Meal Platters Optimization Problem Mark has to buy hamburgers, hot dogs, and pig's feet for an event. The restaurant he is purchasing from offers two Platter options. Platter A comes with 4 hamburgers, 3 hot dogs, and 2 pig's feet. Platter B comes with 3 hamburgers, 4 hot dogs, and 5 pigs feet. Platter A costs 15 and Platter B costs 12. If Mark needs to by no less than 220 hamburgers, 270 hot dogs, and 250 pig's feet, what is the minimum amount he can spend?
I have two questions about this problem. The first is, what is the best way to solve this problem? The second is, what is the best way to solve this problem with only Algebra I knowledge?
 A: Problems involving optimizing a linear function constrained by a set of linear inequalities can be solved by a technique called linear programming.  The constraints define a polygonal region of possible solutions.  It can be proved that if there is an optimum (not infinity), it will be achieved at one of the "corners" of the region.
In this case we know there is an optimum, since we have a lower bound of $0$.  So all we need to do is find the right "corner".
In this case, we wish to optimize the function:
$$f(a,b) = 15a + 12b$$
Subject to three constraints:


*

*$4a+3b≥220$

*$3a+4b≥270$

*$2a+5b≥250$


We can find the intersections of the boundaries of each constraint region by solving systems of each pair of linear equations.  Constraints 1 and 2 intersect at $(10, 60)$, 1 and 3 intersect at $(25, 40)$, and 2 and 3 intersect at $(50, 30)$.  Of these, $(25, 40)$ does not satisfy constraint $2$, so it lies outside the region of interest and we can discard it.
We now compute the value of the function we want to optimize at each of the valid corner points, and choose the best one, which happens to be $(10, 60)$.
