For what values of k straight lines `2x - ky + 3 = 0 and 3x + 2y - 1 =0` are parallel ?

Options

A. `4/3`
B. `(-4)/(3)`
C. `(2)/(3)`
D. `(-2)/(3)`

Answer

To determine the value of \( k \) for which the lines \( 2x - ky + 3 = 0 \) and \( 3x + 2y - 1 = 0 \) are parallel, we need to find the slopes of both lines and set them equal to each other.

### Step 1: Write the first line in slope-intercept form
The first line is given by:
\[ 2x - ky + 3 = 0 \]
Rearranging this equation to solve for \( y \):
\[ -ky = -2x - 3 \]
Dividing by \(-k\):
\[ y = \frac{2}{k}x + \frac{3}{k} \]
From this, we can identify the slope \( m_1 \) of the first line:
\[ m_1 = \frac{2}{k} \]

### Step 2: Write the second line in slope-intercept form
The second line is given by:
\[ 3x + 2y - 1 = 0 \]
Rearranging this equation to solve for \( y \):
\[ 2y = -3x + 1 \]
Dividing by 2:
\[ y = -\frac{3}{2}x + \frac{1}{2} \]
From this, we can identify the slope \( m_2 \) of the second line:
\[ m_2 = -\frac{3}{2} \]

### Step 3: Set the slopes equal to each other
For the lines to be parallel, their slopes must be equal:
\[ m_1 = m_2 \]
Substituting the values of \( m_1 \) and \( m_2 \):
\[ \frac{2}{k} = -\frac{3}{2} \]

### Step 4: Solve for \( k \)
Cross-multiplying to solve for \( k \):
\[ 2 \cdot 2 = -3 \cdot k \]
This simplifies to:
\[ 4 = -3k \]
Now, dividing both sides by \(-3\):
\[ k = -\frac{4}{3} \]

### Conclusion
The value of \( k \) for which the lines are parallel is:
\[ \boxed{-\frac{4}{3}} \]