Diophantines

[Mr. Pedantic]

A bank teller made a mistake today. The teller switched the dollars and cents when they cashed a check for Mrs. Jones, giving her dollars instead of cents and cents instead of dollars.

After buying a newspaper for 5 cents, Mrs. Jones realized that she had remaining exactly twice as much as the original check.

What was the amount of the original check?

So first off, just as an FYI: already figured out the answer. No "do your own homework" posts please.

My final equation for this problem simplifies down to 98x - 5 = 199y, where both x and y are integers below 100.

My problem is that diophantine equations don't really seem to be about elegance and simplicity of method as most other mathematical puzzles seem to be, but more about brute strengthing the answer (which is, of course, made a lot easier by computers).

Is there a good way to figure out x and y without plugging every single possibility into a calculator?

 

[DrPizza]

Sure there's a good way - do it in your head (not with a calculator.)

You know that 98x-5 must be evenly divisible by 199.
You also know that 98x - 5 will always be an odd number.
199y will only be odd numbers for odd values of y.

So, 98x-5 must be 199 or 597 or 995 or...
(That just cut the number of calculations in half.)

I started playing with it for a few minutes; more later, if I have free time. (Photocopy time!)

 

[Mr. Pedantic]

Sure there's a good way - do it in your head (not with a calculator.)

You know that 98x-5 must be evenly divisible by 199.
You also know that 98x - 5 will always be an odd number.
199y will only be odd numbers for odd values of y.

So, 98x-5 must be 199 or 597 or 995 or...
(That just cut the number of calculations in half.)

I started playing with it for a few minutes; more later, if I have free time. (Photocopy time!)

A solution:

So basically the idea is to just keep substituting values until the diophantine equation is so simple you can work it out more easily, and then substitute it back.

It works.