Selecting on the Dependent Variable

Introduction about selecting on the dependent variable:

Independent variable is a variable which does not depends on any other variable. But the dependent variable should depend only on independent variable. The dependent variable’s value depends on direct or indirect variation. In direct variation if the value of independent variable increases then the value of dependent variable also increases and if independent variable decreases then the value of dependent variable also decreases. Here we are going to learn about some example problems of selecting on the dependent variable.

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Simple Example Problems of Selecting on the Dependent Variable.

Example 1:

What is the dependent variable in the function f(x) =4-x?

Solution:

In the function f(x) = 4 - x, the value of f(x) depends on the value of x. So, f(x) is dependent variable.

Example 2:

What is the dependent variable in the function f(x) =10-x?

Solution:

In the function f(x) = 10- x, the value of f(x) depends on the value of x. So, f(x) is dependent variable.

These are the examples of selecting on the dependent variable.



Some more Examples of Selecting on the Dependent Variable:

Example 3:

Check whether the d variable dependent or not in the equation d=8b

Solution:

Step 1: The given equation is d=8b.Here =d is dependent variable and b is independent variable.

Step 2: Now plug different values for b and check the value of d for each value of b.

Step 3: When b = 1 =the value of d is 8.

Step 4: When b = 2 =the value of d is 16.

Step 5: When b = 3 =the value of d is 24.

Step 6: When b = 4 =the value of d is 32.

Step 7: When b = 5 =the value of d is 40.

Step 8: If the value of b increases then the value of d is also an increase.

Step 9: So, d is a dependent variable.

These are the example problems selecting on the dependent variable.

Triangle Pyramid Net

Introduction to triangle pyramid net:

In this article we see about definition of triangular pyramid. Pyramid is a polyhedron 3 – dimensional geometric shape. The pyramid has 4 – vertices. Out of them 3 are base of the pyramid and one is top of the pyramid.

Types of pyramid:

Square pyramid
Rectangular pyramid
Triangular pyramid
Pentagonal pyramid
These are some of the types of pyramid. In this section we see about definition of triangular pyramid.

Definition – Triangle Pyramid Net:

The net definition for triangular pyramid is given below. The base of the pyramid is triangle in shape is said to be triangle pyramid.

Types of triangular pyramid:

Equilateral triangular pyramid – Pyramid base is in the shape of equilateral triangle
Isosceles triangular pyramid – Pyramid base is in the shape of isosceles triangle
Scalene triangular pyramid – Pyramid base is in the shape of scalene triangle
Volume of a triangular pyramid = `1/3` area of the triangle x length

Volume = `1/3` x `1/2 ` x base x height x length

That is volume = `1/6` x b x h x l

Let we workout some of the example problems for triangle pyramid net.



Example Problems – Triangle Pyramid Net:

Example problems 1 – triangle pyramid net:

Find out the volume of triangular pyramid where the base is 3.5 m, height is 11.5 m and length is 12.2m.

Solution:

Given:

Base b = 3.5 m

Height h = 11.5 m

Length l = 12.2 m

Volume of triangular pyramid = `1/6` x b x h x l

= `1/6` x 3.5 x 11.5 x 12.2

= `1/6` 491.05

= 81.84

Answer: Volume of a triangular pyramid = 81.84 cubic meter.

Example problem 2 – triangle pyramid net::

Find out the volume of triangular pyramid where the base is 4 m, height is 7 m and length is 5 m.

Solution:

Given:

Base b = 4 m

Height h = 7 m

Length l = 5 m

Volume of triangular pyramid = `1/6` x b x h x l

= `1/6` x 4 x 7 x 5

= `1/6` 140

= 23.3

Answer: volume of a triangular pyramid = 23.3 cubic meter.

Prime Factorization Chart Fractions

Introduction to Prime Factorization chart fractions:

A Prime Number is a complete number, larger than 1, so as to can be evenly divided just by 1 otherwise itself. "Prime Factorization" is established which prime numbers require to multiply as one to obtain the original number.

A few of the prime numbers are:  1,7,13,19 etc...

The prime factorization of a digit is multiplying prime factors of a digit.

For example

38  = 2x19

Basic Prime Factorization Chart Fractions:

Factors:

"Factors" are the facts you multiply mutually to obtain another number: 45 = 3x3x5. In prime factorization, every factor will be prime numbers. 

Example:

39 = 3 x 13    (use chart to verify the answer)

Here multiplication of 13 x 3 is called as prime factorization.
                
Example Problems for Prime Factorization Chart Fractions:

Problem 1:

What are the prime factorization chart fractions of `1/16` ?

Solution:

It is best to create work with the least prime number, which is `1/2` , so let check:

`1/16`  ÷ `1/2` = `1/8`

But `1/8` is not a prime number, so we need to factor it further:

`1/8` ÷ `1/2` = `1/4`

But `1/4` is not a prime number, so we need to factor it further:

`1/4`  ÷ `1/2` = `1/2`

`1/2`  ÷ `1/2` = `1/1`

And 1 is a prime number,

`1/16`  = `1/2` × `1/2` × `1/2` × `1/2`

All factors are a prime number, so the answer should be correct.

The prime factorization of `1/16` is `1/2` × `1/2` × `1/2` × `1/2`


Problem 2:

What are the prime factorization chart fractions of `1/150` ?

Solution:

It is best to create work with the least prime number, which is `1/2` , so let check:

`1/150` ÷ `1/2` = `1/75`

But `1/75` is not a prime number, so we need to factor it further:

`1/75`  ÷ `1/5` = `1/15`

But `1/15` is not a prime number, so we need to factor it further:

`1/15`  ÷ `1/3` = `1/5`

`1/5`  ÷ `1/5` = `1/1`

And `1/1` is a prime number,

`1/150`  = `1/5` × `1/5` × `1/3` × `1/2`

All factors are a prime number, so the answer should be correct.

The prime factorization of fractions `1/150` is `1/5` × `1/5` × `1/3` × `1/2`

Algebra Foil Method

Introduction 

The algebra is a basic topic in mathematics and it is related with binomial expansion. The binomial expansion is done by foil method. In algebra, the foil method is multiplying the binomials. The foil method is considered as rule and the expansion of foil is first – outer – inner – last. Now we are going to see about algebra foil method.

Explanation for Algebra Foil Method
Algebra in math:

The algebra is a simple topic in math and it is defining the relations, rules and so on. The algebra problems are based on variables. In foil method also the variables are used.

Algebra foil method steps:

Multiplication of binomials process is called as foil method and it is done with variables. Steps for foil method:

Binomial’s first terms are multiplied.
Binomial’s outer terms are multiplied.
Binomial’s inner terms are multiplied.
Binomial’s last terms are multiplied.
The terms are combined with each other.




More about Algebra Foil Method

Example problems for algebra foil method:

Problem 1: Use the foil method and determine the binomial expansion of (x + 11) and (x + 5).

Solution:

The binomials are given as (x + 11) and (x + 5).

We can expand the binomials as.

Binomial’s first terms are multiplied as x^2.
Binomial’s outer terms are multiplied as 5x.
Binomial’s inner terms are multiplied as 11x.
Binomial’s last terms are multiplied as 55.
Combine the terms as x^2 + 5x + 11x + 55.
The result is x^2 + 16x + 55.
Problem 2: Use the foil method and determine the binomial expansion of (x + 8) and (x + 1).

Solution:

The binomials are given as (x + 8) and (x + 1).

We can expand the binomials as.

Binomial’s first terms are multiplied as x^2.
Binomial’s outer terms are multiplied as x.
Binomial’s inner terms are multiplied as 8x.
Binomial’s last terms are multiplied as 8.
Combine the terms as x^2 + x + 8x + 8.
The result is x^2 + 9x + 8.
Exercise problems for algebra foil method:

1. Use the foil method for binomial expansion.

(x – 2) (2x + 3)

Solution: The binomial expansion is 2x^2 – x – 6.

2. Use the foil method for binomial expansion.

(3x + 1) (x + 1)

Solution: The binomial expansion is 3x^2 + 4x + 1.

Opposite Rays Geometry

Introduction to opposite rays:

In geometry ray is a line but it has starting point with no ending point. Two opposite rays join at a single vertex and forms straight angles.  Straight angles are one of the angles in mathematical geometry.Straight angle is a straight line angle it measures 180 degree angle. Therefore we can say that an angle formed by opposite rays is the straight angle.  In this article we are going to learn about how two opposite rays make a straight angle in geometry.

Explanation about Opposite Rays in Geometry

Ray:                      

Ray is the important concept in mathematical geometry. A ray is the one of the part of the line. All angles are lies between two rays. Generally a ray as a starting point but is has no ending point it ends up to infinity level.  Starting point of the ray is mentioned by the dot and then ending point of the ray is mentioned by the arrow.

Opposite rays:
Two rays that share the starting point and it goes up to infinity level on both directions is known as opposite rays.  It looks like a straight line so it is forms an straight angle.

The above diagram AB and BC are two opposite rays they share a common point B and it forms 180 degree straight angle.  Here B is known as the vertex of the straight angle 180 degree.



Description about Pictures and Points of the Opposite Rays

Different pictures of the opposite rays:

Collinear points:

If the point are in the straight line that has been similar to two opposite then that points are called as the collinear points. Simply known as a set of points in the straight line then those points is called as the collinear points.  In the above diagram 1 A, B and then C are the collinear points because they lie in the same straight line and formed by two opposite rays.

Complex matrix inverse

Complex matrix inverse - Introduction

          In algebra, the determinant is a particular number related with some square matrix. The essential geometric significance of a determinant is a scale factor for compute when the matrix is considered as a linear transformation. Thus a 3× 3 matrix with determinant 3 when useful to a place of points with fixed area will convert those points into a place with double the area. Determinants are significant together in calculus, where they go into the replacement rule for various variables, and in multi linear algebra.

          `A = [[A,B],[C,D]] If AD - BD != 0` , then A has an inverse, Denoted` A^-1`

          `A^-1 = 1/(AD-BC) [[D,-B],[-C,A]]`

Complex Matrix Inverse - Examples:

Complex matrix inverse - Example 1:

Find the Complex inverse matrix of the matrix below?

`A = [[4,2],[3, 8]]`

Solution:

Inverse A =` 1/((8*4)-(3*2)) [[8,-3],[-2,4]]`

           = `1/(32-6) [[8,-3],[-2,4]]`

           =` 1/26[[8,-3],[-2,4]]`

          = `1/26 * [[8,-3],[-2,4]] = [[8/26, -3/26],[-2/26,4/26]]`

          = `[[4/13, -3/26], [-1/13, 2/13]]`

Complex matrix inverse - Example 2:

What is the inverse of the matrix below?

          `[[9, 1], [6, 3]]`

Solution:

Inverse of the matrix is the

Inverse A =` 1/((9*3)-(6*1)) [[3,-1],[-6,9]]`

           = `1/(27-6) [[3,-1],[-6,9]]`

           = `1/21[[3,-1],[-6,9]]`

          = `1/21 * [[3,-1],[-6,9]] = [[3/21, -1/21],[-6/21,9/21]]`

         ` [[3/21, -1/21],[-6/21,9/21]] = [[1/7, -1/21],[-2/7,3/7]]`

Complex matrix inverse - Example 3:

What is the inverse of the matrix below?

  A =  ` [[7, 1], [5, 3]]`

Solution:

Inverse of the matrix is the

Inverse A =` 1/((7*3)-(5*1)) [[3,-1],[-5,7]]`

           = `1/(21-5) [[3,-1],[-5,7]]`

           = `1/16[[3,-1],[-5,7]]`

          = `1/16 * [[3,-1],[-5,7]]`

          = `[[3/16, -1/16],[-6/16,9/16]]`

`[[3/16, -1/16],[-6/16,9/16]] = [[3/16, -1/16],[-3/8,9/16]]`




Complex Matrix Inverse - more Examples:

Complex matrix inverse - Example 1:

What is the inverse of the matrix below?

         ` [[8, 2], [6, 4]]`

Solution:

Inverse of the matrix is the

Inverse A = `1/((8*4)-(6*2)) [[4,-2],[-6,8]]`

           `= 1/(32-12) [[4,-2],[-6,8]]`

           `= 1/20[[4,-2],[-6,8]]`

         ` = 1/20* [[4,-2],[-6,8]]`

          `= [[4/20, -2/20],[-6/20,8/20]]`

`[[4/20, -2/20],[-6/20,8/20]]= [[1/5, -1/10],[-3/10,2/5]]`

Complex matrix inverse - Example 2:

Find the inverse Complex matrix of the matrix below?

          `[[9, 3], [7, 5]]`

Solution:

Inverse of the matrix is the

Inverse `A = 1/(9*5)-(7*3) [[5,-3],[-7,9]]`

          `= 1/(45-21) [[5,-3],[-7,9]]`

          `= 1/24[[5,-3],[-7,9]]`

          `= 1/24* [[5,-3],[-7,9]]`

          `= [[5/24, -3/24],[-7/24,9/24]]`

`[[5/24, -3/24],[-7/24,9/24]] = [[5/24, -1/8],[-7/24,9/24]]`

Complex matrix inverse - Example 3:

What is the inverse of the matrix below?

          `[[10, 4], [8, 6]]`

Solution:

Inverse of the matrix is the

Inverse `A = 1/(10*6)-(8*4) [[6,-4],[-8,10]]`

          `= 1/(60-32) [[6,-4],[-8,10]]`

           `= 1/28 [[6,-4],[-8,10]]`

          `= 1/28* [[6,-4],[-8,10]]`

          `= [[6/28, -4/28],[-8/28,10/28]]`

`[[6/28, -4/28],[-8/28,10/28]] = [[3/14, -1/7],[-2/7,5/14]]`

Nonnegative Integers

Introduction for non negative integers:

                      Non negative integers are defined as a numbers greater than zero, numbers at the right side of the zero. Thus the non negative integers are all the integers from zero on upwards, and the non negative real are all the real numbers from zero on upwards. The set of all non-negative integers forms a commutative mono id under addition. Non negative integers always proceed by a non negative sign (+). An integer with or without the non negative integers is always non negative.

Operations and Examples of Non negative Integers:

Operation for non negative integers:

          1. Reduce anything in the parentheses for solving

          2. Reduce the exponents for solve non negative integer.

          3. Multiplication or division for solve non negative integer.

          4. Finally, Addition or Subtraction for solve non negative integer.

Examples of non negative integers:

                   Integers are similar to whole numbers, but they also consist of negative numbers but still fractions are not allowed. So integers can be negative {-1, -2,-3, … }, positive {1, 2, 3, … }, or zero {0}.

Operations on Non negative Integers:

1. Adding non negative integers:

           Non negative + Non negative = Positive: 8 + 4 = 12

           Negative + Negative = Negative: (- 6) + (- 4) = - 10

           Addition of a negative and a non negative integer: Use the sign of the larger number and subtract, examples are,

           (- 5) + 3 = - 2

           7 + (-4) = 3

2. Subtracting non negative integers:

           Negative - Non negative = Negative: (- 4) - 2 = -4 + (-2) = -6

           Non negative - Negative = Non negative + Non negative = Non negative: 6 - (-2) = 6 + 2 = 8

           Negative - Negative = Negative + Non negative = Use the sign of the larger number and subtract (Change double negatives to a non negative)

          (-8) - (-3) = (-8) + 3 = -5

          (-3) - (-8) = (-3) + 8 = 5

3. Multiplying non negative integers:

           Non negative x Non negative = Non negative: 4 x 3 = 12

           Negative x Negative = Non negative: (-3) x (-2) = 6

           Negative x Non negative = Negative: (-3) x 3 = -9

           Non negative x Negative = Negative: 2 x (-5) = -10

4. Dividing non negative integers:

           Non negative ÷ Non negative = Positive: 14 ÷ 7 = 2

           Negative ÷ Negative = Non negative: (-12) ÷ (-4) = 3

           Negative ÷ Non negative = Negative: (-18) ÷ 3 = -6

           Non negative ÷ Negative = Negative: 15 ÷ (-3) = -5