The Ultimate Guide to Algebra

Math Is Super Cool - The Ultimate Guide for Understanding Algebra

By Peta-Gaye Reid

THANK YOU READER

Dear Reader,

Welcome! You are about to embark on a Super Cool journey of learning Algebra.

Thank you very much for choosing MATH IS SUPER COOL – THE ULTIMATE GUIDE FOR

UNDERSTANDING ALGEBRA BOOK 1. I would greatly appreciate your comments and reviews

about this book to help me improve my future books in an effort to serve you better.

Additionally, I am truly grateful that you took time out to read this book and I hope you enjoy it.

Also, I would like to thank all the readers who bought my previous books. Thank you for all the

wonderful reviews, they are truly inspiring and greatly appreciated.

Best wishes,

Peta-Gaye Reid

Table of Contents Chapter 1: Understanding the Basics ............................................................................................................ 1

Chapter 2: Understanding Expressions and Equations ................................................................................. 7

Chapter 3: Rearranging Expressions ........................................................................................................... 21

Chapter 4: Simplifying Expressions ............................................................................................................. 25

Chapter 5: Solving Equations ...................................................................................................................... 35

ANSWERS .................................................................................................................................................... 47

1

Chapter 1: Understanding the Basics Hi everyone and welcome to Math Is Super Cool – The Ultimate Guide for Understanding

Algebra. I hope you are ready to become a rockstar in Algebra. In this book, I am going to show

you how easy, simple and fun Algebra really is. My name is Peta-Gaye but you can call me Peta.

Peta: Are you sure we can’t?

Peta: Well I am sure that by the end of this book you will put Algebra, Simple

and Fun in one sentence.

Peta, I don’t think you are

allowed to put the words

ALGEBRA, SIMPLE and FUN

in the same sentence.

I am pretty sure.

2

Before we start, I would like you to clear your mind of all thoughts that Algebra is hard. Just imagine it as

being easy, simple and fun. Like playing a video game or going to the beach.

Ok, now let’s begin.

Algebra is all about solving equations to find the value of unknowns or variables. Let’s look at the simple

equation below.

is called an unknown or variable. This unknown or variable can be a letter such as (‘ ’, ‘ ’,

‘ ’) or symbol such as (‘ ’, ‘ ’).

is called an equation. Our main aim is to find the value of the unknown, .

When solving equations we have some rules that we must follow. I like to call these rules The

Two Commandments of Algebra because they are very important.

These rules will guide you whenever you are in difficulty with equations.

It will even help you to win a difficult game, get that girl or boy you have a crush on to like you

or make your parents give you 5 bucks just for being you.

SERIOUSLY, THOSE

RULES CAN DO ALL

THOSE STUFF.

3

No…I am just joking. It will help you do well in algebra all the time and maybe your parents

will reward you if you get an A+ [fingers crossed]. I can’t promise you about the crush though

(sorry).

Ok, here are the rules.

MAIN RULES WHEN SOLVING EQUATIONS WITH UNKNOWNS

Now that you know the rules, let’s get to the real juicy stuff.

Let’s use these rules in some simple equations which include addition, subtraction,

multiplication and division.

1. PUT ALL UNKNOWNS ON 1 SIDE OF THE EQUAL SIGN

AND ALL CONSTANTS ON THE OTHER SIDE OF THE

EQUAL SIGN

2. WHEN SOLVING EQUATIONS WHATEVER YOU DO ON

ONE SIDE OF THE EQUAL SIGN YOU HAVE TO DO THE

SAME ON THE OTHER SIDE

4

CALCULATING AN UNKNOWN IN AN EQUATION WITH ADDITION

–

–

EXPLANATION

The first rule of algebra is to get your constants on one side of the equal sign (=) and all the unknowns on the other side.

Looking at the problem we realize that is on one side of the equal sign and is on the other side.

Our aim is to get (the constants) on one side of the equal sign and (the unknown) on the other side.

The second rule which states that whatever you do on one side of the equal sign MUST also be done on the other side.

We realize that if we want to get rid of on the left of the equal sign we need to introduce .

Introducing this will cancel the as – . Also, we MUST introduce on the right of the equal sign

– .

As a result, we get the equation – , which is what

we need.

We can then calculate the to find the value of .

NOTE – (Shortcut)

An easy way to do this is to always remember that when a

number goes over the equal sign the sign changes.

5

CALCULATING AN UNKNOWN IN AN EQUATIONS WITH SUBTRACTION

CALCULATING AN UNKNOWN IN AN EQUATION WITH MULTIPLICATION

EXPLANATION

In this example, we must try to get all the unknowns on one side of the equal sign and all the constants on the other side.

To do this we must introduce in the equation on both sides of the equal sign.

The and on the left side of the equal sign will cancel each other and we can calculate the value on the right side of the equation.

NOTE

An easy way to do this is to always remember that when a

number goes over the equal sign the sign changes.

EXPLANATION

In this example, we must try to get all the constants on one

side of the equal sign and all the unknowns on the other

side.

Our goal is to move the from the left side of the equal sign

to the right side.

To do this, we need to divide both sides of the equation by

.

By dividing by on the left side of the equation, this cancels

the leaving only .

We can now find the answer for by calculating the values

on the right side of the equation.

6

CALCULATING IN UNKNOWN AN EQUATIONS WITH DIVISION

Ok…let’s practise some questions.

Understanding the Basics Practise Questions

Solve the Following.

a)

b)

c)

d)

EXPLANATION

In this example, we must try to get all the constants on one

side of the equal sign and all the unknowns on the other

side.

Our goal is to move the from the left side of the equal sign

to the right side.

To do this, we need to multiply both sides of the equation by

.

By multiplying by on the left side of the equation, this

cancels the in the denominator leaving only .

We can now find the answer for by calculating the values

on the right side of the equation.

7

Chapter 2: Understanding Expressions and Equations After going through all those rules, I know you are probably saying “Yes…I am Ready…Algebra Is

Easy and Fun…Bring it on”.

Ok, let’s do some more stuff.

Before I continue, I want to ask you a question.

Do you know the difference between an Expression and an Equation?

I am still NOT convinced

8

Well, the best way to show this difference is by an example.

Do you see the difference?

The most obvious difference is that an expression does not have an equal sign while an

equation does. When given an equation, your main aim is to find the value of the unknown, in

our case . When given an expression, you are usually asked to simplify or expand it.

Let’s practise some questions which will help in identifying equations and expressions.

Expression and Equation Practise Questions

Indicate if the following is an expression or an equation.

1)

2)

3)

4) –

9

Ok, let’s take it a little step further.

When you are adding, subtracting or multiplying unknowns, there are a few things which you

can do and others cannot do. I have created some rules below so that they will be easier to

understand.

RULES FOR MULTIPLING UNKNOWNS

RULES FOR ADDING UNKNOWNS

1) When an unknown is by itself, it usually has an imaginary 1 in front of it.

Example: is the same as is the same as is the same as

2) When a constant is multiplied by an unknown such as ( ), we calculated this by multiplying the constant ( ) by the number in front of the unknown ( ) to get the answer ( ).

Example: is equal to is equal to is equal to

1) ONLY IDENTICAL UNKNOWNS CAN BE ADDED

Example:

2) UNIDENTICAL UNKNOWNS CANNOT BE ADDED

Example:

Example:

10

RULES FOR SUBTRACTING UNKNOWNS

Now, let’s work a few examples with these rules.

1) ONLY IDENTICAL UNKNOWNS CAN BE SUBTRACTED

Example:

2) UNIDENTICAL UNKNOWNS CANNOT BE SUBTRACTED

Example:

Example:

No need for practise.

Let’s just keep being

cool and move to the

next stuff.

Well Donkey, these

practise questions

will help you to do

well.

11

Come on Peta, I am

sure everyone

understands those

SIMPLE rules.

Well, if you solve

these questions, I

will give you a 5

minutes break to do

whatever you like.

Well Peta, let’s make

it 10 minutes. I need

at least ten minutes

to move to the next

stage of my video

game.

Ok Donkey, let’s go.

12

Example 1

Rules for Multiplying Unknowns

The rules for multiplication states that, when an unknown is by itself such as , , , etc. it has

an imaginary in front of it. This means we can do the following.

is the same as

is the same as

is the same as

is the same as

is the same as

… Just making sure.

The rule also states that a constant multiplied by an unknown ( ) is calculated by

multiplying the constant ( ) by the number ( ) in front of the unknown to get the answer

( ). This means that we can do the following.

is the same as

is the same as

is the same as

Ok, we get the point.

13

is the same as

is the same as *remember that has an imaginary in front of it

is the same as *remember that has an imaginary in front of it

Wait Peta, I have just figured

out something. is the

same as , and that is the

same as .

Yes, that’s correct.

I am a Genius.

14

Example 2

Rules for Adding Unknowns

This rule states that only Identical Unknowns can be added. This means that we can do the

following.

1)

2)

3)

4)

5)

Add the constants in front of the unknown ( and 3) and

write back the unknown ( ). The answer will be

Add the constants in front of the unknown (7 and 3) and

write back the unknown (r). The answer will be

Add the constants in front of the unknown (20 and 3) and

write back the unknown (m). The answer will be 10m

Add the constants in front of the unknown (15, 3 and 7)

and write back the unknown (y). The answer will be

Add the constants in front of the unknown (20, 17, 30

and 40) and write back the unknown (w). The answer will

be

15

The rule also states that you CANNOT add Un-identical unknown. This means that you cannot

add the following.

1)

2)

3)

4)

5)

We CANNOT add and because the unknowns are

different.


different.


different.

We CANNOT add and because does not have an

unknown while does. This means that they are un-

identical.

We CANNOT add and because has an unknown

while does not. This means that they are un-identical.

16

Example 3

Rules for Subtracting Unknowns

This rule states that only Identical Unknowns can be subtracted. This means that we can do the

following.

1)

2) –

3)

4)

5) – –

Subtract the values in front of the unknowns (10 and 3)

and keep the unknown (w). The answer will be 7w.


and keep the unknown (s). The answer will be 7s.


and keep the unknown (m). The answer will be 37m.

Subtract the values in front of the unknowns (29, 6 and 7)

and keep the unknown (y). The answer will be 16y.

Subtract the values in front of the unknowns (40, 17, 5

and 3) and keep the unknown (p). The answer will be 15p.

17

The rule also states that you CANNOT subtract Un-identical unknown. This means that you

cannot subtract the following.

1)

2)

3)

4) –

5) –

We CANNOT subtract 7y and 3r because the unknowns

are different.

We CANNOT subtract 5p and 3w because the unknowns

are different.

We CANNOT subtract 12p and 12m because the

unknowns are different.

We CANNOT subtract 16k and 8 because 16k has an

unknown while 8 does not. This means that they are un-

identical.

We CANNOT subtract 40 and 3e because 40 does not

have an unknown while 3e does. This means that they

are un-identical.

18

[10 minutes break]: This would be a great time to relax, have some fun

or grab a snack.

Peta, remember that you

promised me a 10

minutes break to do

whatever I like.

Ok Donkey, you can have

your 10 minutes break.

OK, your 10 minutes is

up.

19

What! But I am in the middle

of a difficult stage in my video

game.

Just Press the Pause button.

Dude, girls will never

understand video games.

It’s not that simple. I will lose

my focus.

20

Ok guys, it’s time for some practise questions.

Rules for Addition, Subtraction and Division Practise Questions

Calculate the following.

1) is the same as _________

2) is the same as __________

3)

4)

5)

6)

7)

8)

9)

10)

21

Chapter 3: Rearranging Expressions Sometimes in math you will be given long expressions with different unknowns as in the example below.

The question will usually ask you to simplify the expression. Before you can simplify the expression you will need to know how to rearrange it. To do this you will have to put all the identical unknowns together. In the expression above, the unknowns in 10p and 12p are identical, so we can put them together. The unknowns in 4w and 8w are identical so we can put those together.

When the unknowns are rearranged, the expression will be as below.

All the p’s are together and all the w’s are together.

Let’s work some examples.

Rearrange the following Expressions

Example 1

Example 2

22

Example 3

Now it’s your turn.

Rearranging Expression Practise Questions #1

Rearrange the following expressions.

1)

2)

3)

Peta: Did you do the practise questions?

Peta: Are you sure you did it?

Peta: I have a feeling you did not do the question.

MAYBE!

23

Peta: Well just make sure you do the questions before you move to the next

step.

Now we are going to learn how to rearrange expressions that have negative signs.

REARRANGING EXPRESSIONS WHICH INVOLVES SUBTRACTION

Sometimes you may be given an expression which involves subtraction.

For instance, the expression below.

–

Whenever you see these questions always remember to keep calm.

Ok, now that we are calm, let’s attempt the question.

EXPLANATION

To rearrange the expression we will have to put all the

identical unknowns together. This means that we will have

to put 18p and together.

Don’t forget the minus sign in front of the this is very

important. The sign in front of the unknown must move

with the unknown.

We can also put and together. We will rewrite

the expression as

A Penguin

never tells.

24

*Always remember that the sign ( or ) must move with the unknown when it is

rearranged.


Example 1

–

–

Example 2

–

–

Example 3

OK guys, it’s your favourite time again.


Rearrange the following.

1)

2) –

3) –

Put all the identical unknowns together. This means that and

will go together and and together. The expression will now be

–


will go together and and together. The expression will now be

–


will go together and and together. The expression will now

be –

25

Chapter 4: Simplifying Expressions Guys, it’s now time to simplify the expressions.

That is a great question. It means that we are going to put the expression in its simplest form.

To do this, we need to use all the knowledge we have learnt in Chapter 3 and 4.

Let’s work an example.

Simplify the following.

To simplify this expression, the first thing we need to do is to put all the identical unknowns

together. This means that and will go together and and together. The

expression will now be

Now, our main aim is to put the expression in its simplest form.

Peta: Do you see how we can make the expression simpler?

Hold up a second Peta.

What does simplifying

the expressions mean?

26

Peta: You are on the right track, continue.

That is correct. All we need to do is calculate all the identical unknowns. We know that will

give us and is . The expression will now be

Peta: Do you think that we can simplify the expression any further?

Well, I know that

is

equal to …

...and

is equal to .

27

Peta: Wow…you are brilliant.

The expression is now in its simplest form because we cannot add Un-identical Unknowns.

Let’s put it all together so that you can see the flow in simplifying the expression.

That was simple. Let’s work some practise questions.

Ok let’s do some more examples.

Nope. Because we cannot

add Un-identical

Unknowns.

WHAT!!! Time to practise questions

already. No! No! Not yet!! Please do one

more example.

28

Simplify the following expression.

Example 1

Example 2

–

–

Example 3

– –

– –

It’s time for some practise questions guys.

Simplifying Expressions Practise Questions #1

Simplify the following expressions.

1)

2)

3) – –

4) – –

Numbers outside of the Bracket


will go together and and - go together. The expression will

now be . We can now calculate the identical

unknowns. equals and equals .


will go together and and go together. The expression

will now be – . We can now calculate the identical

unknowns. equals and – equals .


will go together and and go together. The expression will

now be – – . We can now calculate the identical

unknowns. – equals and – equals .

29

Sometimes you might be asked to simplify expressions which look like this

Peta: Don’t be scared. I am going to show you how easy it is.

Peta: I am sure.

Let’s start from the basics. Whenever there is a number outside of a bracket – like this 9(5) – we can

multiply the number inside of the bracket by the number outside. This means the 9(5) is equal to 9 x 5.

Let’s look at a few examples.

1.

2.

3.

4.

If we have more than one number (or variable) inside of the bracket, we will multiply what is inside of

the bracket by what is outside of the bracket.

That looks

scary Peta.

Are you sure?

30

For Instance:

We use brackets to hold numbers and variables so that they are easier to calculate. This is why we put

and in brackets.

Peta: ALWAYS REMEMBER THAT WE WORK OUT WHAT IS INSIDE THE BRACKETS FIRST.

Example 1

Example 2

Let’s take it a little step further.

Example 3

Simplify the following.

First, we will multiply the contents of the bracket by

the number outside of the bracket.

This will give us .

The next step is to put and in

brackets so that they will be easier to calculate.

We can now calculate ( ) which is and

( ) which is . We learnt how to do this in

Chapter 2.

31

Example 4

Simplify

There are 2 ways that you can simplify this expression.

Peta: Yes, that’s the beauty of math.

Peta: Yes. I am sure.

Now let’s begin.

Seriously, there are

two ways you can

simplify one

expression.

Beauty of Math! Are

you sure you know

what beauty is Peta?

32

Method 1

Method 2

First, we will multiply the contents of the bracket by the

number outside of the bracket.

This will give us .

The next step is to put and in brackets so

that they will easier to calculate.

We will then calculate ( ) which is and

( ) which is . We learnt how to do this in

Chapter 2.

The sum of and is .

First, we will calculate what is inside the bracket.

will give us .

The expression will now be .

We will then multiply the contents of the bracket by the

number outside of the bracket.

This will give us . We will then calculate

which is . We learnt how to do this in Chapter 2.

33

Now, let’s go back to the scary question.

Simplify 7(y + 2y).

Ok Peta let’s try

to simplify it.

First, we will calculate what is inside the bracket.

will give us .

The expression will now be .

We will now multiply the content of the bracket by the number

outside of the bracket.

This will give us . We will then calculate which is

. We learnt how to do this in Chapter 2.

WHAT! IT WAS THAT

EASY.

Yes. It’s that easy. It’s now

time to practise a few

questions.

34



1) 2) 3) 4) – 5)

Peta, why do we have to

practise all the time?

Well, when you practise you

will get better at math.

35

Chapter 5: Solving Equations

Ok, let’s take a 20 minutes break before we start solving equations.

Take a 20 MINUTES BREAK.

Remember this book is all about math being cool. Relax your brain and have some fun.

[20 MINUTES BREAK]

Ok guys, 20 minutes is up. Time to get back to the lesson.

Now we are going to learn how to solve equations. Remember that an equation has an equal sign and

the main aim is to find the value of the unknown.

Let’s refresh our brain before we start.

Do you remember this diagram from chapter 1? We will have to use these rules when we are solving

equations.

1. PUT ALL UNKNOWNS ON 1 SIDE OF THE EQUAL

SIGN AND ALL CONSTANTS ON THE OTHER SIDE

OF THE EQUAL SIGN

2. WHEN SOLVING EQUATIONS WHATEVER YOU DO ON

ONE SIDE OF THE EQUAL SIGN YOU HAVE TO DO

THE SAME ON THE OTHER SIDE

My brain is tired

Peta. I am going to

take a nap.

36


Example 1

Find the value of c?

I have created some steps so that it will be easier to answer the question.

EXPLANATION

Step 1: Understand the Question.

Ask your yourself ‘What is the questions asking me to do?’ In this case, the

question is asking us to find the value of c.

Step 2: Put all the constants on 1 side of the equal sign and all the

unknowns are on the other side.

We need to put all the unknowns on one side of the equal sign and all the

constants on the other side. In this case, and is already on one side

and 5 (the constant) is on the other side.

Step 3: Simplify the identical unknowns.

The identical unknowns are and . When is simplified we

will get . The equation will now look like this .

Step 4: Find the value of the unknown by calculating the equation.

We can now find the value of by calculating the equation. We can use the

knowledge we have learnt in Chapter 1: Calculating Unknowns with

Multiplication to solve the equation.

37

Example 2

Find the value of p?

–

Let’s apply the steps that we learnt to solve this question.

EXPLANATION



question is asking us to find the value of .

Step 2: Put all the constants are on 1 side of the equal sign and all the



constants on the other side. In this case, and is already on one side

and (the constant) is on the other side.


The identical unknowns are and . When – is simplified we

will get . The equation will now be .



knowledge we learnt in Chapter 1: Calculating Unknowns with


38

Example 3

Find the value of y?

– –


– –

EXPLANATION



question is asking us to find the value of .

Step 2: Put all the constants are on 1 side of the equal sign and all the



constants on the other side. In this case, , and is already on

one side and is on the other side.


The identical unknowns are , and . When – is

simplified we will get . The equation will now look like this



knowledge we learnt in Chapter 1: Calculating Unknowns with


39

Example 4

Find the value of w?

–


–

– –

–

EXPLANATION


Ask your yourself ‘What is the questions asking me to do?’ In this

case, the question is asking us to find the value of .

Step 2: Put all the constants are on 1 side of the equal sign and

all the unknowns are on the other side.

We need to put all the unknowns on one side of the equal sign

and all the constants on the other side. In this case, we have ,

and on one side and is on the other side. We need

to move the to the other side of the equal sign with the .


The identical unknowns are and . When – is

simplified we will get . The equation will now be

Step 4: Find the value of the unknown by calculating the

equation.

We can now find the value of by calculating the equation. We

can use the knowledge we learnt in Chapter 1: Calculating

Unknowns with Multiplication to solve the equation.

40

Example 5

Find the value of n?

–


–

–

EXPLANATION


Ask your yourself ‘What is the questions asking me to do?’

In this case, the question is asking us to find the value of .

Step 2: Put all the constants are on 1 side of the equal sign

and all the unknowns are on the other side.

We need to put all the unknowns on one side of the equal

sign and all the constants on the other side. In this case, we

have , and on one side and is on the

other side. We will have to move the to the other side

of the equal sign with the .


The identical unknowns are and + 10n. When

is simplified we will get . The equation

will now be .


equation.

We can now find the value of by calculating the equation.

We can use the knowledge we learnt in Chapter 1:

Calculating Unknowns with Multiplication to solve the

equation.

41

Example 6



–

EXPLANATION


Ask your yourself ‘What is the questions asking me to do?’ In this

case, the question is asking us to find the value of .

Step 2: Put all the constants are on 1 side of the equal sign and all

the unknowns are on the other side.

We need to put all the unknowns on one side of the equal sign and

all the constants on the other side. In this case, we have and

on one side and is on the other side. We will have to

move the to the other side of the equal sign with the .


We only have one unknown ( ) which can’t be simplified any

further.


equation.

We can now find the value of by calculating the equation. We

can use the knowledge we learnt in Chapter 1: Calculating

Unknowns with Addition to solve the equation.

42

Example 7


–


–

–

EXPLANATION


Ask your yourself ‘What is the questions asking me to do?’ In this case, the question is asking

us to find the value of .

Step 2: Put all the constants are on 1 side of the equal sign and all the unknowns are on the

other side.

We need to put all the unknowns on one side of the equal sign and all the constants on the

other side. In this case, we have , and on one side and is on the other side.

We will have to move the and to the other side of the equal sign with the .


We only have one unknown ( ) which can’t be simplified any further.


We can now find the value of by calculating the equation. We can use the knowledge we

learnt in Chapter 1: Calculating Unknowns with Addition to solve the equation.

43

Is everyone ok?

Just take deep breaths .

Now it’s your turn.

Solving Equations Practise Questions


1)

2)

3) –

4) –

5) –

I am still

recovering.

44

Peta: We are finished Guys.

Peta: Yes we are. Now guys what do you think of algebra?

SERIOUSLY, WE ARE

REALLY FINISHED...YEAH!!!

It’s Easy...

45

I think I need to hear one person say that in a sentence.

...Simple...

...and fun...

She’s talking to

you dude.

46

Peta: Great!!

Please leave comments about this book and the topics you would like me to do next.

You can check out my site http://mathissupercool.com/

Bye Guys. Always remember that math is easy, simple and cool .

Peta.

Ok Peta, you win.

Algebra is easy, simple

and fun.

http://mathissupercool.com/

47

ANSWERS Understanding the Basics Practise Questions

Solve the Following.

a)

Answer

b)

Answer

c)

Answer

d)

Answer

48

Expression and Equation Practise Questions

Indicate if the following is an expression or an equation.

1) - Expression

2) - Equation

3) - Equation

4) – - Expression

Rules for Addition, Subtraction and Division Practise Questions


1) is the same as 4 x y

2) is the same as 1p or 1 x p

3)

4)

5)

6)

7)

8)

9)

10)

We cannot add Un-identical Unknowns

We cannot subtract Un-identical Unknowns

49


Rearrange the following expressions.

1)

2)

3)


Rearrange the following.

1)

2) –

–

3) –

–



1)

2)

3) – –

–

50

4) – –

–



1) =

=

2)

3)

4) – –

–

–

5)

Alternative Method

OR

OR –

Alternative Method

OR

Alternative Method

51

Solving Equations Practise Questions


1)

2)

3) –

4) –

–

–

OR –

–

Alternative Method

52

5) –

Education

The Ultimate Guide to Algebra