It's obvious you meant $1.00 but left off the decimal. I changed it for you.



Eliminate n by multiplying the first equation through by -5



Adding these term by term:



The smallest absolute value of a coefficient is 4,
so write all integers in terms of their nearest multiple
of 4




Divide through by 4



Isolate fractional terms:



The right side is an integer, so let that integer be A, 
set both sides equal to integer A:



Clear the first equation of fractions:



Solve the first equation for d



Substitute that for d in 







Subsitutute  and  in 






So now we have the numbers of coins in terms of integer A



10 dimes makes a dollar so there can't be as many as 10 dimes, since
we have to have 21 coins, so





Add 1 to all three sides:



Divide all three sides by 4




Since A is an integer then A is either 1 or 2, for
they are the only integers between  and 
So there will be two solutions,

If A = 1, then 



So that's one solution 5 pennies, 13 nickels and 3 dimes.

If A = 2, then 



So that's the other solution 10 pennies, 4 nickels and 7 dimes.

Edwin