Integer Linear Problem

Introduction to integer linear problem:

Integer linear problem can be solved under linear algebra category. The linear expression with integer term is called as integer linear function. The linear problem associates with the families of vectors called linear spaces, and the expression has the general form as input one vector and output one vector, based to certain rules.

Linear problem has the demonstration in analytic geometry and their functions with integer can be generalized in operator theory. The following are the examples for linear integer problem.

Example Problems in Linear Integer:

Example 1:

Solve the linear expression -2(c - 3) – 4c - 1 = 3(c + 4) - c

Solution:

Given expression is
-2(c - 3) – 4c - 1 = 3(c + 4) - c

Multiplying the integer terms
-2c + 6 – 4c - 1 = 3c + 12 - c

Grouping the above terms
-6c + 5 = 2c + 12

Subtract 5 on both sides
-6c + 5 - 5 = 2c + 12 -5

Grouping the above terms
-6c = 2c + 7

Subtract 2x on both sides
-7c – 2c = 2c + 7 -2c

Grouping the above terms
-9c = 7

Multiply -1/9 on both sides
C = - 7/9

C= - 7/9 is the solution for the given equation

Example 2:

Solve the linear expression     -5(k + 2) = k + 9

Solution:

Given expression is
-5(k + 2) = k + 9

Multiplying the factors in left term
-5k - 10 = k + 9

Add 10 on both sides
-5k - 10 + 10 = k + 9 + 10

Grouping the above terms
-5k = z + 19

Subtract k on both sides
-5z - k = k + 19 -k

Grouping the above terms
-6k = 19

Multiply -1/6 on both sides
K = -19/6

K = -19/6 is the solution for the given equation

Practice Problems for Linear Integer:

1) Solve the linear expression -5(z - 3) – 2z - 3 = 2(z + 1) – 3z

Answer: z = 13/4 is the solution for the above given equation

2) Solve the linear expression     -7(b - 2) – 2b - 2 = 5(b + 2) – 5b

Answer: b = 2/9 is the solution for the given equation

Mixed Number Percent

Introduction to mixed number percentage:

Numbers are the basic blocks of mathematics. Numbers are of different types, they are natural numbers, whole numbers, rational numbers, irrational numbers, mixed numbers, fraction numbers, decimal numbers etc.  Mixed number percentage is defined as converting a mixed fraction into percentage, for this the first step is to convert the mixed number into improper fraction, then  the improper fraction is converted into a decimal value and multiplied by  a value of hundred.

Problems for Mixed Number Percentage:

Problem 1:

Find mixed number percentage of 4 ½.

Sol:

Step 1: convert mixed number to improper fraction,

Step 2: then improper fraction to decimal value and then multiply by hundred,


4 1/2 = ((4*2) +1)/2 = 9/2 = 4.5

Multiply 100 with this to get the percent,

4.5 X 100 = 450%

So the result is 450%

Problem 2:

Find mixed number percent of 6 ½.

Sol:

Step 1: convert mixed number to improper fraction,

Step 2: then improper fraction to decimal value and then multiply by hundred,


6 1/2 =((6 X 2)+1)/2 = 13/2 = 6.5

Multiply 100 with this to get the percent,

6.5 X 100 = 650%

so the result is 650%

Problem 3:

Find mixed number percent of 8 ½.

Sol:

Step 1: convert mixed number to improper fraction,

Step 2: then convert  improper fraction to decimal value and then multiply by hundred,


8 1/2 = ((8 X 2) +1)/2 = 17/2 =8.5

Multiply 100 with this to get the percent,

8.5 X 100 = 850%

so the result is 850%

More Problems for Mixed Number Percentage:

Problem for mixed number percentage to fraction form:

Ex 1:

Find mixed number percentage of 8 ½.

Sol:

Step 1: convert mixed number to improper fraction,

Step 2: then improper fraction value divides by hundred,


8 1/2 = ((8 X 2) +1)/2 = 17/2 %

Multiply 100 with this to get the percent,

(17/2) * 100)

so the result is 850%

Ex 2:

Find mixed number percentage to fraction:

=7 ½ (mixed number)

Sol:

Step 1: convert mixed number to improper fraction,

Step 2: then improper fraction value divides by hundred,


7 1/2 = ((7 X 2) +1)/2 = 15/2 %

Multiply 100 with this to get the percent,

(15/2) X 100

so the result is 750%

Solve Simultaneous Linear Equations

Introduction :

An equation which has only one variable and its degree (power) is 1 called a simple equation.  A linear equation with only one variable is of the form ax + b=0.  A linear equation includes  two variables is in  the formation of  ax +by +c =0. Here the variables  x and y,and the  constants are a,b,c. When two variables in the linear equations are satisfied by the same pair of values of the variables, the equations are called simultaneous linear equations.

Methods of solving simultaneous linear equations:

(a)   Substitution method

(b)   Elimination method


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Steps for Solving the Simultaneous Linear Equation

Substitution method:

This involves the following steps,

1: Simplify the equations. 

2:  Solve one equation for any variable.

3: Substitute what you get for step 2 into the next equation.

4:  Solve for the next variable.

Example for solving simultaneous Linear equation by using Substitution method:

2x+3y= -4   ------------(1)

y=x-3  

Solution:

Plug y=  x-3 in equation 1

2x+3y= -4

2x+3(x-3)= -4

2x+3x-9= -4

5x-9= -4

5x=-4+9

5x=5

x=5/5

x=1

Plug in x=1 in y=x-3

y=x-3

y=1-3

y= -2

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Elimination Method for Solving Simultaneous Linear Equation:

The second method for  Solving simultaneous  equation  is Elimination method. It is also known as either addition or subtraction method. It is the concept of eliminating any one of the variable in the given equation either by adding or subtracting the equations.

In other words solving the simultaneous equation by making the co-efficient of any one of the variables in both equations has the same value. After adding or subtracting those two equations to form a new equation contains only one variable that is known as the eliminating the variable.

Example:

x+2y=3

2x+3y=4

Solution:

x + 2y  =  3         ---------------------------------(1)

2x + 3y = 4          ---------------------------------(2)

Subtracting equation 2 from equation 1.

(1)*(2)=>     2x + 4y = 6

(2)*(1)=>     2x+ 3y =  4

----------------------------------------------------------

y = 2

Now plug in y=2 in equation (1)

x+2y=3

x+2(2)=3

x+4=3

x=3-4

x=-1

Area of a Semi Circle

Introduction:-
In mathematics (more specifically geometry), a semicircle is a two-dimensional geometric shape that forms half of a circle. Being half of a circle's 360°, the arc of a semicircle always measures 180°. A triangle inscribed in a semicircle is always a right triangle.


The semi circular looks liek the following image.

The formula used to calculate the area of semicircle is `pi/2 r^2` .

Solved Problems:-

Problem 1:-

Calculate the area is semi circle, which has the radius of 6 centimeter.

Solution:-

The formula used to calculate the area of the semi circle is `pi/ 2 r^2` .

Given the radius of the semi circle is 6 centimeter.

Area of semi circle = `pi/ 2 r^2.`

Plug-in the value of radius in the formula we get

`= pi/ 2 6^2` .

`6^2` can be written as 6 * 6 = 36.

So the area of the semi circle is `18 pi` .


Problem 2:-

Calculate the area is semi circle, which has the radius of 12 centimeter.

Solution:-

The formula used to calculate the area of the semi circle is` pi/ 2 r^2` .

Given the radius of the semi circle is12 centimeter.

Area of semi circle =` pi/ 2 r^2` .

Plug-in the value of radius in the formula we get

= `pi/ 2 12^2` .

`12^2` can be written as 12 * 12 = 144.

So the area of the semi circle is `72 pi` .

Problem 3:-

Calculate the area is semi circle, which has the diameter of 4 centimeter.

Solution:-

The formula used to calculate the area of the semi circle is` pi/ 2 r^2` .

Given the diameter of the semi circle is 4 centimeter.

Radius is half of the diameter so the radius is `4 / 2 = 2` .

Area of semi circle = `pi/ 2 r^2.`

Plug-in the value of radius in the formula we get

= `pi/ 2 2^ 2` .

`2^2` can be written as 2 * 2 = 4.

So the area of the semi circle is `2 pi.`

Formula for Volume of a Triangular

Formula for volume of a triangular:
It have three angles and it forms like a triangle is called triangular figure  The volume of a triangular depends on their sides. A triangle pyramid has a triangle for a base. That is shown below.

Triangular prism

Triangular pyramid


Formula for volume of a triangular prism:

Volume = `1/2` * length * width * height.

L –   Length

W –  Width

H –   Height

Formula for volume of a triangular pyramid

Volume of Pyramid = (1/6)abh

a – apothem length

b – Side

h – Height

Example Problems Regarding Formula for Triangular Prism:

Example 1:

The rectangular prism has its length 12 cm , width 9 cm and height 18cm. find the volume of triangular prism.

Solution :

Given that Length  = 12

Width   =  9

Height = 18

Formula for volume of a triangular = `1/2` * length*width*height.

= `1/2` * 12 * 9 * 18

= 0.5 * 1944

= 972

The solution for the triangular prism is = 972

Example 2:

The rectangular prism has its length 10 cm , width 8cm and height 15cm. find the volume of triangular prism.   

Solution :

Given that Length  = 10

Width   =  8

Height = 15

Formula for volume of a triangular = `1/2` * length*width*height.

= `1/2` * 10 * 8 * 15

= 0.5 * 1200

= 600

The solution for the triangular prism is = 600


Example Problem Regarding Formula for Volume of a Triangular Pyramid:

Example 3:

The pyramid has its apothem length 8 cm , base side 4cm and height 12cm. find the volume of triangular pyramid.

Solution:

Given that  a - 8 cm

b – 4 cm

h – 12 cm

Formula for volume of a triangular pyramid = (1/6)abh

a – apothem length

b – Side

h – Height

=  `( 1 / 6 )` * 8 * 4 * 12

= 8 * 4 * 2

=  64

The solution for the triangular pyramid is = 63.744

Example 4:

The pyramid has its apothem length 9 cm , base side 6cm and height 13cm. find the volume of triangular pyramid.

Solution:

Given that  a - 9 cm

b – 6 cm

h – 13 cm

Formula for volume of a triangular pyramid = (1/6)abh

a – apothem length

b – Side

h – Height

=  `( 1 / 6 )` * 9 * 6 * 13

= 9 * 13

=  117

The solution for the triangular prism is = 116.532

Inverse of Non Square Matrix

Introduction to inverse of non square matrix:

Matrix has a list of data. In math matrix is a rectangular arrangement of the elements. The elements are shown in the rows and the columns. In math array elements are put in the parenthesis or square brackets. In math matrix is represented by capital letters for example A, B, C…… Square matrix has equal number of rows and equal number of column. If matrix has not equal number of rows and columns called as non square matrix.

Inverse of Non Square Matrix:

Square matrix:

Square matrix has equal number of rows and equal number of columns.

Example:

`[[a,b],[c,d]]` 

The above matrix has equal number of rows and equal number of columns. So it is called as square matrix. The order of square matrix is represented by n `xx` n or m `xx` m. The order of above matrix is 2 `xx` 2.

Non-square matrix:

Non square matrix has not equal number of rows and columns.

Example:

`[[a,b],[c,d],[e,f]]` 

The above matrix has 3 rows and 2 columns. The number of rows and number of columns of given matrix is not equal so it is a non-square matrix.

Inverse of non square matrix:

Here we see additive inverse of non-square matrix. The additive inverse of matrix X is –X.

In additive inverse put – sign to all the positive numbers in the given matrix and put + sigh to all the negative number in the given matrix. The addition of normal and inverse matrix is zero matrixes.

Additive rules for inverse of non-square matrix.

X + (-X) = (-X) + X = 0

Example:

Matrix A:

A= `[[2,1],[6,4],[9,7]]`

Inverse of matrix A:

-A = `[[-2,-1],[-6,-4],[-9,-7]]`

Example Sums for Inverse of Non Square Matrix:

Example 1:

A= `[[-5,2],[3,-8],[-1,-4]]`

Find inverse of matrix A

Solution:

Given A=  `[[-5,2],[3,-8],[-1,-4]]`

In additive inverse put – sign to all the positive numbers in the given matrix and put + sigh to all the negative number in the given matrix. The addition of normal and inverse of a matrix is zero matrixes.

-  A= -  `[[-5,2],[3,-8],[-1,-4]]`

- A =  `[[5,-2],[-3,8],[1,4]]`

Example 2:

Prove

X + (-X) = (-X) + X = 0

Solution:

Let take X = `[[1,2],[3,4],[5,6]]`

- X = -  `[[1,2],[3,4],[5,6]]`

- X= `[[-1,-2],[-3,-4],[-5,-6]]`

X + (-X) =  `[[1,2],[3,4],[5,6]]` + `[[-1,-2],[-3,-4],[-5,-6]]`

X + (-X) = 0

(-X) + X is similar to X + (-X)

Therefore X + (-X) = (-X) + X = 0 is proved

Introduction to subset and proper subset

Introduction to subset and proper subset:

SET:  A set is a collection of distinct objects, considered as an object in its own right.

Example:   A = { 4,9,6,9 } , B = {blue, green , red}

SUBSET:

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment.

Example : A = { 1,5,3,8,}    , B = { 3,5} ,Here B is subset of A.  That is B `sube`

Proper Subset:

If  A and B are two sets means, A contains all the elements of  B and some additional elements that are not in B.

Example : A = { 3,5,8,10} and B ={ 3,5} .Here B is proper subset of A.

An empty set is always a proper subset of all sets.That is empty set {} is always a proper subset.

This can be denoted as ,  `O/` `subs`

Problems on Subset and Proper Subset :

Problem 1: Find the possible subsets of the set  A = { green ,Yellow,Blue }

Solution:

Given A = { green,yellow,Blue,Black}

We know that empty set is subset of every set.

So subsets of a given set are ,

B = {}

C = {Green,yellow,Blue}

D = { Green,yellow}

E=  { Yellow,Blue}

F = {Green, Blue}

G= {Green}

H = { yellow }

I = {Blue}

The above sets are the subsets of the given set A.

Problem 2 : Find the parent set of the following subsets

B = { 3,8} ,C = { 15,7}, D = { 34,15,8} , E = { 3,7,8}

Solution:

Given B = { 3,8} ,C = { 15,7}, D = { 34,15,8} , E = { 3,7,8}

We know that Subsets are the sets that contains some elements of the Parent set.

So The parent set  might be A = { 3,8,15,7,34,7 }

Problem 3: Express the following sets in-terms of  Venn diagram.

A = { -6 ,8 ,9,0 ,2 } , B = { 0,2,6} , C = { 2,6 } and D = { 34,67,89 }

Solution:

Given A = { -6 ,8 ,9,0 ,2 } , B = { 0,2,- 6} , C = { 2,-6 } and D = { 34,67,89 }

Venn diagram: