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.

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

The Ultimate Guide to Algebra

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The Ultimate Guide to Algebra