SOLUTION: Hello, I have a problem, that I dont know what am I doing wrong with it.The problem is like this: A vendor sends an invoice that totals $6115.68, this invoice contains 2 types o

Question 823594: Hello, I have a problem, that I dont know what am I doing wrong with it.The problem is like this:
A vendor sends an invoice that totals $6115.68, this invoice contains 2 types of products that have different prices.
Product 1: $1.65 per case
Product 2: $2.46 per case.
How many cases of each product is the vendor charging for?
I just cant figure out how to do this, can someone please explain me?

Found 2 solutions by josgarithmetic, KMST:
Answer by josgarithmetic(39620) (Show Source): You can put this solution on YOUR website!
The question lacks some data for a specific answer.

Let x and y be the number of cases for product 1 and product 2 in that order.

Accounting for money, .
You may expect these simple restrictions.
and , because you must have whole numbers for x and y; negative numbers cannot happen.

Additionally, the cost of either x or y must not reach or go beyond 6115.68 dollars. This means
1.65x<6115.68
x<6115.68/1.65

and
2.46y<6115.68
y<6115.68/2.46

With this, you may have several possible solutions. Having any further more specific data may give you a much more specific answer.

The y-intercept is at 2485.77

Do you have ANY case count data to include with the description?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
= number of cases of product 1.
= number of cases of product 2.
= cost in $ for units of product 1.
= cost in $ for units of product 2.
= total cost for units of product 1, and units of product 2.
according to the vendor's invoice, that is $6115.68, so
is our equation.
We have two variables and only one equation.
In general that would mean an infinite number of answers.
However, in this case, only integer, non-negative solutions would work.
That limits the number of solutions.

Once we find one solution, we can increase the found by while decreasing the found by .
That would change both terms by units and in opposite directions, so that they would still add up to .
Since divided by yields a quotient of with a remainder of , there are at least solutions, and maybe .

If , transforms into
, , and .
So one of the possible solutions is .
That is a non-negative integer, so it is allowed to be and .
That solution means units of each product.
It could be that there were 55 less units of product 2,
meaning units of product 2,
but 82 more units of product 1,
meaning .
That would decrease the term by ,
but would increase term by .
The total would still be , with and .
We could repeat the procedure a few more times, as long as the decreasing is still a nonn-egative number.
Since divided by has a quotient of with a remainder of , we can do that decreasing by a total of times until we get to .
That means that besides ordered pair (x,y)=(1488,1488), There are other solutions with lesser values of , going all the way to , .
On the other hand, we could decrease by 82, while increasing by 55.
That would give us another solutions, with decreasing all the way to because yields a quotient of and a remainder of 12.
There is a total of solutions:
units of product 1 with units of product 2,
units of product 1 with units of product 2,
units of product 1 with units of product 2,
units of product 1 with units of product 2,
and so on, all the way to
units of product 1 with units of product 2.
I guess we could write them as
paired with
foe integer, such that
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