Just bought a house and nearly all the faucets had leaks. The faucets appear to be Delta and I replaced the plastic cylinders which go in the hot and cold handles for the kitchen and two bathrooms. Don't know much about plumbing but do know how to do that much.

The drips from all the faucets stopped, but about 3 weeks later the kitchen has started a slow drip again. I was gone for a weekend and it had stopped, but started again as soon as I started using it.

Don't know whether to replace the cylinders again or what to do next; hate to call a plumber for $100/hour. Any suggestions?

Replace the faucets with new ones... its cheaper than a plumber and will solve the problem for many years.

You said you replace the cartridge (cylinders), did you replace the seats and springs?

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Unfortunately I don't know what those are. Guy at hardware store only sold me the plastic cartridges so I simply replaced those in the metal case that holds them. Didn't seem like there was any room for any springs and none were there when I pulled out the old ones.

Does the metal which holds the cartridge pull out as well?

They are under the cartridge. Remove the cartridge and use a small hook to pull them out. Note the direction of the spring! Replace both and that should solve the problem. In Delta faucet like yours, the cartridge only controls flow. Does the cartridge have a metal bottom or just plastic?