What methods did the Old Babylonian society use for solving equations?
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You can see some general survey, like: Victor Katz, A History of Mathematics: An Introduction (3rd edition, 2008), Ch.1.2 MESOPOTAMIA, page 10-on.
There are references to "modern classics":
Otto Neugebauer, The Exact Sciences in Antiquity (2nd ed, 1957), Ch.2 Babylonian mathematics
B.L. Van der Waerden, Science Awakening I (or.ed, 1954), Ch.3 Babylonian mathematics.
More recent and "technical":
Jens Høyrup, Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin (2002),
Eleanor Robson, Mesopotamian Mathematics, 2100–1600 BC: Technical Constants in Bureaucracy and Education (1999).
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