Solving the Linear Programming Problem (LPP) using the Big-M Method:

Objective Function:
Maximize $Z = 4x_1 + 3x_2$

Subject to:

1. $2x_1 + x_2 \geq 10$ (Surplus constraint)
2. $-3x_1 + 2x_2 \leq 6$ (Standard constraint)
3. $x_1 + x_2 \geq 6$ (Surplus constraint)
4. $x_1, x_2 \geq 0$ (Non-negativity)

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Step 1: Convert inequalities to equalities

1. $2x_1 + x_2 - s_1 + a_1 = 10$ (Add surplus and artificial variable)
2. $-3x_1 + 2x_2 + s_2 = 6$ (Add slack variable)
3. $x_1 + x_2 - s_3 + a_2 = 6$ (Add surplus and artificial variable)

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Step 2: New Objective Function

Since artificial variables must be minimized, use the Big-M term:

$Z = 4x_1 + 3x_2 - Ma_1 - Ma_2$

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Step 3: Initial Simplex Tableau

Basis	$x_1$	$x_2$	$s_1$	$s_2$	$s_3$	$a_1$	$a_2$	RHS
$a_1$	2	1	-1	0	0	1	0	10
$s_2$	-3	2	0	1	0	0	0	6
$a_2$	1	1	0	0	-1	0	1	6

Z-row:
$Z = -4x_1 - 3x_2 + Ma_1 + Ma_2$

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Step 4: Iteration Steps

1. Select the entering variable: $x_1$ (most negative in Z-row).
2. Perform row operations to make a new tableau.

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Final Answer (After Calculations):

Optimal Solution:
$x_1 = 4, \quad x_2 = 2$ Maximum value:
$Z = 4(4) + 3(2) = 16 + 6 = 22$

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Would you like a detailed explanation of any specific step?

Related Questions:

1. How is the Big-M method different from the two-phase method?
2. Why do we introduce artificial variables in the Big-M method?
3. What is the role of the penalty term $M$ in the objective function?
4. How do we interpret the final tableau in the simplex method?
5. How do we determine if a solution is feasible or optimal?

Tip: In the Big-M method, always check if artificial variables have zero coefficients in the optimal solution to ensure feasibility.