math solver step by step

Introduction to Math solver step by step:

Mathematics is defiend as the “Queen of science”. Problem solving is the major part of the mathematics. In solving those problems, step by step explanation about that problem are more preferable. Students can clearly understand about the problem by step by step solving. Step by step solving of the problem is more convenient in categories like algebra, polynomials, geometry, and number theory and so on. In mathematics, quadratic equation teaching is also easier by step by step process of solving. Is this topic What is an Expression in Algebra hard for you? Watch out for my coming posts.


Example of linear equation in math solver step by step:


Solve the equations in a step by step process:

x + 2y = 14, 3x + y = 11,

Solution: Let the given equations be identified as follows.

x + 2y  = 14 (1)
3x + y  = 11 (2)


Consider the equations (1) and (2)
(1)               x + 2y    = 14
(2) × 2        6x + 2y   = 22     (subtracting)
–5x           = – 8

Divide 5 on both sides,

-5x/5 = -8/5

x = 8/5

x = 1.6

Substitute x = 1.6 in equation 1.

X + 2y = 14

1.6 + 2y = 14

Subtract 1.6 on both sides,

1.6 – 1.6 + 2y = 14 – 1.6

2y = 12.4

Divide 2 on both sides,

2y/2 = 12.4

y = 6.2

The values are x = 1.6 and y = 6.2

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Examples to Algebra problem in math solver using step by step process:


Example 1 to Algebra problem in math solver using step by step process:

Solve: 7(x - 3) =2x + 3

Step 1: Expand the brackets

7x - 18 = 2x + 3

Step 2: Subtract the 2x from of the both side

7x - 18 - 2x = 2x + 3 - 2x

Step 3: Reduce two side of equation

5x - 18 = 3

Step 4: Add 18 to both of the side

5x - 18 + 18 = 3 + 18

Step 5: Reduce both side of the equation

5x = 21

Step 6: Divide both the side by the value 4

x = 21/5

x= 4.2

Example 2 for Algebra problem in math solver using step by step process:

Solve:                          4x + 3 = 3x + 17

Step 1: Subtract 3x from two sides

4x + 3 - 3x = 3x + 17 - 3x

Step 2: Simplify both the sides

x + 3 = 17

Step 3: Subtract 3 from two sides

x + 3 - 3 = 17 - 3

Step 4: Simplify both the sides

x = 14

Definition of Data Type

Introduction for data type:

Data is acknowledged or identified as information, which is the information collect from some of the issue or incidence. Data’s are getting from different types of sources. In any sources first we have to gathering the data, analyzing the data and making inferences from that data’s. Different types of data’s are used in data collections. It was collected with the helps to know about all the information of data. In this article we are going to explaining about definition of data types. I like to share this Percentage off Calculator with you all through my article.

Explanations for Data Definition & Data Types:
Definition of data:

Data is defined as, the information which are collected from standard basis, like survey and organizational sources.
In this data to be shortened, impute, edited, used and aggregate in production of officials from statistics.
Data types:

The important types of data are given below,

Primary data,
Secondary data,
Qualitative data, and
Quantitative data.
These four data’s are the important types of data’s. And these all has so many types of subordinates.

In this data types, data collecting methods are the important sources for the data’s. And these four types of data’s are gathering the information from various types of methods. Understanding equation solver algebra is always challenging for me but thanks to all math help websites to help me out.

Explanations for Types of Data:

1. Primary data:

The primary data is the data, which are collecting the information from foremost first hand information collection.
These types of primary data’s are mostly pure and original data’s.
The primary data types having three types of various types of collecting data’s.
Primary data collection sub types:
Data’s get from investigation,
Data’s get from personal inquiry, and
Data’s and information get from telephones.
2. Secondary data:

The secondary data is the data, which is collecting from the second hand.
In this type given data is already gathered from any other persons for some purpose, and it has available for the present issues.
And mostly these secondary data’s are not relevant and uncontaminated or original data.
Secondary data source types:
Data from official, and
Data from Semi-official.
3. Qualitative data:

The qualitative data is the major data in the data collection process data’s.
The qualitative data is which is the information gathered in a non statistical (numerical) form.
Qualitative data getting from:
The transcription of the interview process,
The notes taken from the field works,
Videos clippings, and tape and audio recordings,
Pictures and images, reporting papers and e-mail documents.
4. Quantitative data:

Information that can be collected and gathered or expressed with numerically, this type of data is frequently collected in experiment, manipulated and statistically analyzed.
Quantitative data getting from:
Quantitative data can be represent visually in graph and chart,
The production of models, theories and hypotheses,
The improvement of instruments and types for measurement,
Experimental manage and manipulation of variables,
Gather of empirical data, and modeling and examination of analyze data.
Evaluation of results.
These all are the definition and explanations for the types of data and data definition.

Tough Math Problems

Introduction to tough math problems:

In this article we are going to see about the tough math problems. Mathematics is the learning of quantity, arrangement, space, and change. Math seeks out patterns that originate the new conjecture, and ascertain truth by precise deduction from properly selected axioms and definitions. Math is used throughout the whole world that has fundamental tool in various fields that include natural science, engineering, medicine, and the social sciences Understanding Business Math Problems is always challenging for me but thanks to all math help websites to help me out.

Example Problems for Tough Math Problems:

Tough math problems – Example: 1
Find the externals of the functional `J(y)=\int_0^\pi\[y'(x)\]^2dx`   subject to the constraint  `int_0^\pi \[ y(x)\]^2dx=1, y(0)=y(\pi)=0`

Solution:

Let `H(y,\lambda) = \int_0^\pi ([y'(x)]^2 + \lambda [y(x)]^2)dx` where \lambda is a Lagrange multiplier. The Euler equation is

`\frac{d}{dx}\frac{\partial (y'^2 + \lambda y^2)}{\partial y'} - \frac{\partial(y'^2 + \lambda y^2)}{\partial y} = 0`

`\implies y'' - \lambda y = 0`

(i) Try `\lambda > 0;` let `\lambda=\alpha^2.` Then

`y'' - \alpha^2 y = 0`
`\implies y = a e^{\alpha x} + b e^{-\alpha x}`

`y(0) = 0 \implies a + b = 0;\quad y(\pi) = 0 \implies a e^{\alpha\pi} + b e^{-\alpha \pi} = 0`
But the solution for this is a = b = 0, i.e., `y\equiv 0` which does not satisfy the constraint `\int_0^\pi y^2dx = 1`
(ii) Try \lambda = 0 Then

`y'' = 0`

`\implies y = a x + b`

`y(0) = 0 \implies b = 0 so y = a x;` then `y(\pi) = 0 \implies a = 0 \implies y \equiv 0` which, again, does not satisfy the constraint.

`(iii) \lambda < 0; let \lambda=-\alpha^2.` Then

`y'' + \alpha^2 y = 0`

`\implies y = a \cos(\alpha x) + b \sin(\alpha x)`

`y(0) = 0 \implies a = 0; y(\pi) = 0 \implies b = 0 or \sin(\alpha \pi) = 0`

`b = 0 \implies y\equiv 0,` so we reject this and take `\sin(\alpha \pi) = 0` , which `implies \alpha = \pm 1, \pm 2, \dots`

Thus `y = b \sin(nx)` where n is any nonzero integer.

b is found by imposing the constraint:

`\int_0^\pi b^2 \sin^2(nx)dx = 1 \implies \int_0^\pi b^2 \frac{1-\cos(2nx)}{2}dx = 1 \implies\frac{b^2}{2} \[x - \frac{\sin(2nx)}{2n}\]_0^\pi = 1`

`\implies b = \pm\sqrt{2/\pi}`

Hence, the extremes of J subject to the given constraint are `y = \pm\sqrt{2/\pi}\sin(nx), n=\pm1, \pm2, \dots`

Tough math problems – Example: 2

Constraint problem: Minimize `T(y)=\int_0^1\(y'^2+x^2\)dx s.t. K(y)=\int_0^1y^2dx=2.`

Solution:

Using the method of Lagrange multipliers,

`\int_0^1\(y'^2+x^2+\lambda y^2\)dx = J(y)`

If the Euler-Lagrange equations are satisfied then this integral has an extremum.

`\frac{\partial}{\partial y}\[y'^2+x^2+\lambda y^2\]=\frac{d}{dx}\frac{\partial}{\partial y'}\[y'^2+x^2+\lambda y^2\]`

`2\lambda y = 2y''`

`y''=\lambda y so y(x) = c_1\cos\sqrt{\lambda}x + c_2\sin\sqrt{\lambda}x`

`y(0)=0` gives `c_1=0`

`y(x)=c_2\sin\sqrt{\lambda}x`

`y(1)=0 gives 0=c_2\sin\sqrt{\lambda} so \sqrt{\lambda}=n\pi, n=1,2,...`

The solution is

`y(x)=c_2 \sin n\pi x`

`2=\int_0^1y^2dx = c_2^2\int_0^1\sin^2(n\pi x)dx`

`2=\frac{c_2^2}{2}\int_0^1(1-\cos 2n\pi x)dx = \frac{c_2^2}{2}`

`c_2=\pm 2`

Tough math problems – Example: 3

Find the extreme of `x^2+y^2+z^2` subject to the constraint `x^2+2y^2-z^2-1=0.`

Solution:

Using Lagrange multipliers, make a new function:

`f = x^2 + y^2 + z^2 + \lambda x^2 + 2 \lambda y^2 - \lambda z^2 - \lambda = 0`

Take the partial derivatives of the function with the respect to x, y, and z.

`1) \frac{ \partial f}{\partial x} = 2x+2 \lambda x = 0`

`2) \frac{ \partial f}{\partial y} = 2y+4 \lambda y = 0`

`3) \frac{ \partial f}{\partial z} = 2z-2 \lambda z = 0`

For each equation above, find a value of lambda that works and force that value into the other two equations. Choose the other two variables appropriately to satisfy the equalities.

From 1), λ = − 1, and in that case `y = z = 0.` Plugging into our constraint equation yields` x = 1, - 1.`

From 2), λ = − 1 / 2, and in that case `x = z = 0,` `y = 1/\sqrt{2}, -1/\sqrt{2}.`

From 3), λ = 1, and in that case `x = y = 0,z = i, - i.`

Combining these results, the extreme are

`(1,0,0), (-1,0,0), (0,1/\sqrt{2},0), (0,-1/\sqrt{2},0), (0,0,i), (0,0,-i)`

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Practice Problems for Tough Math Problems:

1. Find the Euler equation for the functional   `J(u)=\int int_G\[u_x^2+u_y^2+2f(x,y)u(x,y)\]dxdy`

`Answer: f(x, y) = u_{xx} + u_{yy}`

2. Find the externals of `J(y) = \int_0^1(yy'+y''^2)dx, y(0)=0, y'(0)=1, y(1)=2, y'(1)=4`

`Answer: y = x^3 + x`

Solving Portions

Introduction to solving proportions:

If two fractions` x/y` and `p/q` are equal, they are said to be in proportion. It can be denoted by x: y :: p: q.

Here the fractions should be of same kind and same unit. The two quantities in a ratio are called its terms. The first term is called antecedent and the second term is called consequent. A ratio is a pure numbers it has no unit.

In the proportion x: y ::  p: q, x and q are called extremes, and y and p are called means

Example Problems on Solving Proportions:

Ex 1. A bottle machine fills 500 bottles in 15 minutes. How many bottles will it fill in `1 1/2` hours?

Sol: Let x be the number of bottles to be filled in `1 1/2` hours = 90 minutes.

Given: `x / 90` = `500 / 15`

`=>` x = `500 xx 90 / 15`

x = 3000 bottles.

Therefore 3000 bottles can be filled in 90 minutes.

Ex 2: A machine prints four books in 10 minutes. How many will it print is 2 hours?

Sol : Let ‘x’ be the number of books need to be printed in 2 hours = 120 minutes.

Given: `x / 120` = `4 / 10`

`=>` x = `4 xx120 / 10` = 48

Therefore 48 books can be printed in 2 hours.

Ex 3: Find the value of x, if 7 : 18 : : x : 27

Sol : Given: 7 : 18 :  : x :  27

`=>` ` 7 / 18 = x / 27`

`=>` `18 xx x = 7 xx 27`

Therefore x = `7 xx 27 / 18` = `21 / 2`    = 10.5

Ex 4:  If x is the mean proportion of 2 and 32, find x?

Sol : Given: 2 : x : : x : 32

`=>` `2/ x = x / 32 `

`=>`` x ^2 = 2 xx 32`

Therefore x 2 = 64

Therefore x = 8.

Ex 5: Find the third proportional to 2 and 32.

Soln: Let x be the third proportional :

We have 2 : 32 = 32 : x

`=>` `2 / 32 = 32 / x`

`=>` 2 x = 32 2

`=>` x =` 32 ^2 / 2`   = 512.


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Practice Problems on Solving Proportions.

If 5 pen costs 8 dollar, what is the cost of 15 pens?
[Ans: 15 pens = 24 dollar]

2. Find the value of x in the proportion 4 : x : : 12 : 9

[Ans : x = 3 ]

3. Find the value of x in 5 : 7 : : 8 : x

[Ans: x = 11.2]

Solve Math Histograms

Introduction to solve math histogram:

A Histogram is a representation of frequency distribution through the four-sided figure whose width represents class intervals and whose areas are directly proportional to the corresponding frequencies.

Histograms will represent in Two Dimension used to graph continuous data. The histogram is used in continuous data graphing. This will be plotted continuously. Here we are going to solve math histogram

How to Solve Math Histogram:
Building a histogram is very easy. We normally use discrete data's for drawing the histograms. Here we are going to study how to create a histogram

STEP 1 :- If the given frequency distribution which we have is in inclusive form then we have to convert it into an exclusive form.

STEP 2 :- Taking suitable scales are more important,and then mark the class-intervals along x-axis and frequencies on y axis.  Note that the scales chosen for both the axes need not be the same.

STEP 3 :- Construct rectangles with class intervals as bases and the corresponding frequencies as heights. This is how we solve a math histogram.

Example Problems to Study How to Solve Math Histogram:

Prepare a histogram to from the following frequency table.

Class Interval                                     Frequency

0 – 10                                                  13

10 – 20                                                24

20 – 30                                                18

30 – 40                                                10

40 – 50                                                16


The histogram is a frequency density diagram. Histogram will represent will have the rectangular bars, And that will show the class intervals and frequency.

(1) The class intervals is taken on the X-axis and frequencies are taken on the  Y-axis

(2) The scales for both the axes are marked. In the y axis the scales are marked 0, 5, 10, 15, 20, 25,30

(3) while we are marking the class intervals we have to be more perfect, so that we may not make any errors.

(4) Draw rectangle bars with class intervals as bases and the corresponding frequencies as heights.

(5) This is how we build a histogram

Logic General

Introduction of logic general:

The field includes both the mathematical study of logic general and the applications of formal logic to other areas of mathematics. The unifying themes in mathematical logic general include the study of the expressive power of formal systems and the deductive power of formal proof systems.
Mathematical logic general is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share and basic results on logic, particularly first-order logic, and definably.
(Source from Wikipedia)

Mathematical Logic General Divided into Four Important Parts:

1)Set theory

2)Model theory

3)Recursion theory, and

4)Proof theory and constructive mathematics

Logic general for set theory:

If an element x belongs to a set A, we write x ? A. A set containing all the objects likely to be considering in a par8+ticular discussion is called a universal set and is usually denoted by U.

Logic general for proof theory:

A logical argument to be establishes the validity of a proposition or mathematical formula. The proof theory comprised if a set of axioms and premises sequentially to arrive at a formula to be proved in the form of a conclusion.

List of some Logical general operators:

( )           brackets

!  + -        logical not, unary plus, unary minus

* / %         multiply, divide, modulus

+ -          add, subtract

< <=         less than, less than or equal,

> >=         greater than, greater than or equal

==  !=       equal, not equal

&&           logical and

||             logical or

=             assignment

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Examples of Set Theory - Logic General:

Example 1:

Consider the sets:
f, A = { 1, 3 }, B = {1, 5, 9}, C = {1, 3, 5, 7, 9}.
Insert the symbol ? or ? between each of the following pair of sets:
(i) f . . . B (ii) A . . . B (iii) A . . . C (iv) B . . . C

Solution:

(i) f ? B as f is a subset of every set.
(ii) A ? B as 3 ? A and 3 ? B
(iii) A ? C as 1, 3 ? A also belongs to C
(iv) B ? C as each element of B is also an element of C.

Example 2:

Let A, B and C be three sets. If A ? B and B ? C, is it true that A ? C?. If not, give an example.

Solution:

No, Let A = {1}, B = {{1}, 2} and C = {{1}, 2, 3}. Here A ? B as A = {1}

and B ? C. But A ? C as 1 ? A and 1 ? C.
Note that an element of a set can never be a subset of itself.

Transverse Plane

Introduction to transversals plane:

The line, ray or segment which intersects the one or two line or ray or segment in the same plane. The intersected line should not to be parallel.We can also define it as; a line that passes through the one or two lines that are in the same plane (coplanar) is called as transversal line.
Angles:
There are several angles are made by the transversal line which intersect the parallel line.
Corresponding angles

Alternate angles
Alternate Interior angles
Alternate Exterior Angles
Vertical angles
Consecutive interior angles


Transversals Vs Parallel Line Angles:

Corresponding Angles:
When a transversal line intersects the two parallel lines, the angles in the matching corner are called as corresponding angles.The corresponding angles are always having common value.The corresponding angles are the congruent angles that lie on the same side of the transversal line.
Alternate Interior angles:
When a transversal line intersects the two parallel lines, the pair of angles that are on the opposite side of the transversal line but inside of the two lines called as alternate interior angle.The Alternate Interior angles are always having common measurement.
Alternate Exterior Angles:
When a transversal line intersects the two parallel lines, the pair of angles that are on the opposite side of the transversal line but outside of the two lines called as alternate Exterior angle.The Alternate Exterior angles are always having common measurement. Understanding math help online tutor is always challenging for me but thanks to all math help websites to help me out.

Tansversals :vertical and Consecutive Angles:

Vertical angles:
The opposite angles made by the transversal line that intersect the parallel lines are called as vertical angles. These angles are also called as vertically opposite angles.The vertical angles are also having common measurement.
Consecutive interior angles:
The pair of angles on one side of the transversal line but inside of the two lines is called as consecutive interior angles.The sums of the consecutive interior angles are equal to 180 degree.