Let's solve the problem step by step based on the uploaded image and the provided Linear Programming (LP) details:

Given:

• Objective Function:
  $z = 2x_1 + 3x_2 + 4x_3$

• Constraints:
  $x_1 + 2x_2 \leq 12$ $x_1 + x_3 \leq 8$ $x_1, x_2, x_3 \geq 0$

1. Solve the LP problem:

We aim to maximize $z = 2x_1 + 3x_2 + 4x_3$ under the given constraints. Let's manually or computationally determine the optimal solution.

2. Feasibility Check for $(x_1, x_2, x_3) = (5, 2, 0)$:

Substitute $(x_1, x_2, x_3) = (5, 2, 0)$ into the constraints:

1. $x_1 + 2x_2 = 5 + 2(2) = 9 \leq 12$ ✅
2. $x_1 + x_3 = 5 + 0 = 5 \leq 8$ ✅
3. $x_1, x_2, x_3 \geq 0$: $5 \geq 0, 2 \geq 0, 0 \geq 0$ ✅

Thus, $(5, 2, 0)$ is a feasible solution.

3. Feasibility Check for $(x_1, x_2, x_3) = (6, 3, 4)$:

Substitute $(x_1, x_2, x_3) = (6, 3, 4)$ into the constraints:

1. $x_1 + 2x_2 = 6 + 2(3) = 12 \leq 12$ ✅
2. $x_1 + x_3 = 6 + 4 = 10 \leq 8$ ❌

The second constraint is violated, so $(6, 3, 4)$ is not a feasible solution.

Solve for Optimal Solution:

The solution involves maximizing $z$ while ensuring all constraints are satisfied. I'll calculate it computationally:

scipy.optimize import linprog

Coefficients of the objective function (to maximize)

c = [-2, -3, -4] # Negated because linprog performs minimization

Coefficients of the constraints (LHS)

A = [ [1, 2, 0], # Constraint 1: x1 + 2x2 <= 12 [1, 0, 1] # Constraint 2: x1 + x3 <= 8 ]

Right-hand side values of the constraints (RHS)

b = [12, 8]

Bounds for each variable: x1, x2, x3 >= 0

x_bounds = (0, None) # No upper bound specified, but must be non-negative bounds = [x_bounds, x_bounds, x_bounds]

Solve the LP problem

result = linprog(c, A_ub=A, b_ub=b, bounds=bounds, method='highs')

Extract results

result.x, -result.fun, result.success### Optimal Solution:

• $x_1 = 0$, $x_2 = 6$, $x_3 = 8$
• Maximum $z = 50$

The solution is valid, and the optimization was successful.

Summary:

1. Optimal solution: $(x_1, x_2, x_3) = (0, 6, 8)$ with $z = 50$.
2. $(x_1, x_2, x_3) = (5, 2, 0)$ is feasible.
3. $(x_1, x_2, x_3) = (6, 3, 4)$ is not feasible.

Do you have any questions or want further details?
Here are five related questions to explore:

1. How do you graphically solve an LP problem with two variables?
2. Can constraints be non-linear in LP, and if not, why?
3. How do different optimization methods like simplex compare to computational solvers like linprog?
4. What is duality in Linear Programming?
5. How would adding more constraints affect the feasible region and optimal solution?

Tip: Always verify feasibility first before solving for optimization; it saves time and computation.