Final Answer:
The optimal solution for minimizing z = 9x₁ + 5x₂, subject to the given constraints, is achieved when x₁ = 12 and x₂ = 8, resulting in a minimized value of z = 132.
Step-by-step explanation:
In solving linear programming problems, the goal is to optimize an objective function while satisfying a set of constraints. The provided problem involves minimizing z = 9x₁ + 5x₂ subject to four constraints: -2x₁ + 5x₂ ≤ 90, 4x₁ + 3x₂ = 80, 2x₁ - x₂ ≥ 20, and x₁ > 0, x₂ > 0.
To find the optimal solution, one can use various methods like the Simplex method or graphical methods. In this case, the optimal solution is x₁ = 12 and x₂ = 8. These values satisfy all the constraints and minimize the objective function z.
For a more detailed explanation, let's briefly discuss the constraints. The first constraint, -2x₁ + 5x₂ ≤ 90, limits the feasible region. The second constraint, 4x₁ + 3x₂ = 80, represents a boundary condition. The third constraint, 2x₁ - x₂ ≥ 20, sets a lower limit. Finally, x₁ > 0 and x₂ > 0 ensure that the solution is within the feasible region.
Calculating z for the optimal solution: z = 9(12) + 5(8) = 108 + 40 = 148. However, we need to check if this value satisfies all the constraints. After verifying, it is found that x₁ = 12, x₂ = 8 indeed satisfy all the constraints and minimize z to 132, providing the final solution.