Study Solution Set

Definition

A solution set is the set of values that satisfy a given set of equations or inequalities.

For example, for a set {fi} of polynomials over a ring R, the solution set is the subset of R on which the polynomials all vanish (evaluate to 0).i

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Expressing the Requirements


a solution set is a set of possible values that a variable can take on in order to satisfy a given set of conditions.

When you say that the solution set of two parallel lines.  I assume you mean the intersection between two parallel lines y=m*x+b and y=m*x+c. That is, we seek m*x+b = m*x+c.

Expressing this requirement in the terms of a polynomial f(x) =0, where f(x) =b-c, first, we observe that this has no solution unless b=c. i.e., unless the two parallel lines are in fact one and the same. In this case, the variable x is could not constrained and we have an infinite set of solutions since f(x)=b-c=0 is independent of x. In general, however, there is no solution to f(x)=b-c=0 when b is different from c. No x-value in R will satisfy this condition. This means two parallel lines do not intersect to the irrespective of the x value.



The solution set of the single equation f(x) = x is the set {0}.

1. The solution set of the single equation f(x) = x is the set {0}.

2. For any non-zero polynomial f can be over the complex numbers in one variable, the solution set is made up of finitely many points.

3. However, for that the complex polynomial in more than one variable the solution set has no isolated points.

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If the solution set is empty, then there are no such xi such that

f(x0,...,xn) = c

Becomes true for a given c.

He solution set of an equation in the form ax + by = c with a, b, and c real-valued constants, this forms a line in the vector space R^2. However, it cannot always be easy to graphically depict solutions sets – for example, the solution set to an equation in the form ax + by + cz + dw = k (with a, b, c, d, and k real-valued constants) is a hyper plane

Math Pattern Solutions

Introduction of math pattern solutions:

A pattern, from the French patron, is a type of theme of recurring events or objects, sometimes referred to as elements of a set. These elements repeat in a predictable manner. It can be a template or model which can be used to generate things or parts of a thing, especially if the things that are created have enough in common for the underlying pattern to be inferred, in which case the things are said to exhibit the unique pattern.

(Source:Wikipedia)


Concept of math pattern solutions:


In math, numeric patterns in algebra are patterning prepared from numbers.

The numbers can be in a listing. Any math operation likes addition, subtraction, multiplication, or division, you makes the pattern.

Many of the patterns in math you watch will use addition. The same number will be additional to each number in the list to make the next number in the math list.

Example

2, 4, 6, 8, 10, 12, 14…

The pattern in math is to add 2 every time. The subsequently number is 16, then 18.

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Example for math pattern solutions:


6, 14, 30, 62, …

In the math series on top of, each term after the primary is determined by multiplying the previous term by m and then adding n. What is the value of n?

Solution:

Method 1:

The best way to solve this would be if you see that the pattern:

6 × 2 + 2 = 14
14 × 2 + 2 =28

The value of n is 2.

Method 2:

If you were not able to see the pattern then you can get nearer with two equations and then resolve for n.

6m + n =14             (equation 1)
14m + n = 30          (equation 2)

Use substitution method

n in equation 1

n = 14 – 6m

Substitute into equation 2

14m + 14 – 6m = 30
8m = 16
m = 2

Substitute m = 2 into equation 1

6(2) + n = 14
n = 2

Final solution is n = 2

Balanced Equation Math

Introduction of balanced equation math:

We know that algebra is very important term in math. Algebra has a different type of equations. Normally we can do addition , subtraction, multiplication and division. An algebra equation contains variables, symbols etc. Here we are going to learn about balanced equation in math. It’s also called as verification in math. Using this method we can also verify if the solution is correct or not.

Example problems of balanced equation in math:

Example 1:

Y+9=15

Solution:

Step 1: y-9+9= 15-9 (Subtract 9 on both the sides)

Step 2: After simplification we get y = 6.

Balanced equation:

Step 1: y+9= 15

Step 2: 6+9= 15

Step 3: Therefore 15=15.

Here we get the value of y = 6.Now we need to substitute y value in the question. When we substitute we get, the same value on both the sides. Its called balanced equation in math.



Example 2:

Given:

Solve: 5z-10=130

Solution:

We need to find the z value

Step 1: 5z-10=130 (first we need to add 10 on both the sides)

Step 2: 5z=140 (Now we divide using 5 on both the sides)

Step 3: z=28.

Balanced equation:

Step 1: 5z-10=130

Step 2: 5(28)-10= 130.

Step 3: Therefore 130 = 130.

Here we get the value of z = 28.Now we need to substitute z value in the question. When we substitute we get, the same value on both the sides. Its called balanced equation in math.

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More about balanced equation math:


Example 3:

Solving equations using elimination method

Solve: 4p+3q =21 and -4p-6q=12

Solution:

Step 1: It question is 4p+3q =21 and -4p-6q=12

Step 2: first we need to change the sign and simplify the equations. Here also we have the opposite sign. So we need not change the sign

Step 3: When we simplify we get -3q= 33.So q= -11

Now we need to calculate the value of p.

Step 6: so, we plug in q value in any equation to find p.

Step 7: here the equation one is 4p+3q =21

Step 8: So, 4p+3(-11)=21

Step 9: 4p-33=21 (Add 33 on both the sides).

Step 10:4p-33+33=21+33. 4p=54

Step 10: So the value of p =13.5.

Balanced equation:

For, the past question we have 2 equations. We can take any one of the equation and substitute p and q value.

Step 1: Let us take the first equation, 4p+3q =21

Step 2: 4(13.5)+3(-11)= 21

Step 3:54-33=21

Step 4:21=21

Here we get the value of p = 13.5 and q= -11.Now we need to substitute y value in the question. When we substitute we get, the same value on both the sides. Its called balanced equation in math.

These are the example problems of balanced equation in math.

Math Integer Solver

Introduction to math integer solver:

In mathematics, integer solver is one interesting topics in number representation. Integer has a set of numbers in which positive numbers, negative numbers and zero. It contains complete entity or unit. In integer, there is no fractional part. Integer performs different arithmetic operations such as addition, subtraction, multiplication and division. Let us solve some example problems in math integer solver.

Example for integers:

112, -252, 0 etc,

Positive integer: {1, 2, 3, 4, 5 …..}

Negative integer: {…….., -5, -4, -3, -2, -1}


Example problems for math integer solver:


Let us see some of the example problems for math integer solver.

Example 1:

Perform arithmetic operation addition for the given two integer numbers

256 + 386

Solution:

Given two integer numbers are

256 + 386

Both are two positive integers so, the result is also a positive numbers

Here we add 256 into 386, and then we get the result

256 + 386

642

Solution to the given two integers is 642.

Example 2:

Perform arithmetic operation subtraction for the given two integer numbers

857 and 420

Solution:

Given two integer numbers are

857 - 420

Here we subtract 857 into 420, and then we get the result

857 - 420

437

Solution to the given two integers is 437.



Example 3:

Perform arithmetic operation multiplication for the given two integer numbers

45 × 28

Solution:

Given two integer numbers are

45 × 28

Both are positive integers so, the result is also a positive numbers

Here we multiply 45 into 28, and then we get the result

45 × 28

1260

Solution to the given two integers is 1260.

Example 4:

Perform arithmetic operation division for the given two integer numbers

1250 / 25

Solution:

Given two integer numbers are

1250 / 25

Both are positive integers so, the result is also a positive numbers

Here we divide 1250 by 25, and then we get the result

1250/ 25

10

Solution to the given two integers is 10.

Example 5:

Perform arithmetic operation addition for the given two integer numbers

- 36 + 42

Solution:

Given two integer numbers are

- 36 + 42

- 36 is negative number and 42 is positive number

Here we add - 36 into 42, and then we get the result

- 36 + 42

6

Solution to the given two integers is 6.

Example 6:

Perform arithmetic operation subtraction for the given two integer numbers

22 - 0

Solution:

Given two integer numbers are

22 - 0

22 is positive numbers and 0 has no sign

Here we subtract 22 into 0, and then we get the result

22 – 0

22

Solution to the given two integers is 22.


Practice problems for math integer solver:


Some practice problems for math integer solver are,

Perform arithmetic operation for the given two integer numbers

i). 56 + 23

ii). 85 – 34

iii). 32 × 12

iv). 64 / 8

Solution:

i). 79

ii). 51

iii). 384

iv). 8

Patterns in Math

Introduction for Patterns in Math:

In math, under a certain conditions, the numbers are listed, which is said to be a pattern. Basically patterns in math are classified into three types. Every pattern has some properties. In this article, we shall discuss about various kinds of patterns in math. Also we shall solve some problems regarding pattern in math.

Types of Patterns in math:

Arithmetic Pattern
Alphabetic Pattern
Geometric Pattern
These are the 3 different types of patterns in math.

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Arithmetic patterns:

Typically the patterns are indicated as sequences. There are two types of possibilities for the occurrence of sequence. They are finite and infinite sequences. We can articulate a number pattern using some unique symbols. We can clarify the number patterns in several ways. Let us consider a number pattern {2, 4, 6, 8, 10,….}. For the given pattern, the first term of this pattern is 2; and the second term can be attaining by adding 2 with the first term.

Alphabetic Patterns:

Patterns based on alphabetical lettering are called as alphabetic pattern. In addition, the alphabets in a sequence are the alphabetical pattern for the exacting sequence.

Geometric Patterns:

Whenever the geometric shapes involved in some patterns, then they are said to be geometric patterns. For instance, Ellipses are the geometric shapes. Those ellipses are developed from the basic geometric shape called circles. Is this topic T Test Example hard for you? Watch out for my coming posts.


Examples for patterns in math:


Example 1:

Find the missing terms from the pattern given below.

2, 4, 8, 16, 32, 64, ___, ____

Solution:

The first term of the pattern is 2

The second term of the pattern = 2 * 2 = 4

The third term is 4 * 2 = 8

The fourth term is 8 * 2 = 16

Similarly, the missing terms can be determined as follows.

The seventh term will be 64 * 2 = 128

The eighth term is 128 * 2 = 256

So the correct pattern is 2, 4, 8, 16, 32, 64, 128 and 256.

Example 2:

Find the next two terms in the pattern given below.

1, 3, 5, 7, 9, 11, ___, ____

Solution:

The first term of the pattern is 1

The second term of the pattern = 1 + 2 = 3

The third term is 3 + 2 = 5

The fourth term is 5 + 2 = 7

Similarly, the missing terms can be determined as follows.

The seventh term will be 11 + 2 = 13

The eighth term is 13 + 2 = 15

This is an odd sequence of number.

So the exact pattern is 1, 3, 5, 7, 9, 11, 13 and 15.

Math Problems Quizzes

Introduction of math problem quizzes:

In this lesson we can show the math problem quizzes. It very helpful and you can develop your math skills. These quizzes are really exercising your brain at the same time carefully handle the number terms in word problems. The quizzes are showing word problem, fraction, decimal, ect. Let we see math quizzes.

Math problem quizzes:

1) At restaurant, the men drunk 4 1/2 cup of coffee and women drunk 5 ½ cup of coffee. How many coffees were drunk at restaurant?

(Answer:10)

2) Yesterday I bought 6.25ml paint. And I used 1.50ml paint. Now, how much paint I have? Or Find rest of paint?

(Answer: 4.75)

3) There were 40 questions on a science test. William got 70% of them correct. How many questions did he answer correctly?

(Answer: 28)

Find Missing number:

4)  --- , 5 , 4 ,10,--- , 15, 8 ,20, 10 ,25, ---- ,---- , ----- ,----, 16 ,40, 18 ,45, 20 ,----, ----,----,

24,60 26, ---- ,----,

(Answer: 2,6,12,30,14,35,50,22,55,65,28)

5)  Suppose 5 dogs take 5 minutes to eat 5 bones.

How many minutes would it take 4 dogs to eat 4 bones?

How many bones would it take to eat 24 bones in 24 minutes?

(Answer: 4 and 24)

To find with formula:

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6) A pillar 8 feet tall casts a shadow 4 feet long on the ground. If the pillar was 26 feet tall, how many feet in length would the shadow be?

(Answer: 13)

7) Which of the following are the prime factors of 96?

(Answer: 2 and 3)

8) A wall has an area of 96 square inches and a width of 8 inches. What is its length?

(Answer: 12inches)

9) A florist buys roses at $1.50 a piece and sells them for $3.00 a piece. If there are no other expenses, how many roses should be sold in order to make a profit of $300?

(Answer: 200)

10) If the average person throws away 2.5 pounds of garbage every day, how much garbage would the average person throw away in one week?

(Answer: 17.5 pounds)                                                                                                                                                                                                                                                                         

Learning Solution Set

Introduction for learning solution set:

Theory:

Set is an achievement of our recent mathematics. It appears in all branches.  It originated when mathematicians attempted to axiomatize mathematics within the frame work of logic. Mathematicians developed the theory of sets and solutions of set. It becomes a milestone in the growth of mathematics. Now we proceed to introduce the concept of a set. We are learning the solution of set by operations. Let us see about learning solution set in this article. Please express your views of this topic Union Set Theory by commenting on blog.


Learning Solution Set : Set Operations


We shall now study about the set operations

(i) Union of two sets

(ii) Intersection of two sets

(iii) complements

(i) Union of two sets

Let A and B be two given sets. The set of all elements that belong either to A to either to B or to A or to both is called the union of A and B. We denote the union of A and B by A U B. We are learning the union by following expression:

A U B = { x | x ∈ A or x ∈ B or x ∈ A and B}.

We write A U B = {x| x ∈ A or x ∈ B} where it is understood that the word or is used in the inclusive sense; that is, x ∈ A or x ∈ B stands for x ∈ A or x ∈ B or x ∈ A and B.


(ii) Intersection of two sets

Let A and B be two given sets. The set formed by the elements that are common to both A and B is called the intersection of A and B. We denote the intersection of A and B by A ∩B. We are learning intersection by following expression:

A ∩ B = {x | x ∈ A and x ∈ B}.

(iii) Complements:

Subtract the given sets A and B. Learning solution by A - B.

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Learning Example Solutions for Set Operations:


Find the solution for following examples:

1). If A = {1, 2, 3, 4} and B = {3,4, 6}, find A UB.

Solution:

List the elements of both A and B and avoid duplication.

Thus, A UB = {1, 2, 3, 4, 6}.

2). If A = {4,5,6,7} and B = {5,6,8}, find A∩B.

Solution:

All elements in A and B: 4,5,6,7,5,6,8.

Common elements in A and B:5,6

∴ A ∩B = {5,6}.

3). If A  = {2,3,4,5}, B={3,4,5}, find A - B.

A - B = {2}.