In history of algebra, who was the first to add one equation to another equation? Someone gave me the name of an Italian mathematician of Renaissance period, but I lost the email. I wish to make it a bit more precise: adding two equations (hence involving unknowns), not just equalities: the latter was apparently dealt with by Euclid.
For sure, the technique was used by Leonhard Euler (15 April 1707 – 18 September 1783).
See : Elements of Algebra (English transl., 1822), page 208:
Since the two equations are,
if we add the one to the other, we have $2x=a+b$.
The first German edition of the Elements was in 1770.
An earlier example seems to be in Gerolamo Cardano (24 September 1501 – 21 September 1576)'s Ars Magna (1545), but we have to consider that we are reading a modern English translation: the original Latin text can be quite different (see Caput IX (page 21)).