 & a & 6 & | & 8 \\ 1 & 2 & b & | & 5 \\ 1 & 1 & 3 & | & 4 \end{bmatrix} \]

### Step 2: Row Reduction to Echelon Form
We will perform row operations to reduce this matrix to echelon form.

1. **R2 = R2 - (1/2)R1**:
\[ R2 = \begin{bmatrix} 1 & 2 & b & | & 5 \end{bmatrix} - \frac{1}{2} \begin{bmatrix} 2 & a & 6 & | & 8 \end{bmatrix} \]
This gives us:
\[ R2 = \begin{bmatrix} 0 & 2 - \frac{a}{2} & b - 3 & | & 1 \end{bmatrix} \]

2. **R3 = R3 - (1/2)R1**:
\[ R3 = \begin{bmatrix} 1 & 1 & 3 & | & 4 \end{bmatrix} - \frac{1}{2} \begin{bmatrix} 2 & a & 6 & | & 8 \end{bmatrix} \]
This gives us:
\[ R3 = \begin{bmatrix} 0 & 1 - \frac{a}{2} & 0 & | & 0 \end{bmatrix} \]

Now the augmented matrix looks like this:
\[ \begin{bmatrix} 2 & a & 6 & | & 8 \\ 0 & 2 - \frac{a}{2} & b - 3 & | & 1 \\ 0 & 1 - \frac{a}{2} & 0 & | & 0 \end{bmatrix} \]

### Step 3: Analyze Consistency
For the system to be consistent, we need to analyze the conditions on \( a \) and \( b \).

1. **From Row 3**: The second column gives us \( 1 - \frac{a}{2} = 0 \) which implies \( a = 2 \).

2. **Substituting \( a = 2 \) into Row 2**:
\[ R2 = \begin{bmatrix} 0 & 2 - 1 & b - 3 & | & 1 \end{bmatrix} \Rightarrow \begin{bmatrix} 0 & 1 & b - 3 & | & 1 \end{bmatrix} \]
This leads to the equation:
\[ (b - 3)z = 1 \]
For this equation to hold true for some \( z \), we must have \( b - 3 \neq 0 \) (i.e., \( b \neq 3 \)).

### Conclusion
Thus, the conditions for the system of equations to be consistent are:
- \( a = 2 \)
- \( b \neq 3 \)

### Final Answer
The conditions for the equations to be consistent are:
- \( a = 2 \)
- \( b \neq 3 \)