problem: Hillary has eleven coins, all dimes and nickels. In total, the coins are worth 75 cents. How many nickels does she have?
solution: Let the number of dimes Hillary has be $d$ and the number of nickels she has be $n$. We have the two equations \begin{align*}
  d+n&=11\\
  10d+5n&=75
  \end{align*} (The last equation is in terms of cents.) To make the second equation nicer, we divide both sides by 5 to get $2d+n=15$. From the first given equation, we have $d=11-n$. Substituting this into the simplified second given equation to eliminate $d$, we get $2(11-n)+n=15\Rightarrow n=7$. Thus, Hillary has $\boxed{7}$ nickels.