The provisions of law relating to the term of office of members of boards of education, including method and time of qualification and of taking office shall apply to every school district in this state. In the event no candidate is elected to a position on the board of education of a school district in a regular election, or if an elected member moves out of the school district after such member is elected and before such member takes office the second Monday in January following the election, or if such member becomes ineligible to serve for any other reason during that period of time, the holdover member shall continue to sit as a voting member of the board of education of such school district until an eligible successor is appointed by the board of education to fill the position.