To solve this problem, we need to set up a system of equations that represents the interdependence between the industries: farming, building, and clothing. The external demand for each industry must be satisfied while also considering the intermediate purchases between industries.

Step 1: Define Variables

Let the total production in each industry be represented by the following variables:

• $x_1$: Total production of food (farming) in millions of dollars.
• $x_2$: Total production of building in millions of dollars.
• $x_3$: Total production of clothing in millions of dollars.

Step 2: Set Up the Equations

The total production in each industry is influenced by the demand from external sources and the purchases made by other industries. For each industry, we will write an equation based on the total production of that industry, considering both internal and external demand.

Food (Farming) Equation:

The total demand for food consists of:

• External demand of $290 million.
• Purchases made by the builders ($0.18 per dollar of food production) and tailors ($0.06 per dollar of food production).

Thus, the equation for food production is: $x_1 = 290 + 0.11x_1 + 0.07x_2 + 0.07x_3$

Building Equation:

The total demand for building consists of:

• External demand of $170 million.
• Purchases made by the farmers ($0.07 per dollar of building production) and tailors ($0.18 per dollar of building production).

Thus, the equation for building production is: $x_2 = 170 + 0.18x_1 + 0.05x_2 + 0.12x_3$

Clothing Equation:

The total demand for clothing consists of:

• External demand of $190 million.
• Purchases made by the farmers ($0.06 per dollar of clothing production) and builders ($0.12 per dollar of clothing production).

Thus, the equation for clothing production is: $x_3 = 190 + 0.15x_1 + 0.18x_2 + 0.07x_3$

Step 3: Simplify the Equations

We can simplify each equation to isolate the variables on one side.

1. For Food (Farming): $x_1 = 290 + 0.11x_1 + 0.07x_2 + 0.07x_3$ Subtract $0.11x_1$ from both sides: $x_1 - 0.11x_1 = 290 + 0.07x_2 + 0.07x_3$ $0.89x_1 = 290 + 0.07x_2 + 0.07x_3$ Now divide by 0.89: $x_1 = \frac{290 + 0.07x_2 + 0.07x_3}{0.89}$

2. For Building: $x_2 = 170 + 0.18x_1 + 0.05x_2 + 0.12x_3$ Subtract $0.05x_2$ from both sides: $x_2 - 0.05x_2 = 170 + 0.18x_1 + 0.12x_3$ $0.95x_2 = 170 + 0.18x_1 + 0.12x_3$ Now divide by 0.95: $x_2 = \frac{170 + 0.18x_1 + 0.12x_3}{0.95}$

3. For Clothing: $x_3 = 190 + 0.15x_1 + 0.18x_2 + 0.07x_3$ Subtract $0.07x_3$ from both sides: $x_3 - 0.07x_3 = 190 + 0.15x_1 + 0.18x_2$ $0.93x_3 = 190 + 0.15x_1 + 0.18x_2$ Now divide by 0.93: $x_3 = \frac{190 + 0.15x_1 + 0.18x_2}{0.93}$

Step 4: Solve the System of Equations

Now that we have three equations in terms of $x_1$, $x_2$, and $x_3$, we can solve them either by substitution or using matrix methods.

Would you like me to continue solving the system step-by-step, or would you prefer an alternative approach?