write the first equation in slope-intercept form: 5y=7x+4, y=7/5x+4/5
you see that these two equation have the same slope 7/5, they are parallel.
the distance is the line segment perpendicular to both lines.
perpendicular lines have slopes that are negative reciprocals to each other, in this case, the two parallel lines' slope is 7/5, so the perpendicular line has a slope of -5/7, the equation is y=(-5/7)x+b
pick a point on either line for this perpendicular lien to go through. I'll pick (0, -10) on the second line Y=(7/5)x-10 (when x=0, y=-10)
the perpendicular line going through (0,-10) is y=(-5/7)x-10
Next, find out the where this perpendicular line meet the other line y=(7/5)x+4/5 by solving the system of equations:
y=(7/5)x+4/5
y=(-5/7)x-10
x=-729/37, y=-6235/259
these numbers are ridiculous, I've tried other points, no better results.
I give up.
the last step is to find the distance between the point (0,-10) and the point (-729/37, -6235/259)
