Parallelogram diagonal and line equation

The diagonals of a parallelogram bisect each other. Midpoint of the given opposite vertices $(2,1)$ and $(-3,-4)$ is: M = $\left(\frac{2 - 3}{2}, \frac{1 - 4}{2}\right) = \left(-\frac{1}{2}, -\frac{3}{2}\right)$.

The other two vertices also form a diagonal whose midpoint is M. Hence, the line containing the other two vertices must pass through M.

Substitute $\left(-\frac{1}{2}, -\frac{3}{2}\right)$ into $x + 9y + c = 0$: $-\frac{1}{2} + 9\left(-\frac{3}{2}\right) + c = 0 \implies -\frac{28}{2} + c = 0 \implies -14 + c = 0 \implies c = 14$.