I have a 1976 Lincoln Mark IV that I recently got off craigslist. I know the heater core is bypassed (the mechanic who went with me told me), but my knowledge on cars is just starting to grow, and as such I’m not totally certain what needs to be done (beyond replacing the heater core). I’ve attached a picture of what I think are two pipes that lead to the heater core, but I’m not sure once I do get the heater core replaced what I need to attach these to and what kind of hose I need. Any help would be much appreciated!
IMHO your best bet is to get a repair manual for the car and follow its descriptions and diagrams. There are companies on the internet that specialize in supplying repair manuals for old vehicles.
@the_same_mountainbik I have the electrical shop manual for my car, which covers HVAC, but it doesn’t seem to talk about the heater core beyond how to bleed it. I’ve looked through a lot of it and I’ve yet to find a basic diagram for what needs to get connected where.
You’ll probably need the electrical manual at some point, but I was referring to the mechanical shop manual.
I have volumes 2 (Engine), 3 (Electrical), and 4 (Body) of the 1975-1976 Ford Car Shop Manual. The electrical covers HVAC, but the engine doesn’t cover anything related except for stuff about the radiator.
The horse shoe shaped hose going around the oil dipstick is the “bypass”. The connection on the intake manifold is the supply and the connection on the water pump is the return. The person that replaces the heater core should reconnect these to the two pipes to check the heater operation and for leaks.
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Just curious, what are your intentions with the car? Daily driver? I am curious about it, how many miles, story behind it, stuff like that.
For some reason those old 70’s boats captivate me. I love the story behind cars like that.
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IMHO the old mid-'70s Lincolns were very classy cars. As were the mid-'70s Caddys. Big, extraverted, and “personal” were luxury hallmarks for the two manufacturers in those days.
But I don’t know enough about them to help the OP. I was driving Toyotas back then. I was one of the “have nots”. 
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@WhoSaidRick Yeah it’s my daily driver. It was a one-owner car before me. Came with 47k miles. The guy barely drove it, and he died 4 or 5 years ago, and his son didn’t have the space/ability to store it so he sold it. I’m loving it so far. Pretty much everything about the car is original (even the spark plug wires were the originals from '76), and the only issues I’ve had was a power steering/brake booster line exploding, a slight radiator leak, and needing to replace the thermostat. Once I loaded it with some refrigerant, even the A/C worked! The compressor sounds like it’s nearing the end, but it works great!
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@Nevada_545 thank you very much!
You need to get a fresh set of plug wires
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@db4690 don’t worry I did . I replaced the wires, plugs, and distributor cap.
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Sweet car! It seems as if you are giving it a great loving home. I have never seen plug wires that old that were still functioning properly, that must be a record.
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In regards to the AC
I know your car came with R12 refrigerant, and it’s no surprise the system was empty
If you haven’t already done so, I recommend adding some ultraviolet dye, because those old systems were known for leaking. And then you’ll at least be able to see the green glow
R12 is expensive, not available everywhere, and there’s a lot of counterfeit stuff for sale. If you have to start replacing leaking components every week, you might want to consider retrofitting
@WhoSaidRick Thanks! 
It gets me lots of looks and compliments, though some are on my parking (it isn’t easy to park a 19-foot-long boat) . The body is in nearly perfect shape, too.
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@db4690 I retrofitted it to R-134a, and added dye, but my UV light seems to have disappeared, so I need to pick up another one.
So which pipe should connect to which? Is the large pipe the return line or the supply line?
The heater pipes appear to be two different sizes, 5/8" and 3/4". The fittings on the engine should also be two different sizes, match the same size connections.
@Nevada_545 The two connectors are the engine are about 1 inch on the pump, and around .9 inches on the other connector. You are correct with the two pipe diameters.
The motivation to bypass the heater core is usually that it is leaking and dripping water on the passenger compartment floor. You should make sure the floor is sound, and not so rusted w/holes it is about to fall apart. Pull up the carpet for a look-see. You want your passengers to stay in the car with you, not fall out through a hole in the floor. 
I wouldn’t replace the heater core until I knew for sure that it leaked, and if it did leak, what exactly was leaking. It could be just the hose connection at the heater core has split. My truck’s heater core of similar era is still working fine. It’s probably a copper core and should be pretty tough. The hoses running to the heater core are usually pretty obvious to find, as they are the only coolant hoses that go into and out of the fire wall.
I have a Ford truck w/302 V8 of similar era. You probably have a larger engine, but will probably experience similar challenges as I do w/ my truck keeping it going. FWIW, keeping it on the road is definitely doable, Detroit iron is pretty tough stuff. But expect from time to time to develop engine vacuum leaks (the engine will start to idle poorly all of a sudden), brake problems, water pump failures, and the carb will likely need to be rebuilt at some point. That will probably be the second or third rebuild on it since new. Suggest to get a copy of the owner’s manual and follow the suggested maintenance schedule. Most of the problems I’ve had on my truck were caused by deferred maintenance. For example, part of your heater core diagnosis and fix should include replacing the coolant with fresh stuff. Best of luck.