Solve Linear Programming Problem using Simplex Method

Published on June 30, 2026

Linear-Algebra

Step-by-step solution to a maximization problem using the Simplex method, including converting to standard form and tableau iterations.

Question

Answer

Based on the provided image, here is the step-by-step solution for Question One.

Problem Statement

Maximize the objective function:

Subject to the constraints:

Step 1: Convert to Standard Form

To use the Simplex method, we introduce slack variables

to convert the inequalities into equalities:
Maximize

Subject to:

Step 2: Initial Simplex Tableau

The initial basic variables are

						RHS
2	3	1	1	0	0	460
4	1	0	0	1	0	420
2	1	1	0	0	1	430
-2	-3	-5	0	0	0	0

Pivot Selection:

Entering variable: (most negative coefficient in -row: ).
Leaving variable: (smallest positive ratio: ).
Pivot element: 1 (at row , col ).

Step 3: First Iteration

Perform row operations to make the pivot column

						RHS
0	2	0	1	0	-1	30
4	1	0	0	1	0	420
2	1	1	0	0	1	430
8	2	0	0	0	5	2150

Final Result

Since all coefficients in the

-row are non-negative (

), the optimal solution has been reached.

Optimal Values:

Maximum Profit:

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