At today's prices, you can buy a 1 oz. gold american eagle for about $450.
The $50 comes out of thin air, actually. It's no different than when the Mint takes 2 cents worth of metal and stamps it into a dime. When that dime is delivered to the Federal Reserve, it magically becomes worth 10 cents. That "profit" gets credited to the US Mint, and is called seigniorage.
The reason the gold and silver american eagles have a face value is that it is more attractive to buyers, even though the face value is far below the value of the metal content of gold or silver. And the face value is purposely set very low because the government doesn't want to see gold and silver become part of regular circulating coinage. With a 1 oz. gold american eagle at $50 face value, it's very unlikely anyone will ever spend one for $50.