For what values of k is the system of equations
` k x + 3y = k - 2`
`12 x + ky = k `
inconsistent ?

Options

A. `k =pm 6`
B. `k = -6` only
C. `k = 6` only
D. none of these

Answer

To determine the values of \( k \) for which the given system of equations is inconsistent, we need to analyze the equations:

1. \( kx + 3y = k - 2 \)
2. \( 12x + ky = k \)

### Step 1: Identify the coefficients
From the first equation, we can identify the coefficients:
- \( A_1 = k \)
- \( B_1 = 3 \)
- \( C_1 = k - 2 \)

From the second equation, we have:
- \( A_2 = 12 \)
- \( B_2 = k \)
- \( C_2 = k \)

### Step 2: Set up the condition for parallel lines
For the lines represented by these equations to be parallel (and thus inconsistent), the following condition must hold:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} \quad \text{and} \quad \frac{A_1}{A_2} \neq \frac{C_1}{C_2} \]

Substituting the coefficients, we get:
\[ \frac{k}{12} = \frac{3}{k} \]

### Step 3: Cross-multiply to solve for \( k \)
Cross-multiplying gives:
\[ k^2 = 36 \]

### Step 4: Solve for \( k \)
Taking the square root of both sides, we find:
\[ k = 6 \quad \text{or} \quad k = -6 \]

### Step 5: Check the condition for inconsistency
Now we need to ensure that these values lead to the lines being parallel but not coincident. We check:
- For \( k = 6 \):
- \( C_1 = 6 - 2 = 4 \)
- \( C_2 = 6 \)
- Here, \( \frac{C_1}{C_2} = \frac{4}{6} \neq \frac{1}{2} \) (which is \( \frac{A_1}{A_2} = \frac{6}{12} \)).

- For \( k = -6 \):
- \( C_1 = -6 - 2 = -8 \)
- \( C_2 = -6 \)
- Here, \( \frac{C_1}{C_2} = \frac{-8}{-6} = \frac{4}{3} \neq \frac{1}{2} \) (which is \( \frac{A_1}{A_2} = \frac{-6}{12} \)).

Both values satisfy the condition for inconsistency.

### Final Answer
The values of \( k \) for which the system of equations is inconsistent are:
\[ k = 6 \quad \text{or} \quad k = -6 \]