1 nickle = 5 cents

1 dime = 10 cents

1 quarter = 25 cents

let

n = the number of nickels, n>0, n is integer

d = the number of dimes, d>0, d is integer

q = the number of quarters, q>0, q is integer

I have $6.50 = 650 cents:

5n + 10d + 25q = 650 divide both sides by 5

n + 2d + 5q = 130

I have 7 more quarters than dimes

q = d + 7

and 5x as many nickles as quarters

n = 5q

by solving the system of equations

n + 2d + 5q = 130

q = d + 7

n = 5q

we find

n = 60 nickles

d = 5 dimes

q = 12 quarters

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I have 60 nickles, 5 dimes and 12 quarters.