Originally posted by @kquinn909
What is the absolute pressure in a piping system when the vacuum gage reads 20" and the barometer reads 30.71" of mercury?

What are the units on the vacuum gauge? When you find that information ( it's probably in Water ) the Barometer is reading the atmospheric pressure and the gauge is reading vacuum gage pressure (measurement relative to atmosphere). You need to convert to consistent units ( whichever pressure unit suits you ).

Then,

P_abs = P_atm - P_vacuum

Originally posted by @kquinn909
Not an answer to the question. I know the answer.

It looks to me like a 20 inch reading indicates a partial vacuum. And at 20 " it would seem that pressure would indicate it being 1/3 of full pressure so my guess is the pressure in the closed system is about 5 PSI.
Since we are clearly talking about a closed system using some metal for the parts there would be no connection of the barometric pressure which is outside air, the fact the pipes are strong says at 14.7 PSI there would be essentially zero compression of the piping and so the air is of little consequence. But to compare apples to oranges, there would be about a 10 PSI differential but the pressure inside the pipes will still be 5 PSI which would mean if a valve was opened to the internal pressure, air would be forced inside at about 10 PSI till they equalized.
With good seals, that 5 PSI would still be 5 PSI a month later.

Originally posted by @kquinn909
you win

If you already knew the answer why did you ask the question? Just to see how dumb we are?