Suppose the economy of a primitive society has only three occupations, farming (F), manufacturing of tools (M), and weaving/sewing of clothes (C). This society has no monetary system, so all goods and services are bartered. The table below summarizes how the barter system works. The farmers keep half of the produce for themselves and give one-fourth of the produce to the manufacturers and one-fourth to the clothiers. The manufacturers divide their goods evenly between the three groups, and the clothiers split things similarly to the farmers.

	F (giving)	M (giving)	C (giving)
F (receiving)	1/2	1/3	1/2
M (receiving)	1/4	1/3	1/4
C (receiving)	1/4	1/3	1/4

As the size of the tribe grows, bartering can become too inefficient, so the tribe wants to create a currency to make things run smoother. They need to set prices for the goods in the new currency which reflect the current relative value of the goods in the barter system. Let 1,2,3x1,x2,x3 represent the prices for the products produced by F, M, and P respectively. Then the first row of the table says that the value of goods received by the farmers is 121+132+12312x1+13x2+12x3, but since the trade is fair and the farmers produce goods worth 1x1 dollars, the first row of the table gives us the equation:

121+132+123=112x1+13x2+12x3=x1

Use the other rows to determine two more equations and give the corresponding augmented matrix.

Solve the resulting SLE. If we decide to value a unit of clothing at 3 dollars (i.e. 3=3x3=3) what are the prices for the remaining goods?

1=x1=

2=x2=

3=3

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