The Arlington County Board yesterday approved a $6.7 million financing plan that will permit the construction of the Washington area's first county-wide cable television system. The 30-channel system is expected to begin broadcasting next April.

The Board, which licenses cable television in Arlington, unanimously approved a financing plan for the system under which Northern Virginia stockholders would retain a majority interest in the Arlington Tele Communications Corp (ARTEC), according to Thomas W. Richards, the president.

The Board approved the sale of $2 million of television company stock to a Delaware subsidiary of the General Electric Corp. It also gave ARTEC approval to take out a $4.7 million loan from the Chemical Bank of New York to finance construction.

An Arlington ordinance requires that any major shift in financing must be approved by the Board, which appoints one-third of Aors. ARTEC officials and board members said they were "very pleased" that the proposed cable television network could now begin construction after being under consideration since 1971.

The Arlington Board also heard a report by County Manager W. Vernon Ford and community affairs director William L. Hughes on the sewer back-ups that have been plaguing some residents of the recently renovated Fairlington Villages condominiums.

Hughes told the Board that because of a backlog in inspections, caused in part by a shortage of inspectors, 200 units were currently being occupied although certificates of occupancy have not nbebeen issued.

In response to questions from Chairman Joseph S. Wholey and Vice Chairman John W. Purdy, Hughes acknowledged that occupancy without an appropriate permit is technically illegal.

However, Hughes noted and the Board agreed that the 200 homeeowners would be required to vacate their homes only in an extreme case involving a menace to public health.

Hughes said later, "The county hasn't enforced the certificates vigorously unless it involved a major safety issue," such as the partial occupancy of a building under construction. He also said that the couunty does not normally issue occupancy certificates for single-family dwellings. "We have gone to this process because of the major renovations at Fairlington" and 300 citizen complaints about raw sewage backups," he said.

Hughes told the Board that he directed county inspectors last week to post fluorescent orange stickers on the doors of 400 unsold and unoccupied units. The stickers advise prospective buyers to ask to see a certificate before completing a sale.

Hughes said his staff is investigating the incidence of sewage backups in some of the 200 occupied units before his staff issues occupancy permits. They are attempting to determine if there is a public health issue involved. CBI Fairmac Corp., the sales agent for Fairlington, has told Hughes it has lost its data on past sewage problems. Hughes said he is presently trying to track down previous tenants and contact current residents to see if they have experienced problems. Collecting data this way "is the most frustrating thing in the world," he said.