Question

Consider the linear programming problem

Maximize P=60x+50y.

x+y≤80

5x+10y≤560

50x+20y≤1600

x≥0

y≥0

Answer 1

http://www.wolframalpha.com/widget/widgetPopup.jsp?p=v&id=1e692c6f72587b2cbd3e7be018fd8960&title=Linear%20Programming%20Calculator&theme=blue

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3220 at (12,50)
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maximize | function | 60 x + 50 y\ndomain | x + y<=80 && 5 x + 10 y<=560 && 50 x + 20 y<=1600 && x>=0 && y>=0
Global maximum:
max{60 x + 50 y|x + y<=80 && 5 x + 10 y<=560 && 50 x + 20 y<=1600 && x>=0 && y>=0} = 3220 at (x, y) = (12, 50)

Answer 2

therefore at the intersection of
5x+10y≤560
and
50x+20y≤1600

Answer 3

If you want to do it manually, on graph paper, plot these lines:

x+y=80
5x+10y=560
50x+20y=1600
x=0 ; y=0
The acceptable solution region will be on the border or in the enclosed area. But there is a nice theorem in Math that tells us the maximum will lie on the border, at one of the intersecting border lines. So test the corners with P=60x+50y, and you will find the maximum in a jiffy.