This section of the NCERT Solutions for Class 12 Maths Chapter 12 explains the formulation of a linear programming problem using a real-world example of a furniture dealer trying to maximise profits by trying different combinations while purchasing chairs and tables.

Translating real-world problems into solvable mathematical equations is known as the mathematical formulation of a linear programming problem.

The following steps are used to formulate a linear programming problem:

• Step 1: Identify the decision variables that control the output.
• Step 2: Find the set of constraints on the decision variables and translate them into linear equations or inequalities. This gives the region within which the function can be optimised.

Note: All the decision variables must be positive (follow the non-negativity rule) because in the real world, decision variables only exist as positive values.

• Step 3: Transform the objective function in the form of a linear equation.
• Step 4: The objective function can now be optimised either graphically or mathematically.

The furniture dealer problem is then solved using the aforementioned steps.

Here are some of the different types of linear programming problems covered in the NCERT Solutions Class 12 Chapter 12:

• Manufacturing problems: These types of problems involve finding the number of units of different products sold by a company, where each product has constraints such as fixed number of workers, fixed number of work hours, machine usage, etc. in order to make the maximum profit (highest value of the function).
• Diet problems: These problems require finding out how much of each nutrient is needed in a product in such a way that at least some amount of each nutrient is utilised while ensuring that the cost of the product is minimised.
• Transportation problems: These types of problems involve finding a transportation schedule for factories transporting different products in such a way that the cost of the transportation and logistics is minimised.

Some key features of the NCERT Solutions for Class 12 Maths Chapter 12 are:

• The solutions will help students gain confidence and bridge any gap in knowledge about the topics in Chapter 12 Linear Programming.
• Students will get familiar and comfortable with all the important theorems and definitions, and be able to apply them while solving problems.
• The solutions provide methodical answers and explanations for a variety of problems related to linear programming, enabling students to practice questions thoroughly.
• The solutions are created by subject-matter experts, making this a trustworthy source for students to study from.

1 . How many problems do the NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming cover?

The NCERT Solutions for Class 12 Maths Chapter 12 has three exercises. Exercise 12.1 has 10 questions, Exercise 12.2 has 11 questions while the miscellaneous exercise has 10 questions covering all topics in the chapter. The problems are carefully curated and follow CBSE guidelines.

2 . What is meant by Linear Programming in mathematics?

Linear programming in mathematics is the optimisation of a function that is subjected to certain constraints. The optimised value may be the highest or the lowest value required for the function, and the constraints are represented as linear inequalities.

3 . Where can I find Class 12 Maths Chapter 12 Solutions online?

The NCERT Solutions for Class 12 Maths Chapter 12 is available on the MSVgo website and mobile app, and is free to download. Download the solutions from the MSVgo website or app to understand the chapter better and prepare well for the exams.

Linear graphs can be used in our daily life to represent the relationship between the different quantities. For example, if a school hires more teachers, then the number of students taking admission increases as well and vice-versa. 

This relationship between the quantities can be in direct or indirect proportions. The relationship can be represented in a graphical manner called linear graphs.

Linear equations can help solve real-world problems efficiently, and for that, it is necessary that the real-life problems are converted to mathematical expressions. The expression should represent the relationship between the variables and all the necessary information. 

Certain procedures are involved when converting the situation to a mathematical statement. The steps are as follows:

• To convert the problem to a mathematical statement, the algebraic expression should portray the problem efficiently. 
• The unknown values must be assigned variables. 
• It is necessary to obtain the data, keywords, and phrases by reading the problem multiple times. 
• The information obtained must be organized sequentially. 
• With the information obtained, you can frame your equation and solve it using systematic techniques. 
• To make sure that the solution is correct, retrace it to the problem statement.

Linear EquationsNon- Linear EquationsThe equation is represented by a straight line in the graph.The equation is represented by a curve in the graph.The degree of variables is 1. The variables can have degrees of 2 or more. The equation for linear equation is:

Where, x, y= variables

m= slope

c= constant

The equation for non- linear equation is 

Where, x, y= variables

a, b, c= constant 

1. What is the difference between the linear and non-linear equation?

A linear equation is represented by a straight line in the graph, while a non-linear equation is represented by a curve. 

2. What are the advantages of linear programming?

Linear programming has various advantages:

• Help get the insight into the real-world problems.
• Helps in solving multi-dimensional problems.
• Helps in finding the best possible solution to a problem with given available options.

3. What are the different types of linear programming?

There are many different types of linear programming, and depending on the situation, you can choose the suitable method:

• Simplex method
• R method 
• Graphical method
• Open solver method