I find it easier to identify the vertices of the feasible region if I graph it. You can find them by solving the equations pairwise. The vertices are ...
(0, 0), (0, 3), (1.5, 3.5), (5, 0)
The corresponding objective function values are 0, -12, -5, 30.
The maximum value of the objective function is 6·5-4·0 = 30 at (x, y) = (5, 0).
