To determine which lines are parallel, we need to look at their slopes. Lines are parallel if they have the same slope.
Let's analyze the lines one by one:
A. \( y = \frac{1}{2}x + 3 \) and \( y = \frac{1}{2}x - 5 \)
- Both lines have a slope of \( \frac{1}{2} \). Therefore, these lines are parallel.
B. \( y = \frac{2}{3}x - 6 \) and \( y = -\frac{3}{2}x - 6 \)
- The slopes are \( \frac{2}{3} \) and \( -\frac{3}{2} \), which are not equal. So, these lines are not parallel.
C. \( y = \frac{1}{3}x + 4 \) and \( y = \frac{1}{3}x + 4 \)
- Both lines are identical, with a slope of \( \frac{1}{3} \). Therefore, these lines are also parallel.
D. \( y = -\frac{1}{2}x + 6 \) and \( y = \frac{1}{2}x - 8 \)
- The slopes are \( -\frac{1}{2} \) and \( \frac{1}{2} \), which are not equal. So, these lines are not parallel.
In conclusion, the two lines that are parallel are:
- A: \( y = \frac{1}{2}x + 3 \) and \( y = \frac{1}{2}x - 5 \)
- C: \( y = \frac{1}{3}x + 4 \) and \( y = \frac{1}{3}x + 4 \) (they are identical, thus parallel).