Answer:
P = 18
Explanation:
You want the maximum value of P = 3x +2y given the constraints ...
- 5x +y ≤ 16
- 2x +3y ≤ 22
- x ≥ 0
- y ≥ 0
Graphical solution
The four inequalities define a quadrilateral solution region with vertices ...
(0, 0), (0, 7.333), (2, 6), (3.2, 0)
The value of the objective function P is maximized at (x, y) = (2, 6). Its value there is ...
P(x, y) = 3·2 +2·6 = 6 +12 = 18
The maximum value of P is 18.
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Additional comment
When the solution region is the overlapping solution spaces of several inequalities, it can be easier to see the solution region if that region is white, rather than four overlapping colors.
In the attached graph, we reversed each of the inequalities so that the solution space is white, and the shaded regions are excluded from the solution space. The boundary lines are all included.
The blue line is the objective function. It is maximized when the line is as far as possible from the origin. We like to plot this line because its slope tells us that we have located the vertex that maximizes P.