jmc

Let $c$ and $b$ be the number of pounds of corn and beans Shea buys, respectively. We can turn the givens into a two-variable, linear system: $$\begin{array}{rl}b+c& =24\\ 45b+99c& =1809\end{array}$$ We can multiply the first equation by 45 and subtract it from the second to get $(99-45)c=1809-45(24)$, which reduces to $54c=729$ or $c=13.5$. So Shea buys $\boxed{13.5\text{ pounds}}$ of corn.