--- title: "Simplex method max:Z=3x₁+5x₂" subject: "Mathematics III" chapter: 5 key: "semester_3/mathematics_iii:simplex-method-max-z-3x-5x-two" source: https://www.bctengineer.com/q/semester_3/mathematics_iii/simplex-method-max-z-3x-5x-two site: BCT Engineering --- # Simplex method max:Z=3x₁+5x₂ Mathematics III · Chapter 5: Linear Programming - Asked in: 2075 Asoj, 10 marks - Video: [LPP WITH SIMPLEX METHOD||MAXIMIZATION MODEL WITH 3 CONSTRAINTS](https://www.youtube.com/watch?v=bmUXoXDpMrc) — SwarupaLR Vlogs - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/simplex-method-max-z-3x-5x-two - Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-5 ## Question Solve the following linear programming problem by the simplex method: Maximize Z = 3x₁ + 5x₂ subject to 3x₁ + 2x₂ ≤ 18, x₁ ≤ 4, x₂ ≤ 6, x₁, x₂ ≥ 0. ## Short answer Introducing slack variables and performing two simplex iterations gives the optimal solution x₁ = 2, x₂ = 6, with maximum Z = 36. ## Answer **Step 1: Standard form.** Every constraint is of the ≤ type, so add one slack variable to each (s₁, s₂, s₃ ≥ 0) to turn it into an equation. Slack variables have zero cost. ```text Maximize Z = 3x₁ + 5x₂ subject to 3x₁ + 2x₂ + s₁ = 18 x₁ + s₂ = 4 x₂ + s₃ = 6 ``` **Step 2: Initial basic feasible solution.** Put x₁, x₂ = 0, so the slack variables are the basic variables: s₁ = 18, s₂ = 4, s₃ = 6. Then Z = 0. Rules used: the entering variable is the one with the largest positive Cj − Zj; the leaving variable is the one with the smallest non-negative ratio b/column; the solution is optimal when every Cj − Zj ≤ 0. **Step 3: Simplex tables** **Table 1** ```text Cj 3 5 0 0 0 Basis CB x₁ x₂ s₁ s₂ s₃ b Ratio ------------------------------------ s₁ 0 3 2 1 0 0 18 9 s₂ 0 1 0 0 1 0 4 - s₃ 0 0 1 0 0 1 6 6 ------------------------------------ Zj 0 0 0 0 0 0 Cj-Zj 3 5 0 0 0 Entering: x₂ Leaving: s₃ Pivot = 1 ``` **Table 2** ```text Cj 3 5 0 0 0 Basis CB x₁ x₂ s₁ s₂ s₃ b Ratio ------------------------------------ s₁ 0 3 0 1 0 -2 6 2 s₂ 0 1 0 0 1 0 4 4 x₂ 5 0 1 0 0 1 6 - ------------------------------------ Zj 0 5 0 0 5 30 Cj-Zj 3 0 0 0 -5 Entering: x₁ Leaving: s₁ Pivot = 3 ``` **Table 3** ```text Cj 3 5 0 0 0 Basis CB x₁ x₂ s₁ s₂ s₃ b ---------------------------------- x₁ 3 1 0 1/3 0 -2/3 2 s₂ 0 0 0 -1/3 1 2/3 2 x₂ 5 0 1 0 0 1 6 ---------------------------------- Zj 3 5 1 0 3 36 Cj-Zj 0 0 -1 0 -3 ``` **Step 4: Optimality test.** In the last table every Cj − Zj is ≤ 0, so the solution is optimal. **Answer:** x₁ = 2, x₂ = 6, and Maximum Z = 36. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In Table 1, Cj - Zj = 3 for x1 and 5 for x2. Which enters? A. x1 B. s1 C. x2 D. s3 Answer: C. The largest positive Cj - Zj is 5, so x2 enters. (Key point: Entering variable) 2. For x2 entering, the column entries are 2, 0, 1 with b = 18, 4, 6. Which leaves? A. s3, ratio 6 B. s1, ratio 9 C. s2, ratio 4 D. None Answer: A. s2 has a 0 entry so has no ratio; of 9 and 6 the smaller is 6, so s3 leaves. (Key point: Leaving variable) 3. In Table 2, x1 enters. Ratios are 6/3 = 2 and 4/1 = 4. Which leaves? A. s2 B. s1 C. x2 D. s3 Answer: B. The minimum ratio is 2, in the s1 row, with pivot 3. (Key point: Second iteration) 4. What is the optimal solution? A. x1 = 4, x2 = 6, Z = 42 B. x1 = 0, x2 = 6, Z = 30 C. x1 = 6, x2 = 0, Z = 18 D. x1 = 2, x2 = 6, Z = 36 Answer: D. Check: 3(2) + 2(6) = 18, x1 <= 4, x2 <= 6, and Z = 6 + 30 = 36. (Key point: Final result) ## Close variants set in other exams - [Simplex method max:P=15x+10y](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asimplex-method-max-p-15x-10y) — Mathematics III, chapter 5, asked 17× - [Simplex method max:Z=4x+3y](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asimplex-method-max-z-4x-3y) — Mathematics III, chapter 5, asked 17× - [Simplex method max:P=20x₂−5x₁](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asimplex-method-max-p-20x-5x) — Mathematics III, chapter 5, asked 17× - [Big-M method max:Z=2x₁+x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Abig-m-method-max-z-2x) — Mathematics III, chapter 5, asked 17× - [Simplex method max:Z=10x₁+x₂+2x₃](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asimplex-method-max-z-10x-x) — Mathematics III, chapter 5, asked 18× - [Simplex method max:Z=15x₁+10x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asimplex-method-max-z-15x-10x) — Mathematics III, chapter 5, asked 15× - [Big-M method max:Z=4x₁+5x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Abig-m-method-max-z-4x) — Mathematics III, chapter 5, asked 15× - [Big-M method max:F=2x₁+x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Abig-m-method-max-f-2x) — Mathematics III, chapter 5, asked 17× - [Two-phase simplex method max:Z=3x₁−x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Atwo-phase-simplex-method-max-z) — Mathematics III, chapter 5, asked 15× - [Big-M method max:P=2x₁+5x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Abig-m-method-max-p-2x-two) — Mathematics III, chapter 5, asked 15× - [Big-M method max:P=2x₁+x₂+3x₃](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Abig-m-method-max-p-2x) — Mathematics III, chapter 5, asked 17× ## Related in Mathematics III - [Simplex method max:Z=3x₁+5x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asimplex-method-max-z-3x-5x) — Mathematics III, chapter 5 - [Simplex method max:P=−x₁+2x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asimplex-method-max-p-x-2x) — Mathematics III, chapter 5, asked 8× - [Big-M method max:P=2x₁+5x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Abig-m-method-max-p-2x-four) — Mathematics III, chapter 5, asked 3× - [Graphical method max:Z=5x₁+3x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Agraphical-method-max-z-5x-3x) — Mathematics III, chapter 5, asked 12× - [Graphical max/min min:Z=5x₁+3x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Agraphical-max-min-min-z-5x) — Mathematics III, chapter 5, asked 12×