The zoning board of appeals shall fix a reasonable time for the hearing of the appeal or other matter referred to it pursuant to section 14-409 and give due notice of such hearing to the parties and decide such appeal or other matter within a reasonable time. Upon the hearing, any party may appear in person or by agent or by attorney. The zoning board of appeals may reverse or affirm, wholly or partly, or may modify the order, requirement, decision, or determination appealed from and shall make such order, requirement, decision, or determination as in its opinion ought to be made in the premises, and to that end shall have all the powers of the officer from whom the appeal is taken. Where there are practical difficulties or unnecessary hardships in the way of carrying out the strict letter of such ordinance, the zoning board of appeals shall have the power in passing upon appeals, to vary or modify the application of any of the regulations or provisions of such ordinance relating to the use, construction, or alteration of buildings or structures or the use of land, so that the spirit of the ordinance shall be observed, public safety and welfare secured, and substantial justice done.