https://file.aiquickdraw.com/mathbot/file/1730462508870wp6j9tnk.jpg

The image shows a linear programming problem with constraints and the solution process for Phase 1 using surplus and artificial variables.

### Problem Setup
The initial constraints are:
1. \( 2X_1 + X_2 \geq 4 \)
2. \( X_1 + 2X_2 \geq 3 \)
3. \( X_1, X_2 \geq 0 \)

### Solution - Phase 1
To convert these inequalities into equalities, surplus variables \( S_1 \) and \( S_2 \), and artificial variables \( A_1 \) and \( A_2 \) are added:
1. \( 2X_1 + X_2 - S_1 + A_1 = 4 \)
2. \( X_1 + 2X_2 - S_2 + A_2 = 3 \)

The initial basic feasible solution is:
- \( A_1 = 4 \)
- \( A_2 = 3 \)

### Objective Function
The objective function is defined as:
\[
\text{Maximize } Z = -A_1 - A_2
\]
This form is typically used in the first phase of the two-phase method to drive artificial variables out of the basis.

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### Further Questions
1. How are surplus and artificial variables used in solving linear programming problems?
2. What is the two-phase method in linear programming, and why is it used?
3. How does the addition of artificial variables affect the initial solution?
4. What is the significance of maximizing \( Z = -A_1 - A_2 \) in Phase 1?
5. How is the final solution found once the artificial variables are removed?

### Tip
When dealing with inequalities in linear programming, converting them to equalities with surplus or slack variables allows using matrix-based methods like the Simplex method effectively.

Convert the inequalities into equations by adding surplus and artificial variables to solve the linear programming problem.

Learn to solve a linear programming problem using the two-phase method by adding surplus and artificial variables. This solution includes steps to convert inequalities to equalities, introducing surplus and artificial variables, and maximizing the objective function Z in Phase 1. Suitable for university-level students.

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