Chapter 29 Section 5 Mutual Multiplication and Cancellation Method
The direct division method to eliminate elements is cumbersome, and Liu Hui created the method of mutual multiplication and elimination in the equation chapter "Cows and Sheeps Straight (Equal Value) Gold".This question is "Today there are five cattle and two sheep, and ten taels of direct gold;
Liu Hui multiplied the coefficient of right-moving cattle by 5 and multiplied the coefficient of left-moving cattle by 2, and then multiplied the coefficient of left-moving cattle by 2 to obtain
Subtract the two lines to get 21y=20, y=20/21, which is the mutual multiplication and cancellation method.Liu Hui said that pushing the big from the small is "although the four and five elements are the same", that is to say, it is a common method.
It is a pity that Liu Hui's advanced thought has not attracted the attention of mathematicians for more than 700 years.Jia Xian used the method of mutual multiplication and elimination more often for the title of the equation chapter. Sometimes after mutual multiplication, he first reduced and then eliminated. Ascent, there are ten packs of grain at the moment; fifteen packs of the next crop, five liters of lost grain, and five packs of the upper grain. How much is the real grain of the upper and lower grains?"
List the equations:
"Multiplying two lines with each other, both of them are about ten", using positive and negative techniques to eliminate each other, 4y=12, y=3, x=8
But in some problems, Jia Xian still uses the straight division method.Because in these equations, it is easier to use the direct division method, and we must adapt measures to the problem.
Qin Jiushao completely abolished the straight division method, and all used the mutual multiplication and elimination method.Sometimes he finds the common divisor of the two-phase multiplier before multiplying each other, and then multiplies each other after reduction, which is more simple.For example, "Shushu Nine Chapters" asks to list the equations
If you want to use the first line to cancel the coefficient 525 of the third line x, and 525 and the coefficient 5 of the first line x have the same number 5, then use 5 to approximate 525, and get 105, then multiply the first line by 105 and add the third line ,have to
The coefficients of x in the first and fourth rows have a common factor of 5, and the coefficients of y in the second and third rows have a common factor of 3, but the former is larger than the latter, so the coefficient of the latter is approximated first.Calculate 3 to 33 to get 11, 105 to get 35, multiply the third row by 11, multiply the second row by 35, subtract the third row, and get
Then find the value of z.Like this, the values of y, x, and u are calculated in turn.
Subtract the two lines to get 21y=20, y=20/21, which is the mutual multiplication and cancellation method.Liu Hui said that pushing the big from the small is "although the four and five elements are the same", that is to say, it is a common method.
It is a pity that Liu Hui's advanced thought has not attracted the attention of mathematicians for more than 700 years.Jia Xian used the method of mutual multiplication and elimination more often for the title of the equation chapter. Sometimes after mutual multiplication, he first reduced and then eliminated. Ascent, there are ten packs of grain at the moment; fifteen packs of the next crop, five liters of lost grain, and five packs of the upper grain. How much is the real grain of the upper and lower grains?"
List the equations:
"Multiplying two lines with each other, both of them are about ten", using positive and negative techniques to eliminate each other, 4y=12, y=3, x=8
But in some problems, Jia Xian still uses the straight division method.Because in these equations, it is easier to use the direct division method, and we must adapt measures to the problem.
Qin Jiushao completely abolished the straight division method, and all used the mutual multiplication and elimination method.Sometimes he finds the common divisor of the two-phase multiplier before multiplying each other, and then multiplies each other after reduction, which is more simple.For example, "Shushu Nine Chapters" asks to list the equations
If you want to use the first line to cancel the coefficient 525 of the third line x, and 525 and the coefficient 5 of the first line x have the same number 5, then use 5 to approximate 525, and get 105, then multiply the first line by 105 and add the third line ,have to
The coefficients of x in the first and fourth rows have a common factor of 5, and the coefficients of y in the second and third rows have a common factor of 3, but the former is larger than the latter, so the coefficient of the latter is approximated first.Calculate 3 to 33 to get 11, 105 to get 35, multiply the third row by 11, multiply the second row by 35, subtract the third row, and get
Then find the value of z.Like this, the values of y, x, and u are calculated in turn.