Alan wants to bake blueberry muffins and bran muffins for the school bake sale. for a tray of blueberry muffins, alan uses 1/3 cup of oil and 2 eggs. for a tray of bran muffins, alan uses 1/2 cup of oil and 1 egg. alan has 4 cups of oil and 12 eggs on hand. he sells trays of blueberry muffins for $12 each and trays of bran muffins for $9 each. alan wants to maximize the money raised at the bake sale. let x represent the number of blueberry muffins and y represent the number of bran muffins alan bakes.

what are the constraints for the problem?

So we know there are 12 eggs and they will be distributed like this:

12=x2+y1
and 4 cups of flour:
4=1/3x+1/2y

let's see if we can find a solution in which all the materials are used:
12=x2+y1
4=1/3x+1/2y//*6

12=2x+y
24=2x+3y//*6

substract from each other:
12=2y
y=2
12=2x+y
12=2x+2
10=2x
x=5

So if nothing is wasted, he can make 5 blueberry muffins (making 12*5=60) and 2 bran muffins (2*9=18), together 78 dollars,

would it be better if instead of the bran muffins he did blueberry muffins (which are more expensive?)
for the branberry muffins he will used the double amount needed for 1: so 1 cup of oil and 2 eggs. From this he can only make 1 tray of blueberry muffins (and some oil will be left over), which will cost 12 dollars, which is less than 18

so he better make 6 blueberry and 2 bran trays