The equation for line g is given by 5y=2x-20. suppose line g is parallel to like r and line e is perpendicular to line g. point (10,-8) lies on both line r and line

Sounds as tho' all three lines are parallel.  Is that correct?

If so, all three lines have the same slope!

What is the slope of   5y=2x-20  ?     Solving for y, y = (2/5)x - 4

The slope is (2/5).

Now find the equation of the line with slope (2/5) that passes thru (10,-8).

It is    y - [-8] = (2/5)(x - 10).

We can simplify this, with the result   y + 8 = (2/5)x - 4.  This, in turn, can be simplified to y + 8 = 2x/5 - 4,                  or   y = 2x/5 - 12.  Mult. all terms by 5 to eliminate the fraction:

5y = 2x - 60

Note how this has the exact same form as the given  5y = 2x - 20, EXCEPT that the constant at the end is different.

5y = 2x - 60    and    5y = 2x - 20 are parallel lines with the same slope but different y-intercepts.