In history of algebra, who was the first to add one equation to another equation?

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.

• Do you mean taking linear combinations of simultaneous equations?
– J.G.
Jun 18 '18 at 6:35
• Yes, in modern terms it is a linear combination of simultaneous equation. In old times it was likely to be adding one of two simultaneous equation to, or subtracting from, another one with the simpler to get a simpler equtions. Jun 18 '18 at 10:06
• Is the question about any use of $f = 0 = g \Rightarrow af+bg = 0$ for polynomials $f,g,$ or only the special case where it is used to eliminate a variable? Jun 18 '18 at 14:02

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,

$x+y=a$, and

$x-y=b$;

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)).