How did solving simultaneous equations lead Chinese mathematicians to use negative numbers?

In early Chinese mathematics, the development of negative numbers was a direct consequence of the methods used to solve systems of simultaneous linear equations, as detailed in The Nine Chapters.1

Chinese mathematicians organized data from these equations into a table on a computing surface, where coefficients were arranged in columns. To solve these systems, they employed an algorithm—now known in the West as Gauss elimination—which relied on column-to-column subtraction to reduce variables and eliminate coefficients.1

Because this process required subtracting values from one another to reach an equivalent configuration, it necessitated the use of negative numbers. Consequently, The Nine Chapters provided detailed methods for performing calculations with both positive and negative coefficients, marking what is considered the first occurrence of negative numbers in the history of mathematics.1