Final answer:
Lines are classified as parallel, perpendicular, or neither based on their slopes. Line 2 and Line 3 are parallel as both have slopes of -2/3. Line 1, with a slope of 3/2, is neither parallel nor perpendicular to Lines 2 and 3.
Step-by-step explanation:
To determine the relationship between the given pairs of lines, we need to find their slopes. Lines are parallel if they have the same slope, perpendicular if the product of their slopes is -1, and neither if they do not meet the criteria for being parallel or perpendicular.
For Line 1: 6x - 4y = -8, we can rewrite it in slope-intercept form (y = mx + b) to find its slope (m). Dividing the entire equation by -4, we get y = (3/2)x + 2, which means the slope is 3/2.
For Line 2: y = -(2/3)x - 8, the slope is already given as -2/3.
For Line 3: 3y = -2x + 7, rewriting it gives us y = (-2/3)x + 7/3, which means the slope is -2/3.
To compare:
- Line 1 and Line 2 have slopes of 3/2 and -2/3, respectively. To be perpendicular, the slopes should be negative reciprocals of one another. Here, 3/2 is not the negative reciprocal of -2/3, so they are neither parallel nor perpendicular.
- Line 1 and Line 3 have slopes of 3/2 and -2/3, respectively. The same reasoning applies, so they are also neither parallel nor perpendicular.
- Line 2 and Line 3 both have the same slope of -2/3, which means they are parallel to each other.
Therefore, the only pair of lines that have a specific relationship are Line 2 and Line 3, which are parallel. Line 1 does not share a perpendicular or parallel relationship with Lines 2 or 3.