15 items in total. By using a system of linear equations, I can buy

a) sneaker=7, wallet=6, watch=2
b) sneaker=6, wallet=5, watch=4
c) infinitely many solutins
d) None.

A) works, it comes out to exactly $82. But I was wondering could there be other combinations? Should I even consider C)?
Please explain why or why not.

There are 3 unknowns ie 3 quantities to buy and 2 equations ie one from total quantity and the other from toal amount.
You need one more eqn which is not available. Hence guess work is the only way out. You did good.