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The problem described involves an economy based on three sectors: agriculture, manufacturing, and energy. Each sector has interdependencies, with one sector requiring inputs from the others to produce output. The final demand for each sector's products is specified, and we are asked to find the total output of each sector required to satisfy this demand.

Let's define the variables as follows:
- Let $ x_1 $ be the output of the agriculture sector.
- Let $ x_2 $ be the output of the manufacturing sector.
- Let $ x_3 $ be the output of the energy sector.

The input-output coefficients (how much input from one sector is required to produce a dollar of output in another) are given:
- Agriculture requires 0.20 from agriculture, 0.40 from manufacturing, and 0.20 from energy.
- Manufacturing requires 0.30 from agriculture, 0.20 from manufacturing, and 0.20 from energy.
- Energy requires 0.20 from agriculture, 0.30 from manufacturing, and 0.30 from energy.

The final demand for each sector's output is:
- $80 billion for agriculture.
- $29 billion for manufacturing.
- $37 billion for energy.

We need to solve the following system of equations based on these relationships:

\[
x_1 = 0.20x_1 + 0.40x_2 + 0.20x_3 + 80
\]
\[
x_2 = 0.30x_1 + 0.20x_2 + 0.20x_3 + 29
\]
\[
x_3 = 0.20x_1 + 0.30x_2 + 0.30x_3 + 37
\]

I will now solve this system of equations.The outputs of the three sectors, rounded to three decimal places, are:

- The output of the agricultural sector ($ x_1 $) is **234.960** billion dollars.
- The output of the manufacturing sector ($ x_2 $) is **172.880** billion dollars.
- The output of the energy sector ($ x_3 $) is **194.080** billion dollars.

Would you like further details or explanations? Here are some related questions to consider:

1. How would the outputs change if the final demand for energy increased?
2. Can you explain how input-output matrices represent economic systems?
3. What happens if the economy becomes more self-reliant in agriculture?
4. How would this model change with more sectors, like services or technology?
5. How could you interpret the outputs in terms of policy decisions?

**Tip:** Always double-check the interpretation of input-output coefficients in economic models to ensure accurate calculations!

An economy is based on three sectors, agriculture, manufacturing, and energy. Production of a dollar’s worth of agriculture requires inputs of $0.20 from agriculture, $0.40 from manufacturing, and $0.20 from energy. Production of a dollar’s worth of manufacturing requires inputs of $0.30 from agriculture, $0.20 from manufacturing, and $0.20 from energy. Production of a dollar’s worth of energy requires inputs of $0.20 from agriculture, $0.30 from manufacturing, and $0.30 from energy. Find the output for each sector that is needed to satisfy a final demand of $80 billion for agriculture, $29 billion for manufacturing, and $37 billion for energy.

Learn how to solve a linear system representing an economy with three interdependent sectors: agriculture, manufacturing, and energy. The system is based on input-output coefficients and final demand, providing a detailed step-by-step solution. Suitable for advanced high school or university students.

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