You have ten U.S. coins that add up exactly to $1.00

Hi, my name is Andy. I am currently taking linear algebra. My teacher assigned us a problem but I've come to an impasse.

(4 points) You have ten U,S. coins in current circulation that add up to exactly $1.00. Find the solution that has the most types of coins. (Current circulation denominations: $1.00, 50 cents, 25 cents, 10 cents, 5 cents, 1 cent.)

(variables are represented by the first letter of their coin name)

If you multiply the first equation by 10, and the second equation by 100, you can then set both equations equal to each other.

However, I'm not quite sure what to do from here. Isolate a variable? This worked with 3 variables (chicken, hen, rooster problem), but 5 variables is too much to do this with! or is it? RREF form? hmmmm....

The intended way to solve this was to isolate the variable with the highest value, F. I'm lost here.

Can anyone give me the next step? I'm not looking for a total answer, but just a lead in the right direction.