For what value of `k`, the two lines `3x - 5y = 4` and `6x - ky = 8` are parallel to each other?

Options

A. `k = 1` only
B. `k = 0` only
C. `k` can have any real value
D. No value of `k` is possible

Answer

To determine the value of \( k \) for which the lines \( 3x - 5y = 4 \) and \( 6x - ky = 8 \) are parallel, we need to use the concept of slopes of lines. Two lines are parallel if their slopes are equal.

### Step 1: Write the equations in slope-intercept form
We start with the equations of the lines:

1. \( 3x - 5y = 4 \)
2. \( 6x - ky = 8 \)

We will rearrange both equations to the form \( y = mx + b \), where \( m \) is the slope.

### Step 2: Rearranging the first equation
For the first equation \( 3x - 5y = 4 \):

\[ -5y = -3x + 4 \]
\[ y = \frac{3}{5}x - \frac{4}{5} \]

The slope \( m_1 \) of the first line is \( \frac{3}{5} \).

### Step 3: Rearranging the second equation
Now for the second equation \( 6x - ky = 8 \):

\[ -ky = -6x + 8 \]
\[ y = \frac{6}{k}x - \frac{8}{k} \]

The slope \( m_2 \) of the second line is \( \frac{6}{k} \).

### Step 4: Setting the slopes equal
For the lines to be parallel, their slopes must be equal:

\[ \frac{3}{5} = \frac{6}{k} \]

### Step 5: Cross-multiplying to solve for \( k \)
Cross-multiplying gives:

\[ 3k = 30 \]

### Step 6: Solving for \( k \)
Now, we divide both sides by 3:

\[ k = 10 \]

### Conclusion
Thus, the value of \( k \) for which the two lines are parallel is \( \boxed{10} \).

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