Final answer:
To determine which set of lines will always be equidistant or parallel, compare the slopes of the lines. If two lines have the same slope, they will be parallel.
Step-by-step explanation:
To determine which set of lines will always be equidistant or parallel, we need to compare the slopes of the lines. If two lines have the same slope, they will be parallel.
Let's compare the slopes of each set of lines:
A) ay = 2x - 5 and y = 5x - 5
Slope of ay = 2x - 5 is 2 and slope of y = 5x - 5 is 5. Since the slopes are different, these lines are not parallel.
B) by = (3/5)x - 3 and 5y = 3x - 10
Slope of by = (3/5)x - 3 is 3/5 and slope of 5y = 3x - 10 is 3/5. Since the slopes are the same, these lines are parallel.
C) cy = (5/6)x - 6 and x + 5y = 4
Slope of cy = (5/6)x - 6 is 5/6 and slope of x + 5y = 4 is -1/5. Since the slopes are different, these lines are not parallel.
D) dy = 7x + 2 and x + 7y = 8
Slope of dy = 7x + 2 is 7 and slope of x + 7y = 8 is -1/7. Since the slopes are different, these lines are not parallel.
Therefore, the answer is option B) by = (3/5)x - 3 and 5y = 3x - 10.