Attempting to find all combinations of 3 product subtotals given a certain sales tax rate.

[shatterstar6457]

Ok, so the research I've done:

Googled multi-variable equation calculator, system of equations calculator/solver, multi-variable combinatorics, etc., but none of those searches have what I'm looking for. I do a ton of couponing and I am looking to reduce my out of pocket as much as possible.
scenario:

I have 3 $8 coupons, but the subtotal must be greater than the total of my coupons ($24) and the number of coupons cannot be greater than the number of items in my transaction, my tax rate 7%. I have one item priced at $19.99, so that leaves item 2 (lets call it x) and item 3 (lets call it y) + the sales tax (7%) = $29 (the maximum of the transaction amount so that I can use reward points to pay the remaining taxes). I think the equation would look something like this:
(((19.99+x+y))*0.07)+((19.99+x+y)(8*3))+v =29

x=item 1 subtotal

y=item 2 subtotal

v=reward points (can be used to pay off taxes, maximum is $5 worth of points, henceforth $29 being the max transaction)

So first, we calculate the tax for 3 items, given that one of them is $19.99,
(((19.99+x+y))*0.07)

then we add that tax back to the original subtotal,
+((19.99+x+y)

then we add our coupons ($24 worth)
+(8*3))

and lastly, we calculate points
+v

I hope this wasn't too confusing, thanks in advanced.

 

[tkhunny]

the subtotal must be greater than the total of my coupons ($24)
the number of coupons cannot be greater than the number of items in my transaction
(lets call it y) + the sales tax (7%) = $29
maximum of the transaction amount so that I can use reward points to pay the remaining taxes

"the equation"? Please note that your constraints are NOT generally equalities. They are INequalities. What do you know of Linear Programming?

 

[shatterstar6457]

next to nothing, do I need to create a program to solve this? If so, please point me in the right direction.

 

[tkhunny]

Define your variables carefully.

Maybe:
$$I$$ = # of items in the transaction
$$C$$ = # of coupons

Now, write a single equation for EACH necessary relationship.

Here's the easiest one: "... the number of coupons cannot be greater than the number of items in my transaction."

$$C \le I$$