--- title: "Graphical method max:Z=7x₁+10x₂" subject: "Mathematics III" chapter: 5 key: "semester_3/mathematics_iii:graphical-method-max-z-7x-10x-two" source: https://www.bctengineer.com/q/semester_3/mathematics_iii/graphical-method-max-z-7x-10x-two site: BCT Engineering --- # Graphical method max:Z=7x₁+10x₂ Mathematics III · Chapter 5: Linear Programming - Asked in: 2083 Baishakh, 5 marks - Video: [Graphical Method for LP Problem | Graphical + Corner Point Method | Operations Research | MBA](https://www.youtube.com/watch?v=fWHzKpqZctE) — KN Prasad Academy (NITian) - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/graphical-method-max-z-7x-10x-two - Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-5 ## Question Applying the graphical method, maximize Z = 7x₁ + 10x₂ subject to the constraints 3x₁ + x₂ ≤ 9, x₁ + x₂ ≤ 8, x₁, x₂ ≥ 0. ## Short answer Plot lines 3x₁+x₂=9, x₁+x₂=8; feasible region corners: (0,0) Z=0, (3,0) Z=21, (0.5,7.5) Z=78.5, (0,8) Z=80. Maximum Z=80 at x₁=0, x₂=8. ## Answer **Step 1: Draw the constraint lines.** Replace each inequality by an equation, find the intercepts and decide which side of the line is allowed (test the origin). | Constraint | Line | Meets x-axis | Meets y-axis | Allowed side | |---|---|---|---|---| | (1) 3x₁ + x₂ ≤ 9 | 3x₁ + x₂ = 9 | (3, 0) | (0, 9) | towards the origin | | (2) x₁ + x₂ ≤ 8 | x₁ + x₂ = 8 | (8, 0) | (0, 8) | towards the origin | Also x₁ ≥ 0 and x₂ ≥ 0, so the region lies in the first quadrant. **Step 2: Feasible region.** The region satisfying all constraints is the shaded polygon below (dots), with corner points marked A, B, C, …. ```text 9| | |D. |.....C. |......... |........... |............. |............... |................. |................... |..................... |....................... |......................... |........................... |............................. |............................... 0|A...............................B +-------------------------------------------- 0 4 ``` **Step 3: Corner points and the objective value.** The optimum of a linear function over a convex polygon occurs at a corner point. | Point | Coordinates | Lines meeting there | Z = 7x₁ + 10x₂ | |---|---|---|---| | A | (0, 0) | x₁ = 0, x₂ = 0 | 0 | | B | (3, 0) | (1), x₂ = 0 | 21 | | C | (1/2, 15/2) | (1), (2) | 157/2 | | D | (0, 8) | (2), x₁ = 0 | 80 | **Step 4: Conclusion.** The largest value of Z is 80, at D(0, 8). **Answer:** Maximum Z = 80 at x₁ = 0, x₂ = 8. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What are the intercepts of x1 + x2 = 8? A. (8, 0) and (0, 8) B. (8, 0) and (0, 4) C. (4, 0) and (0, 8) D. (9, 0) and (0, 8) Answer: A. Equal coefficients give equal intercepts. (Key point: Intercepts) 2. Where do the lines meet? A. (2, 3) B. (1, 7) C. (1/2, 15/2) D. (3, 0) Answer: C. Subtract: 2x1 = 1, x1 = 1/2, x2 = 15/2. (Key point: Intersection C) 3. Why is the maximum at D(0, 8) and not C? A. Z = 80 at D beats 157/2 = 78.5 at C B. C is infeasible C. Z at C is 44 D. D lies on the first line Answer: A. 157/2 < 80. (Key point: Comparison) 4. What is the maximum Z? A. 72 at (0, 7.2) B. 21 at (3, 0) C. 80 at x1 = 0, x2 = 8 D. 78.5 at (1/2, 15/2) Answer: C. 7(0) + 10(8) = 80. (Key point: Answer) ## Related in Mathematics III - [Graphical method max:Z=7x₁+10x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Agraphical-method-max-z-7x-10x) — Mathematics III, chapter 5, asked 12× - [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 method max:z=2x₁+3x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Agraphical-method-max-z-2x-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× - [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× - [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×