z = 6x + 7y would define a plane in 3-space, so the maximum value will occur somewhere on the boundary of the region. But all the boundaries of the region are straight lines, so the maximum value will occur at one of the vertices.
The vertices of the region are (0,3), (0,9), (4,0), (6,0). The corresponding values of f(x,y) are
21, 63, 24, 36. The largest is 63.
And that's the answer, 63. Achieved when x = 0 and y = 9.