x/2 + y/5 = 1/2,    (1)
3x  - y   = -8.     (2)

Multiply equation (1) by 10 (both sides) to rid off denominators. You will get

5x + 2y = 5         (3)

instead of (1), and it is good. Now you have the system of two equations 

5x + 2y =  5,       (3)
3x -  y = -8.       (2)

Now multiply equation (2) by 2 (both sides) and then add to equation (3).
By doing this, you eliminate the unknown y and get a single equation for the unknown x:

5x + 6x = 5 + 2*(-8),   or

11x = -11.

Hence, x = -1.

Now from (2) you have y = 3x + 8 = 3*(-1) + 8 = 5.

The system is solved. The solution is x = -1, y = 5.