Linear Programming

Given the linear programming problem:

To find the number of basic solutions:

Step 1: Convert inequalities to equations by adding slack variables :

Step 2: The system now has 5 variables ( ) and 2 equations.

Step 3: The number of basic solutions is equal to the number of ways to choose 2 basic variables from 5, which is:

Step 4: However, we must consider only feasible solutions (non-negative values). All 10 combinations need to be checked for feasibility, but the total possible basic solutions is 10.

The correct answer is (C) 10.

The correct formulation must satisfy these requirements:

1. All variables must have non-negativity constraints (x ≥ 0)
2. The constraints must form a convex feasible region
3. The objective function must be linear
4. Constraints must be properly balanced (≤ for maximization, ≥ for minimization typically)

Evaluating the options:

(A) Invalid because x₂ ≤ 0 violates non-negativity

(B) Invalid because x₁, x₂ ≤ 0 violates non-negativity

(E) Correct because:

• Minimization problem with proper constraints
• All variables satisfy x ≥ 0
• Constraints use ≤ appropriately

(D) Invalid because x₁ ≤ 0 violates non-negativity

(C) Invalid because x₁, x₂ < 0 violates non-negativity

The correct answer is (E) Min Z = x₁ + 5x₂; subject to 2x₁ + 5x₂ ≤ 10; x₁ + 3x₂ ≤ 9; x₁, x₂ ≥ 0.

Given: The given LPP is as follows: Maximize

Subject to the constraints: x_1 + 2x_2 ≤ 40 , 3x_1 + 2x_2 ≤ 60 , x_1, x_2 ≥ 0

We can graph the feasible region defined by these constraints in the plane to visualize the problem:

Plot the lines and .

Then,

Shade the region below both lines (since they are inequalities, the feasible region is below the lines).

Then,

The feasible region is bounded by the and axes, as both variables are non-negative .

The feasible region will look like a triangular area in the first quadrant.

Now, we need to find the optimal solution. To do that, we evaluate the objective function (z = 60x_1 + 50x_2) at each corner point (vertex) of the feasible region, as there are only a finite number of corner points.

Corner points of the feasible region:

- The origin

- The intersection of -axis and the first constraint

- The intersection of -axis and the first constraint

- The intersection of the two constraints

Now, we calculate z for each corner point:

The max. value of occurs at point , where

Now, since the objective function has a unique maximum value at a specific point, and the feasible region is bounded (as shown by the graph), we can conclude that the LPP has a unique optimal solution.