The oldest known simultaneous equations solved using something we would recognize as
matrix methods are these (from the Chinese document “The nine chapters on the
mathematical art” circa 200 BCE ). The first problem of Chapter 8 is:
3 bundles of high-grade grain, 2 bundles of medium-grade grain and 1 bundle of low-
grade grain give 39 units, 2 bundles of high-grade grain, 3 bundles of medium-grade
grain, and 1 bundle of low-grade grain give 34 units. 1 bundle of high-grade grain, 2
bundles of medium-grade grain, and 3 bundles of low-grade grain yields 26 units. How
many units does one bundle of each grain yield?
Certainly! Let’s solve the given system of equations using matrix methods. The problem involves finding the unit values of high-grade (), medium-grade (), and low-grade () grain bundles based on the provided equations.
Given System of Equations:
Step 1: Represent the System as an Augmented Matrix
Step 2: Perform Row Operations to Achieve Row-Echelon Form
a. Eliminate from the second and third rows:
R2’ = R2 - (2/3)R1
R3’ = R3 - (1/3)R1
b. Update the Matrix:
c. Eliminate from the third row:
Multiply R2’ by 4 and R3’ by 5 to make the coefficients of the same:
R3’’’ = R3’’ - R2’'
d. Update the Matrix:
Step 3: Back Substitution
a. From the third row:
b. Substitute into the second row:
c. Substitute and into the first row:
Final Solution:
Or in mixed numbers:
Verification:
Let’s verify by substituting the values back into the original equations:
✔️
✔️
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All equations are satisfied.
Therefore, the unit values are:
High-grade grain bundle: units (9¼ units)
Medium-grade grain bundle: units (4¼ units)
Low-grade grain bundle: units (2¾ units)
Answered by UpStudy AI and reviewed by a Professional Tutor
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The ancient Chinese were way ahead of their time when it came to solving equations! In “The Nine Chapters on the Mathematical Art,” they utilized a method similar to modern matrix algebra to tackle problems involving simultaneous equations. This method allowed them to systematically organize and manipulate equations to find unknowns, paving the way for later mathematical developments.
Now, onto a real-world application—think of farmers! If they need to figure out the yield of different types of grain, solving similar equations can help them understand the profitability of various crops. By determining how much each bundle yields, farmers can make smart, data-driven decisions on what to plant, ultimately boosting their efficiency and income!