Given two straight lines `3x - 2y = 5` and `2x + ky + 7 =0`. Find the value of k for which the given lines are :
parallel to each other.

Answer

To find the value of \( k \) for which the given lines \( 3x - 2y = 5 \) and \( 2x + ky + 7 = 0 \) are parallel, we need to follow these steps:

### Step-by-Step Solution

1. **Identify the equations of the lines**:
- The first line is given as \( 3x - 2y = 5 \).
- The second line is given as \( 2x + ky + 7 = 0 \).

2. **Convert the first line to slope-intercept form**:
- Rearranging \( 3x - 2y = 5 \) to the form \( y = mx + c \):
\[ -2y = -3x + 5 \]
\[ y = \frac{3}{2}x - \frac{5}{2} \]
- The slope \( m_1 \) of the first line is \( \frac{3}{2} \).

3. **Convert the second line to slope-intercept form**:
- Rearranging \( 2x + ky + 7 = 0 \) to the form \( y = mx + c \):
\[ ky = -2x - 7 \]
\[ y = -\frac{2}{k}x - \frac{7}{k} \]
- The slope \( m_2 \) of the second line is \( -\frac{2}{k} \).

4. **Set the slopes equal for parallel lines**:
- For the lines to be parallel, their slopes must be equal:
\[ m_1 = m_2 \]
\[ \frac{3}{2} = -\frac{2}{k} \]

5. **Cross-multiply to solve for \( k \)**:
- Cross-multiplying gives:
\[ 3k = -4 \]
- Solving for \( k \):
\[ k = -\frac{4}{3} \]

### Final Answer
The value of \( k \) for which the given lines are parallel is \( k = -\frac{4}{3} \).

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