Resisting the temptation to make even a token tax cut, the Fairfax County Board of Supervisors approved a budget last night that could result in a $3.4 million surplus at the end of the next fiscal year.

The sentiment was overwhelming against a cut of 3 cents in the present tax rate of $1.74 for each $100 of real estate valuation. When Chairman John F. Herrity (R) made the motion, it died without being seconded.

Supervisor Marie B. Travesky (R-Springfield) seemed to sum up that sentiment when she said during the deliberations: "I think the taxpayers would view us as more responsible if we kept the tax rate steady instead of yo-yoing it up and down year after year."

The total budget of the next fiscal year, which runs from July 1 to June 30, 1978, rounds out at about $500 million. The general fund budget - spending supported by county taxes and not federal, state and other outside revenues - accounts for $313.2 million of the total, an 11 per cent increase over last year.

By comparison to last year - when the supervisors slashed $14 million from proposed spending and still had to increase taxes by 13 per cent - this year's budget making process was relatively unevenful, although, typically, it was extended virtally up to the deadline.

While the tax rate remains the same, Fairfax homeowners will pay higher taxes this year. That will happen because the value of most houses, according to the assessors, went up. The tax bill for the owner of a median-priced house, valued at $63,000, will be $1,096 this year, about $10 more than last year.

While the effective tax rate will not change, homeowners do have now figures to work with because beginning this year, the rate is based on 100 per cent valuation instead of 40 per cent valuation. Thus a tax rate of $4.35 per $100 at the old 40 per cent valuation translates into the new $1.74 rate. The end result in the amount of tax paid is the same.

In putting together the new budget, the supervisors made no drastic reductions in expenditures, and only modest additions.

Schools lost only about $1.5 million from the $14.6 million the school board had requested, and only 20 funded but vacant position in the county government were eliminated.

The police got $600,00 more than County Executive Leonard Whorton had requested in his proposed budget - an amount that translates into about 26 addition officers. Fire and rescue got $100,000 more, which will mean nine new fire fighters.

The supervisors appropriated $1.3 million for sidewalk and drainage construction. Some of that money will provide relief to many homeowners whose low-lying property and basements are chronically flooded during rains. Another $264,000 was added for tax relief for the disabled.

Perhaps the biggest loser yesterday was chairman Herrity, who not only saw his 3-cent tax cut proposal die unseconded but lost on every one of his cost-cutting motions. After he was forced to withdraw one motion to cut $22,700 from a miscellaneous fund, he grumbled from his chair, "I haven't accomplished a helluva lot."

Herrity also suffered two big defeats on Metro. The supervisors, by a 7-to-2 vote, refused to support his idea for a new formula that would drastically reduce Fairfaxs share of making up subway operating deficits. He was also overwhelmingly defeated when he sought to wipe cut Fairfax's share of regional liability for revenue bonds.