Question 1119561
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Tutor @ikleyn has provided a thorough solution along with a clear explanation of how to solve the problem using the standard linear programming method.<br>
However, you can refine the standard method to allow you to reach the final answer with less work.<br>
The three constraint lines determine the boundaries of the feasibility region.  The maximum value of the objective function has to occur at a corner of the feasibility region, or perhaps at any point along one edge of the feasibility region.<br>
The standard solution process says you should evaluate the objective function at all corners of the feasibility region to determine the maximum value of the objective function.  In fact, that is not true.<br>
Furthermore, you don't even have to find all of the intersection points of the constraint lines; you can tell where the maximum value of the objective function is going to be by comparing the slope of the objective function to the slopes of the three constraint lines.<br>
The slopes of the three constraint lines in this problem are 4, -2, and -4; the slope of the objective function is -8/5.  Since -8/5 is between 4 and -2, the maximum value of the objective function is going to be where the constraint lines with slopes 4 and -2 intersect.<br>
So after you have the slopes of the three constraint lines and the objective function, you only need to find one corner of the feasibility region and evaluate the objective function at that corner.<br>
To understand why this shortcut works, look at the graph in tutor @ikleyn's response and imagine various lines with the slope of the objective function, -8/5.  You want a line with slope -8/5 that just touches a corner of the feasibility region; that will be at the corner where the slope of -8/5 is between the slopes of the two intersecting constraint lines.