1. The problem is to find the coefficients of each variable $x_1, x_2, x_3, x_4, x_5$ in the objective function and constraints of the given linear optimization problem. 2. The objective function to maximize is: $$0.09x_1 + 0.12x_2 + 0.15x_3 + 0.08x_4 + 0.06x_5$$ Here, the coefficients are the numbers multiplying each variable. 3. The constraints are: - $x_1 + x_2 + x_3 + x_4 + x_5 \leq 100$ - $3x_1 + 6x_2 + 8x_3 + 2x_4 + x_5 \leq 5(x_1 + x_2 + x_3 + x_4 + x_5)$ - $x_1 \geq 0.2(x_1 + x_2 + x_3 + x_4 + x_5)$ - $x_1 + x_2 \leq 0.5(x_1 + x_2 + x_3 + x_4 + x_5)$ - $x_2 + x_3 \leq x_1$ - $x_1, x_2, x_3, x_4, x_5 \geq 0$ 4. Extracting coefficients from the constraints: - For $x_1 + x_2 + x_3 + x_4 + x_5 \leq 100$, coefficients are all 1. - For $3x_1 + 6x_2 + 8x_3 + 2x_4 + x_5 \leq 5(x_1 + x_2 + x_3 + x_4 + x_5)$, rearranged: $$3x_1 + 6x_2 + 8x_3 + 2x_4 + x_5 \leq 5x_1 + 5x_2 + 5x_3 + 5x_4 + 5x_5$$ which simplifies to: $$-2x_1 + x_2 + 3x_3 - 3x_4 - 4x_5 \leq 0$$ - For $x_1 \geq 0.2(x_1 + x_2 + x_3 + x_4 + x_5)$, rearranged: $$x_1 - 0.2x_1 - 0.2x_2 - 0.2x_3 - 0.2x_4 - 0.2x_5 \geq 0$$ which is: $$0.8x_1 - 0.2x_2 - 0.2x_3 - 0.2x_4 - 0.2x_5 \geq 0$$ - For $x_1 + x_2 \leq 0.5(x_1 + x_2 + x_3 + x_4 + x_5)$, rearranged: $$x_1 + x_2 - 0.5x_1 - 0.5x_2 - 0.5x_3 - 0.5x_4 - 0.5x_5 \leq 0$$ which simplifies to: $$0.5x_1 + 0.5x_2 - 0.5x_3 - 0.5x_4 - 0.5x_5 \leq 0$$ - For $x_2 + x_3 \leq x_1$, rearranged: $$-x_1 + x_2 + x_3 \leq 0$$ 5. Summary of coefficients: - Objective function: $x_1:0.09$, $x_2:0.12$, $x_3:0.15$, $x_4:0.08$, $x_5:0.06$ - Constraints coefficients as above. This completes the identification of coefficients for each variable in the problem.