Fairfax Board Chair Sharon Bulova (D). (Tom Jackman)

The Fairfax County Board of Supervisors on Tuesday passed a new cap on property taxes that would allow homeowners’ bills to increase by an average of $304 in Virginia’s largest jurisdiction.

The 7-to-3 vote means that the board can increase the property tax rate by no more than 4 cents, to $1.13 per $100 of assessed value, when it adopts its fiscal 2017 budget in April. Four cents is the amount recommended by County Executive Edward L. Long Jr. last month.

The board agreed to advertise that rate after a sometimes passionate debate over the merits of raising property taxes even higher to better meet the increasing demands of the county school system while preserving other services that have long made Fairfax County a magnet for upper-middle-class families.

A motion to raise property taxes by as much as 5 cents failed in a 5-to-5 tie. Another motion for a cap that is 6 cents higher than the current rate of $1.09 per $100 of assessed value got little support.

“That motion carries, and that is unfortunate,” Sharon Bulova (D), chairman of the Board of Supervisors, said about the 4-cent proposal. Bulova had supported a 5-cent hike.

The debate about the tax rate in Fairfax pitted worries about declining schools and other services against the frustrations of homeowners who argue that living in Fairfax is becoming too expensive.

School Superintendent Karen Garza argues that the quality of one of the nation’s top school systems is in jeopardy without extra funding, as more teachers leave for higher-paying districts and students crowd into classrooms with as many as 35 pupils.

On Tuesday, Garza, who is seeking $68 million more than the $1.88 billion that county officials have proposed for next year, expressed her frustration with the 4-cent cap on the tax-rate increase.

“Today’s decision is discouraging for the thousands of community members who have reached out to the School Board and the Board of Supervisors this year to advocate for a voice in this very important conversation about values and priorities,” Garza said in a statement posted on the school system’s website.

Other county officials warned that property taxes are already too high, particularly in areas such as Tysons Corner, where property owners pay an extra 6 cents per $100 of assessed value to cover the cost of services there.

“The folks in Tysons right now are not millionaires, for heaven’s sake,” said Supervisor Linda Q. Smyth (D-Providence), who represents a portion of that area. “They’re really average working people, and they’re seeing their taxes and their assessments go up.”

After the vote Tuesday, frustration was palpable among some board members and the teachers in the audience.

“This advertised tax rate will not come close to adequately funding the needs of our county,” said Kimberly Adams, head of one of the county’s two teachers unions.

Several supervisors agreed that the county needs to find new ways to fund schools and other services in the county.

“We need to think about not just this year but several years ahead and find something that’s sustainable,” said Supervisor John C. Cook (R-Braddock), who was advocating for budget cuts.

Supervisor Catherine M. Hudgins (D-Hunter Mill) proposed that the county revisit the idea of asking voters to approve a tax on restaurant meals, which restaurant owners have fiercely opposed in the past.

The board passed a motion by Hudgins to study the idea.