Chapter 8 of the 'Chiu chang suan shu' describes a method of solving systems of linear equations as simple and very similar to that used today. In fact, the Chinese solution procedure is virtually identical to the method of Gaussian elimination and is presented in matrix form.
For example look at problem one of the chapter, "There are three classes of grain, of which three bundles of the first class, two of the second, and one of the third make 39 measures. Two of the first, three of the second and, one of the third make 34 measures. And one of the first, two of the second and three of the third make 26 measures. How many measures of grain are contained in one bundle of each class?" (Katz, 1993, p16).
Today we would represent this problem as :
3x + 2y + z = 39
2x + 3y + z = 34
x + 2y + 3z = 26
The Chinese would have arranged these equations vertically instead, with the first equation on the left, the second in the middle and the third on the right. This arrangement is presented in matrix form below:
_1 _2 _3
_2 _3 _2
_3 _1 _1
26 34 39
Next the text states; "With the first class on the right column multiply currently the middle column and directly leave out" (Katz, 1993, p16). In English, for our specific example this means multiply the middle column by 3 and subtract from it 2 lots of the right column. This will reduce the middle column to 0. The same operation is then performed with respect to the left column. The result is:
_0 _0 _3
_4 _5 _2
_8 _1 _1
39 24 39
Next perform the same operations using the middle column and the left column:
_0 _0 _3
_0 _5 _2
36 _1 _1
99 24 39
This diagram is equivalent to:
3x + 2y + z = 39
____5y + z = 24
______36z = 99
The final results are then found by 'back substitution' as we do today with z = 99/36.
There is no reason given why this method works. It can be guessed that the Chinese discovered subtracting multiple equations from other equations produces a new system with the same results. Clearly this method is similar to that we use toady except for we place the equations horizontally, in order to perform row operations rather then column.
Another difference is that the Chinese only dealt with systems with the same number of equations and unknowns, up to 5. Today we can perform such operations with an unlimited amount of equations and unknowns, not necessarily even.
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