Question

If you sell two cows and five sheep and you buy $13$ pigs, you gain $1000$coins. If you sell three cows and three pigs and buy nine sheep, you break even. If you sell six sheep and eight pigs and you buy five cows, you lose$600$ coins. What is the price of a cow, a sheep, and a pig, respectively? (Nine Chapters, Chapter 8, Problem 8)

Short answer

$1200$coinsare the price of a cow, $500$coinsis the price of a sheep and $300$is the price of a pig.

Step-by-step solution

Represent the buying and selling of cows, sheep,and pigsin terms of a system of equations.

The price of a cow, a sheep,and a pig can be found using the augmented matrix.

There is a gain of 1,000 coins if we sell two cows and five sheep and buy 13 pigs.

$2C+5S-13P=1000\mathrm{......}\left(1\right)$

We gain 0 coins if we sell three cows and three pigs and buy 9sheep.

$3C-9S+3P=0\mathrm{......}\left(2\right)$

We lose 600 coins if we sell six sheep and eight pigs and buy 5 cows.

$-5C+6S+8P=-600\mathrm{......}\left(3\right)$

Consider the equations (1), (2) and (3). The system of equations is,

$\left|\begin{array}{l}2C+5S-13P=1000\\ 3C-9S+3P=0\\ -5C+6S+8P=-600\end{array}\right|$

Represent the above equations in terms of a matrix.

$\left|\begin{array}{cccc}2& 5& -13& 1000\\ 3& -9& 3& 0\\ -5& 6& 8& -600\end{array}\right|$

Perform the row operations on the matrix. 

Consider the matrix and reduce it into row echelon form.

$\mathbf{rref}\left|\begin{array}{cccc}2& 5& -13& 1000\\ 3& -9& 3& 0\\ -5& 6& 8& -600\end{array}\right|\to \left|\begin{array}{cccc}1& 0& 0& 1200\\ 0& 1& 0& 500\\ 0& 0& 1& 300\end{array}\right|$

The values are, $C=1200,S=500,P=300$.

Hence, $1200$coins are the price of a cow, $500$coins is the price of a sheep and $300$is the price of a pig.