Final answer:
The lines represented by the equations 5x - 8y = 3 and 10y + 16x = 1 are perpendicular to each other as their slopes are negative reciprocals (5/8 for the first line and -8/5 for the second line).
Step-by-step explanation:
To determine whether the lines represented by the equations 5x - 8y = 3 and 10y + 16x = 1 are parallel, perpendicular, or neither, we'll need to find the slopes of the lines by converting each equation into slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
For the first equation 5x - 8y = 3, we solve for y to get the slope-intercept form:
- -8y = -5x + 3
- y = (5/8)x - 3/8
The slope of the first line is 5/8.
For the second equation 10y + 16x = 1, we also solve for y:
- 10y = -16x + 1
- y = (-16/10)x + 1/10
- y = (-8/5)x + 1/10
The slope of the second line is -8/5.
Since the slopes of the two lines are negative reciprocals of each other (5/8 and -8/5), the lines are perpendicular to each other. If the slopes were equal, the lines would be parallel, and if the slopes were neither equal nor negative reciprocals, the lines would be neither parallel nor perpendicular.