The Fairfax County Board of Supervisors approved tax rate cuts yesterday on real estate and cars, softening the blow to residents who have complained of sharply higher assessments.

The board voted unanimously to reduce the real estate tax rate by 4 cents to $1.35 per $100 of assessed value for fiscal 1987. Under the new rate, annual taxes on an average house -- assessed at $109,597 -- will increase by about $18 to $1,480.

If the board had maintained the current tax rate of $1.39, taxes for the same property would have been about $1,523, an increase of $61.

The board also voted unanimously to cut the tax rate on vehicles by 20 cents to $4.60, which will trim $8 from the annual tax bill on an average automobile in Fairfax, valued at $4,040.

The reduction in the real estate tax rate for fiscal 1987, which begins July 1, was the third enacted by the board in as many years. Most of the supervisors credit the decrease to the county's economic development program, a position echoed in other area jurisdictions where the value of real estate has increased.

The assessment for an average house in Fairfax in 1984 was $100,207, compared with $109,597 for 1987. The tax rate has dropped 12 cents since 1984.

"There's no question in my mind that it's directly and almost totally attributable to the increase in commercial development in the county over the past 10 years," board Chairman John F. Herrity said of the tax rate cuts.

The change was partly overshadowed, however, by the board's decision to reject the demands by the county's 7,000 public schoolteachers for a pay increase beyond a cost-of-living raise.

The board adopted a carefully worded resolution, introduced by Supervisor Thomas M. Davis III (R-Mason), "to ensure that appropriate resources will be available when requested by the School Board" next year.

County board members said they are waiting for the recommendations of a commission that is studying the issue of teacher pay. The commission, appointed by School Superintendent Robert Spillane, is expected this summer to recommend some form of teacher pay increases.

The resolution that passed yesterday, county board members agreed, seemed tantamount to a promise to give the School Board what it requests for teacher salaries.

But some supervisors cautioned that the county board's commitment to spend money to increase teacher salaries also might, in effect, reverse the recent pattern of tax rate cuts.

"It could cost an awful lot of money" to increase teacher pay, said Supervisor Audrey Moore (D-Annandale).

Herrity insisted, however, that the board will continue to reduce tax rates no matter what the commission recommends on teacher salaries.

Donna Caudill, president of the Fairfax Education Association, which represents most of the county's teachers, said, "It is very clear the board [is] trying to find some way to satisfy teachers. They have never . . . made the promise before."

She added, however: "Teachers are very frustrated this year, and the morale is very low. This motion is a promise. It doesn't put 1 cent into teachers' salaries for next year."

Teachers asked for an 8.8 percent across-the-board raise, but the board granted them only a 4 percent cost-of-living increase that was given to all county employes.

One education association official last week threatened that teachers might strike if their demands were not met by the county board. Strikes by public employes are illegal in Virginia.

There were few surprises in the 2 1/2 hours it took the supervisors to mark up the budget. The supervisors had said last year, when they enacted a 7-cent real estate tax rate reduction, that this would be the year for a personal property tax cut, which many residents have demanded.

The reduction was the first change in the property tax rate since 1977.

To make room for the tax changes in the budget, the board cut spending by $19.5 million, eliminating, among other proposed items, a $7 million reserve fund to offset the potential loss of federal and state funds caused by the Gramm-Rudman-Hollings deficit reduction act. County Executive J. Hamilton Lambert had proposed the reserve fund, but supervisors said there were enough contingency funds in the budget without it.

The nine-member county board is to make the tax rates final and adopt a budget of $1.4 billion at its next session, April 28.