To solve the given Linear Programming Problem (LPP) using the Simplex Method, we need to formulate the problem and solve it step by step. Here's the detailed approach:

Objective Function

Maximize $Z = 5x + 4y$

Constraints

1. $3x + 5y \leq 78$
2. $4x + y \leq 36$
3. $x, y \geq 0$

Step 1: Convert Inequalities to Equalities (Slack Variables)

Introduce slack variables $s_1$ and $s_2$ to convert the inequalities into equalities:

1. $3x + 5y + s_1 = 78$
2. $4x + y + s_2 = 36$

Slack variables $s_1, s_2 \geq 0$.

Step 2: Initial Simplex Table

We write the initial simplex table using the augmented form of the constraints and objective function:

Basic Variable	$x$	$y$	$s_1$	$s_2$	RHS
$s_1$	3	5	1	0	78
$s_2$	4	1	0	1	36
$Z$	-5	-4	0	0	0

Step 3: Identify Entering and Leaving Variables

• The entering variable is the one with the most negative coefficient in the objective function row ($Z$). Here, $x$ (coefficient = -5) is the entering variable.
• Calculate the pivot row using the formula $\text{RHS} / \text{Coefficient of entering variable}$:
  • For $s_1$: $78 / 3 = 26$
  • For $s_2$: $36 / 4 = 9$

The pivot row is for $s_2$, as it has the smallest ratio.

Step 4: Perform Row Operations

Using the pivot element (4 in the $s_2$ row under $x$), perform row operations to make the pivot column unitary and update the table.

New Simplex Table:

After row operations, the updated table becomes:

Basic Variable	$x$	$y$	$s_1$	$s_2$	RHS
$s_1$	0	11/4	1	-3/4	45
$x$	1	1/4	0	1/4	9
$Z$	0	-3/4	0	5/4	45

Step 5: Repeat Process

• The next entering variable is $y$ (most negative coefficient in the $Z$ row, $-3/4$).
• Compute pivot row again:
  • For $s_1$: $45 / (11/4) = 16.36$
  • For $x$: $9 / (1/4) = 36$

The pivot row is for $s_1$.

Perform operations to update the table:

Final table:

Basic Variable	$x$	$y$	$s_1$	$s_2$	RHS
$y$	0	1	4/11	-3/11	16.36
$x$	1	0	-1/11	7/11	7.64
$Z$	0	0	1/11	1/11	58.18

Step 6: Optimal Solution

From the final table:

• $x = 7.64$
• $y = 16.36$
• Maximum $Z = 58.18$

Summary

The optimal solution is:

• $x = 7.64$, $y = 16.36$
• Maximum $Z = 58.18$

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Would you like a step-by-step explanation of any part, or assistance with graphing? Here are five related questions:

1. How can the graphical method validate this solution?
2. What happens if an additional constraint is added?
3. Can this problem be solved using Python or Excel Solver?
4. How does the simplex method handle unbounded solutions?
5. What are the limitations of using slack variables?

Tip: Always double-check the pivot element and row operations to ensure accuracy in simplex calculations