SOLUTION: I am lost with this questions, can someone explain it to me? Given the restraints {x+y {<=}5; y>_x; x>_0 and the objective Function C == x + y, find the maximum value?

Question 549801: I am lost with this questions, can someone explain it to me?
Given the restraints {x+y {<=}5; y>_x; x>_0
and the objective Function C == x + y, find the maximum value?

Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
In this kind of problem, you have two variables, with restrictions, and a function of those two variables that you want to optimize (maximize or minimize). Your restrictions define a feasibility region. You can graph the inequalities and see what that region looks like. In the type of problem you will be given, your restrictions set boundaries that are straight lines. Your feasibility region ends up being some sort of polygon. Your function to optimize is also a linear function of both variables. Life may throw you a curve, but your math course just hands you linear functions for this type of problem.
For your problem:
is the y-axis, and is the y-axis plus all the points to the right.
is the x-axis, and is the x-axis plus all the points up from there.
is the line that goes through (5,0) and (0,5), as you can see by substituting. is that line plus the points down and to the left.
The three restrictions together determine a feasibility region that is the triangle with vertices at (0,0), (5,0), and (0,5).
The linear function will have constant values along paraller lines, like , , , etc. To you increase the value of you have to move from one line to the next, and you can do that until you get to the boundaries of your feasibility region. In general you can get to the maximum at one of the corners (vertices) of your feasibility region, or you may get to the maximum all along a boundary line.
In this case, you can see that for the point (0,0), as you substitute the coordinates x=0 and y=0 into , you get C=0.
For (5,0) and (0,5) you get C=5. That means that the maximum is C=5 and happens all along the boundary line .
So it does not matter what your x and y values are. As long as they are at the total maximum of 5 set in your restrictions, you get the maximum for C.
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