How to Pass Algebra

What is Algebra?

A branch of math that replace with letters for numbers. An algebraic equation represent a scale, what is done on one side of the scale with a number is also done to the other side of the scale. The numbers are the constants. Algebra can include matrices, vectors, real numbers, complex numbers etc. Moving from Arithmetic to Algebra will see something like this: Arithmetic:8+4=8+4 in Algebra. It would look like: x + y = y + x


Instructions: Learn Algebra


Step 1

Know that you can succeed. Anyone who can earn a college degree can know the concepts of algebra. Learning is repetition. So you can do this. Even if you fall short the first time and need to take the class again (and again), you can do this.

Step 2

Embrace the concept of variables. Some people get thrown off by trying to use an X and a Y instead of a number like 2 or 20. Variables make sense because they permit you to 'vary' things, and that means the equations you're solving can be used many times over with different inputs.



Step 3


Use fun examples to study. Nobody cares about a problem from a book where Jimmy has ten apples and Jane has twenty. But if you can use real world examples to teach yourself, it's fun. For example, you can use algebra to forecast your prospect net worth, figure out sporting statistics, or plan the expenses of a vacation.

Step 4

Again you’re studying over and over but with breaks of several days in between. The repetition embeds the understanding in your long term memory if you learn it, study it again within about a week, and then review the material about a month later.

Step 5

Do your homework and pass your tests!

Gaussian Elimination

Introduction of Gaussian Elimination:
In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations, finding the rank of a matrix, and calculating the inverse of an invertible square matrix. Gaussian elimination is named after German mathematician and scientist Carl Friedrich Gauss. Elementary row operations are used to reduce a matrix to row echelon form. Gauss–Jordan elimination, an extension of this algorithm, reduces the matrix further to reduced row echelon form. Gaussian elimination alone is sufficient for many applications.


Solving Gaussian Elimination:

•    Solve three-variable, three-equation linear systems is more complex, at least in the beginning, than solving the two-variable systems, because the computations concerned are messier.

•    You will require being very efficient in your functioning, and you should outline to use lots of scratch paper.

•    The method for explain these systems is an extension of the two-variable solving-by-addition method, so make sure you identify this process well and can use it consistently properly.

•    Though the way of explanation is based on addition/elimination, demanding to do real addition tends to get very messy, so there is a systematized process for solving the three-or-more-variables systems. This method is called "Gaussian elimination".

Example problems of Gaussian Elimination:

1) Solve: 3x - 4y = 20 …(i)

5x + 6y = 8 …(ii)

Solution:

Multiply (i) by 3 and (ii) by 2:

9x – 12y =  60
10x + 12y = 16
-------------------
19x         = 76

x = 76 / 19

= 4

Substituting x = 4 in (ii), we get

5(4) + 6y = 8

6y = 8 - 20

6y = -12

y = -2

The solution is x = 4 and y = -2.

2) solve the following system using Gaussian Elimination method.

3x + y = 9

3x – y = 15

Solution:

If add down, the y determination cancel out. So sketch an "equals" bar below the system, and add down:

3x + y =  9
3x – y = 15
--------------
6x = 24
x = 24 / 6

x = 4

At the present divide from side to side to solve for x = 4, and then back-solve, using either of the original equations, to find the value of y. The first equation have lesser facts, so back - explain in that one:

2(4) + y = 10

8 + y = 10

y = 2

Then the solution is (x, y) = (4, 2)

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Example of Gaussian Elimination calculator:


Consider the system of equations,

`[[x,-3y,z],[2x,-8y,8z],[-6x,3y,-15z]]`  = `[[4],[-2],[9]]`

• To solve for x, y, and z we should eliminate some of the unknowns from some of the equations. regard as adding -2 times the first equation to the second equation and also adding 6 times the first equation to the third equation. The result is

`[[x,-3y,z],[0x,-2y,6z],[0x,-15y,-9z]]`  = `[[4],[-10],[33]]`

• To remove the y term in the last equation, multiplication the second equation by -5 and add it to the third equation:

`[[x,-3y,z],[0x,-y,2z],[0x,0y,-18z]]`  = `[[4],[-5],[36]]`

-18z = 36

z = 36/-18

z = -2

Then plug the value z=-2, in equation 2,

-y+2z = -5

-y+2(-2) = -5

-y = -5+4

-y = -1

Y = 1

Now plug the values (y=1, z=-2), in equation 1,

x-3y+z = 4

x-3(1)-2 = 4

x-5 = 4

x = 9

• The third equation says z=-2. Substituting this into the second equation yields y=1. Using both of these result in the first equation gives x=9. The method of gradually solving for the unknowns is called back-substitution.

• This is the essence of Gaussian elimination. However, we may clean up the notation in our worked by using matrices.

Scale Factor in Math

Definition of Scale Factor in math

In math the ratio of two corresponding lengths in two similar figures or squares  are called as Scale Factor.

In other words the scale factor also called as the ratio of the length of the scale picture to the parallel length of the actual object is called as Scale Factor.

The scale factor going from the large figure to the small figure compare to the scale factor of the small figure to the large figure. I like to share this Polynomials Factoring with you all through my article.


Math Scale Factor - Properties:


In math  scale factor is a number used as a multiplier in scaling.

In math  scale factor is used to identify the shapes in 1, 2, or 3 dimensions.

In math Scale factor can be found in the following scenarios:

1. Size Transformation: In size transformation, the scale factor is the ratio of expressing the amount of magnification.

2. Scale Drawing: In scale drawing, the scale factor is the ratio of measurement of the drawing compared to the measurement of the original figure.

3. Comparing Two Similar Geometric: The scale factor when comparing two similar geometric figures, is the ratio of lengths of the corresponding sides. Having problem with math tutoring online for free keep reading my upcoming posts, i will try to help you.


Math Examples of Scale Factor:


1) Find the scale factor from the both rectangles, if the two rectangles are similar. 24 X 12 and 20 X 10

The Choices are:

A. 5:1

B. 5:6

C. 6:5

D. 6:7

Correct Answer: C

Solution:

Step 1: If we multiply the length of the first side of the larger rectangle by the scale factor we get the length of the corresponding side of the smaller rectangle.

Step 2: Dimension of larger rectangle × scale factor = dimension of smaller rectangle

Step 3: 24 × scale factor = 20 [Substitute the values.]

Step 4: Scale factor = [Divide each side by 24.]

Step 5: Scale factor = = 5:6 [Simplify.]

Step 6:Scale factor of two rectangle is = 5:6.

The scale factor of larger one to smaller one rectangle is = 6:5.

Related terms for Scale Factor:

Dimension

Length

Multiplier

2) WXYZ and IJKL are similar polygons. Then the scale factor of polygon WXYZ to polygon IJKL is the ratio of the lengths of the corresponding sides.

so the Scale factor is = XY: JK.

MTAP Math Challenge Questions

Challenge questions:

MTAP math challenge is to teach and let the children learn the basic mathematical skills. This also helps the parents to review their children to prepare for any quiz in mathematics. Here we will see some example problems for mtap math challenge. Challenge questions are not exams or test preparations have to be passed. It is the way of improving the skills and quality of work of children. I like to share this Mixed Fraction Calculator with you all through my article.


Example mtap math challenge questions


Mtap math challenge question 1:

What is the value of 8 in 8761?

Answer: 8000
Mtap math challenge question 2:

What is the place value of 9 in 9575?

Answer: Nine thousand

Mtap math challenge question 3:

Write 4 ones, 5 hundreds and 2 tens in symbols.

Answer: 524

Mtap math challenge question 4:

Rachael took 56 minutes to do his assignment. Rosa took 18 minutes less than Rachael. How many minutes did Rosa take to do his assignment?

Answer: 38 mins

Mtap math challenge question 5:

Which is the biggest among 889, 899, 988, 989, 998, 987?

Answer: 998

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Few more example questions for mtap math challenge


Mtap math challenge question 6:

Michelle has `$` 38. Mike has twice as much money as Michelle. How much money do Michelle and Mike have together?

Answer: `$` 114

Mtap math challenge question 7:

What is the perimeter of the given figure?

Answer: 30 m

Mtap math challenge question 8:

The area of square is 64 cm2. What is the side of square?

Answer: 8 cm

Mtap math challenge question 9:

Bruce had $140. He gave an equal amount to each of his 4 friends. He
had $20 left. How much did each friend have?

Answer: $ 30

Mtap math challenge question 10:

In 5 489, which digit is in the thousands place?

Answer: 5

Mtap math challenge question 11:

Mercy bought 2 dozen of chocolates. She gave 3 chocolates to each of her kids. How many kids does she have?

Answer: 8 kids.

Mtap math challenge question 12:

Peter  found out he can do 4 problems in 15 minutes. About how long will it take Peter finish 12 problems?

Answer: 45 minutes.

Mtap math challenge question 13:

Mathew has $120 and Allen has $240. If they buy their favorite toy that worth $500. How much more money do they need?

Answer: $140

Mtap math challenge question 14:

How many numbers between 22 and 88, have 0 as their ones digit?

Answer: 6

Mtap math challenge question 15:

What number is missing in 120 + ____ + 25 = 350?

Answer: 205

Online Math Help

Introduction to online math help:

Online math help is a service that provides mathematics help on online. It includes reviewing lessons, homework help, learning aids, games, puzzles and test preparation for kids, teens, schools, home schools. Here all the tutors are well qualified and they explain the topic in detailed.  All grade students are used to get any help from mathematics. For example Grade 1 to Grade 12 and it is also possible for college student. This is a process how we get help in math through online. All the topic doubts are cleared by the experienced tutors (Algebra, calculus, probability, geometry, trigonometry ,Number system etc…).Online math help provides  two different  services such as free and  payment system. Is this topic Half Angle Formula Examples hard for you? Watch out for my coming posts.


Features of online math help:

Online math help provide live and fully interactive one to one tutoring. Master basic, advanced math concepts and also get help with homework, practicing problems, assignment test and etc…In this service parents, students, teachers are interact with the tutors. Different types of sites are available in the online math help. All the sites provide online math help with fully interacting, interesting, and qualified math problem help. Online math help services s are unlimited and get step by step detailed help and take back to back sessions.  Online math help also help in a step by step procedure in different ways so that students can follow the concept and ask any doubt on any particular step.

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Benefits of online math help:

In online math help, all formulas, Practicing problems, home work problems, graphs, geometric diagrams are explained by the well expert teachers. The first requirement is to admit the problem and then it needs to be understood that many student who had math anxiety did manage to overcome it. Success requires guidance and work. Online math help can be viewed from a couple of perspectives that might be helpful. One is that math is a language and students need to learn some examples and their definitions very well before they can start doing the problems.

Binary Relations

Introduction of binary relations:-

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subset of A × B. The terms dyadic relation and 2-place relation are synonyms for binary relations.(Source: Wikipedia).

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Example problems for binary relation:-


Problem 1:-

Find four binary relations from `{a,b}` to `{x,y}` that are not functions from `{a,b}` to `{x,y}`

Solution:-


R = `{(a, x), (a, y), (b, x)} ` is not a function from `{a, b}` to `{x, y}` because two values related to a
R = `{(a, x)} ` is not a function from `{a, b}` to `{x, y}` because there are no values related to b

It able to find another two easily. In fact, of the 16 binary relations from `{a, b}` to `{x, y}` , only 4 are functions, and 12 are not functions.



Problem 2:-

Find four binary relations from `{a,b,c}` to `{x,y,z}` that are not functions from `{a,b,c}` to `{x,y,z}`

Solution:-

R = `{(a,x),(a,y),(a,z),(b,x),(b,y),(b,z),(c,x),(c,y),(c,z)} `  is not a function from` {a,b,c}` to `{x,y,z}` because three values related to a.

R = `{(a, x)}` is not a function from `{a, b, c}` to `{x, y,z}` because there are no values related to b & c.


It able to find another two easily. In fact, of the 36 binary relations from `{a, b, c}` to `{x, y, z}` , only 6 are functions, and 30 are not functions. Please express your views of this topic What is Rational Number by commenting on blog.


More example problems for binary relation:-


Problem 1:-

Do you understand the following calculations?

X = `{1,2,3}`

|x| = 3.

Solution:-

X x X = `{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}`

| X x X | = 9

|P(X x X)| = `2^9`
Because a binary relation on X is any subset of pairs in X x X there are `2^9` possible binary relations on X.
(Some authors say` 2^9 -1` because of not liking an empty relation.)



Problem 2:-

Do you understand the following calculations?

X = `{1,2,3,4}`

|x| = 4.

Solution:-

X x X = `{(1,1),(1,2),(1,3),(1,4)(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)}`

| X x X | = 16

|P(X x X)| = `2^16`
Because a binary relation on X is any subset of pairs in X x X there are` 2^16` possible binary relations on X.
(Some authors say `2^16 -1` because of not liking an empty relation.)

What is Statistics in Math

Introduction to statistics in math:

What is statistics in math?

In math, Statistics is the science of making effective use of numerical data relating to groups of individuals or experiments. Mathematical statistics deals with all aspects of this, including not only the collection, analysis and interpretation of such data, but also the planning of the collection of data, in terms of the design of surveys and experiments. I like to share this Data Sets for Statistics with you all through my article.

Statistics is specifically useful in drawing general conclusions about a set of data from a sample of the data. Statistics can either be singular or plural. In the singular form, Statistics is a quantity calculated from a set of data, whereas in plural form statistics is the mathematical science discussed in their article


Statistical definitions in math:

What is statistical inference?

It makes use of information from a sample data to formulate conclusions about the population from which the sample was taken.

What is experiment in statistics?

Experiment is a process which results in the collection of data, the outcome of which is unknown

What is Sampling unit in statistics?

Sampling unit is a unit in which an aggregate is divided for the purpose of sampling, each unit being regarded as individual and indivisible when the selection is made.

What is population in statistics?

Population represents the collection of people, plants, animals or thing from which we may collect data.

Sample:

A group of units selected from a larger group is called sample.

Parameter:

Parameter is an unknown value which has to be estimated used to represent a certain population characteristic

Sampling distribution:

Sampling Distribution explains the probabilities associated with a statistic when a random sample is drawn from a population .Understanding Double Box and Whisker Plot is always challenging for me but thanks to all math help websites to help me out.


What are the benefits and applications of statistics in math

Benefits:

Statistics provides a presentation of collected and organized data through the figures, charts, diagrams and graph.

Statistics also provides more critical analyses of information

Application:

Bio statistics
Chemo metrics
Demography
Actuarial science
Econometrics
Epidemiology
Geo statistics
Population ecology
Statistical finance
Statistical mechanics
Statistical physics

Volume calculation formula

Introduction

Volume is how much three-dimensional space a substance (solid, liquid, gas, or plasma) or shape occupies or contains, often quantified numerically using the SI derived unit, the cubic meter.

There are arithmetic formulas to find the volume of simple solid shapes, like regular straight edged or circular shapes. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space. (Source: From Wikipedia).


Different types of shapes and formulas for calculating volume


1. Cube = a^3 cubic units, Where a = side of the cube

2. Cuboids = lbh cubic units, where l, b, h are the length, breadth, and height respectively

3. Cylinder = `pi` r^2 h cubic units, where r, h are the radius and height of the cylinder.

4. Pyramid = `(1/3)` bh  cubic units, where b = base area of the pyramid, and h is the height.

5. Cone = `(1/3)`  `pi` r^2h cubic units, where r and h are radius and height of the cone.

6. Sphere = `(4/3)` `pi` r^3 cubic units, where r = radius of the sphere

7. Ellipsoid = `(4/3)` `pi` r1 r2 r3 cubic units, where r1, r2, r3 are the radii of the ellipsoid.


Example for Calculation of the Volumes:


Example 1: Find the volume of a cylindrical canister with radius 10 cm and height 18 cm.

Solution:

Formula for volume calculation: V= ?r^2h

= 3.14 x 102 x 18

= 5652

So, the volume of the canister is 5652 cm^3.


Example 3: Find the volume of a brick 50 cm by 40 cm by 30 cm.

Solution:

Formula for volume calculation: V = lbh

= 50 x 40 x 30

= 60000

So, the volume of the brick is 60000 cm^3

Million Dollars

Introduction  :

Million dollars are nothing but the quantity. we can express the million dollars in numerical form. Numerical form is nothing but the conversion of words into numbers. For example, one million dollar can be expanded in numerical form as 1,000,000 or 106 dollars. In this article million dollars, we are going to discuss about conversion of various values of million dollars into numerical form through example problems and practice problems.

Example Problems on Numerical Form for Million Dollars:

Example 1:

Write thousand nine hundred million dollars in numerical form.

Solution:

Step 1: one million dollar is equal to $ 1000,000.

Step 2: Thousand nine hundred million dollars will be equal to $ 1,900,000,000.

Step 3: Hence, the numerical form is $ 1,900,000,000

Example 2:

Write two thousand five hundred million dollars in numerical form.

Solution:

Step 1: one million dollar is equal to $ 1000,000.

Step 2: two thousand five hundred million dollars will be equal to $ 2,500,000,000.

Step 3: Hence, the numerical form is $ 2,500,000,000

Example 3:

Write four thousand three hundred million dollars in numerical form.

Solution:

Step 1: one million dollar is equal to $ 1000,000.

Step 2: four thousand three hundred million dollars will be equal to $ 4,300,000,000.

Step 3: Hence, the numerical form is $ 4,300,000,000

Example 4:

Write six thousand two hundred million dollars in numerical form.

Solution:

Step 1: one million dollar is equal to $ 1000,000.

Step 2: six thousand two hundred million dollars will be equal to $ 6,200,000,000.

Step 3: Hence, the numerical form is $ 6,200,000,000

Example 5:

Write two thousand seven hundred million dollars in numerical form.

Solution:

Step 1: one million dollar is equal to $ 1000,000.

Step 2: two thousand seven hundred million dollars will be equal to $ 2,700,000,000.

Step 3: Hence, the numerical form is $ 2,700,000,000

Example 6:

Write eight thousand four hundred million dollars in numerical form.

Solution:

Step 1: one million dollar is equal to $ 1000,000.

Step 2: eight thousand four hundred million dollars will be equal to $ 8,400,000,000.

Step 3: Hence, the numerical form is $ 8,400,000,000




Practice Problems on Numerical Form for Million Dollars:

1) Write four thousand five hundred million dollars in numerical form.

Answer: Numerical form is $ 4,500,000,000

2) Write seven thousand six hundred million dollars in numerical form.

Answer: Numerical form is $ 7,600,000,000

Formula to Figure Slant Height

Introduction about cone:

The slant height of a circular cone is the distance from any point on the circle to the apex of the cone. The slant height of a cone is given by the formula sqrt(r^2+h^2)  where r is the radius of the circle and h is the height from the center of the circle to the apex of the cone. If we know the value of radius and height of the circle, the slant height of cone can be finding using that formula. In this article we shall see how to calculate the slant of the cone .some example problems and practice problems. I like to share this Acute Angled Triangle with you all through my article.

(Source-Wikipedia)

Formula to Figure Slant Height of the Cone with Example Problems:

Slant height of the cone =  `sqrt (r^2+h^2) `

Where r - the radius of the circle

h - the height from the center of the circle to the apex of the cone

Problem 1:

Find the slant height of cone, which has the radius 6 cm and altitude14 cm.

Solution:

Given:

Radius (r) =6 cm.

Altitude (h) =14cm

Formula to figure slant height:

Slant height of the cone = `sqrt (r^2+h^2)` units

=`sqrt(6^2+14^2)`

=`sqrt(36 + 196)`

=`sqrt(232)`

= 15.23

Slant height of the cone = 15.23 cm

Problem 2:

Find the slant height of cone, which has the radius 7 cm and altitude15 cm .

Solution:

Given:

Radius (r) =7 cm.

Altitude (h) =15cm

Formula to figure slant height:

Slant height of the cone = `sqrt (r^2+h^2)`  units

=`sqrt(7^2+15^2)`

=`sqrt(49+ 225)`

=`sqrt(274)`

= 16.55

Slant height of the cone = 16.55 cm

Problem 3:

Find the slant height of cone, which has the radius 8 cm and altitude 16 cm.

Solution:

Given:

Radius (r) = 8 cm.

Altitude (h) =16 cm

Formula to figure slant height:

Slant height of the cone = `sqrt (r^2+h^2)` units

=`sqrt(8^2+16^2)`

=`sqrt(64+ 256)`

=`sqrt(320)`

=17.88

Slant height of the cone = 17.88 cm


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Practice Problems to Find the Slant Height:

1. Find the slant height of cone, which has the radius 9 m and altitude 17 m.

Answer: Slant height of the cone = 19.23 cm

2. Find the slant height of cone, which has the radius 10 cm and altitude 18 cm.

Answer: Slant height of the cone = 20.59 cm

3. Find the slant height of cone, which has the radius 11 m and altitude 19 cm.

Answer: Slant height of the cone = 21.954 cm

Loci Locus

Introduction to locus :

The word loci is the plural of the word locus.  This is a Latin word meaning "place". Though it has several meanings let us discuss the  mathematical meaning of locus or loci in this article. In mathematics, locus is the set of points satisfying a particular condition forming a geometrical figure. Let us discuss locus in detail.

Definition of locus

Locus of a point is the path traced by the point which moves under certain geometrical conditions. More precisely, locus is the set f all points which satisfy a given geometrical conditions or conditions.

Locus - Loci - Examples
The set of all points in a plane such that the distance of each point from a fixed point in the plane is a constant is a locus. We know that this locus is a circle whose centre is the fixed point and whose radius is the constant distance.

Consider the locus of a point which moves so that it is equidistant from two given points A and B. It is the perpendicular bisector of the line AB.

Consider two lines AOB and COD intersecting at O. A point moves so that it is equidistant from those of the two lines. Clearly the path traced by the point is either the internal or external bisector of the angle between AOB , COD.  So, here the locus consists of two straight lines bisecting the angles between the lines AOB ,  COD.

How to Find the Equation of a Locus ( Loci)

Any geometrical condition satisfied by a point ( a , b ) on a locus can be expressed algebraically as a relation between a and b. The relation thus obtained is called the equation of the locus. To find the equation of a locus, we take a point ( a , b ) on the locus and use the given conditions to obtain the relation between a and b . The equation of the locus thus obtained is satisfied by the co-ordinates of every point on the locus and by the coordinates of no other point.