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Could you run your in-house electricity on a 12V system since all your appliances and radios and TVs mostly have a built-in transformer to reduce the current to a much lower voltage?

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You wouldn't want to. The Watts are the reason why.

If you needed to power 2000 watts worth of appliances and you have a 120v system, you'd need a little under 17 amps to do it. A relatively thin 12-gauge wire will handle that much power. To power the same load with 12v current, you need 170 amps--and wire 3/8" in diameter.
The built-in transformer you speak of requires 110 volts. They're known as step-down transformers and are a ratio of the incoming voltage. For example: a 10:1 transformer will drop 100 volts to 10 volts. If you gave it 12V you'd end up with 1.2V.
In short: No to your question.

There are many appliances -- such as Google's servers -- that are designed to run on a 12 VDC supply. Appliances designed for recreational vehicles (RVs) and boats often run on 12 VDC.
In theory, a technician with some knowledge of electronics could convert an appliance that is designed to require 110 VAC and convert it to require only 12 VDC. If you are lucky, the appliance already has a transformer and rectifier that converts input power to 12 VDC, and a technician can simply attach a 12 VDC power plug, cable, and fuse directly in to that point. In practice, you would need some system to prevent accidentally plugging the 12 VDC power plug into a 110 VAC outlet, which would probably destroy the appliance, and also prevent accidentally plugging in both the 12 VDC power plug and the 110 VAC power plug, which risks destroying the 12 V power supply.

A few devices use the AC cycle to derive a time base (50 Hz or 60 Hz), and so cannot work on 12 VDC power alone, unless you use an "inverter" that generates that frequency.

Some devices -- such as air conditioners -- require a lot of energy. Given a particular energy requirement, reducing the voltage by 1/10 means increasing the current by a factor of 10, which requires power lines with 10 times as much copper. Increasing the voltage allows us to reduce the amount of copper used -- which is one reason some automobiles have a "42 V automotive system".
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