1. **State the problem:** We have three commodities: agriculture (A), manufacturing (M), and transportation (T). Each requires certain units of the others to produce one unit. We want to find how many units of each commodity must be produced to satisfy a demand of 1000 units each. 2. **Set up variables:** Let $x$, $y$, and $z$ be the total units produced of agriculture, manufacturing, and transportation respectively. 3. **Write the production equations based on input requirements:** - To produce 1 unit of agriculture, requires $\frac{1}{4}$ unit of manufacturing and $\frac{1}{2}$ unit of transportation. - To produce 1 unit of manufacturing, requires $\frac{1}{2}$ unit of agriculture and $\frac{1}{4}$ unit of transportation. - To produce 1 unit of transportation, requires $\frac{1}{4}$ unit of agriculture and $\frac{1}{4}$ unit of manufacturing. 4. **Formulate the system of equations including demand:** $$ x = \frac{1}{4}y + \frac{1}{2}z + 1000 $$ $$ y = \frac{1}{2}x + \frac{1}{4}z + 1000 $$ $$ z = \frac{1}{4}x + \frac{1}{4}y + 1000 $$ 5. **Rewrite equations to isolate variables on one side:** $$ x - \frac{1}{4}y - \frac{1}{2}z = 1000 $$ $$ y - \frac{1}{2}x - \frac{1}{4}z = 1000 $$ $$ z - \frac{1}{4}x - \frac{1}{4}y = 1000 $$ 6. **Express as matrix form:** $$ \begin{cases} x - \frac{1}{4}y - \frac{1}{2}z = 1000 \\ -\frac{1}{2}x + y - \frac{1}{4}z = 1000 \\ -\frac{1}{4}x - \frac{1}{4}y + z = 1000 \end{cases} $$ 7. **Solve the system:** Multiply the second and third equations to clear fractions for easier calculation: - Multiply second equation by 4: $$ -2x + 4y - z = 4000 $$ - Multiply third equation by 4: $$ -x - y + 4z = 4000 $$ 8. **Rewrite the system:** $$ \begin{cases} x - \frac{1}{4}y - \frac{1}{2}z = 1000 \\ -2x + 4y - z = 4000 \\ -x - y + 4z = 4000 \end{cases} $$ 9. **From first equation, multiply both sides by 4 to clear fractions:** $$ 4x - y - 2z = 4000 $$ 10. **Now system is:** $$ \begin{cases} 4x - y - 2z = 4000 \\ -2x + 4y - z = 4000 \\ -x - y + 4z = 4000 \end{cases} $$ 11. **Solve for $y$ from first equation:** $$ y = 4x - 2z - 4000 $$ 12. **Substitute $y$ into second and third equations:** - Second equation: $$ -2x + 4(4x - 2z - 4000) - z = 4000 $$ Simplify: $$ -2x + 16x - 8z - 16000 - z = 4000 $$ $$ 14x - 9z = 20000 $$ - Third equation: $$ -x - (4x - 2z - 4000) + 4z = 4000 $$ Simplify: $$ -x - 4x + 2z + 4000 + 4z = 4000 $$ $$ -5x + 6z + 4000 = 4000 $$ $$ -5x + 6z = 0 $$ 13. **Solve the two equations:** $$ 14x - 9z = 20000 $$ $$ -5x + 6z = 0 $$ From second: $$ -5x + 6z = 0 \Rightarrow 6z = 5x \Rightarrow z = \frac{5}{6}x $$ Substitute into first: $$ 14x - 9 \times \frac{5}{6}x = 20000 $$ $$ 14x - \frac{45}{6}x = 20000 $$ $$ 14x - 7.5x = 20000 $$ $$ 6.5x = 20000 \Rightarrow x = \frac{20000}{6.5} = 3076.92 $$ 14. **Find $z$:** $$ z = \frac{5}{6} \times 3076.92 = 2564.10 $$ 15. **Find $y$:** $$ y = 4x - 2z - 4000 = 4(3076.92) - 2(2564.10) - 4000 $$ $$ y = 12307.68 - 5128.20 - 4000 = 3679.48 $$ 16. **Final answer:** - Agriculture units $x \approx 3077$ - Manufacturing units $y \approx 3680$ - Transportation units $z \approx 2564$ These are the units to produce to satisfy the demand of 1000 units each considering the input requirements.