There are 23 coins in a bank. all the coins are dimes and quarters. the total value of the coins is $3.80. how many dimes are there? how many quarters?

write the system of equations that would be used to solve this problem.

The answer is 13 dimes and 10 quarters.

Let d be the number of dimes and q the number of quarters.

Since there are 23 coins (all dimes and quarters) in a bank, then the equation is:
d + q = 23
Since the value of 1 quarter is $0.25 and the value of 1 dime is $0.10, if the total value is $3.80, then the equation will be:
0.10d + 0.25q = 3.80 

Now, we have the system of equations:
d + q = 23
0.10d + 0.25q = 3.80 

Rearrange the first equation:
d = 23 - q

Substitute d from the first equation into the second equation:
0.10 * (23 - q) + 0.25q = 3.80

0.10 * 23 - 0.10 * q + 0.25 q = 3.80
2.30 + 0.15q = 3.80
0.15q = 3.80 - 2.30
0.15q = 1.50
q = 1.50 / 0.15
q = 10

Now when we know the number of quarters, it is easy to calculate the number of dimes using the first equation:
d = 23 - q
d = 23 - 10
d = 13