Find the best tutors and institutes for Class 12 Tuition
Q2:
Solve the following Linear Programming Problems graphically:
2. Minimise $Z = – 3x + 4y$ subject to $x + 2y \leq 8$, $3x + 2y \leq 12$, $x \geq 0$, $y \geq 0$.
Solution :
Step 1: Establishing Constraints and Feasible Region
We are given the objective function $Z = -3x + 4y$ which is to be minimized, subject to the following system of linear inequalities:
- $x + 2y \leq 8$ [Constraint 1]
- $3x + 2y \leq 12$ [Constraint 2]
- $x \geq 0, y \geq 0$ [Non-negativity constraints]
Step 2: Identifying Boundary Lines
To determine the feasible region, we solve for the boundary lines by converting inequalities to equalities:
- Line 1: $x + 2y = 8$. Intercepts: $(8, 0)$ and $(0, 4)$.
- Line 2: $3x + 2y = 12$. Intercepts: $(4, 0)$ and $(0, 6)$.
Step 3: Determining the Intersection Point
Solving the system of linear equations to find the vertex where the two constraint lines meet:
$x + 2y = 8$ ... (Eq. i)
$3x + 2y = 12$ ... (Eq. ii)
Subtracting (Eq. i) from (Eq. ii): $(3x - x) + (2y - 2y) = 12 - 8 \implies 2x = 4 \implies x = 2$.
Substituting $x = 2$ into (Eq. i): $2 + 2y = 8 \implies 2y = 6 \implies y = 3$.
The point of intersection is $(2, 3)$.
Step 4: Evaluating the Objective Function at Vertices
The feasible region is bounded by the vertices $(0, 0)$, $(4, 0)$, $(2, 3)$, and $(0, 4)$. We evaluate $Z = -3x + 4y$ at each point:
| Corner Point $(x, y)$ | $Z = -3x + 4y$ |
|---|---|
| $(0, 0)$ | $-3(0) + 4(0) = 0$ |
| $(4, 0)$ | $-3(4) + 4(0) = -12$ |
| $(2, 3)$ | $-3(2) + 4(3) = -6 + 12 = 6$ |
| $(0, 4)$ | $-3(0) + 4(4) = 16$ |
Step 5: Determining the Minimum Value
Comparing the values obtained: $0, -12, 6, 16$. The minimum value is $-12$, which occurs at the point $(4, 0)$. [Per the Corner Point Theorem for Linear Programming].
Final Solution: The minimum value of $Z$ is $-12$ at the point $(4, 0)$.
More Questions from Class 12 Mathematics Linear Programming EXERCISE 12.1
- Q1: Solve the following Linear Programming Problems graphically: 1. Maximise $Z = 3x + 4y$ subject to the constraints : $x + y \leq 4$, $x \geq 0$, $y \geq 0$.
- Q3: Solve the following Linear Programming Problems graphically: 3. Maximise $Z = 5x + 3y$ subject to $3x + 5y \leq 15$, $5x + 2y \leq 10$, $x \geq 0$, $y \geq 0$.
- Q4: Solve the following Linear Programming Problems graphically: 4. Minimise $Z = 3x + 5y$ such that $x + 3y \geq 3$, $x + y \geq 2$, $x, y \geq 0$.
- Q5: Solve the following Linear Programming Problems graphically: 5. Maximise $Z = 3x + 2y$ subject to $x + 2y \leq 10$, $3x + y \leq 15$, $x, y \geq 0$.
- Q6: Solve the following Linear Programming Problems graphically: 6. Minimise $Z = x + 2y$ subject to $2x + y \geq 3$, $x + 2y \geq 6$, $x, y \geq 0$. Show that the minimum of $Z$ occurs at more than two points.
- Q7: Solve the following Linear Programming Problems graphically: 7. Minimise and Maximise $Z = 5x + 10y$ subject to $x + 2y \leq 120$, $x + y \geq 60$, $x – 2y \geq 0$, $x, y \geq 0$.
- Q8: Solve the following Linear Programming Problems graphically: 8. Minimise and Maximise $Z = x + 2y$ subject to $x + 2y \geq 100$, $2x – y \leq 0$, $2x + y \leq 200$; $x, y \geq 0$.
- Q9: Solve the following Linear Programming Problems graphically: 9. Maximise $Z = – x + 2y$, subject to the constraints: $x \geq 3$, $x + y \geq 5$, $x + 2y \geq 6$, $y \geq 0$.
- Q10: Solve the following Linear Programming Problems graphically: 10. Maximise $Z = x + y$, subject to $x – y \leq –1$, $–x + y \leq 0$, $x, y \geq 0$.
CBSE Solutions for Class 12 Mathematics Linear Programming
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Linear Programming
Top Tutor
G.T. Road, Ballabgarh
7 yrs of
Exp
9 students
389 hrs of classes completed
500
per hour
Online Classes
I have done MTECH in ECE from YMCA university faridabad and BTECH from DIT university Dehradun. I Have been teaching BTECH ECE students for last 8 Years and Physics to students of class XI and XII for last 7 years. Physics has always excited me and its wonderful to be associated with physics & Besides, Teaching Physics imparts a sense of satisfaction to me. I have been giving home/online tutions to students of various schools in faridabad region. I also teach Maths to students of class IX an X. I have Published research papers based on Routing in Cognitive Networks in International Journals . Besides being an Educator I am Mentor/Counselor in the same College.
Vivek sir is excellent in his way of teaching various concepts in physics ,he will clear all our doubts very patiently and in a way which u will understand it in an easy way
Top Tutor
Indian Institute of Technology Roorkee, Roorkee
5 yrs of
Exp
1 student
13 hrs of classes completed
600
per hour
Online Classes
Student's Home
I am working as an engineer and hold a PhD degree from IIT Roorkee. Last year, I successfully taught 6 students from this plateform.
Explain the topic very well . Understand the needs of students. Gave sufficient time to student for proper learning and understanding the subject.
Super Tutor
Village Mandoli, Delhi
20 yrs of
Exp
44 students
3635 hrs of classes completed
600
per hour
Online Classes
I have taught maths with more than 24 years experience,I tech maths in my own Coaching Institute .I prepare class 12 students for JEE main so that they can clear JEE exam without dropping a year. My last year 12th student Ashu scored 97.5 percentile in maths in JEE-2024. My student Kartik has scored 100 out of 100 in CBSE class 12th examination.
Thank you for your dedication and support in teaching Mathematics online. Your clear explanations, patience, and engaging teaching methods have greatly helped in improving understanding and confidence in the subject. The classes are well-structured, interactive, and easy to follow. I truly appreciate your efforts and commitment to making learning enjoyable and effective. Your exceptional teaching skills, commitment, and ability to simplify complex mathematical concepts have inspired me to develop a genuine interest in Mathematics. Your hard work and encouragement are truly appreciated.
Top Tutor
Gadarpur, Gadarpur
5 yrs of
Exp
6 students
144 hrs of classes completed
500
per hour
Online Classes
I am a teacher with 7 years of experience in teaching Accountancy at school (ISC & CBSE). I have completed my graduation (Bcom) from Kurukshetra University and Post Graduation (Mcom) from IGNOU with a diploma in International Business Operations.
Explains everything as many times as you ask for. solves question with you properly and also adjust with timing.
Top Tutor
Sector 54 Suncity, Gurgaon
8 yrs of
Exp
1185 hrs of classes completed
800
per hour
Online Classes
Tutor's home
I have a varied experience of over 16 years and have enjoyed all roles as a writer, editor, banker, business analyst, teacher and marketer. I love a challenge, and during the course of my career, I took up English Language classes for working professionals. My classes are learner-centric and interactive, packed with activities, games and exercises that draw learners out of their shells and infuse them with confidence. Students are encouraged to think, raise questions, absorb new vocabulary and finally adopt the language as their own. My ability to connect and engage with students from all sections make my classes friendly, approachable and non-intimidating. My varied work experience across sectors further adds to my understanding of job and situation requirements.
