2009 Fermat Problem 23

Suppose that a, b, c, and d are positive integers that satisfy the equations \begin{array}{rcl} ab + cd &=& 38 \\ ac + bd &=& 34 \\ ad + bc &=& 43 \end{array} What is the value of a + b + c + d?

\textbf{(A)}\ 15\quad \textbf{(B)}\ 16\quad \textbf{(C)}\ 17\quad \textbf{(D)}\ 18\quad \textbf{(E)}\ 19

If there are no answer choices shown, enter a numerical answer.

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