To find the equation of the line that passes through the point (5,-8) and is perpendicular to the line 5x-4y=16, we need to follow these steps:
1: Rewrite the given equation in slope-intercept form:
5x - 4y = 16
-4y = -5x + 16
y = (5/4)x - 4
So the slope of the given line is 5/4.
2: The slope of a line perpendicular to this line will be the negative reciprocal of 5/4, which is -4/5.
3: Use the point-slope form of the equation of a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line, to find the equation of the line:
y - (-8) = (-4/5)(x - 5)
y + 8 = (-4/5)x + (4/5)*5
y + 8 = (-4/5)x + 4
y = (-4/5)x - 4
Therefore, the equation of the line that passes through the point (5,-8) and is perpendicular to the line 5x-4y=16 is y = (-4/5)x - 4.