Francis A. Teti, a 40-year-old Rockville accountant going through a divorce, has a large house and two school-age children. What he didn't have somebody at home to look after his son and daughter when they got out of class.

Hairdresser Theresa Gibson, 29, is a single mother with a newborn son. She wanted to stay home with him, but somehow she had to earn their living.

They face a common dilemma -- how to resolve the conflicting demands of home and job -- confronting the growing number of single-parent families in our society.

In their case, they both took their problem to Operation Match, a Montgomery County program aimed at helping people willing to share a house or an apartment find each other.

Other Washington jurisdictions are seriously considering adopting a similar program, says housing planner Roger Wentz of the Metropolitan Washington Council of Governments. Home-sharing programs are springing up around the country in response to the high cost of housing.

Teti told the organization he was looking for someone to help clean the house -- a sprawling, four-bedroom rambler -- cook the evening meal and look after his children from 3 to 6 p.m. But "a housekeeper was more than I could afford."

His offer, in exchange, was a private bedroom and bath in the basement, along with free meals and a small weekly salary. To Gibson, who lacked the means for her own apartment "that I have always wanted," the deal seemed a good one.

Since she and her son moved in a few weeks ago, she's changed her mind about an apartment. "I like this better. It gives me a chance to be home with my child, and there's a yard for him to play in."

Says Teti: "It works out great in my situation. I'm a CPA working late at night. It's a big relief when I have someone there."

The arrangement also gives him more time with his children. "I'm not bogged down with laundry and housecleaning."

Operation Match, now 3 years old, was started initially to link elderly homeowners struggling with the high cost of maintaining a house with low-income elderly who needed an inexpensive place to live. "We're astonished by the low incomes of the elderly," says founder Marjorie Levenson.

But almost immediately, the program's scope was broadened to include all age groups. Finding a place to live, as Levenson points out, is becoming increasingly difficult "as low and moderate-income households are faced with escalating rents, condominium conversion and reduction in the availablity of affordable housing." More than 90 percent of the organization's applicants earn less than $13,000 a year.

"Our population," adds staff member Gretchen Stromberg, "is heavy at each end. The younger and the older seem to be most affected by inflation." The young often are newcomers to Washington stunned by the high cost of rentals. A very large percentage of the organization's clients -- 77 percent -- are women.

"We're getting white-collar women who don't want to pay condo fees and are trying to find comparable apartments." Even at $350 a month, this is proving difficult.

So far, say Levenson and Stromberg, they have matched almost 500 people, either having a room available or seeking one. The arrangement can involve a simple sharing of rent, to the barter of room and board (and wages) for household help. They use the word "share," because full use of the house, including kitchen, is generally available.

Harry Bracken, 88, of Silver Spring, who is deaf, wanted someone to cook his evening meals in exchange for room and board in his small bungalow. That suited Michael Israel, 50, a retired crane operator with a small disability check. He came here recently from Washington state "because for many years I wanted to see Congress and how the government works."

They have shared the two-bedroom home now for five months, and so far, says Israel, the arrangement "is not too bad at all. I'm a little stubborn and he's a little stubborn."

Frances Jackson, 27, a medical technician, "needed a roommate" to rent a three-bedroom townhouse in Rockville. Currently, she shares the rent and utilities "50-50" with a 26-year-old woman. Because of Jackson's varied working hours, "I hardly see her. It's like having a house to yourself."

A Washington woman, 58, bought a two-bedroom condominium and then a few months later was forced to retire on disability. Her mortgage and condo fees were higher than her monthly check, and she had to dip into a small savings to pay them. She grew anxious, and her health began to fail.

Operation Match paired her with a widow her age who had moved out of Washington to cut living expenses and then returned because she missed her friends and church. She'd been looking for a $ 200 apartment in her old neighborhood, but couldn't find one.

"The home-seeker's new affordable housing is within a block of her church, convenient to her doctor's office and her old friends," says Levenson. c"The owner is comforted by the additional income she receives while finding someone pleasant with whom to share her retirement. Her home has been saved."

Though some matches can be arranged within a week, one homeowner father waited six months for a woman his daughter could look up to as a role model. A former nun with "not a great deal of money," says Stromberg, "met the requirements."

An elderly woman sought the company of another woman needing inexpensive housing. "I just need someone to be here, especially at night," she told Operation Match. She said she was afraid of falling down, perhaps in the tub. "I just want someone to be able to call for help."

A divorced man paying alimony -- and unable to afford much rent -- lives in the home of a woman who wanted a "fix-it man" to paint. "We told him to go slow," says Levenson, so he could keep the room.

Levenson and Stromberg see Operation Match as a "clearinghouse." A staff of eight, including volunteers, screens both the home providers and the home seekers, pinpointing preferences in roommates. "Smoking and nonsmoking is crucial," says Levenson. "It can throw off an otherwise perfect match."

Pets are another important consideration, and furniture can be a stumbling block. "Most homes are completely furnished," says Levenson. "The renter can't bring any furniture."

Outlying locations such as Poolesville and Burtonsville may be a problem because of transportation difficulties.

The staff visits the homes it lists for sharing. Currently, about 100 residences -- including houses, townhouses and apartments, all with at least one extra bedroom -- are available.

Home seekers must have a face-to-face interview at Operation Match's Silver Spring office, and references are checked. That's a big reason medical technician Jackson turned to them. "They screen, and I didn't have time."

Staff members meet in conference each Wednesday to see which applicants they can link up. The more "flexible" a person's requirements, the easier the match. After that, it's up to the two parties to work out the details.

For people unaccustomed to strangers in their house, they suggest starting out on a week-by-week trail basis. "An underlying sense of humor, to see things lightly," says Stromberg, also helps smooth the relationship. "It's like a marriage; you're betting on an impression."

So far as home providers who may expect too much from a housemate, Levenson say, "We don't want them to believe that they can get a fulltime slave for just room and board." They rougly equate room and board with 15 hours of work a week.

Some matches last; others are set up as temporary from the beginning, and some householders have had three or four housemates in the past three years. A few matches break up quickly for a variety of reasons. As one renter complained: "There's so much loud music in the house, I can't stand it."

Operation Match, a part of Montgomery County's Housing Opportunities Commission, is funded by government agencies. Unlike commercial roommate firms, it is free. Matches are made throughout the Washington area, but services are limited to individuals and couples over 18 and single parents with one child.

"Home-sharing," says Levenson, "implies a loss of the privacy so prized in our culture and a necessary adjustment to another's personality and life style." But for those willing or needing to give it a try, "there are significant tradeoffs."