A siphonic flush toilet system and method of priming the same having a toilet bowl assembly comprising at least one jet flush valve assembly and at least one rim valve; and bowl having a rim and a jet defining at least one jet channel, the at least one jet channel having an inlet port and a jet outlet port configured for discharging fluid to a sump area, wherein the sump area is in fluid communication with a trapway. The bowl has a closed jet pathway including the jet channel and extending from the jet flush valve assembly outlet to the jet channel outlet port to maintain the jet channel in a primed state with fluid from the jet flush valve assembly so as to assist in preventing air from entering the closed jet pathway. Flush valves are also disclosed having back-flow preventer mechanisms and/or at least partly flexible valve covers for use with the toilet systems and methods herein.