To solve the given linear programming problem, we will first define the objective function and constraints clearly and then proceed to find the solution using methods like the Simplex Method or by graphical analysis if applicable.

Objective Function: $\text{Maximize } Z = 3x_1 + 2x_2 + x_3$

Subject to:

1. $-3x_1 + 2x_2 + 2x_3 = 8$
2. $-3x_1 + 4x_2 + x_3 = 7$

Where $x_1, x_2, x_3 \geq 0$.

Step 1: Express the constraints and objective function

This is a system of two linear equations (involving three variables), so it can be approached by solving it as a system of equations or using optimization techniques.

Step 2: Solve the system of equations

We have two equations with three unknowns, and since the number of equations is less than the number of variables, this suggests that there are infinitely many solutions, and we need to find an optimal solution in the feasible region.

We'll use a substitution method to solve for the variables:

1. $-3x_1 + 2x_2 + 2x_3 = 8$
2. $-3x_1 + 4x_2 + x_3 = 7$

We can solve these equations systematically. Let’s eliminate one variable and solve for the remaining two.

Step 3: Subtract the equations

Let’s subtract equation (2) from equation (1) to eliminate $x_1$.

$(-3x_1 + 2x_2 + 2x_3) - (-3x_1 + 4x_2 + x_3) = 8 - 7$ $0x_1 - 2x_2 + x_3 = 1$ $-2x_2 + x_3 = 1 \quad \text{(Equation 3)}$

Step 4: Solve for $x_3$ in terms of $x_2$

From equation (3), solve for $x_3$: $x_3 = 1 + 2x_2$

Step 5: Substitute $x_3$ into one of the original equations

Now, substitute $x_3 = 1 + 2x_2$ into either equation (1) or (2). Let’s use equation (1):

$-3x_1 + 2x_2 + 2(1 + 2x_2) = 8$ $-3x_1 + 2x_2 + 2 + 4x_2 = 8$ $-3x_1 + 6x_2 + 2 = 8$ $-3x_1 + 6x_2 = 6$ $x_1 = 2x_2 - 2$

Step 6: Substitute $x_1$ and $x_3$ into the objective function

Now we can substitute $x_1 = 2x_2 - 2$ and $x_3 = 1 + 2x_2$ into the objective function $Z = 3x_1 + 2x_2 + x_3$:

$Z = 3(2x_2 - 2) + 2x_2 + (1 + 2x_2)$ $Z = 6x_2 - 6 + 2x_2 + 1 + 2x_2$ $Z = 10x_2 - 5$

Step 7: Maximize the objective function

The objective function $Z = 10x_2 - 5$ increases as $x_2$ increases. Since $x_2 \geq 0$ and $x_1 \geq 0$, we must ensure that the constraints are satisfied.

To ensure non-negativity:

• From $x_1 = 2x_2 - 2 \geq 0$, we get $x_2 \geq 1$.
• From $x_3 = 1 + 2x_2 \geq 0$, we get $x_2 \geq -\frac{1}{2}$, but the more restrictive condition is $x_2 \geq 1$.

Thus, $x_2 \geq 1$.

Step 8: Check for the optimal solution

If $x_2 = 1$:

• $x_1 = 2(1) - 2 = 0$
• $x_3 = 1 + 2(1) = 3$

Now, check if these values satisfy the original equations:

1. $-3(0) + 2(1) + 2(3) = 8$, which is true.
2. $-3(0) + 4(1) + 3 = 7$, which is also true.

Thus, the solution is: $x_1 = 0, \quad x_2 = 1, \quad x_3 = 3$

The maximum value of $Z$ is: $Z = 3(0) + 2(1) + 3 = 5$

Final Answer:

The solution to the linear programming problem is: $x_1 = 0, \quad x_2 = 1, \quad x_3 = 3 \quad \text{with} \quad Z = 5$

Would you like more details or have any further questions?

Further Questions:

1. How would you solve a linear programming problem with more than two constraints?
2. Can the Simplex method be used here for optimization? How would you set up the initial simplex tableau?
3. What are the graphical methods for solving linear programming problems with two variables?
4. What is the role of slack variables in linear programming?
5. How does the dual problem relate to the primal problem?

Tip: Always check that the final solution satisfies all constraints before concluding the optimal solution.