Dennis mowed his next door neighbor's lawn for a handful of dimes and nickles, 80 coins in all. Upon completing the job he counted out the coins and it came to $6.60. How many of each coin did he earn?

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

• [tex]d+n=80[/tex] is an equation representing the total number of coins

• [tex]0.10d+0.05n=6.60[/tex] is an equation representing the total value in money based on the number of coin. 0.10 and 0.05 come from the value of a dime and nickel individually.

We write the first equation in terms of n by subtracting it across the equal sign to get [tex]d=80-n[/tex]. We now substitute this for d in the second equation.

[tex]0.10(80-n)+0.25n=6.60\\8-0.10n+0.05n=6.60\\8-0.05q=6.60[/tex]

After simplifying, we subtract 8 across and divide by the coefficient of n.

[tex]8-0.05q=6.60\\-0.05q=-1.4\\n=28[/tex]

We now know of the 80 coins that 28 are nickels. To find the total number of dimes, we subtract 28 from 80 to get 52 dimes.