The Fairfax County Board of Supervisors formally adopted a budget yesterday that will leave the $1.54 real estate tax rate unchanged but still mean a 13 percent higher bill for the average homeowner.

The supervisors last week rejected Acting County Executive J. Hamilton Labert's bigger budget calling for a four-cent increase in the tax rate. But the spending plan they unanimously approved yesterday will mean $137 more in the annual real estate taxes for the owner of a median-priced house. Taxes for that house will go from $1,063 to $1,200 because the value increased from $69,025 to $77,902 between 1979 and this year.

The county's $443.3 million budget includes 8.5 percent raises for school-teachers and other public employes and a $6.5 million reserve as a cushion against higher energy prices and expected cutbacks in federal aid.

This year's budget generated little controversy among board members. To keep the tax rate at $1.54 per $100 of assessed valuation, the supervisors reduced Fairfax's subsidy for Metro transit deficits by $2 million and ordered a total of $1.37 million in agency cuts -- or less than 1 percent trimmed from the entire budget. Only $1.24 million was cut from the school system's $217.4 million operating budget.