For the first one, graph the lines x+5y = 16 and 5x+y = 8. They intersect at (1,3). Then our region is bounded by the two lines and the two coordinate axes. This region has 4 vertices. One at the origin, (0,0), one at the point of intersection (1,3), one at (0,16/5), and one at (8/5,0). Plugging all these into the objective function, we see that the function is maximized at (1,3) with a value of 7.
For the second one, graph the lines x+4y = 27 and 7x + y = 27. They intersect at (3,6). Now our region is still bounded by two lines and the coordinate axes, but now the region is unbounded to in the positive x and y directions. Our region, then has only 3 vertices. (27,0), (3,6), and (0,27). Plugging these into the objective function we see that we get a minimum value of 15 at (3,6). If we were asked to maximize the function, we would not get an answer as the objective function is unbounded from above in the region.
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