To determine whether the lines [tex]\( y = -\frac{2}{7}x - 7 \)[/tex] and [tex]\( y = \frac{7}{2}x + 3 \)[/tex] are parallel, perpendicular, or neither, we need to carefully consider their slopes.
### Step-by-Step Solution:
1. Identify the slopes of each line:
- For the first line [tex]\( y = -\frac{2}{7}x - 7 \)[/tex], the slope is the coefficient of [tex]\( x \)[/tex], which is [tex]\( -\frac{2}{7} \)[/tex].
- For the second line [tex]\( y = \frac{7}{2}x + 3 \)[/tex], the slope is the coefficient of [tex]\( x \)[/tex], which is [tex]\( \frac{7}{2} \)[/tex].
2. Check if the lines are perpendicular:
- Two lines are perpendicular if the product of their slopes is [tex]\( -1 \)[/tex]. Let's check that:
[tex]\[
\left(-\frac{2}{7}\right) \times \left(\frac{7}{2}\right) = -\frac{2 \times 7}{7 \times 2} = -\frac{14}{14} = -1
\][/tex]
- Since the product of the slopes is [tex]\( -1 \)[/tex], the lines are perpendicular.
3. Check if the lines are parallel:
- Two lines are parallel if their slopes are equal. Clearly,
[tex]\[
-\frac{2}{7} \neq \frac{7}{2}
\][/tex]
- Hence, the slopes are not equal, and the lines are not parallel.
4. Summary:
- Since the product of the slopes is [tex]\( -1 \)[/tex], the lines are perpendicular.
- And since the slopes are not equal, the lines are not parallel.
Therefore, we conclude that the lines are perpendicular.
Answer: A. perpendicular
