find the distance between the lines with equations 3x -4y=8 and y=3x/4+4.

Answer 1

lines are
y = (3/4) x + 4 through (0,4)(-16/3,0)
and
y = (3/4) x - 2 through (0,-2)(8/3,0)
so they are indeed parallel (same slope)
sketch the lines
the lower line goes up (2 while it goes right (8/3)
so the tangent of its angle to the y axis is (8/3)/2 = 8/6 = 4/3
or tan T = 4/3
of course the hypotenuse of a triangle with tangent = 4/3 is 5
so sin T = 4/5
and cos T 3/5
now if we draw a line perpendicular to the upper line from point (0,4) to the lower line it is of our unknown length d.
then sin T = d/6 (the distance between lines on the y axis)
4/5 = d/6
d = 24/5