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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.

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  • $\begingroup$ Do you mean taking linear combinations of simultaneous equations? $\endgroup$ – J.G. Jun 18 '18 at 6:35
  • $\begingroup$ 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. $\endgroup$ – user97019 Jun 18 '18 at 10:06
  • $\begingroup$ 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? $\endgroup$ – Bill Dubuque Jun 18 '18 at 14:02
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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)).

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