To find the equilibrium quantity of good X we have to equate the supply and demand of good X because the equilibrium quantity is determined where demand equals the supply of a good.
$\begin{array}{rcl}\mathrm{QDX}& =& 100\u20132\mathrm{PX}\u20134\mathrm{PY}+.05\mathrm{M}+.1\mathrm{AX}\\ \mathrm{QSX}& =& 4\mathrm{PX}\u201310\\ \mathrm{PY}& =& \$3\\ \mathrm{M}& =& \$24,000\\ \mathrm{AX}& =& \$500\\ \mathrm{Put}\mathrm{QSX}& =& \mathrm{QDX}\\ 4\mathrm{PX}\u201310& =& 100\u20132\mathrm{PX}\u20134\mathrm{PY}+.05\mathrm{M}+.1\mathrm{AX}\\ 4\mathrm{PX}\u201310& =& 100\u20132\mathrm{PX}\u2013(4\times 3)+(0.05\times 24,000)+(0.1\times 500)\\ 4\mathrm{PX}\u201310& =& 100\u20132\mathrm{PX}\u201312+1,200+50\\ 4\mathrm{PX}+2\mathrm{PX}& =& 100\u201312+1,200+50+10\\ 6\mathrm{PX}& =& 88+1,200+50+10\\ 6\mathrm{PX}& =& 1,288+50+10\\ 6\mathrm{PX}& =& 1,338+10\\ 6\mathrm{PX}& =& 1,348\\ \mathrm{PX}& =& \frac{1,348}{6}\\ \mathrm{PX}& =& 224.67\\ \mathrm{Put}\mathrm{PX}& =& 224.67\mathrm{in}\mathrm{supply}\mathrm{equation}\mathrm{to}\mathrm{find}\mathrm{equilibrium}\mathrm{quantity}.\\ \mathrm{QSX}& =& 4\mathrm{PX}\u201310\\ \mathrm{QSX}& =& 4\times 224.67\u201310\\ \mathrm{QSX}& =& 898.68\u201310\\ \mathrm{QSX}& =& 888.68\\ \mathrm{The}\mathrm{equilibrium}\mathrm{quantity}\mathrm{of}\mathrm{good}\mathrm{X}& =& 888.68\end{array}$
Therefore, the equilibrium quantity of good X is 888.68 units.