For what value of k, do the equations `3x - y + 8 = 0` and `6x - ky = -16` represent coincident lines ?

Options

A. `(1)/(2)`
B. `-(1)/(2)`
C. `2`
D. `- 2`

Explanation

To find the value of \( k \) for which the equations \( 3x - y + 8 = 0 \) and \( 6x - ky = -16 \) represent coincident lines, we need to use the condition for coincident lines. The lines will be coincident if the ratios of the coefficients of \( x \), \( y \), and the constant terms are equal.

### Step-by-Step Solution:

1. **Rewrite the equations in standard form**:
- The first equation is already in standard form:
\[ 3x - y + 8 = 0 \implies 3x - y = -8 \]
- The second equation can be rewritten as:
\[ 6x - ky = -16 \implies 6x - ky + 16 = 0 \]

2. **Identify coefficients**:
- From the first equation \( 3x - y + 8 = 0 \), we have:
- \( A_1 = 3 \)
- \( B_1 = -1 \)
- \( C_1 = 8 \)
- From the second equation \( 6x - ky + 16 = 0 \), we have:
- \( A_2 = 6 \)
- \( B_2 = -k \)
- \( C_2 = 16 \)

3. **Set up the ratio condition for coincident lines**:
- The condition for the lines to be coincident is:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \]
- Substituting the values:
\[ \frac{3}{6} = \frac{-1}{-k} = \frac{8}{16} \]

4. **Simplify the ratios**:
- Simplifying the ratios:
\[ \frac{3}{6} = \frac{1}{2}, \quad \frac{8}{16} = \frac{1}{2} \]
- Therefore, we have:
\[ \frac{1}{2} = \frac{1}{k} \]

5. **Cross-multiply to solve for \( k \)**:
- Cross-multiplying gives:
\[ 1 \cdot k = 2 \cdot 1 \implies k = 2 \]

### Final Answer:
Thus, the value of \( k \) for which the lines are coincident is:
\[ \boxed{2} \]