The long-pursued vision of Washington and its sprawling suburbs tied together by Metrorail took a leap forward Tuesday as Loudoun County committed to support Metro’s Silver Line extension to Dulles International Airport and into eastern Loudoun.

After months of heated debate and tense negotiations, the 5 to 4 vote by the Board of Supervisors appeared to end the latest, and perhaps the last, round of uncertainty over the fate of the Silver Line.