Chapter 25 Chapter 6 Solutions of Linear Equations

For ancient equations, people often interpret the word "fang" as a square.In fact, the original meaning of "Fang" is to combine two boats and tie their bows together, which was called "Fang" in ancient times; and "Cheng" means standard, and as a verb, it is to seek its standard.Therefore, juxtaposing the quantitative relations of a group of items and finding the quantitative standard of each item is an equation.Liu Hui said: "Groups of things are always miscellaneous, and there are numbers in each line. In general, it is true. Let each behavior rate, two things have another journey, and three things have three journeys. They are all like the number of things. They are listed side by side, so it is called an equation. "("Nine Chapters Arithmetic · Equation Chapter Note") This is the clear definition of the equation.Obviously, if there are several unknowns, there must be several equations.It is worth noting that Liu Hui's idea of ​​"ordering the rate of each row" is to regard a row of an equation as an ordered or directional array, which is roughly equivalent to the concept of a row vector in modern linear algebra theory.The equation is expressed by the separation coefficient method, and each row is arranged from top to bottom (as opposed to horizontal rows and vertical columns in ancient times, it is usually horizontal and vertical rows in ancient times), there is no need to write the name of the unknown, and the constant item is placed at the bottom.For example, the first question in the Equation Chapter of "Nine Chapters": "Now there are three parcels of upper grain, two parcels of middle grain, and one parcel of lower grain, which is thirty-nine dou; two parcels of upper grain, three parcels of middle grain, and one parcel of lower grain Thirty-four buckets of real grain; one pack of upper grain, two packs of middle grain, three packs of lower grain, and twenty-six buckets of real grain. Ask the geometry of each pack of upper, middle, and lower grain?" The equation is listed as (a) shown (p. 89), which is equivalent to