Kevin and Randy Muise have a jar containing 73 ?coins, all of which are either quarters or nickels. The total value of the coins in the jar is ?$ 8.45. How many of each type of coin do they? have?

This situation has two unknowns - the total number of nickels and the total number of quarters. Because we have two unknowns, we will write a system of equations with two equations using the two unknowns.

n+q=73 is an equation representing the total number of coins
0.05n+0.25q=8.45 is an equation representing the total value in money based on the number of coin. 0.05 and 0.25 come from the value of a nickel and quarter individually.

We write the first equation in terms of q by subtracting it across the equal sign to get n=73-q. We now substitute this for n in the second equation.

0.05(73-q)+0.25q=8.45

3.65-0.05q+0.25q=8.45

3.65+0.20q=8.45

After simplifying, we subtract 3.65 across and divide by the coefficient of q.

0.20q=4.8

q=24

We now know of the 73 coins that 24 are quarters. To find the number of nickels, we subtract 24 from 73 and get 49 nickels.