Linear Growth Formula

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Linear growth way that it grows by the similar quantity in all time step.If an individual, residents or anything topic to growth starts absent at early range P0 and grows by a permanent amount d throughout each equal-length growth stage then its size Pn at the end of the of the nth growth period will be given by:

Pn = P0 + nd
Two Types of Linear Growth Function and their Formula:

Using the notion of vector-valued functions relating linear growth formula, a action of homogenous linear systems of difference and differential equations with stable coefficients is obtainable. This advance allows the source of the universal solution for a system of differential equations as the permanent analogue of the 'obvious' solution of the discrete case.

1) Analytic geometry function

2) Vector spaces function

Analytic geometry function:

The term of the linear growth formula is sometimes used to the mean a first-degree polynomial function of one variable.These functions are recognized as "linear function" since they are exactly that the functions are whose graph in the Cartesian coordinate plane ...
... is a straight line.

Such a linear growth function(formula) can be written as

f(x) = mx + b

Vector spaces function:

In advanced math a linear growth function means a function that is a linear graph, that is, a graph between the two vector spaces that the preserves of the vectors are addition and scalar multiplication of the functions.

For example if x and f(x) are represented to as the coordinate vectors linear function, then the linear functions are those functions f that can be expressed as

f(x) = Mx,

Where M is a matrix. A function(formula)

f(x) = mx + b

Is a linear map if and only if b = 0. For other values of b this cascade in the additional universal class of affine maps
Linear Models on Linear Growth Formula

Linear models are minimized or maximized which is used to place a linear constraints. Objective function variable, position of constraints for linear, set of choice variables for linear are the significant mechanism of linear models. the majority of the actual world difficulty is leads to linear models mechanism. the majority of the actual world problem in linear algebra can be approximated by using linear models.

Linear Programming:- It is the method used in decision making in business for obtaining the maximum or minimum values of a linear expression subject to satisfying certain given inequations.

The linear expression is known as an objective function and the linear inequations are known as linear constraints.

Linear Constraints:- In business or industry we want to make the best use of our limited resources like money, labour, expenses, etc.

The limitations on the resources can often be expressed in the form of linear inequations, known as linear constraints.

Objective Function:- A linear function of the involved variables, which we want to maximize or minimize subject to the given linear constraints, is known as an objective function.

Optimal Value Of An Objective Function:- The maximum or minimum value of an objective function is known as its optimal value.

Feasible Solution:- A set values of the variables satisfying all the constraints is known as a feasible solution.

Optimal Solution:- A feasible solution which leads to the optimal value of an objective function is known as an optimal solution.

Optimization Techniques:- The processes of obtaining the optimal values are called optimization techniques.

A linear Programming Problem(LPP):- A general linear programming problem consists of maximizing or minimizing an objective function subject to certain given constraints.

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Formulation of a linear programming problem(LPP)
Working Rules of Linear Constrint Optimization

Step1. Identify the unknowns in the given LPP. Denote them by x and y.

Step2. Formulate the objective function in terms of x and y. Be sure whether it is to be maximized or minimized.

Step3. Translate all the constraints in the form of linear inequations.

Step4. Solve these inequations simultaneously. Mark the common area by shaded region. This is the feasible region.

Step5. Find the coordinates of all the vertices of the feasible region.

Step6. Find the value of the objective function at each vertex of the feasible.

Step7. Find the values of x and y for which the objective function z = ax + by has maximum or minimum value (as the case may be).

Solved Examples for Linear Constraint Optimization

Ex 1: Find the maximum and minimum values of 5x + 2y subject to the constraints

-2x – 3y = -6, i.e., 2x + 3y = 6; x – 2y = 2;

6x + 4y = 24; -3x + 2y = 3;

x = and y = 0.

Sol: 2x + 3y = 6

x/3 + y/2 = 1

This line meets the axes at (3,0) and (0,2). Join these points to obtain the line 2x + 3y = 6.

x/2 + y/-1 = 1

This line meets the axes at (2,0) and (0,-1). Join these points to obtain the line x – 2y = 2.

6x + 4y = 24

x/4 + y/6 = 1.

This line meets the axes at (4,0) and (0,6). Join these points to obtain the line 6x + 4y = 24.

-3x + 2y = 3

x/-1 + y/3,2 = 1.

This line meets the axes at (-1,0) and (0,3/2). Join these points to obtain the line -3x + 2y = 3.

Also, x = 0 is the y-axis and y = 0 is the x-axis.

Thus, all these lines include the quadrilateral ABCD, shown by the shaded region.

Thus shaded region represents all possible solutions of the system. Now, we may use the following two methods for obtaining the maximum and minimum values of 5x + 2y.

linear constraint optimization

Ex 2 :

A furniture dealer deals in only two items tables and chairs. He has Rs 5000 to invest ad a space to store at cost 60 pieces. A table costs him Rs 250 and a chair Rs 50. He can sell a table at a profit of Rs 50 and a chair at a profit of Rs 15. Assuming that he can sell all the items that he buys, how should he invest his money in order that he may maximize his profit?

Sol : This problem can be formulated as under

Let x and y be the required number of tables and chairs respectively

Then, clearly we have.

x 0, y 0; x + y 60;

250x + 50y 5000, i.e., 5x + y 100

Let P be the profit function. Then, P = 50x + 15y

Now, we have to maximize P

Now, x + y = 60 ? x/60 + y/60 = 1

This line meets the axes at (60, 0) and (0, 60). Plot these points and join them to get the line x + y = 60

Also, 5x + y = 100 ? x/100 + y/100 = 1

This line meets the axes at (20,0) and (0, 100). Plot these points and join them to get the line 5x + y = 100

Also, the line x = 0 is y-axis and the line y = 0 is the x-axis

These our straight lines enclose the quadrilateral OABC

The coordinates of the points O, A, B, C are (0, 0), (20, 0), (10, 50) and (0, 60) respectively

At these points, the corresponding values of P = 50x + 15y are 0, 1000, 1250 and 900 respectively

Clearly, it is maximum at B(10,50)

So, for a maximum profit, the dealer should purchase 10 tables and 50 chairs.

linear constraint optimization x

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