First, to write the slope-intercept equation for a line passing through (3, -2) that is parallel to the line 3X + 4Y = 9, we need to find the slope of the given line. We can rewrite the line equation in slope-intercept form:
4Y = -3X + 9
Y = (-3/4)X + 9/4
The slope of this line is -3/4. Since the line we want is parallel to this line, it will have the same slope. We can use the point-slope form of a line to write the equation:
Y - (-2) = (-3/4)(X - 3)
Simplifying this equation, we get:
Y = (-3/4)X + 7/2
This is the slope-intercept equation for the line passing through (3, -2) that is parallel to the line 3X + 4Y = 9.
Next, to write the equation for a line passing through (3, -2) that is perpendicular to the line 3X + 4Y = 9, we again need to find the slope of the given line. We can rewrite the line equation in slope-intercept form:
4Y = -3X + 9
Y = (-3/4)X + 9/4
The slope of this line is -3/4. Since the line we want is perpendicular to this line, it will have a slope that is the negative reciprocal of -3/4. The negative reciprocal is 4/3. We can use the point-slope form of a line to write the equation:
Y - (-2) = (4/3)(X - 3)
Simplifying this equation, we get:
Y = (4/3)X - 2/3
This is the slope-intercept equation for the line passing through (3, -2) that is perpendicular to the line 3X + 4Y = 9
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