For what value of k, are the lines x + 2y + 9 = 0 and kx + 4y - 5 = 0 parallel?

Options

A. 2
B. -1
C. 1
D. 0

Answer

To determine the value of \( k \) for which the lines \( x + 2y + 9 = 0 \) and \( kx + 4y - 5 = 0 \) are parallel, we need to find the slopes of both lines and set them equal to each other, as parallel lines have the same slope.

### Step-by-Step Solution:

1. **Find the slope of the first line**:
The first line is given by the equation:
\[ x + 2y + 9 = 0 \]
Rearranging this equation to the slope-intercept form \( y = mx + c \):
\[ 2y = -x - 9 \]
\[ y = -\frac{1}{2}x - \frac{9}{2} \]
From this, we can see that the slope \( m_1 \) of the first line is:
\[ m_1 = -\frac{1}{2} \]

2. **Find the slope of the second line**:
The second line is given by the equation:
\[ kx + 4y - 5 = 0 \]
Rearranging this equation to the slope-intercept form:
\[ 4y = -kx + 5 \]
\[ y = -\frac{k}{4}x + \frac{5}{4} \]
From this, we can see that the slope \( m_2 \) of the second line is:
\[ m_2 = -\frac{k}{4} \]

3. **Set the slopes equal to each other**:
Since the lines are parallel, we set their slopes equal:
\[ -\frac{1}{2} = -\frac{k}{4} \]

4. **Solve for \( k \)**:
To eliminate the negative signs, we can multiply both sides by -1:
\[ \frac{1}{2} = \frac{k}{4} \]
Now, cross-multiply to solve for \( k \):
\[ 1 \cdot 4 = 2 \cdot k \]
\[ 4 = 2k \]
Dividing both sides by 2:
\[ k = 2 \]

Thus, the value of \( k \) for which the lines are parallel is \( \boxed{2} \).