I'm assuming your constraints contain ≤ and ≥, rather than < and >.
The expression (-2x+y) will be at a minimum when x is as large as possible and y is as small as possible. The smallest y can be is 0. Since 5x+2y<20, the largest x can be is 4. Therefore, c=-2x+y has its minimum at (4,0), which is c=-8.