The answer is likely very obvious but I know pretty much nothing about this so:
Can I make an alloy, using an alloy? That is, could I make an alloy of steel (in itself an alloy of iron and carbon) and silver?
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An alloy is purely a mixture of metals (and sometimes non-metals) which are non-chemically bonded.
So yes, you could mix steel and silver to create an alloy. There would be nothing stopping you from doing this, but I cannot say whether or not it would produce any desirable results. Mixing alloys with other materials is merely creating another alloy. Stainless steel is actually just regular steel combined with Chromium.
An alloy is usually created to increase certain properties of its constituents. For example, iron is mixed with carbon in order to create steel, which is far stronger and can be used to manufacture items such as girders. Or, in other cases, Chromium may be added, if the steel is to be used in, say, cutlery.
Silver and Iron are almost immiscible- 1% in the solid and liquid phase. Steel which is iron plus perhaps only 0.03% carbon and some manganese. You aren't going to able to mix much steel into silver or much silver to steel as a liquids or after it freezes. Any more than 1% of either and it will just separate.