1 penny = 1 cent

1 nickle = 5 cents

1 dime = 10 cents

let

p = the number of pennies

n = the number of nickels

d = the number of dimes

Diana has $7.37 = 737 cents in pennies, dimes, and nickels in her change purse

p + 5n + 10d = 737

she has one more than three times as many nickels as pennies

n = 1 + 3p

and six fewer dimes as nickels

d = n - 6

by solving the system of equations

p + n + d = 737

n = 1 + 3p

d = n - 6

we find

p = 17 pennies

n = 52 nickles

d = 46 dimes

click here to see the step by step solution of the system of equations