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Why is the simultaneous-equation method in The Nine Chapters compared with Gaussian elimination?

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East Asian mathematics Counting boards and markers, or counting rods, were used in China to solve systems of linear equations. This is an example from the 1st century ce.

The simultaneous-equation method described in The Nine Chapters is compared to Gauss elimination because both rely on the systematic reduction of variables to solve systems of linear equations.

In The Nine Chapters, data is arranged on a computing surface in a table where coefficients are organized by column, effectively utilizing a place-value notation. The core procedure involves column-to-column subtraction to reduce coefficients to zero, thereby eliminating variables until an equivalent, solvable configuration is reached. This algorithmic approach—using column reduction to isolate unknowns—is functionally identical to the method known in modern mathematics as Gaussian elimination.

This ancient Chinese technique was highly sophisticated, as it also necessitated the development of methods for computing with positive and negative numbers to handle the results of these subtractions, marking a significant milestone in the history of mathematics.1