problem: Given $f(x)=cx^3-9x+3$ and $f(2)=9$, find the value of $c$.
solution: Plugging $x=2$ into the expression for $f(x)$, we find $f(2)=c(2^3)-9(2)+3=8c-18+3=8c-15$. Since we know that $f(2)=9$, \begin{align*} f(2)&= 9
  \\\Rightarrow\qquad8c-15&=9
  \\\Rightarrow\qquad8c&=24
  \\\Rightarrow\qquad c&=\boxed{3}
  \end{align*}