Tim, I asked to have the following problem solved: Now, here is a real challenge problem. This particular problem is associated with a class of problems called linear programming problems. We already know that there are an infinite number of points that satisfy the labor and material constraints. So, there are an infinite number of possible mixes of type X and type Y frames that Ryan can produce during the upcoming week. Now, suppose that each frame type X contributes a profit of $2.00, and each Frame type Y contributes a profit of $3.00. Now, assuming the company wishes to maximize profit, how many of each type frame should Ryan make during the upcoming week? And you gave me this solution:We have to maximize the profit, so if Type X produces x and Type Y produces y frames then 2x+3y has to be maximized given conditions 2x+y<=4000 and x+2y<=5000 and also x , y>0 as profits are non-negative so just assume 2 slack variables x1=4000-2x-y and x2=5000-x-2y the intersection is (1000 , 2000) then 1000 of X type and 2000 of Y type so the maximum profit is 2*1000+3*2000=8000 Now I need some clarification as to What are x1 and x2, and why do they equal 4000-2x-y and 5000-x-y, respectively. I have no idea what those expressions represent. And, given those expressions where did the point (1000, 2000) come from.

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