Final answer:
The equation of the line parallel to the given line 5x - 4y = -2 and passing through (6, 6) is y = ⅔x - 1. The equation of the line perpendicular to the given line and passing through (6, 6) is y = -⅔x + 12.
Step-by-step explanation:
To find the equation of the line that is parallel to the given line 5x - 4y = -2 and passes through the point (6, 6), we need to keep the same slope as that of the given line. First, we rewrite the given equation in slope-intercept form to find its slope:
4y = 5x + 2
y = ⅔x + ½
The slope is ⅔. The parallel line must have the same slope, so its equation will have the form y = ⅔x + b. To find b, the y-intercept, we substitute the coordinates of the point (6, 6) into the equation:
6 = ⅔(6) + b
After solving, we find b = -1. So the equation of the parallel line is y = ⅔x - 1.
To find the equation of the line perpendicular to the given line that passes through the point (6, 6), we use the fact that the product of the slopes of two perpendicular lines is -1. The slope of the perpendicular line is therefore the negative reciprocal of ⅔, which is -⅔. The equation of the perpendicular line has the form y = -⅔x + b. We again substitute the point (6, 6):
6 = -⅔(6) + b
After solving, we find b = 12. So the equation of the perpendicular line is y = -⅔x + 12.