You have have a jar containing 72 coins, all of which are either quarters or nickels. the total value of the coins in the jar is $8.40. how many of each type of coin do you have?

Total number of coins: T=72
Number of quartes ($0.25): q=?
Number of nickels ($0.10): n=?

Number of quarters + Number of nickels = Total number of coins
q+n=72  (Equation 1)

Total Value of the coins in the jar: V=$8.40
Value of the quarter's coins: $0.25q
Value of the nickel's coins: $0.05n

Value of the quarter's coins + Value of the nickel's coin=Total value of the coins in the jar
$0.25q+0.05n=$8.40
0.25q+0.05n=8.40 (Equation 2)

We have a system of 2 equations and 2 unknowns (q and n):
(1) q+n=72
(2) 0.25q+0.05n=8.40

Using the method of substitution:
Let's isolate q in the first equation:
(1) q+n=72
(1) q+n-n=72-n
(1) q=72-n

Let's replace q by 72-n in the second equation:
(2) 0.25q+0.05n=8.40
0.25(72-n)+0.05n=8.40
Applying the distributive property:
(0.25)(72)-0.25n+0.05n=8.40
Multiplying the constant terms and adding similar terms of the variable n:
18-0.20n=8.40
Isolating n:
18-0.20n-18=8.40-18
-0.20n=-9.6
(-0.20n)/(-0.20)=(-9.6)/(-0.20)
n=48

Replacing n by 48 in the first equation:
(1) q=72-n
q=72-48
q=24

We have 24 coins of quarters and 48 coins of nickels