Introduction to Algebra Mantu

7/28/2019 Introduction to Algebra Mantu

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Introduction to Algebra

Variables

Expressions

Equations

Solution of an equation

Simplifying equations

Combining like terms

Simplifying with addition and subtraction

Simplifying by multiplication

Simplifying by division

Word problems as equations

Sequences

Math Contests

School League Competitions Contest Problem Books


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Variables

A variable is a symbol that represents a number. Usually we use letters such as n,

t, or x for variables. For example, we might say that s stands for the side-length of

a square. We now treat s as if it were a number we could use. The perimeter ofthe square is given by 4 s. The area of the square is given by s s. When working

with variables, it can be helpful to use a letter that will remind you of what the

variable stands for: let n be the number of people in a movie theater; let t be the

time it takes to travel somewhere; let d be the distance from my house to the

park.

Expressions :An expression is a mathematical term or a sum or difference ofmathematical terms that may use numbers, variables, or both.

Example:

The following are examples of expressions:

2 x

3 + 7

2 y + 5

2 + 6 (4 - 2)

z + 3 (8 - z)

Example:

Roland weighs 70 kilograms, and Mark weighs k kilograms. Write an expressionfor their combined weight. The combined weight in kilograms of these two people

is the sum of their weights, which is 70 + k.

Example:


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A car travels down the freeway at 55 kilometers per hour. Write an expression for

the distance the car will have traveled after h hours. Distance equals rate times

time, so the distance traveled is equal to 55 h..

Example:

There are 2000 liters of water in a swimming pool. Water is filling the pool at the

rate of 100 liters per minute. Write an expression for the amount of water, in liters,

in the swimming pool after m minutes. The amount of water added to the pool after

m minutes will be 100 liters per minute times m, or 100 m. Since we started with

2000 liters of water in the pool, we add this to the amount of water added to the

pool to get the expression 100 m + 2000.

To evaluate an expression at some number means we replace a variable in an

expression with the number, and simplify the expression.

Example:

Evaluate the expression 4 z + 12 when z = 15.

We replace each occurrence of z with the number 15, and simplify using the usual

rules: parentheses first, then exponents, multiplication and division, then addition

and subtraction.

4 z + 12 becomes

4 15 + 12 =

60 + 12 =

72

Example:Evaluate the expression (1 + z) 2 + 12 3 - z when z = 4.

We replace each occurrence of z with the number 4, and simplify using the usual

rules: parentheses first, then exponents, multiplication and division, then addition

and subtraction.


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(1 + z) 2 + 12 3 - z becomes

(1 + 4) 2 + 12 3 - 4 =

5 2 + 12 3 - 4 =

10 + 4 - 4 = 10.

Equations

An equation is a statement that two numbers or expressions are equal. Equations

are useful for relating variables and numbers. Many word problems can easily be

written down as equations with a little practice. Many simple rules exist for

simplifying equations.

Example:

The following are examples of equations:

2 = 2

17 = 2 + 15

x = 7

7 = x

t + 3 = 8

3 n +12 = 100

w + 4 = 12 - w

y - 1 - 2 - 9.3 = 34

3 (d + 4) - 11 = 321 - 23

Example:

Translate the following word problem into an equation:

My age in years y plus 20 is equal to four times my age, minus 10.

The first expression stands for "my age in years plus 20", which is y + 20.


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This is equal to the second expression for "four times my age, minus 10", which is

4 y - 10.

Setting these two expressions equal to one another gives us the equation:

y + 20 = 4 y - 10

Solution of an Equation

When an equation has a variable, the solution to the equation is the number that

makes the equation true when we replace the variable with its value.

Example: We say y = 3 is a solution to the equation 4 y + 7 = 19, for replacing

each occurrence of y with 3 gives us

4 3 + 7 = 19 ==>

12 + 7 = 19 ==>

19 = 19 which is true.

Examples:

x = 100 is a solution to the equation x 2 - 40 = 10

z = 12 is a solution to the equation 5 (z - 6) = 30

Counterexample:

y = 10 is NOT a solution to the equation 4 y + 7 = 19. When we replace each y

with 10, we get

4 10 + 7 = 19 ==>

40 + 7 = 19 ==>

47 = 19 not true!

Counterexamples:

x = 200 is NOT a solution to the equation x 2 - 40 = 10

z = 20 is NOT a solution to the equation 5 (z - 6) = 30


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Simplifying Equations

To find a solution for an equation, we can use the basic rules of simplifying

equations. These are as follows:

1) You may evaluate any parentheses, exponents, multiplications, divisions,

additions, and subtractions in the usual order of operations. When evaluating

expressions, be careful to use the associative and distributive properties properly.

2) You may combine like terms. This means adding or subtracting variables of the

same kind. The expression 2x + 4x simplifies to 6x. The expression 13 - 7 + 3

simplifies to 9.

3) You may add any value to both sides of the equation.

4) You may subtract any value from both sides of the equation. This is best done

by adding a negative value to each side of the equation.

5) You may multiply both sides of the equation by any number except 0.

6) You may divide both sides of the equation by any number except 0.

Hint: Since subtracting any number is the same as adding its negative, it can be

helpful to replace subtractions with additions of a negative number.

Example:

This problem illustrates grouping like terms and dealing with subtraction in an

equation.

Solve x - 12 + 20 = 37.

Replacing the -12 with a +(-12), we get

x + (-12) + 20 = 37.

Since addition is associative, the two like terms (the integers) may be combined.

(12) + 20 = 8

The left side of the equation becomes


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x + 8 = 37.

Now we may subtract 8 from each side of the equation, (we will actually add a -8

to each side).

x + 8 + (-8) = 37 + (-8)

x + 0 = 29

x = 29

We can check this solution in the original equation:

29 - 12 + 20 = 37x + 0 = 29

17 + 20 = 3737 = 37 so our solution is correct.

Example:

This problem illustrates the proper use of the distributive property.

Solve 2 (x + 1 + 4) = 20.

Grouping like terms in the parentheses, the left side of the equation becomes

2 (x + 1 + 4) ==> 2 (x + 5).

Using the distributive property,

2 (x + 5) ==> 2 x + 2 5.

Carrying out multiplications,

2 x + 2 5 ==> to 2x + 10.

The equation now becomes

2x + 10 = 20.

Subtracting a 10 (adding a -10) to each side gives us

2x + 10 + (-10) = 20 + (-10) ==>


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2x + (10 + (-10)) = 20 - 10 ==>

2x + 0 = 10 ==>

2x = 10.

Since the x is multiplied by 2, we divide both sides by 2 to solve for x:

2x = 10 ==>

2x 2 = 10 2 ==>

(2x)/2 = 5 ==>

x = 5.

We can check this solution in the original equation:

2 (5 + 1 + 4) = 20 ==>

2 10 = 20 ==>

20 = 20 so our solution is correct.

Combining like terms

One of the most common ways to simplify an expression is to combine like terms.Numeric terms may be combined, and any terms with the same variable part may

be combined.

Example:

Consider the expression 2 + 7x + 12 - 3x - 5. The numeric like terms are the

numbers 2, 12, and 5. The variable like terms are 7x and 3x. Combining the

numeric like terms, we have 2 + 12 - 5 = 14 - 5 = 9. Combining the variable like

terms, we have 7x - 3x = 4x, so the expression 2 + 7x + 12 - 3x - 5 simplifies to 9 +

4x.

Simplifying with addition and subtraction

We can use addition and subtraction to get all the terms with variables on one side

of an equation, and all the numeric terms on the other.


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The equations 3x = 17, 21 = y, and z/12 = 24 each have a variable term on one side

of the = sign, and a number on the other.

The equations x + 3 = 12, 21 = 30 - y, and (z + 2) 4 = 10 do not.

We usually do this after simplifying each side using the distributive rules,eliminating parentheses, and combining like terms. Since addition is associative, it

can be helpful to add a negative number to each side instead of subtracting to avoid

mistakes.

Examples:

For the equation 3x + 4 = 12, we can isolate the variable term on the left by

subtracting a 4 from both sides:

3x + 4 - 4 = 12 - 4 ==>

3x = 8.

For the equation 7y - 200 = 10, subtracting the 200 on the left side is the same as

adding a -200:

7y + (-200) = 10.

If we add 200 to both sides of the equation, the 200 and -200 will cancel each

other:

7y + (-200) + 200 = 10 + 200 ==>

7y = 210.

For the equation 8 = 20 - z, we can add z to both sides to get 8 + z = 20 - z + z

==> 8 + z = 20. Now subtracting 8 from both sides,

8 + z - 8 = 20 - 8 ==>

z = 12, so we get a solution for z.

Simplfying by multiplication


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When solving for a variable, we want to get a solution like x = 3 or z = 2001.

When a variable is divided by some number, we can use multiplication on both

sides to solve for the variable.

Example:Solve for x in the equation x 12 = 5.

Since the x on the left side is being divided by 12, the equation is the same as x

1/12 = 5. Multiplying both sides by 12 will cancel the 1/12 on the left side:

x 1/12 12 = 5 12 ==>

x 1 = 60 ==>

x = 60.

Simplifying by division

When solving for a variable, we want to get a solution like x = 3 or z = 2001.

When a variable is multiplied by some number, we can use division on both sides

to solve for the variable.

Example:

Solve for x in the equation 7x = 133. Since the x on the left side is being multipliedby 7, we can divide both sides by 7 to solve for x:

7x 7 = 133 7 ==>

(7x)/7 = 133 7 ==>

x/1 = 19 ==>

x = 19.

Note that dividing by 7 is the same as multiplying both sides by 1/7.

Word problems as equations


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When converting word problems to equations, certain "key" words tell you what

kind of operations to use: addition, multiplication, subtraction, and division. The

table below shows some common phrases and the operation to use. Word

Operation Example As an equationsum addition The sum of my age and 10 equals 27. y + 10 = 27

difference subtraction The difference between my age and my younger sister's

age, who is 11 years old, is 5 years. y - 11 = 5

product multiplication The product of my age and 14 is 168. y 14 = 168

times multiplication Three times my age is 60. 3 y = 60

less than ,subtraction Seven less than my age equals 32. y - 7 = 32

total addition The total of my pocket change and 20 dollars is $22.43. y + 20

= 22.43

more than addition Eleven more than my age equals 43. 11 + y = 43

Sequences

A sequence is a list of items. We can specify any item in the list by its place in the

list: first, second, third, fourth, and so on. Many useful lists have patterns so we

know what items occur in each place in the list. There are 2 kinds of sequences. A

finite sequence is a list made up of a finite number of items. An infinite sequence is

a list that continues without end.

Examples:

The following are examples of finite sequences.

The sequence 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 is the sequence of the first 10 odd

numbers.


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The sequence a, e, i, o, u, is the sequence of vowels in the alphabet.

The sequence m, m, m, m, m, m is the sequence of 6 m's.

The sequence 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0 is the sequence of 12 alternating 1's and

0's.

The sequence 1, 2, 3, 4, ..., 9998, 9999, 10000 is the sequence of the first tenthousand integers.

The sequence 0, 1, 4, 9, 16, 25, 36, 49 is the sequence of the squares of the first 8

whole numbers.

Examples:

The following are examples of infinite sequences.

The sequence 2, 4, 6, 8, 10, 12, 14, 16, ... is the sequence of even whole numbers.

The 100th place in this sequence is the number 200.

The sequence a, b, c, a, b, c, a, b, c, a, b, ... is the sequence of the letters a, b, c,

repeating in this pattern forever.


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The 100th place in this sequence is the letter a. The 300th place in this sequence is

the letter c.

The sequence -1, 2, -3, 4, -5, 6, -7, 8, -9, ... is the sequence of integers withalternating signs. The 10th place in this sequence is 10. The 100th place in this

sequence is 100. The 101st place in this sequence is -101.

The sequence 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, ... is a sequence of 1's separated

by 1 zero, then 2 zeros, then 3 zeros, and so on. The 100th place in this sequence is

a 0. The 105th place in this sequence is a 1.

The sequence 1, 3, 6, 10, 15, 21, 28, 36, 45, ... is the sequence of places the 1

occurs in the sequence of 1's and 0's above! If this sequence seems strange, note

the difference between pairs of numbers next to one another:

3 - 1 = 2

6 - 3 = 3

10 - 6 = 4

15 - 10 = 5

21 - 15 = 6


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28 - 21 = 7

Checking these differences makes the pattern clearer.

1, 1, 1, 1, 1, 1, ... is the sequence where every item in the list is the number 1.

1, 2, 3, 4, 5, 6, 7, ... is the sequence of counting numbers. Each item in the list is its

place number in the list.

a, b, a, b, a, b, a, b, ... is the sequence of alternating letters a and b. The a's occur in

odd-numbered places, and the b's occur in the even-numbered places.

1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, ... is the sequence of reciprocals of the whole

numbers.

1, 4, 9, 16, 25, 36, 49, 64, 81, ... is the sequence of squares of the whole numbers.

a, e, i, o, u, a, e, i, o, u, a, e, ... is the repeating sequence of vowels in the alphabet.

4, 7, 10, 13, 16, 19, 22, 25, ... is the sequence of numbers beginning with the number 4, and each

number in the list is 3 more than the number before it.

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Explore Education and Science (66,958) Physics (868)

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How to Pass a Physics Exam

See all 3 photos

A calculator and a pencil: essential tools for passing a physics exam

Source: AliciaC

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Writing a physics exam doesnt have to be a daunting experience! Im a high school science teacher and

I've taught physics, chemistry and biology for many years. Each year I help senior students prepare for

their graduation exams. Ive found that my students do well in their physics exams If they prepare for

the examination throughout their physics course and follow a few important strategies when theyre

actually writing the exam.

The students who do best in physics use an organized filing system for their assignments and study

materials, do all their assigned work, ask me lots of questions to make sure that they understand

everything, take notes even when theyre not asked to, study regularly and solve many practice

problems, including ones that I assign, ones that I recommend and ones that they find themselves.

When they answer questions on exams they make sure that they work carefully, show their reasoning

clearly when solving problems and check their work before handing in the exam. Usually the students

who get the best exam marks are the ones who don't leave the exam room early but stay in the room

until they are required to hand in their exams.

Use an Agenda or Planner

Organize Your Information, Time and Study Area

Your physics exam will last for just a few hours on one particular day, but your preparation for the exam

should begin when youre buying your school supplies before you attend your first physics class.

There are a number of decisions that you need to make before purchasing school supplies. Where are

you going to store your notes, the handouts that your teacher gives you, your assignments, your lab

reports, your practice problems and your practice exam questions? Are these all going into the same

binder, or will it be more efficient to separate them into more than one binder or notebook? Do you

need dividers? Where will you store miscellaneous information like useful web addresses, names of

other resources that your teacher recommends, and test and exam hints? Do you have an agenda or

planner to record important dates, facts, a to-do list and your study schedule? (You are going to make a

study schedule, arent you?!)


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You also need to create a neat and organized study area at home, with adequate lighting and no

distractions. Your desk or table needs to have enough space for your writing supplies and calculator,

your open textbook, your notebook, binder or paper, and your agenda and study schedule.

Use a Computer as a Learning Tool

A computer can be a very useful learning resource. In addition, some people use an agenda on their

computer or iPod and create their study schedule on one on these devices too. Back up your data if you

do this, and only turn on the computer or iPod while you're studying if you're self disciplined. Its very

easy to get distracted by the entertainment that computers offer!

Searching for physics information and practice problems on the Internet is an excellent idea, but do this

outside of your scheduled study time. You'll find that there are many physics resources online, including

facts, explanations, videos, experiment demonstrations, podcasts, example problems and practice

problems. Bookmark useful sites when you find them and organize your bookmarks folder on your

computer so that you can quickly visit a specific site again. If you don't have a computer at home, make

sure that you look at physics resources on a school or public library computer.

Magnetism is a topic in many physics courses. In this photo a magnet is attracting iron filings.

Source: Oguraclutch, CC BY-SA 3.0, via Wikimedia Commons

Work Effectively During Your Physics Course

Even if you use good exam-writing strategies during your physics examination youre unlikely to get a

good result if you havent gathered information during the physics course and studied effectively. Here

are some tips for gathering information.

Attend all of your physics classes.

If you have to miss a class due to unavoidable circumstances, get the information or assignment that

you missed from your teacher.

Complete all your assignments during the course.

When your receive your marked assignments, correct any errors that you made.

If you don't understand something, ask your teacher or another knowledgeable person for help, or

check a reference source.

Copy example problems that are shown on the blackboard, white board or overhead projector.


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Make notes about the information that your teacher presents. You won't be able to write down

everything that the teacher says or shows, so use point form and abbreviations, writing down just the

key points. If a teacher is showing you a web page write down the address so that you can visit the site

later. Check your notes on the same day as the lecture, filling in any gaps, clarifying them and rewriting

them.

File all the information that you collect in the appropriate place and keep it organized in order to make

studying efficient.

Become very familiar with how to use your calculator, as well as your backup calculator if you are

allowed to take it into the exam room.

Don't simply copy answers from your calculator. Always take a quick moment to decide if the answer

seems reasonable. If it's a ridiculous answer then you've know that either you've used the calculator

incorrectly or the calculator is damaged.

Teach and Learn!

Use Good Study Techniques

Study frequently for short periods instead of occasionally for long periods.

Create and follow a study schedule.

Most physics exams contain a lot of word problems. It's therefore very important to do active studyingin physics. You need to solve problems, then check an answer key to see what your errors are, if any,

and correct your solutions. Simply reading through problem solutions (passive studying) is useful, but

active studying is essential if you want to do well on your physics exam.

Collect practice problems to solve. Look in your textbook for problems, search on the Internet and ask

your teacher where you can get extra problems.

Don't forget to solve complex problems as well as easy ones. Working with harder problems is excellent

training for your brain and gives you confidence that you can deal with whatever problems appear on

the real exam.

If a practice exam contains multiple choice questions, don't simply circle the correct answers but write

down the solution method or relevant facts beside the questions so that the exam becomes a study

resource.

If you are able to get copies of previous exams, once you have studied all the material write mock exams

with the same time limit as the real exam.


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There will be some facts to memorize even in a problem-solving course. Make notes about these facts

based upon what you learn in class or what you read in your textbook, and study these notes.

Active studying is more helpful than passive studying when learning factual information. Try making up

questions about the information in your notes and then answering the questions without looking at the

notes. In addition, try explaining some information that you have just read without looking at theinformation. Talk aloud even if you are on your own.

Solving Physics Word Problems

More Study Skills

Add group study time with your friends to your individual study time. Helping each other solve physics

problems is a great learning strategy! However, in order for group study to be successful you need to

make sure that the group works on physics problems instead of socializing.

Try teaching a topic to your friends. Teaching something is another great way to learn.

If your school offers academic help time, tutorial classes or homework classes, make sure you attend

these events if you need help with physics.

Create diagrams to help you study. For example, draw flow charts that show the sequence of events in

solving specific types of problems. Practice drawing graphs that show relationships. Draw sketches to

represent facts, laws and rules.

If you will be given a formula sheet on your exam, make sure that you can use each formula not only as

it's written on the sheet but also in its rearranged forms.

Sometimes a teacher may let you bring one sheet of information into a physics exam. Start preparing

material for this sheet well in advance of the exam date so that it can be changed and fine-tuned before

you enter the exam room. Study this sheet even though you're allowed to have it with you during the

exam.

More Hints For Solving Word Problems

Prepare for Your Physics Exam


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Pack what you will need during the exam (such as writing utensils, an eraser, a ruler, a geometry set and

a calculator) the night before the exam. Make sure that your calculator is in good working order and has

a fresh battery if it needs one, or take a spare battery with you. Put out the comfortable clothes and

shoes that youll wear the next day. Pack other things that you might need during the exam and are

allowed to bring into the exam room, such as a water bottle.

Try to get a good night's sleep before the exam and for several nights leading up to the exam. Don't get

up very early on the exam day to cram. You will most likely be tired and mentally confused when you

enter the exam room if you do this.

Dont try a new food or drink right before the exam. Eat your usual breakfast or lunch, but dont eat or

drink anything that you know will cause problems while you are writing the exam. For example, dont

eat or drink anything that will make you want to visit the washroom frequently.

On the day of the exam make sure that you leave home early in case you face a traffic jam or an

unforeseen transportation problem. You need to arrive at school with enough time to go the washroom

and gather your thoughts before the exam starts.

Preparing for a Multiple Choice Exam

Taking Multiple Choice Exams

Physics Practice Problems

Physics Exam Questions and Keys

Multiple Choice Questions

College Board Physics B

College Board Physics C

Writing the Physics Exam

Make sure that you only take approved electronic devices into the exam room. Don't forget to leave

your cell phone or iPod outside the room, especially if you're used to carrying it around in a pocket!


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Read the exam instructions carefully before starting the examination so that you don't make procedural

errors. This will also give you time to organize your thoughts and calm you down if you're nervous.

Answer the questions that you can do first. If you're spending a long time trying to answer one particular

question, don't get discouraged. Leave the question and come back to it later after you have completed

the rest of the exam. By then you may have realized how to answer the question that seemed difficultwhen you first read it.

Work carefully as you answer the exam questions, but keep track of the time so that you know when

you're taking too long to complete a section of the exam. Some exams give suggested time limits for

each section. Be aware of these limits.

Try to answer multiple choice questions in your mind before looking at the list of possible answers, and

then choose the answer from the list that best matches yours.

If you're having trouble deciding on the correct answer to a multiple choice question, try eliminating the

wrong answers.

Remember the basic steps for solving word problems: draw a diagram to represent the situation

whenever possible; label the diagram with the data given in the problem, or list the data if you haven't

drawn a diagram; decide what information you are being asked to find; choose an appropriate formula

or formulas to find the required information based on the given data; substitute the data in the formula

or formulas; and solve for the required information.

For word problems that require a written response, show all your calculation steps clearly and in the

order in which they're performed so that the marker can follow your reasoning. This will aid you in

several ways: it will help you obtain the maximum number of marks for the problem if you complete the

answer; it will increase the likelihood that you will get at least partial marks for the problem if you get

stuck half way through the answer; and writing the start of the solution may help you to think of the rest

of the solution!

If you are given blank paper to use for rough work, make use of it. If you can't think of how to solve a

problem, "play" with data, formulas and facts, or use brainstorming techniques. These steps may help

you to think of a solution for the problem.

X-rays are studied in many physics courses. This image shows polydactyly, a condition in which a person

has extra fingers or toes.

Source: Drgnu23, CC BY-3.0, via Wikimedia Commons

More Tips For Writing Physics Exams


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Write in all measurement units, not only in the final answer but in the calculations steps too. You will

likely lose marks if you don't do this, plus if you write in all the units you are more likely to notice when

you have to do a unit conversion in order to get the correct answer.

Make sure that you use significant figures if they are required.

Draw graphs neatly, using your ruler for the axes, and don't forget to label the axes and state the

measurement scale that you are using.

Never leave blank spaces on your answer sheet. If time is running out and you have no idea what the

correct answer for a multiple choice question is, circle any of the answers. If there are four possible

answers you have a 25% chance of being right! If you can eliminate the obviously wrong answers your

chance of choosing the right answer increases.

If you can't solve a word problem, list the data, draw a graph or a diagram that you think might be

relevant or write a formula or fact that you think might be related to the problem. You might get partial

marks for your answer.

Check all your answers before you hand in your exam. When you're writing the exam, make a note

beside problems that you leave out so that you know you have to come back to them at the end.

If you have to answer multiple choice questions by shading in circles on a computer scan sheet, make

sure that you've marked the circles that correspond with your intended answers.

If you discover that you've made an error in a multiple choice question, change the answer very clearly,

especially if the answer is written on a computer scan sheet. Erase any stray marks on the answer sheet.

If you're not completely certain about how to answer a multiple choice question, it's probably a good

idea to go with the first answer that you chose instead of second guessing yourself.

A physics exam will be much less intimidating if you prepare for it throughout the course instead of

thinking about it shortly before the exam date! Working conscientiously and efficiently from the start of

the course will give you the best possible chance of understanding your physics curriculum and having a

good exam experience. While many people feel a little tense when they start an exam, if you've

prepared properly your nervousness should soon fade and you will be able to not only pass your physicsexam but also get a good mark to reward your efforts throughout the course.

This Hub was last updated on May 13, 2012


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Explore Education and Science (66,958) Physics (868)

by dipless

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Science: What is Physics?

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What is Physics: By definition

Firstly let's take a look at the definitions of physics; reference.dictionary.com states:

The branch of science concerned with the properties of matter and energy and the relationships

between them. It is based on mathematics and traditionally includes mechanics, optics, electricity and

magnetism, acoustics, and heat. Modern physics, based on quantum theory, includes atomic, nuclear,

particle, and solid-state studies. It can also embrace applied fields such as geophysics and meteorology

Physical properties of behaviour: the physics of the electron

Archaic natural science or natural philosophy

If we break down from the definition and look at it in a broader sense. It is a study and analysis of

nature, which is done in order to work out how the universe behaves.


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What is physics?

See all 3 photos

A chalkboard with physics equations and doodles.

What is Physics: A History

We will now look at the history of physics in broad strokes. It is one of the oldest scientific disciplines. It

was first formally studied in archaic Greece between 650BC and 480BC, during this time it was known as

natural philosophy. It was during the first part of the 5th Century that the theory of atomisation was

developed, which talked about everything being made of inadvisable elements called 'atoms'.

Aristotle was the first credited with calling the study of natural laws 'physics'. It developed into a verycontentious subject and often contradicted the church which at the time, were seen as the leading

scholars. A famous example of this is the heliocentric theory which put the Sun at the centre of the solar

system and not the Earth which at the time was seen as close to blasphemy.

The 17th Century saw a major advancement with many famous names such as Francis Bacon, William

Gilbert, Robert Hooke and of course Galileo Galilei, who was christened by Stephen Hawkins the father

of modern physics. This was a time that mathematical descriptive schemes were adopted for such fields,

such as mechanics and astronomy which could actually model universally valid characterizations of

motion

Galileo Sun Centred model

Isaac Newton

One of the most famous physicists of all time!

Source: http://www.newton.cam.ac.uk

From the late 17th into the early 18th century a famous Cambridge university physicist, Isaac Newton

published the iconic Mathematical Principles of Natural Philosophy which described the motion with


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beautiful mathematical proofs. Later in the 18th century when Newtons work was applied to rotational

mechanics it became known as 'classical physics.

Moving into the later 18th and early 19th Century we reach an era where experimental physics took a

front foot looking at prisms, electricity and f rational mechanics began to be applied to experimental

phenomena. As we move further into the 19th century branches of physics including

Thermodynamics, statistical mechanics, and electromagnetic theory were developed,it was a change of

rapid developments and challenges on classical ideas, mathematical analysis of many phenomenon were

applied the most famous of which the introduction of a new concept of the 'field' and the publication of

Maxwells 1873 Treatise on Electricity and Magnetism.

From here we move into modern physics and famous scientists which most of the world have heard of

including Einstein who is credited with the special and general theories of relativity which fixed the

anomalies in Newtons classical physics models. Also the branches of quantum physics have been and

continue to be developed.

Famous Physicists Physicist Date Biggest Contribution

Aristotle BC384322 Physicae Auscultationes

Archimedes BC287212 On Floating Bodies

Alhazen 9651040 Book of Optics

Copernicus 14731543 1543 On the Revolutions of the Celestial Spheres

Galilei 15641642 632 Dialogue Concerning the Two Chief World Systems

Newton 16431727 1687 Mathematical Principles of Natural Philosophy

Maxwell 18311879 1873 Treatise on Electricity and Magnetism

Einstein 18791955 1905 On the Electrodynamics of Moving Bodies

These are just some of the famous physicists through time. There are so many moreI could have

mentioned including Bohr, Hawkins, Heisenburg

What is Quantum Physics


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It is the physics of the very small, what happens on atomic and smaller scales. Fact is stranger than

fiction!

Source: http://sandraoles.com/

What is Physics today?

At the present time the main areas of development are quantum physics which is the physics of the very

small, you can read about quantum physics in my series of hubs on the subject. Currently we are looking

at string theory, M-brane theory many of which only work if we are in a universe of 10 or 11 dimensions.

There is also a very active area in GUT (grand unified theory) which is trying to bring together the

theories of quantum mechanics and relativity this is a very hotly anticipated development as it will

potentially give us a more complete understanding of nature and the universe.

Physics is a fascinating subject and one which I encourage you to read up on. Often the reality is a lot

stranger than you can imagine. Can you imagine a cat which is both dead and alive, you being able to

put your hand straight through a table and speeds which make the Bugatti veyron look like a snail.

Below are some links where you can find out much more about the wonderful world of physics.

Further information

physics.org | Home

Your guide to what is physics on the web. physics.org is the place to be if you have a burning physics

question, or if you just want to browse articles and interactive features about physics

Physics-online.com for A-level, AS-level and GCSE Physics | Physics-online.com

Physics-online.com is a unique new online service that which provides a searchable library of 1500 great

interactive resources for teaching physics.

physicsworld.com homepage

physicsworld.com - news, views and information for the global physics community from Institute of

Physics Publishing

This Hub was last updated on June 12, 2012


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Useful {3} Funny Awesome Beautiful Interesting {1}

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Discover What Other People Are Reading

Quantum Physics - Erwin Schrodinger and his equation

Quantum Physics - The Double Slit Experiment (wave - particle duality)

Quantum Physics - Werner Heisenberg: Uncertainty Principle

Quantum Physics - Niels Bohr atomic theory

Quantum Physics - Subatomic particles

Quantum Physics

Follow (4)

Comments 8 comments

Go to last comment

Ibrahim Hany 12 days ago from Alexandria, Egypt

That was useful, but how can a cat be dead and alive, I did not get that part?


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Karmallama 12 days ago from minneapolis, minnesota Level 1 Commenter

very fascinating! My favorite form of physics is quantum but I suppose that is a whole other ball game.

Again, Great job

dipless 12 days ago from Manchester Hub Author

@Nell indeed good old Schrodinger, him and his crazy thoughts :p I too am rather partial to the

quantum world as you have probably noted from my other hubs. Haha oh yeah never made that

connection, but I have now, thanks for commenting.

@karmallama thanks for your comments, indeed it is mine too, I love it when reality is stranger than

fiction :)

Green Lotus 11 days ago Level 6 Commenter

I'm fascinated by Quantum Physics and seem to get my head around it quicker than I ever did with

vanilla Physics back when I was in school!

Schrodinger's cat is a bit of a quandary until you accept the fact (?) that nothing really exists until we

observe it. Mind blowing.

dipless 11 days ago from Manchester Hub Author


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It is pretty mind blowing, I think often when you are e a kid they try and spoon feed you information and

as you get older you develop an affinity with things you find interesting which makes the learning and

understanding process a lot easier. :)

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Home Lessons Calculators Worksheets Resources Feedback Algebra Tutors

near New York, NY

Timothy M.

New York, NY

$70/hr


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New York, NY

$45/hr

Powered by WyzAnt Tutors

Basics of Algebra

Algebra is a division of mathematics designed to help solve certain types of

problems quicker and easier. Algebra is based on the concept of unknown values

called variables, unlike arithmetic which is based entirely on known number

values.

This lesson introduces an important algebraic concept known as the Equation.

The idea is that an equation represents a scale such as the one shown on the

right. Instead of keeping the scale balanced with weights, numbers, or constants

are used. These numbers are called constants because they constantly have the

same value. For example the number 47 always represents 47 units or 47

multiplied by an unknown number. It never represents another value.

The equation may also be balanced by a device called a variable. A variable is an

an unknown number represented by any letter in the alphabet (often x). The

value of each variable must remain the same in each problem.

Several symbols are used to relate all of the variables and constants together.

These symbols are listed and explained below. ??? Multiply

* Multiply

/ Divide


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+ Add or Positive

- Subtract or Negative

( ) Calculate what is inside of the parentheses first. (also called grouping

symbols)

Basics of the Equation

The diagram on the right shows a basic equation. This equation is similar to problems which you may

have done in ordinary mathematics such as:

__ + 16 = 30

You could easily guess that __ equals 14 or do 30 - 16 to find that __ equals 14.

In this problem __ stood for an unknown number; in an equation we use variables,

or any letter in the alphabet.

When written algebraically the problem would be:

x + 16 = 30

and the answer should be written:


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x = 14 Solving Equations

These equations can be solved relatively easy and without any formal method. But,

as you use equations to solve more complex problems, you will want an easier way

to solve them.Pretend you have a scale like the one shown. On the right side there are 45 pennies

and on the left side are 23 pennies and an unknown amount of pennies. The scale is

balanced, therefore, we know that there must be an equal amount of weight on

each side.

As long as the same operation (addition, subtraction, multiplication, etc.) is done to

both sides of the scale, it will remain balanced. To find the unknown amount of

pennies of the left side, remove 23 pennies from each side of the scale. This action

keeps the scale balanced and isolates the unknown amount. Since the

weight(amount of pennies) on both sides of the scale are still equal and the

unknown amount is alone, we now know that the unknown amount of pennies on

the left side is the same as the remaining amount (22 pennies) on the right side.

Solving Equations

Take a look at the equation below. As you can see, after the variable is subtracted

from the left and the constants are subtracted from the right, you are still left with

2x on one side. Initial Equation / Problem x + 23 = 3x + 45

Subtract x from each side x - x + 23 = 3x - x + 45

Result 23 = 2x + 45

Subtract 45 from each side 23 - 45 = 2x + 45 - 45

Result -22 = 2x

Switch the left and right sides of the equation 2x = -22

This means that the unknown number multiplied by two, equals -22. To find the

value of x, use the process "dividing by the coefficient" described on the next page.


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Identifying and Using Coefficients

The coefficient of a variable is the number which the variable is being multiplied

by. In this equation, 2 is the coefficient of x because 2x is present in the equation.

Some additional examples of coefficients:

Term Coefficient of x

2x 2

0.24x 0.24

x 1

-x -1

Note that in the last two examples, the following rules are applied

If the variable has no visible coefficient, then it has an implied coefficient of 1.

If the variable only has a negative sign, then it has an implied coefficient of -1.

Equation Basics Worksheet

Enter an answer in each box, then click the "Check Worksheet" button at the

bottom of the page to automatically check each answer. You may also check your

answers manually by referring to the Answer Sheet.

If you need assistance with a particular problem, click the " step-by-step" link for

an in depth solution.

S1

x + 1 = 9

x =

step-by-step


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1. x + 3 = 5

x =

2. x + -6 = 9

x =

3 . -32 = x + 3

x =

4. 29 + -1x = 13

x =

5. 46 = 47 + -1x

x =

6. 12 = -1x + 1

x =

7. 4x = 16

x =

8. 2x = 10

x =

9. 10x = 130

x =

10. 14 = -2x

x =

11. -3 + 2x = 11

x =

12.4x + 6 = -10


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x =

13.x + 9 = 18 + -2x

x =

14. 2x + 6 = 4x + -2

x =

15. -1x + -1 = 221 + 2x

x =

16. 15 + 5x = 0

x =

17. 17x + -12 = 114 + 3x

x =

18. 2x + -10 = 10 + -3x

x =

19. 12x + 0 = 144x =

20. -10x + -19 = 19 + -8x

x =

Prime Numbers Chart


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Display the first primes progressing with columns. 2 3 5 7 11 13 17

19 23 29

31 37 41 43 47 53 59 61 67 71

73 79 83 89 97 101 103 107 109 113

127 131 137 139 149 151 157 163 167 173

179 181 191 193 197 199 211 223 227 229

233 239 241 251 257 263 269 271 277 281

283 293 307 311 313 317 331 337 347 349

353 359 367 373 379 383 389 397 401 409

419 421 431 433 439 443 449 457 461 463

467 479 487 491 499 503 509 521 523

Perfect Squares Chart

Display the first perfect squares progressing with columns. n n2 n n2 n n2

n n2 n n2

0 0 1 1 2 4 3 9 4 16

5 25 6 36 7 49 8 64 9 81

10 100 11 121 12 144 13 169 14 196

15 225 16 256 17 289 18 324 19 361

20 400 21 441 22 484 23 529 24 576

25 625 26 676 27 729 28 784 29 841

30 900 31 961 32 1024 33 1089 34 1156

35 1225 36 1296 37 1369 38 1444 39 1521

40 1600 41 1681 42 1764 43 1849 44 1936

45 2025 46 2116 47 2209 48 2304 49 2401


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50 2500 51 2601 52 2704 53 2809 54 2916

55 3025 56 3136 57 3249 58 3364 59 3481

60 3600 61 3721 62 3844 63 3969 64 4096

65 4225 66 4356 67 4489 68 4624 69 4761

70 4900 71 5041 72 5184 73 5329 74 5476

75 5625 76 5776 77 5929 78 6084 79 6241

80 6400 81 6561 82 6724 83 6889 84 7056

85 7225 86 7396 87 7569 88 7744 89 7921

90 8100 91 8281 92 8464 93 8649 94 8836

95 9025 96 9216 97 9409 98 9604 99 980

Introduction to Algebra Mantu

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