The LPP given in the image is:

Objective function:
Maximize Z=12x1+16x2
Maximize Z=12x1​+16x2​

Subject to constraints:
10x1+20x2≤120
10x1​+20x2​≤120
8x1+8x2≤80
8x1​+8x2​≤80
x1,x2≥0
x1​,x2​≥0

The given Linear Programming Problem (LPP) is structured as follows:

### Objective Function:
Maximize \( Z = 12x_1 + 16x_2 \)

### Subject to the constraints:
1. \( 10x_1 + 20x_2 \leq 120 \)
2. \( 8x_1 + 8x_2 \leq 80 \)
3. \( x_1 \geq 0, x_2 \geq 0 \) (non-negativity constraints)

### Steps to solve this LPP:
1. **Plot the constraints**: Convert the inequalities into equalities to find the boundary lines.
2. **Find the feasible region**: This is the region where all the constraints overlap.
3. **Identify the corner points**: These are points where the boundary lines intersect.
4. **Evaluate the objective function at each corner point**: Compute the value of \( Z \) at each of these points.
5. **Find the maximum value of \( Z \)**: The corner point giving the highest \( Z \) is the solution.

Would you like me to proceed with plotting the constraints and solving it step-by-step?

Here are 5 related questions for further exploration:
1. How do we find the corner points in a feasible region?
2. What is the graphical method for solving a linear programming problem?
3. Can you explain the simplex method for solving LPPs?
4. What is the difference between bounded and unbounded feasible regions?
5. How do we solve LPPs with more than two variables?

**Tip:** Always check whether the feasible region is bounded, as this guarantees a maximum or minimum solution for the objective function.

The LPP given in the image is: Objective function: Maximize Z=12x1+16x2 Subject to constraints: 10x1+20x2≤120 8x1+8x2≤80 x1,x2≥0

Explore how to solve a Linear Programming Problem (LPP) where the objective is to maximize Z = 12x1 + 16x2, subject to constraints like 10x1 + 20x2 ≤ 120 and non-negativity conditions. Learn about the optimization process using the graphical method, suitable for Grades 10-12.

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