A very quick way of solving this problem is to recognize and use the fact that a line parallel to 5x-2y=2 has an equation which is exactly the same as 5x-2y=2 except for the constant.
Replace that 2 (on the right side of the equation) with C, and x with -4, and y with -3:
5(-4)-2(-3)=C
and find C. C = -20 + 6 = -14.
Therefore, the equation of the line parallel to 5x-2y=2 passing thru the point (-4, -3) is 5x-2y= -14. Notice how only the constant changes.
