Let n = #nickels, d = #dimes, q = #quarters.

Then your equations are

  n +   d +   q = 25,     (1)
 5n + 10d + 25q = 275     (2)    (cents)
10n + 25d +  5q = 375.    (3)

To solve it, express n = 25 - d - q from (1), then substitute it into (2) and (3), replacing n in each of these two equations. You will get

 5*(25 - d - q) + 10d + 25q = 275     (2')
10*(25 - d - q) + 25d +  5q = 375.    (3').

Simplify to get the standard form of a 2x2 system.

Then solve it by any method you know to get n = 10, d = 10 and q = 5.


It is just technique.