Let 'x' be the number of quarters and 'y' be the number of dimes. Also x and y are non-negative integer numbers. We need to find x and y.

We know that

1 quarter = 25 cents

1 dime = 10 cents

$1 = 100 cents

Follwoing the conditions we have

x + y = 15 => x = 15 - y

25 x + 10 y = 270 ( since $2.70 = 2.70 * 100 cents = 270 cents)

By replacing x from the first equation in the second we have

25 (15 - y) + 10 y = 270

375 - 25 y + 10 y = 270

25 y - 10 y = 375 - 270

15 y = 105

y= 105 / 15

y = 7 dimes

x = 15 - y

x = 15 - 7

x = 8 quarters

Amy has 8 quarters and 7 dimes.