Dear Dave,
this is probably not the solution you are looking for, and it is neither the shortest nor the most elegant one. It is also a Mathematica solution and not a WolframAlpha one.
In Mathematica there are symbols with no built-in meaning. One of these is CirclePlus. You can type it in as Esc + c+ + Esc. Your equations then look like this:
ClearAll["Global`*"]; Clear[CirclePlus]
eqns = {5 \[CirclePlus]6 == 16, 6 \[CirclePlus] 4 == 16, 10 \[CirclePlus]1 == 21, 1\[CirclePlus]10 == 12}
The typesetting in the notebook is nicer:
TableForm[eqns]
To solve it we can look at all the numbers in the equations:
data = Transpose[{eqns[[All, 1, 1]], eqns[[All, 1, 2]], eqns[[All, 2]]}]
(*{{5, 6, 16}, {6, 4, 16}, {10, 1, 21}, {1, 10, 12}}*)
Then you can fit a linear model:
Chop[Normal[LinearModelFit[data, {x, y}, {x, y}]]]
(*2. x + 1. y*)
This suggest that your solution is the function:
CirclePlus[x_, y_] := 2 x + y
With that if you evaluate eqns again you get:
{True, True, True, True}
The symbol @ that you use is in fact used in Mathematica all the time for something else.
Best wishes,
Marco