Jeremy has a collection of dimes and quarters. Overall, he has 51 coins and their total value is $10.35. What is the total value of ONLY the quarters?

This situation has two unknowns - the total number of dimes 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.

[tex]d+q=51[/tex] is an equation representing the total number of coins
[tex]0.10d+0.25q=10.35[/tex] is an equation representing the total value in money based on the number of coin. 0.10 and 0.25 come from the value of a dime and quarter individually.

We write the first equation in terms of q by subtracting it across the equal sign to get [tex]d=51-q[/tex]. We now substitute this for d in the second equation.

[tex]0.10(51-q)+0.25q=10.35\\5.1-0.10q+0.25q=10.35\\5.1+0.15q=10.35[/tex]

After simplifying, we subtract 5.1 across and divide by the coefficient of q.

[tex]5.1+0.15q=10.35\\0.15q=5.25\\q=35[/tex]

We now know of the 51 coins that 25 are quarters. To find the total value of the quarters, we multiply 35 by 0.25 and find 8.75.