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 determine the value of \( k \) for which the equations \( 3x - y + 8 = 0 \) and \( 6x - ky = -16 \) represent coincident lines, we will follow these steps:

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

### Step 2: Identify coefficients
For the first equation \( 3x - y + 8 = 0 \):
- Coefficient of \( x \) (denoted as \( a_1 \)) = 3
- Coefficient of \( y \) (denoted as \( b_1 \)) = -1
- Constant term (denoted as \( c_1 \)) = 8

For the second equation \( 6x - ky + 16 = 0 \):
- Coefficient of \( x \) (denoted as \( a_2 \)) = 6
- Coefficient of \( y \) (denoted as \( b_2 \)) = -k
- Constant term (denoted as \( c_2 \)) = 16

### Step 3: Set up the condition for coincident lines
For the lines to be coincident, the following ratios must be equal:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]
Substituting the coefficients:
\[ \frac{3}{6} = \frac{-1}{-k} = \frac{8}{16} \]

### Step 4: Simplify the ratios
The left side simplifies to:
\[ \frac{3}{6} = \frac{1}{2} \]
The right side simplifies to:
\[ \frac{8}{16} = \frac{1}{2} \]
Thus, we have:
\[ \frac{1}{2} = \frac{1}{k} \]

### Step 5: Cross-multiply to solve for \( k \)
Cross-multiplying gives:
\[ 1 \cdot k = 2 \cdot 1 \]
This simplifies to:
\[ k = 2 \]

### Step 6: Conclusion
The value of \( k \) for which the equations represent coincident lines is:
\[ \boxed{2} \]