Start writing the corresponding system of equations taking into account that each dime is equal to $0.1 and each quarter is equal to $0.25, let d be the number of dimes and q the number of quarters.
then,
[tex]\begin{cases}0.1d+0.25q=11.70 \\ d=3q+7\end{cases}[/tex]insert the second equation into the first one
[tex]\begin{gathered} 0.1\cdot(3q+7)+0.25q=11.70 \\ \text{solve for q} \\ 0.3q+0.7+0.25q=11.70 \\ 0.55q+0.7=11.7 \\ 0.55q=11.7-0.7 \\ q=\frac{11}{0.55} \\ q=20 \end{gathered}[/tex]find the number of dimes using the number found and the second equation
[tex]\begin{gathered} d=3q+7 \\ d=3\cdot20+7 \\ d=60+7 \\ d=67 \end{gathered}[/tex]prove the solution using the first equation
[tex]\begin{gathered} 0.1d+0.25q=11.7 \\ 0.1\cdot67+0.25\cdot20=11.7 \\ 6.7+5=11.7 \\ 11.7=11.7 \end{gathered}[/tex]There are 67 dimes and 20 quarters.
