Study Randomness

Introduction to study randomness:

Randomness is a concept of non-order or non-coherence in a sequence of symbols or steps, such that there is no intelligible pattern or combination. Randomness implies a lack of predictability with the concepts of chance, probability, and information entropy. More formally, in statistics, a random process is a repeating process whose outcomes follow no describable deterministic pattern, but follow a probability distribution.


Studying Events on study randomness:

The studying of randomness are as follows:

Event

An outcome of an experiment is called Event. When two or more events occur in connection with each other the joint occurrence is called a compound event. Events are classified into

1) Independent

2) Dependent

3) Mutually Exclusive event.

Independent: An event A is said to be independent of another event B when the actual happening of A does not influence in any degree the probability of the happening of B.

Dependent: If another probability of the happening of B is dependent or influenced by the previous happening, then B is said to be dependent on A.

Mutually Exclusive: Two events B and A are to be mutually exclusive when through the occurrence of one of them, says A, the other event B cannot take place and vice versa, or when two events cannot happen at the same time.


Applications on study randomness


The studying applications of Randomness are as follows:

1) Games: Randomness is used in many games such as gambling, casino games. Randomness is generating unpredictable numbers. Dice, cards use in games of chance because of randomness.

2) Cryptography: Cryptography concept is developed for secure the data from the attacker or unauthorized users. Cryptography has the two events encryption and decryption using algorithms.

3) Music and art: Randomness is used for generating randomness music. We can use randomness for arranging item to exhibit.

4) Simulation: Simulation is used by many engineering field such as neutron transport in nuclear fission

5) Analysis: Many physics experiments are rely on statistical randomness analysis such as analysing x-rays

6) Statistical sampling: Many statistical process based on Randomness concept.

7) Science: Randomness is used in Physics, Operation research and engineering fields

Study Proposition

Definition of proposition:

A proposition is a standpoint that you will create, defend or destroy. It should be worded as a declarative sentence that definitely expresses your position.

A proposition can be the major point of your position. It can also be a single supportive part. It can also be an opposite proposition that you will disprove.

A statement that affirm or denies something.
The meaning expressed in such a statement, as opposite to the way it is expressed.

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Types of proposition


There are three types of proposition that is fact, value and policy.

Proposition of Fact

A proposition of fact is a statement in which you meeting point largely on belief of the listeners in its truth or falsehood. Your arguments are thus expected at getting your listeners to accept the statement as being true or false.

Proposition of Value

In a proposition of values, you make a report where you are asking your listeners to make an evaluative opinion as to whether the report is morally good or bad, right or wrong. This may be done by comparing two things and asking them which is better.

Propositions of Policy

A proportion of policies advocate a course of exploit In this, you ask your listeners to back a policy or to assign themselves to a particular action.

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Example problems


Example 1:

Given:  x/4 = 75 /100

Solution:

Step 1: We need to find x value.

Step 2: Here we are using cross multiplication method to solve x

Step 3: So, when we cross multiply we get 100x= 300

Step 4: Now we divide using 100 on both the sides

Step 5: The value of x= 3.

Example 2:

Given: 15/60 =75/p

Solution:

Step 1:  From this question we need to find p value

Step 2: Using cross multiplication we get, 15x= 4500

Step 3: Now we dived e using 15 on both the sides

Step 4: x= 4500/15

Step 5: x= 300

Study About Trigonometry

Introduction to study about trigonometric

Trigonometry (from Ancient Sanskrit "Trikona" or triangle and "Mati" or measure. Also from Greek trigōnon "triangle" + metron "measure") is a branch of mathematics that studies triangles, particularly right triangles. Trigonometry deals with relationships between the sides and the angles of triangles, and with trigonometric functions, which describe those relationships and angles in general, and the motion of waves such as sound and light waves. Trigonometry is usually taught in secondary schools either as a separate course or as part of a precalculus course. A branch of trigonometry, called spherical trigonometry, studies triangles on spheres, and is important in astronomy and navigation. (Source from Wikipedia)

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Study about basic trigonometry ratios:


There are six basic trigonometric ratios they are following,

sin `theta`= opposite /Hypotenuse

cos `theta`= adjacent / Hypotenuse

tan `theta` = opposite /adjacent

cot `theta` = adjacent / opposite

Sec `theta` = hypotenuse / adjacent

Cosec `theta` = hypotenuse / opposite.


Example problem :


Study about trigonometry word problem:

If the distance of a person from a pyramid is 200 m and the angle subtended by the top of the pyramid with the ground is 30o, what is the height of the pyramid in meters?

Solution:

First we have to plot the diagram to represent the given problem, and mark the date carefully.

Choose which trigonometric function represents the ratio of the side.(200)

Let as assume xy=Distance between man and pyramid.(h)

Similarly yz=Height of the pyramid we have to find

Here we have using the trigonometric function tan x here x=30 degree

Therefore tan x =opposite/adjacent

tan x =yz / xy

tan 30 = h/200

Height=tan 30 degree *200

Tan 30 degree=0.577

=0577*200

Therefore height of the pyramid =115.47

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Study about application of trigonometric:

Trigonometric used large number of application. It is mainly used in astronomy and geography.

Similarly trigonometric used many fields they are,

Electronics
Statics
Number theory
Acoustics
Computer graphics.
Meteorology
Economics
Navigation and aircraft

Study For Patterns Exam

Study for patterns:

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

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More about patterns for exam:


Study for patterns exam, you need to know the numeric patterns. Number patterns in algebra are patterns made from numbers. Study for patterns exam, we need know the basic operations like addition, subtraction, multiplication and division. So you can prepare numbers can be in a list. Any math operation used to make the patterns.

Study for patterns exam, we know about more information about patterns. Many of the patterns in algebra you see will use addition. The same number will be added to each number in the list to make the next number in the list in study for patterns exam.

Example study for patterns exam:

1, 3, 5, 7, 9, 11, 13…

The pattern in algebra is to add 2 every time. The next number is 15, then 17.


Example study for patterns exam:


Example 1:

2,5,8,11,14,17…

Solution:

The pattern in algebra is to add 3 every time. The next number is 20, then 23.

Example 2:

2,4,6,8,9,10,12,14,…

Solution:

If you guessed that the rule is counting by 2 using even numbers, you are correct.

Example 3:

1,1,2,3,5,8,13,21,34,55,…

Solution:

Look carefully and think!!!

If you concluded that the answer to this pattern in algebra is that you need to add number together to get the next number in line, you are correct.

1+1=2,2+1=3,3+2=5,5+3=8,…

This is a famous pattern in algebra or sequence known as the Fibonacci sequence.

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Practice problem study for pattern exam:

Problem 1:

A store gives a door prize to its third customer and to every fifth customer after that. Which of the following customers will get a door prize?

The 62nd

The 71st

The 79th

The 98th

Answer: D

Problem 2:

To find the next two number of pattern with proper explanation                                                                                            1,3,5,6,9,11,13,15,…

Answer: 17,19

Study Calculus Placement Exam

Introduction to Study Calculus Placement Exam

This is for the students who are preparing to take one or two Advance Placement Examinations in Mathematics offered by the College Entrance Examination Board, and for their teachers.

The Courses

Calculus AB and BC are both full-year courses in the calculus of function of a single variable. Both courses emphasize:

Student understanding of concepts and applications of calculus over manipulation and memorization
Developing the student’s ability to express function, concepts problems, and conclusion analytically, graphically, numerically, and verbally, and to understand how these are related.
Using a graphing calculator as a tool for mathematical investigations and problem-solving.

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Studying Topics on Calculus Placement Exam


Both courses are intended for students who have already studied college-preparatory mathematics. The main difference between the two is that BC calculus tests some more theoretical aspects of calculus and it covers a few additional topics. The AB exam usually tests straight forward problems in each topic. The BC asks harder questions but neither is tricky.

The Calculus AB and BC Examinations and the course descriptions are prepared by committees of teachers from colleges or universities and from secondary schools. The examinations are intended to determine the extent to which a student has mastered the subject matter of the course.

Each examination is 3 hours and 15 minutes long.

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The Calculus Placement Exam


First, In Part A there are 28 multiple choice questions for which 55 minutes are allowed. Calculators are not permitted. In Part B there are 17 multiple choice questions for which 50 minutes are allowed and a graphing calculator is permitted.

Second, the free response section ahs twp parts each have three questions each of which requires you to write out the solutions and the steps by which you solved it. You are permitted to use calculators for the first three questions and 45 minutes are allowed for each part.

Study Arithmetic Test Prep

Introduction to study of arithmetic test prep:

Here we will study arithmetic progression for test prep

An Arithmetic Progression study prep, or AP, is a series where each new term after the first is obtained by adding a constant d, called the common difference study, to the preceding term.If the starting term of the sequence is 'a', then the arithmetic progression  is, a + d, a + 2d, a + 3d, . . .where the nth term is a + (n - 1)d.Few test problems prep for study arithmetic is given in the following sections.


Study example problems for arithmetic test prep:


Arithmetic Sequence Problem:

Problem:Find the sum of all positive integers, from 7 to 1764 inclusive, which are divisible by 7.

Solution:study the Sequence of first few elements positive integers divisible by 7 are given by 7, 14, 21...
The above sequence has a first element equal to 7 and a common difference d = 7. We need to know the rank of the term 1764. We use the following formula for the nth term
an = a1 + (n - 1 )d

1765 = a1 + (n - 1 )d
Substitute a1 and d by their values

1764 = 7 + 7(n - 1)
Solve for n to obtain ,      n = 252
1764 is the 252th term, we can use the following formula to find sum
sn = `(n(a_(1) +a_(n)))/(2)`
s252 = `(252 (7 + 1764)) / 2`
= 223146

Study to solve arithmetic word problem for test prep:

Problem:Three apples and four oranges cost $400. The cost of the orange is half the cost of the apple. What is the cost of apple?

Solution: Take cost of the apple is x and cost of the orange is y then test.
From the statement has given we can test as follows
3x + 4y = 400              (1)
x (`1/2` ) = y or x = 2y    (2)
Solving these two equations and test
3(2y) + 4y = 400
6y + 4y = 400
y = 40
Substitute  value and study  y in equation (2) and test
x = 80
Therefore, cost of the apple is $80.

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Study practice problems for Arithmetic Test prep:


Pro1: Find a21 given that a4 = 12 and a9 = 27

Ans: a20 = 63

Pro 2:A factory makes 4249 number of bulbs. There were three kinds of bulbs – serial, power saver, and fancy. The number of fancy was 715 more than the number of power saver. The number of serial was 5 times the number of power saver. How many fancy did the company make test?

Ans: 607

Study Collection of Data

INTRODUCTION:

Statistics is used to make effective use of numerical data for groups of individuals or experiments. It deals with collection of data, analysis and interpretation of data.

It improves quality of data by the design of experiments and survey sampling. It applies in government, business, variety of academic disciplines and social sciences.

It has several methods and which can be used to describe or summarize a collection of data. It is called descriptive statistics which very useful in research.

Source:- Wikipedia

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Overview for collection of data


Data collection techniques use us to systematically collect information about our objects of study like people, phenomena and about the settings which they occur.

If the data are collected randomly, it will be very difficult to answer our research questions in a way.

There are various data collection techniques can be used now days. They are,

Using available information
Observation
Face to face interview
Administering questionnaires
Group discussions
Projective techniques, scaling, mapping.

Methods of Data collection


There are four main methods for collection of data. They are following,

Census                          -  It is a study which collects data from every one of a population.  It requires cost and time.
Sample survey               -  It is a study which collects data from a subset of a population.
Experiment                    - It is a controlled study for understanding cause and effect relationships. Here the researcher controls how subjects are assigned to groups and how many treatments each group receives. Compares some dependent variable.
Observational study       -  Similar to experiments, It also used to understand cause and effect relationships, but here the researcher is no need to control how subjects are assigned and treatments each group receives.

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Each Method of data collection has own advantages and disadvantages. They are following,

Resources                                                   - when the population is too large, a sample survey has a big advantage over a census.
The Generalize of data collection           -  It requires random selection to a larger population. It uses to the appropriateness of      applying findings from a larger population.
The Causal inference of data collection   -  Here cause and effect relationships can be tested when subjects are randomly assigned to groups.

Study About Axes

Introduction to Study About Axes:

Study about axes is of great importance in co-ordinate geometry.Co-ordinate geometry is mainly based on the X and Y axes.study about axes include ,the study of the x and y axes ,in co-ordinate plane.The X and Y axes are the lines placed in the plane perpendicular to each other. Normally the X axis is the line horizontal and Y axis is the line in vertical.

Study about axes may also include the plotting of points in the co-ordinate plane.Normally the x co-ordinate is placed first in the set of ordered pairs.And the Y co-ordinate is placed in second. so a ordered pair will be in the form(x,y).

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Each reference line is called a coordinate axis or  axis of the system, and the point where they meet is its origin. The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as a signed distances from the origin

One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, one can specify a point in a space of any dimension n by use of n Cartesian coordinates, the signed distances from n mutually perpendicular hyperplanes

While talking about study of axes,the quadrants are of great omportance.Mainly there are 4 quadrants.1st,2nd,3rd,and 4th quadrants.In the 1st quadrant both the x and y co-ordinates will be positive.Example:(1,2) is apoint on 1st quadrant.

In the second quadrant  X is negative and y is positive.Example :(-1,3)

In the 3rd quadrant both X and Y co-ordinates are negative. Example:(-4,-2)

In the 4th quadrant X is positive and Y is negative. Example:(4,-8)


Study of axes - Example


Example1:

Plot the point (2,3)

Here the x co-ordinate of the point is 2 and y co-ordinate of the point is 3.so the point we have to mark 2 units away from x  axis and  3 units away from y axis.

Plot the point (-2,3)

Here the point will be in the second quadrant. it will be again 2 units away from the y axis to the left and 3 units from the x axis.

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Importance of Study of axes in co-ordinate geometry


Study of axes in co-ordinate geometry helps us to get the relative position of points in  a plane.and it will also helps us to find the distance between the points in the plane.

Study About Countable

Introduction to study about countable:
Countable is natural number sets.It is denote how many number of variables are present.If there is no countable values means it is called as uncountable.In countable set, we can count the numbers  only once.Injective function is exist in set.It is called countable set.It is defined by f:S`->` N.N={0,1,2,….}.Some theorems are used to derive the contable set.

Theorem: In this theorem, consider a set as S and following conditions should be equal.

1). S is countable. That is f:S`->` N.

2). If subjective function is exist in set or empty means  g:N`->` S.

3). If bisection is exist in set of finite means  h:N`->` S.

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study about countable-Formal introduction


For more information, we study the the some theorems.

Study the countable by formal introduction:

Theorem 1: Cartesian product of finitely many countable sets is countable.

Repeated mapping is generate the finite vectors to natural numbers.Natural numbers easily  map the positive rational numbers.One or more than mapping method is used.

Theorem 2: This define countable set is consider allevery subset.

Prime number set is countable. For example, 2 maps to 1 and 3 maps to 2.

Theorem 3: Rational numbers are countable.

All fractions set are defined.Integers are represented by a and b.For example, 0 maps to (0,1,0) and 3 amps to(3,1,0).

Theorem 4: Natural number sequence is finite length set.It has length sequence 1,2,3.

Theorem 5: All finite subsets are countable.Order the elements when we have a finite subset.

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Study the properties of countable:

Basic theorem: It has three conditions.The basic theorem differentiate the sets as finite and infinite.

Corollary: Consider S and T sets.

1). F:S`->` T is injective ,where countable is T and then countable is S.

2). G:S`->` T is surjective and countable is S and then countable is T.

Study the Cantor’s enumeration is defined collection of countable sets.It is also called denumerable set.If we want to resolve the problem means we can use at most countable.We can study the orders of countable by using some methods.Well orders is used to ordered totally.It has another ways that is normal order of integers of rational numbers.

Study About Grade Measure

Introduction of study about grade measure:
Grade measurement is the geodetic determination of the local radius of curvature of the figure of the Earth by determining the difference in astronomical latitude between two locations on the same meridian, the metric distance between which is known.

The first known grade measurement was performed by Eratosthenes (240 BC) between Alexandria and Syene in what is now Egypt, determining the radius of the Earth with remarkable correctness. The Dutch geodesist Snellius (~1620) repeated the experiment between Alkmaar and Bergen op Zoom using more modern geodetic instrumentation.

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More about the grade measure:


In trigonometry, following three systems are used for measuring an angle according to different units employed:

Sexagesimal system.

Centesimal system

Circular system.

Centesimal system is used to measure a grade.

Centesimal system:

In this system, an angle is measured in terms of grades, minutes and seconds. Thus we have

1 right angle    =          100 grades (100g).

So        90 angle           =          100 grades (100g).

1 angle             =          1.111 grade (1.111g).

1 grade             =          100 minutes (100’).

1 minute          =          100 seconds (100’’).

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Example problem for grade measure:


Find the degree grade corresponding to the following time and angle measures:

a)      270 angle

b)      1 hour 40 minutes

c)      3 hour 20 minutes

d)     1 hour 40 minutes

e)      120 angle

Pro 1:    270o angle = grade?

Sol :     270 angle         =270 x 1.111  (where 1 = 1.111)

=299.97

After routing resultant number, we get

=300

270o angle or right angle = 300 grade (200g)

Pro 2 :      3 hour 20 minutes

Sol :               1 hour = 60 minutes

So 3 hour 20 minutes = 180 minutes+20 minutes

We know 1 grade = 100 minutes, so we can rewrite the given as

3 hour 20 minutes = 180 minutes + 40 minutes

= `300/100` grade

= 3 grades

=3 grades (3g)

3 hour 20 minutes is equal to 3 grades (1g)

Pro 3:          1 hour 40 minutes

Sol :            1 hour = 60 minutes

So 1 hour 40 minutes = 60 minutes+40 minutes

We know 1 grade = 100 minutes, so we can rewrite the given as

1 hour 40 minutes = 60 minutes + 40 minutes

= 100/100 grade

= 1 grade

=1 grade (1g)

1 hour 40 minutes is equal to 1 grade (1g)

Pro 4:       Find the 120 angle into grades?

Sol :       120 angle         =120*1.111

=133.32

After routing resultant number, we get

=133 grades (133g)

120 angles are equal to 133 grades (133g)

Study About Compression

Introduction to study compression:-

In this article we are going to study about compression and its techniques. In graphing, a technique which is used to get smaller graph towards x axis or y  axis is recognized as compression technique. Compression technique is use to compact the given graphical equation. Compression comes below the topic of transformations. A compression technique is of two types, they are Horizontal compression, Vertical compression.

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Study about compression types:-


Horizontal Compression:-

•    Horizontal compression is one of the solidity technique uses to compress the graphical equation. In horizontal compression, the solidity is done towards y – axis.

•    If the function is y = f (x), the horizontal stretching or compressing of the function is the function f (ax).

•    If 0 < b < 1, the graph is stretched horizontally by a reason of a units.

•    If b > 1, the graph is compression horizontally by a reason of “b” unit.

•    If “b” is to be negative, the horizontal compression or stretch of the graph is done by a reflection of the graph across the y-axis. Horizontal compression techniques decrease the graph from its unique position.

Vertical Compression:-

•    Vertical compression is one of the compression technique uses to compress the graphical equation. In vertical compression, the solidity is done towards x – axis.

•    If the function is y = f (x), the vertical stretch or compressing of the occupation is the function f (ax).

•    If 0 < b < 1, the graph is stretched vertically by “b” units.

•    If b > 1, the graph is compressed vertically by “b” units.

•    If “b” is to be negative, the vertical compression or stretch of the graph is done by a reflection of the graph across the x-axis. Vertical compression technique reduces the graph from its innovative position.

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Example problems about study compression:-


Study compression problem1:-

Write the graph of

f(x) = 3x ^2

With a vertical compression by a cause of 1/2

Solution:-

For the given function, f(x), draw the graph,

Take f(x) = y

y = 3x ^2

For the given function the graph will be a parabolic function,

y= 3x ^2/2

2y = 3x ^2

This graph is compressed by a cause of 1/2. The compression of original graph is done towards x – axis by a factor of 1/2.

Study compression problem2:-

Write the graph of

f(x) = 2x ^2

With a horizontal compression by a cause of 1/2

Solution:-

For the given function, f(x), draw the graph,

Take f(x) = y

y = 2x ^2

For the given function the graph will be a parabolic function,

y/2 = 2x ^2

y = 4x ^2

This graph is compressed by a cause of 1/2. The compression of original graph is done towards y – axis by a cause of 1/2.

Study About Composite

Introduction:

A complete integer that has factors other than 1 and the digit itself is one of study about the Composite Number.  A study about composite quantity is a positive numeral which has a cooperative divisor other than itself. In other words, the study about composite numbers consider if n > 0 is numeral and there is some integers 1 < a, b < n such that n = a × b then n is composite. By classification, each integer is better than one either a prime number or a composite number.


Definition:


The study about composite numbers is one of the sessions in the mathematics. The digit one is a unit and it is specified as neither prime number nor composite. For illustration, the integer 14 is composite number because it can be factorized as 2×7. Equally, the study integers 2 and 3 are not yet a composite numbers because each of them can only separated by one itself. Composite are the study about the numbers are the numbers other than prime numbers.


Classification of composite numbers:


1. By counting the number of prime factors we can ale to classify the composite numbers.

2. A composite number with two or further prime factors is a semi prime.

3. A composite number with three different prime factors is called as sphenoid number.

4. One more way is to classify the composite facts is by with the amount of the divisors.

5. Every composite numbers consists of at least two or three divisors.


Examples:


Some of the examples of composite numbers are given below,

4 – Factors of   4 are 2 and 2,4and1.

6 – Factors of   6 are 3 and 2, 6 and 1.

8 – Factors of   8 are 4 and 2, 8 and 1.

9 – Factors of   9 are 3 and 3,9 and 1.

10 – Factors of 10 are 5 and 2,10 and 1.

12 – Factors of 12 are 6 and 2,4 and 3,12 and 1.

14 – Factors of 14 are 7 and 2, 14 and 1.

15– Factors of 15 are 5 and 3, 15 and 1.

16 – Factors of 16 are 8 and 2, 4 and 4, 16 and 1…..

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Practice problem:

Identify the composite number from the following.

Choices:

A.5

B.19

C.27

D.59

Correct answer: c

Step 1: Here, only 27 have factors other than 1 and itself.

Step 2: The factors of 27 are 1, 3, 9, and 27. So, 27 is a composite number.

Study Algebra Edition

Introduction to study algebra edition:

Algebra comes from the Arabic word al-gebra. It represents as the “reunion of the broken words”. In mathematics, Letters like a, b, c, x are been used to represent the numbers, values, complex numbers, integers and these unite with the general operations to form the algebraic expressions. If two algebraic expressions are equated we get the algebraic equations.


Learn study algebra edition:


The algebra edition mainly divides in to two topics they are Algebra1, Algebra2.The algebra 1 further classified into exponents, graphing linear equations, and quadratic equations and factoring. The algebra 2 consists of probability, inequality, trigonometry, parabolas and logarithmic functions. Algebra edition includes all the topics mentioned above.

Algebra 1:

Algebra1 is one of the most important branches in mathematics. Algebraic expression is the collection or of algebraic terms like constants and variables linked by mathematical symbols .In algebra 1 the unknown variables are represented by using letters.

Algebra 2:

Algebra2 get online help of solving equations, graphs and function, quadratic equations, slope of a line, trigonometric identities, etc.

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Example problems for study algebra edition:


Problem 1:

Solve the equations: x + 2y + 3z = 14, 3x + y + 2z = 11, 2x + 3y + z = 11.

Solution:

x + 2y + 3z = 14 (1)

3x + y + 2z = 11 (2)

2x + 3y + z = 11 (3)

Consider the equations (1) and (3)

(1) ?           x + 2y + 3z = 14

(3) × 3 ? 6x + 9y + 3z = 33       (subtracting)

–5x – 7y = –19

5x + 7y = 19 (4)

Consider the equations (2) and (3)

(2) ?           3x + y + 2z = 11

(3) × 2 ?   4x + 6y + 2z = 22    (subtracting)

–x – 5y = –11

x + 5y = 11 (5)

Consider the equations (4) and (5)

(4) ?          5x + 7y  = 19

(5) × 5 ?   5x + 25y = 55   (subtracting)

–18y = –36

?   y = 2

Substitute y = 2 in (5) we get

x + 5(2) = 11

x + 10 = 11

x + 10 - 10 = 11 - 10

? x = 1

Substitute x = 1, y = 2 in (3) we get

2(1) + 3(2) + z = 11

2 + 6 + z = 11

8 - 8 + z = 11 - 8

? z = 3

The solution is x = 1, y = 2, z = 3.


Problem 2:

Solve the equation 7x – 12 = 3x – 24

Solution:

7x – 12 = 3x – 24                    (Adding 12 on both sides)

7x – 12 + 12= 3x – 24 + 12

7x = 3x – 12                                     (Subtract 3x on both sides)

7x – 3x = 3x – 3x – 12

4x = -12                                   (Divide by 4 on both sides)

`4x / 4 = - 12 / 4`

x = -3

So, the answer for x is -3.