Algebra Toolbox Part 1

8/13/2019 Algebra Toolbox Part 1

1/66


2/66

The Algebra Toolbox

Contents

Section Topics Page1. Mastering Minuses a. Multiplication, division 2

b. Addition, subtraction 6c. Signs side by side 9d. Cut n paste 11e. Combinations 12

2. Moving into Algebra a. Adding, subtracting with like terms 13

b. Cut n paste with algebra 21c. Multiplying with algebra 23d. Order of Operations 25e. Power notation 26f. Multiplying with powers 28

3. Brackets in Algebra a. Expanding 1 pair of brackets 30 b. 34

c. Factorising 36d. Expanding - binomials 39

4. Solving equations a. Introduction 41 b. Cut n Paste Numbers 43c. Cut n Paste Algebra 45d. Multiply, divide - numbers 52e. Multiply, divide - Algebra 55f. Advanced equations 60

Expanding & simplifying 2 or more pairs of brackets


3/66

1. MASTERING MINUSES

Skill in working with positive and negative numbers is important to be able

to do algebra. Negatives (or minuses) create many headaches for students.Often my students ask me, quite bewildered, Dont two minuses make a plus?. A fair question!! Read on.

The answer to this depends on which of these 2 categories the question is in:

(a) Multiplication & division(b) Addition & subtraction

(a) MULTIPLICATION & DIVISION

This is the easier of the two categories to understand. 3 simple rules.

Rule 1: Two pluses make a plus!These are the sums you have been doing since Year 1!!

+2 +3 = +6 +6 +3 = +2

4 3 = 12 simply means (+4) (+3) = +12

6 2 = 3 or26 = 3 simply means (+6) (+2) = +3

These should bring back happy memories from early childhood!!

The Algebra Toolbox. 2003 R Bowman. All rights reserved3

Remember:4 and +4 are theSAME THING!!


4/66

Now we look at what happens when we multiply or divide twominuses (negatives):

Rule 2: Two minuses make a plus!

2 3 = +6 8 2 = +4 So 3 5 = +15 2 8 = +16 4 6 = 24

20 10 = 2

428

=

etc!!!

So in these cases, 2 minuses DO make a plus!!

But what if theres one of each?

Rule 3: When multiplying or dividing, one of eachmakes a minus!

4 +5 = 20 and 20 +5 = 4 This means that one of each (a minus and a plus, any order) gives aminus!!

3 +9 = 27+5 6 = 308 4 = 32

6 2 = 12

3 4 2 = 12 2 = 6(First we do the 3 4 to get the 12 !)

40 +5 = 833 11 = 3

98

72=

6954 =


Remember that

6 3 and36

are

the same thing!Both e ual 2.


5/66

Practice Exercises 1 (Try to do without a calculator!):

1 4 7 = 11 16 4 =

2 20 2 = 12 9 9

3 3 4 = 13 45 9

4 20 5 = 14 8 3 =

5 8 6 = 15 55 5

6 16 2 = 16 3 4 5 =

7 9 7 = 17 20 4 1 =

8 35 5 = 18 7 5 1 =

9 60 1 = 19 6 4 12 =

10 30 5 = 20 12 3 2 =

Click here for answers!



6/66

(b) ADDITION & SUBTRACTION

And the good news is.Some of this you have been doing since Year One!!

8 + 7 = 15. Same thing as +8 + 7 = +156 2 = 4. Same thing as +6 2 = +4

Youve also done harder ones like this, where we work from left toright one step at a time (or better still get your calculator to do it!):

8 + 5 2 4 + 7= 13 2 4 + 7 First doing the 8 + 5 to get 13

= 11 4 + 7 Then doing the 13 2 to get 11= 7 + 7 Then doing 11 4 to get 7= 14

Try :

13 2 + 5 8 4 + 6

Did you get 10 ? Hope so!!



7/66

But what about

(1) 6 9?(2) 3 7?

(3) 4 + 5?(4) 8 + 3?

I think the best way to tackle these (without a calculator) is usingtemperatures.

If the temperature is 12 , and goes down by 15 , the newtemperature will be 3 ! This is represented mathematically by

12 15 = 3 (THINK: down is minus)

If the temperature is 5 , and goes down a further 6 , the newtemperature is 11 . This is represented mathematically by

5 6 = 11 (THINK: down is minus)

If the temperature is 3 , and goes up by 8 , the new temperaturewill be +5 , or simply 5 . This is represented mathematically by

3 + 8 = + 5 (THINK: up is plus)

So take Question 1 :

6 9 = ? Think: Temperature starts out at + 6 (6 above zero) and goesdown 9 ( 9 means down 9). What is the new temperature?(Count down 9 spaces beginning at + 6:(5,4,3,2,1,0,-1,-2,-3)

Answer is 3 !! ANS IS WHERE YOUEND UP! 3.

Question 2 : Think: Temperature starts at 3 and drops 7. Newtemperature?

3 7 = ? Begin at 3 & count down 7 spaces. End up 10. (-4,-5,-6,-7,-8,-9 ,-10 )

Answer is 10 !!



8/66

Question 3: Think: Temperature starts at 4 and goes up 5. Newtemperature?

4 + 5 = ? Begin at 4 & count up 5 spaces(-3,-2,-1,0,+1)

Answer is +1 !

Question 4: Think: Temperature starts at 8 and goes up 3.New temperature?

8 + 3 = ? Counting up: (-7,-6 ,-5 )

Answer is 5.

Lets test your skill!! Calculate

(a) 3 + 8.

Begin at 3. Count UP 8 spaces: -2, -1, 0, 1, 2, 3, 4, 5

ANSWER IS 5.

(b) 2 5 + 3 8 4.

Working left to right one step at a time,

2 5 + 3 8 4= 3 + 3 8 4 ( 2 5 = 3 ) = 0 8 4 ( 3 + 3 = 0) = 8 4 ( 0 8 = 8 )= 12

If using temperatures: Begin at +2. Down 5. Up 3. Down 8. Down 4. End up at 12!

Rule 4: When working with addition and subtraction ofpositives and negatives, use temperatures!



9/66

(c) WHEN SIGNS ARE SIDE BY SIDE

Occasionally you might get situations where you have two signs side by side. These are easy!

Two minuses side by side make a plus .This is the other occasion when two minus make a plus!

2 3= 2 + 3 = 5.

There might be brackets.

2 ( 3). Do the same thing as above.= 2 + 3 = 5

One of each side by side: + or + makes a minus

2 + 7= 2 7 = 5 (using our temperature method!)

1 + 3= 1 3 = 2 (temperatures again)

4 (+ 5)= 4 5 = 9 (Yep. temperatures again!)



10/66

Practice Exercises 2: Use notes on Parts (b) and (c). Try to dowithout a calculator!

1 4 + 7 = 11 16 + 4 =

2 20 2 = 12 9 + (9) =

3 3 4 = 13 4 9

4 20 (5) = 14 2 13 =

5 8 6 = 15 11 + 5 =

6 16 + 12 = 16 3 4 5 =

7 2 7 = 17 3 4 + 11 =

8 5 15 = 18 9 5 + 3 8 =

9 3 + 8 = 19 6 + 4 + 12 20 =

10 10 15 = 20 12 3 + 2 5 =

Click here for answers !



11/66

(d) A USEFUL TRICK CUT N PASTE

Did you know that in addition and subtraction sums, you cancut n paste things?

For example, 5 + 8 can be rewritten by cutting the + 8 and pastingit in front of the 5. So,

5 +8

= +8 5= 8 5= 3

The +8 can be cut from the end and pasted in front ofthe 5. This is a legal step in any expression with + and but cannot be done with or

Lets try cutnpaste in a harder question:

There are 5 terms in the expression 2 8 + 3 5 + 10.

Any of these can be cutnpasted to anywhere else in the expression. Onlyone thing to remember, and thats to keep each sign (+ or -) with thenumber that comes after it. A good way is to rewrite the original sum withspaces in front of each and + sign..like this:

+2 8 +3 5 + 10.We can cut n paste terms to wherever we like, but usually its toget all the + terms together (the +2, +3 and +10) and all the together ( 8 & 5) . (Here we cut the +3 and +10 and pastedthem straight after the +2) The 8 and 5 are automaticallypushed to the end. Later in Algebra we cut n paste to get termswith the same letters together.

= +2 +3 +10 8 5.

Now just work left to right:

Thesemake 15

Thesemake -13

= 15 13

= 2 !!


Cut n pasting is a handyskill to know, especiallywhen we are usingalgebra!!


12/66

(e) COMBINATIONS OF +, , ,

In primary school you learned (maybe??) that there is an ORDER OFOPERATIONS that has to be observed. It goes something like this:

Brackets get done firstThen come and , working your way left to rightFinally comes + and , again left to right.

A bit of a mystery? Heres a couple of examples:

(1) Work out 5 3 12 2 + 10 5.

The process:

5 3 12 2 + 10 5No brackets, so pick out the x and and do first.Well bracket them to show they must be done first!Insert an opening bracket right in front of each of the 5,the 12 and the 10 (because theyre the numbersfollowed by x and signs).

= (5 3) (12 2) + (10 5)

Now work these out using the rules for x and

= 15 6 + 2

Now fix the double minus in front of the 6. This gives a+. Fix the + in front of the 2 to give a

= 15 + 6 2

Now its only + and signs left, we work left to right

= 9 2= 11



13/66

2. MOVING INTO ALGEBRA

(a) ADDING & SUBTRACTING(THIS ONLY WORKS WITH LIKE TERMS)

First the good news!!The rules governing algebra are exactly the same as those governingnumbers! After all, the letters merely represent numbers!

Like and unlike terms no longer a mystery!

Like terms are mates or look-alikes . They have the same letters (and powers if they are there). 5ab and 6ab are like terms. But 5a and 6ab arenot! Neither are 5a and 6a 2. 5 and 6 are like terms. 2ab and 3ba are liketerms. 3abc and -2bca are like terms. 5at and 3at and ta are all liketerms.

Remember that 3ab means 3 a band that 3ba means 3 b a, and these are the same!!

because 3 5 6 = 3 6 5 !!!

Heres a batch of 20 algebraic terms

5 x 3a -2b ab 2 6 7ab 9t -y4ba 8 5a 2 6a 2 5a3 6b2t a x 7b2a 2a 3 8

Can you put them into pairs of like terms? (like the x pair)

5 x 3a 2b ab 2 6 7ab 9t y 5a 2 5a 3

x a 6b 7b 2a 8 4ba 2t 8y 6a 2 2a 3



14/66

The key point is that only pairs of like terms can be added or subtracted!

Why is it so? Why can only like terms be added or subtracted?

Read on..

Do you agree that 20 + 15= 35?

YES!!!!

This can be rewritten as 4 5 + 3 5 = 7 5

Now the number that reoccurs HERE is a 5 but it can be anything.Maybe 8?

4 8 + 3 8 = 7 8 Is this true?

32 + 24 = 56

YES!!!!

The two boxed statements have 5 and 8 being repeated. There must be amore convenient way of writing this rule so it applies to all numbers, not just5 and 8. We can write

4 a + 3 a = 7 a

or

4a + 3a = 7a

a can be any number you like.



15/66

And..

We dont have to stick with the 4, 3 and 7 either!

Try making up other rules like

6a + 4a = 10a or 9b 5b = 4b.

Like terms can be added and subtracted

Now make a some number. Say 7. Substitute it in and see if it works.Does 6 7 + 4 7 = 10 7 ??

Now try replacing the b in the other box.(Lets make b something weird like 3 - just for fun!).

Does 9 3 5 3 = 4 3?? [Use your calculator!!]



16/66

Now..

So far weve only looked at like terms. 6a + 4a = 10a9b 5b = 4b etc.

What about 7a + 4b ?? Is there a shorter answer for this??

Remember a and b can be replaced with any numbers.

So lets makea equal to 3

b equal to 5

so 7a will be 21, and 4b will be 20

This means 7a + 4b = 21 + 20 = 41.

I cant think of anything that 41 is equal to (involving a sign)

So to cut a long story short, there is no easy way of writing 7a + 4b, otherthan 7a + 4b!

7a + 4b = 7a + 4b!! Unlike terms cant be added or subtracted

So in summary

Like terms can be added and subtractedUnlike terms cannot!!



17/66

The good news is you can use temperatures when adding and subtractingwith like terms, just like we were doing earlier with numbers!Some examples:

5a + 8a = 13a

2p 9p = 7p

8p + 11p = 3p or +3p

2 x 5y = 2 x 5y. It stays the same!! (because x and y are unlike)

2a2 + 5a = 2a 2 + 5a (because a 2 and a are unlike they cant be added)

5ab + 3ba = 8ab (remember ab and ba are like terms so can be added!)



18/66


19/66

Practice Exercises 3:

Which of these questions can be done? (The ones where the terms are like)Which ones cant? (Theyre the ones where the terms are different).

Remember x and y are unlike and cannot be added or subtracted.a and a 2 are unlike, and cannot be added or subtracted

ab and ba are like, and can be added or subtracted.

Youll need to pick out the like terms before you can add or subtract. Firstrevise your work on negative numbers earlier in the book. Click here for answers !

1 3a + 7a = 21 8a + 2ab =

2 2a 9a = 22 7rs 9sr =

3 a + 8a = 23 7rs 9sr 2 =

4 t 4t = 24 7rs 7r =

5 6ab + 7ab = 25 7rs 7s =

6 2ay 9ya = 26 7r 7s =

7 3a + 7b = 27 7r 7 =

8 2a 5ab = 28 2ab + 8ab =

9 6a + 8 = 29 2y 9y =

10 ab + 9ba = 30 6abc 7 abc =

11 3wx 2xw = 31 6abc + 3bca =

12 6a2 3a = 32 8a 2b 3c =

13 6a2 5a 2 = 33 4a + 5a 9b =

14 ab + ab = 34 2p + 5 + 6 =

15 ab ab = 35 3a 7 8 =

16 2xy 5yx = 36 a2 + 10a

2 =

17 2x 3x 2 = 37 5a3 2a 3 =

18 4ab 4 = 38 6a2 + 3a 3 =

19 4ab 4a = 39 6a4 + a 4 =

20 4ab 4a 2 = 40 a 2a 5 =



20/66

Remember good ol cut n paste??

Remember how were allowed to write 5 + 8 as +8 3 (or just 8 3) bycutting the +8 and pasting it in front of the 5?

Remember were also allowed to rewrite longer things like

6 + 3 2

in many different ways. Instead of 6 + 3 2, we could have written it as

6 2 + 3 or+3 6 2 or+3 2 6 or

2 + 3 6 or 2 6 + 3

All these are the same sum!! They all give an answer of 5 !!

Cut n paste is soooo handy! To do itcorrectly, all you have to remember isto keep each sign with the numberfollowing it!

E.g. in 5 + 8 when you cut n paste,the stays with the 5 and the + stayswith the 8. So 5 + 8 can become+8 5, or just 8 5, which is 3!!



21/66

(b) CUT N PASTE: ADDING AND SUBTRACTINGWITH ALGEBRA

Ever been put off by questions like:

Simplify 5a 2b + a 3b ???This causes headaches for many a Year 8 or 9 or 10 or?

Bowman. All rights reserved21

Then worry no more!!! we use cut n paste!!

Heres the process:

5a 2b + a 3b

= 5a 2b +a 3b First step : split it up with a space in front of each + and - sign

= 5a +a 2b 3b Second step : cut n paste the +a to put it next to its mate (5a)

= 6a 5b Final step : do the like terms: 5a + a = 6a and 2b 3b = 5b

Answer 6a 5b !!

Another one?? Yes!!

Simplify b 3a + 2b + a ab 6b + 5ba

The process:

b 3a + 2b + a ab 6b + 5ba Step 0: Groan!

= b 3a +2b +a ab 6b +5ba Step 1: split up with spaces

= b +2b 6b 3a +a ab +5ab Step 2: cut n paste to getlike terms ( b, a , ab ) together

= 5b 2a +4ab Step 3: simplify the like terms

Answer is 5b 2a + 4ab or 4ab 5b 2a or any other arrangement!

The Algebra Toolbox. 2003 R


22/66

Practice Exercises 4 :

1 3a + 5b + 4a + 3b = 11 7 7a + 3 + 10a =

2 7a + 2b 3a b = 12 w + 5t w 9t =

3 4 x + 2y 6 x + 5y = 13 yz + 3y 5zy y =

4 6a 2b + a 5b = 14 2a 2 + 3a 7 =

5 a + b 5a + 7b = 15 4ab + a 3b + 7b a 2ab

6 2a + 3 4a + 6 = 16 3a + 5a 2 7a a 2 =

7 a 5b 2a 9b = 17 3 + 3a 5a a2

=

8 3ab 2a + 8a + 5ba = 18 4ab 2a 2 ba + 5a 2 =

9 xy + 3x xy 2x = 19 3a + 3ab + 3b + 3ba =

10 4a + b 2ab b 4a = 20 k 2 5k 3 + 2k 3 + k 2 =




23/66

(c) MULTIPLYING with ALGEBRA

This is easier than addition & subtraction for two main reasons :

You dont have to worry about like terms.You can always multiply any terms together

First revise the 3 multiplication rules involving negative numbers:

1. Two positives make a positive . 7 8 = 562. Two negatives make a positive . 7 8 = 563. One of each makes a negative . 7 8 = 56 or 7 8 = 56

Now youre ready for action!!

First youll need some basic tools: Usually we writeanswers inal habetical order!a b = ab

b a = abc b a = abc

a a = aa or a 2

a a a = aaa or a 3

a a b = aab or a 2 b

a b b = abb or ab 2

Powers (indices) saverepeating letters!

Numbers go in front ofletters!Numbers are multipliedon their own, letters ontheir own!

a 2 = a2 (but 2a is better)a 2 b = 2ab

a 3 4 = 12aa 2 4 b = 12ab

a 3 a 5 b = 15 aab or 15a 2 bTHINK: Numbers are 3 -5 = -15 .

Letters are a a b = a 2b.Combine together to get 15a 2b



24/66

2a 5b = 10ab This is the same as if the sum read 2 a 5 b.

3a 5c = 15ac This is the same as 3 a 5 c

4a 2b a 5c = 40aabc = 40a 2 bc The quick way is to multiply all thenumbers together, and then all theletters: 4 2 5 = 40,and a b a c = a 2bc

Ready to try some?? Remember you can always do multiplication & donthave to worry about like terms as you do in addition & subtraction


1 x y z = 11 2a 5b =

2 2 a 3 = 12 3a - 7b 2 =

3 -3 b 5 = 13 4a 3 2b =

4 2 a 4 c = 14 a 2a b 4b =

5 3 a b 2 c = 15 -3a 2a 5 a =

6 4 2 a b = 16 -a 2a 3a -3 =

7 4 5 x x = 17 2ab 3ab 2a =

8 -2 -a b = 18 4ab 3abc -2c =

9 -3 a -4 a = 19 abc abc abc =

10 2 a 3 b a = 20 -3ab -2abc -5ac =

Click here for answers


Phew! Multiplication is so mucheasier because you can always doit.

Not like adding and subtractingwhere you have to make sure youhave like terms before you can do it!


25/66

(d) ORDER OF OPERATIONS

Multiplication must be done before you do addition or subtraction

Now well apply this to algebra!

Example 1:

Simplify 8ab + 2a 3b

Process: 8ab + 2a 3b= 8ab + 6ab Remember we do the first. 2a 3b = 6ab

Think: Can I add the 8ab and 6ab? YES!!

= 14ab Can add 8ab and 6ab as theyre like terms!

Example 2:

Simplify 4a 5ab 3 2ab 2a

Process: (4a 5ab) (3 2ab 2a)= 20a 2 b 12a 2 b Doing the multiplications first

Think: Can I subtract these? YES!!(Because theyre both a 2b and so liketerms!)

= 8a 2 b

Example 3:

Simplify 10ab 2 5a 3b

Process: 10ab 2 5a 3b= 10ab 2 15ab Doing the multiplication first

Think: Can I subtract these?NO!! (ab 2 and ab are not like terms )



26/66

(e) POWER NOTATION FOR MULTIPLICATION

Often you come across multiplications where the same term isrepeated. There is a shorthand way of writing these.

We already know that

a a = a 2 and

b b b = b 3 and so on.

This can be extended to bigger things being repeated:

Examples

ab ab can be written as (ab) 2.The brackets simply mean that everything inside them is repeated.

yz yz yz can be written in shorthand as ..??Did you get (yz) 3 ? Youd be right!!!Of course it can also be written as y 3z3.

What do you think (2a) 3 would be if we wrote it the long way?(2a) 3 = 2a 2a 2a which can then become 8a 3.

What about ( 6ab) 2?(6ab) 2 = 6ab 6ab = +36a 2 b2

Can you rewrite ( 3ab) 4 in longer notation? 3ab 3ab 3ab 3ab. This becomes +81a 4 b4



27/66

More examplesWrite these in expanded notation and then simplify.

(ab) 3 = ab ab ab (expanded notation)

= a3

b3

(simplifying)

(3ab) 2 = 3ab 3ab= 9a 2 b2

(2t) 3 = 2t 2t 2t= 8t 3

(ab) 2 (-3a) 3 = ab ab 3a 3a 3a= 27a 5 b2

Some exercises. Good luck!!Write in expanded (longer) form and then simplify.


1 (5ab) 2 = 6 (3a) 2 (2b) 2 =

2 (3ab) 3 = 7 (-3a) 2 (5a) 2 =

3 (-4a) 2 = 8 (-2ab) 4 =

4 (-2ab) 3 = 9 (5abc) 3 =

5 (-5a) 2 + (4a) 2 = 10 (ab) 5 =




28/66

(f) MULTIPLYING WITH POWERS

Youve probably come across this and it may have caused confusion!Here we meet some new rules, which are just really shortcuts .

Examples

Simplify a 2 a3. There are two approaches to this:

Long way Shorter waya2 a3 = a a a a a

= a5

a2 a3 = a 2 + 3 (adding powers)

= a5

In this first example, it doesnt look like theres much difference. The shorter wayis not that much shorter at all! But read on..

Simplify a 5 a6.Long way Shorter way

a5 a6 = a a a a a a a a a a a = a 11

a5 a6 = a 5 + 6 (adding powers)= a 11

As you can see, once the powers become bigger, the shorter way becomes

much shorter!!

When multiplyingpowers of the sameletter, just add thepowers !! a

u a v = a u+v

Must be the SAME LETTER forthis to work (Here its all as)



29/66

What if there are numbers in front? No problem!

2a3 5a 4 = 10a 7 Using the shorter way (of course!!) You could have written it outthe long way, 2 a a a 5 a a a a and stillobtained the same answer!

3w 7 5w 9 = 15w 16

3c 6 2c 5 = 6 c 11

But what if the letters are different?

2a5 3b 2 means 2 a a a a a 3 b bWorking this out gives 6aaaaabb, or just 6a 5 b2

4a2 3w 6 = 12a 2w6

5a 7 2j3 = 10a 7 j3

Now lets test your memory: What about 3a 5 2b 5 ?This is just 6a 5 b5, but can you remember another way of writing it?

Its 6(ab) 5. Remember this is an alternative you can use if the powers are the same!



30/66

3. WORKING WITH BRACKETS

(a) EXPANDING (Getting rid of brackets) 1 pair ofbrackets.

What do you think is meant by8 (5 + 3) ?

This can be done two ways:

The obvious way A longer (but more important!) way8(5 + 3)= 8 (5 + 3)= 8 8= 64

8(5 + 3)= 8 (5 + 3)= 8 5 + 8 3= 40 + 24= 64

Vital step! The 8multiplies the numbersinside the brackets oneat a time!

Now try this with an x instead of the 5, using our two ways above :

The obvious way The longer way

8( x + 3)= 8 ( x + 3)

and we cant go any further becausewe cant add the x and the 3Remember theyre unlike terms andwere not allowed to add unlike

terms!) The obvious way has itslimitations because it will only workwith numbers.

8( x + 3)= 8 ( x + 3)= 8 x + 8 3= 8 x + 24

This way is important! When weredealing with algebra, we can use thismethod to get further than we couldusing the method on the left



31/66

Some more examples.

Example 14( x 5)

= 4

x 4

5Note the 4 repeats

= 4 x 20

Example 23(a + 2b + 7)= 3 a + 3 2b + 3 7 Note the 3 repeats= 3a + 6b + 21

Example 32y(5y 4)= 10yy 8y Note the 2y repeats= 10y 2 8y

but what if theres a negative number in front of the brackets???

3 (4 + 5)

The obvious way The longer way 3 (4 + 5)

= 3 (4 + 5)= 3 9= 27

3 (4 + 5)

= 3 4 + 3 5= 12 + 15

= 12 15= 27

Watch these steps closely!The 3 multiplies boththe 4 and the 5. The +carries down

Remember when wehave a + together,they just make a



32/66

What if theres algebra?

Example 4

Expand 5 (2a 3b) The 5 multiplies the 2aand then multiplies the 3b.

The minus in the brackets iscarried down to the next line(the arrow).

= 5 (2a 3b)

= 5 2a 5 3b= 10a + 15bOR 15b 10a (cutting n pasting the 15b to the front)

Example 5

Expand 6 (2a + 3b)

6 (2a + 3b) Can you see whats happened?

The 6 appears twice (Line 2)because it has to multiply both the2a and the 3b

The + is carried down from Line 2to Line 3.

The + in Line 3 becomes a in line 4.

= 6 2a + 6 3b (Line 2)

= 12a + 18b (Line 3)

= 12a 18b (Line 4)



33/66

Some for you to try! Aim for 10/10 and take your time!!


Expand

1 3(2a 5) 6 4 (a b)

2 4y(3a 2p) 7 3(2a + 1)

3 3a(b 1) 8 x( x + 5)

4 2a(3a 7) 9 2ab(a 3b)

5 5ab(2a + 3b) 10 4p(y p + 5t)


Remember: A negative number multiplying in front ofbrackets will change every sign inside those brackets. E.g.

a(x + 2y z + 4)

becomes

ax 2y + az 4a



34/66


35/66

Example 4

Expand and simplify 3 5p(p + 1) 4p + 3(2p 1)

Here there are two lone terms, unattached to any brackets:the 3 at the front, andthe 4p

Lone terms are unaffected by the expanding process. They stay as they are.

Answer:

3 5p(p + 1) 4p + 3(2p 1)

Note the + becomes a . WHY??(because the on the 5p makes the signinside the brackets change!)

Important point!!!

By now you might be noticing that a negative number in front ofbrackets changes any signs inside the brackets to the opposite!

3 (a + b) = 3a 3b (+ becomes )

5 (x y) = 5x + 5y ( becomes +)

4 (a + b c) = 4a 4b + 4c(+ becomes and becomes +)You can always use this shortcut to do them quickly!

= 3 5p 2 5p 4p + 6p 3 = 3 3 5p 2 5p 4p + 6p Cut n pasting like terms

together (Theres 3 types of liketerms in this question!)= 0 5 p

2 3p= 5p 2 3p



36/66

(c) FACTORISING (Making brackets) 1 pair ofbrackets

Factorising is the reverse of expanding

2(a + b) = 2a + 2b This is expanding2a + 2b = 2(a + b) This is factorising

2(a + b) 2a + 2bExpanding

Factorising

Example 1

Factorise ab + ac .

Find what you can see in both terms . The a !!Place the a in front of brackets like this:

a ( )

Put the sign in

a( + )

Now insert the b and c in the right places to make it equal ab + ac

a ( b + c ). ANSWER!!

Now check by mentally expanding a(b+c). Does a(b + c) give us ab + ac ?It does, so we must be right!



37/66

Example 2

Factorise 2abx abcd

ab is common to both, so write

ab ( )

Now, what has to go with the ab to make the first term 2abx? 2x

THINK: If ab x ? = 2ab x then ? must be 2 x

If ab x ? = abcd. then ? must be cd

And, what has to go with the ab to make the second term abcd? cd

These red terms are now put into the bracket:

ab( 2x cd) . Answer!

Example 3

Factorise 4ab 12abc

What is in common to both terms? a, b and 4 (because 4 divides 4 and 12)So put 4ab in front of the brackets

4ab ( )

What has to multiply 4ab to make 4ab? 1What has to multiply 4ab to make 12abc? 3c

These red terms are now put into the bracket:

4ab (1 3c). Answer!

4 is a better choice than 2, eventhough both divide 4 and 12. If youhave a choice, always pick thebiggest number!



38/66


39/66

(d) EXPANDING 2 pairs of brackets together

These are sometimes called binomial products.

It works like this:

Suppose you were asked to work out

(3 + 4) (5 + 6)

The quick n easy way (which, sadly, works when there are numbers only) is to do

(3 + 4) (5 + 6)= 7 11= 77.

The longer way , which you need to know , is to work it out like this:

Learn this step! It will serve you well.The first bracket is broken up and the 2 nd bracket is repeated.

(3 + 4) (5 + 6)= 3 (5 + 6) + 4 (5 + 6)= 15 + 18 + 20 + 24= 77

Now lets try this with letters..

Example 1

Expand (a + b)(c + d)

(a + b)(c + d)= a (c + d) + b (c + d) See how the a+b has been split up and c+d written twice? = ac + ad + bc + bd. (answer)

This is where we leave it, because all 4 of these terms are unlike!



40/66

Example 2

Expand (a + 3)(4 a)

(a + 3) (4 a)= a(4 a ) + 3 ( 4 a ) Splitting the first brackets and repeating the 2 nd brackets (4 a)= 4a a 2 + 12 3a = 4a 3a a 2 +12 (Cut n paste the 3a putting it next to 4a) = a a 2 + 12

Example 3

Expand (2a 5)(3a + 1)

(2a 5)(3a + 1)= 2a(3a + 1) 5 (3a + 1)= 6a 2 + 2a 15a 5 Dont forget the sign change because of the 5 = 6a 2 13a 5


Expand and simplify these:

1. (a + 2)(b + 3) 7. (2y 3)(2y + 5)2. (a + 5)(a + 6) 8. (y 6)(y + 6)

3. (a 3)(b 1) 9. (3a + 4)(2a 5)

4. (a + 4)(2a + 5) 10. (a + 5) 2

5. (3a + 4)(2a + 5) 11. (2a + 7) 2

6. (a + 6)(a 6) 12. (3a 4) 2




41/66

4. SOLVING EQUATIONS

(a) INTRODUCTION

An equation is either

a statement of fact , for example6 + 3 = 96 3 = 189 = 11 2

or..

a question where you have to find the value of a letter, for example

a + 7 = 103a = 9a/4 = 53a 2 = 7 a

This book deals with the second type, where we have to find values ofletters. But first, we need to understand some background which willhelp you understand the rules and processes involved.

Background

The rules of equations are easiest to understand if we begin by dealing onlywith numbers. We will then learn how to work with letters.

RULE 1 An equation remains true if the same number is added to both sides (of

the equals sign) , or subtracted from both sides.

Example 1Begin with 6 + 3 = 9 (True?)



42/66

Add 4 to both sides: 6 + 3 + 4 = 9 + 4 (Still true. 13 = 13!)

Example 2Begin with 6 + 3 = 9 (True?) Subtract 2 from both sides 6 + 3 2 = 9 2 (True. 7 = 7 !)

Example 3Begin with 8 5 = 3Add 5 to both sides 8 5 + 5 = 3 + 5 (True. 8 = 8)

Now Subtract 12 from both sides:8 5 + 5 12 = 3 + 5 12 (True. 4 = 4)

Now add 7 to both sides:8 5 + 5 12 + 7 = 3 + 5 12 + 7 (True?)

Etc etc.. we can add/subtract whatever we like as long as we do thesame thing to both sides in the same step.

Q: What have we learned here?A: That if we begin with a true statement, then as long as we add or

subtract the same number to both sides, it will remain a truestatement.



43/66

(b) CUT N PASTE TO REWRITE EQUATIONS

Now.. a shortcut alternative to this rule is cut n paste. Follow thisexample:

Begin with 10 4 = 6Lets add 4 to both sides. What happens?

IMPORTANT!Try to followthis!

10 4 + 4 = 6 + 410 = 10. TRUE!

Instead, try doing it this way.

Again, begin with10 4 = 6.

Now, cut the 4 from the left side and paste itonto the right side as + 4. What happens?10 4 = 6

10 = 6 + 4 10 = 10. TRUE!

Another example:

9 + 5 = 14Cut n paste the + 5 from the left to the right, remembering to changethe + sign to a sign.

9 = 14 5 . True!

Example 3

8 + 7 = 15Cut n paste the 8 (which really is a +8) from the left to the right.Remember to change the +8 into a 8.

+8 + 7 = 15



44/66


45/66

(c) SOLVING EQUATIONS using CUT N PASTE

Example 1.

Solve x + 3 = 8

Aim: to get x alone on one side of the = sign, so our answerreads x =

Best method:

x + 3 = 8.

To get x alone, we need to remove the +3. Most sensible way is to cut

n paste it from the left to the right:

x = 8 3

Tidy up the right. 8 3 is equal to 5.

x = 5 ANSWER!

Example 2.

Solve 9 x = 7

Aim: To get x alone on one side, so our answer reads x =

Best method:9 x = 7

To get x alone, we need to remove the 9, remembering its really +9.Most sensible way is to cut n paste it from the left to the right: (blue)

x = 7 9 x = 2

Theres an unwanted negative sign in front of the x. We have to get rid ofit!The quickest way is to switch all signs in front of every term. (same asmultiplying everything by 1)



46/66


47/66

Example 3

Solve 2x + 5 = x 9

Aim: to get x alone on one side of the = sign.

Handy hint: When there is more than one x & they appear on different sides,cut n paste the xs to the side where most of them are to begin with . In thiscase, aim to move the xs to the left because 2x (which is on the left) is biggerthan 1x (which is on the right).

Best method

2x + 5 = +x 9We decide the xs will end up on the left side, so begin by

moving the +x on the right over to the left. (blue)

2x x + 5 = 9

A golden rule of equation solving is to tidy up terms wheneveryou can. So, tidy up the two x terms on the left. 2x x = x

x + 5 = 9

Now get rid of the +5 by cut n pasting it to the right. Rememberit must become 5 (red)

x = 9 5

Remember from earlier work on negatives that 9 5 = 14(Temperatures?)

x = 14 ANSWER!



48/66

Example 4

Solve 3a 8 = 4a 11

Aim is to get answer a = ..So we begin by getting all as to one side, and all non-as to the other.

Best method

Seeing that 4 is bigger than 3, move the as to the right . Automatically the non-as (the 8 and 11) will go to the left.

+3a 8 = 4a 11

8 = 4a 3a 11 3a moves to right and becomes 3a. Put itnext to its mate, the 4a

8 = a 11 Tidying up. 4a 3a = a.

+ 11 8 = a Removing the 11 from right to left.

3 = a Tidying up. 11 8 = 3

a = 3 ANSWER! Writing it backwards. 3 = a means exactly thesame as a = 3



49/66

Example 5

This is important!Solve 5 x = x 7

Best method:

Noticing there are two xs here (one on each side), we have to cut n paste oneof them so it moves over with its mate! Both xs need to be on the same side!Which side has the bigger x term? +1x (on the right) is bigger than 1x (whichis on the left), so aim to move the xs to the right. This means the non-xs (the 5and 7) will move to the left.

5 x = x 7Cut n paste the -x from the left to the right: Put it next to its mate, the other x

5 = x + x 7

Tidy up the right (remember your earlier skills??) x + x 7 = 2x 7

5 = 2x 7

Now cut n paste the 7 from right to left (Green)

5 + 7 = 2x

Tidy up the left. 5 + 7 = 12

12 = 2x

Remembering 2x really means 2 multiplied by x, this gives us

x = 6. ANSWER!

More on this in thenext section.

It would be illegal totry and cut n pastethe 2 as it isconnected to the xby a multiply sign.Cut n paste canonly be done onterms which areconnected to each

other by + or signs!



50/66

Example 6

Solve 3(x 5) = 2(2x + 3)

Best method:

Here we have brackets but we know how to get rid of them!! Thats what wehave to do. Get rid of the brackets before you do anything!

Remember from before that 3(x 5) really means 3 times x minus 3 times 5 ?So 3(x 5) = 3x 15.and 2(2x + 3) = 4x + 6 similarly.

So..3(x 5) = 2(2x + 3) becomes

3x 15 = 4x + 6Now we proceed as we did in previous examples. Cutn paste the xs to the sidewhere the bigger lot of xs are..THE RIGHT!! (4x is of course bigger than 3x).This means the numbers without xs (the 6 and 15) will end up on the LEFT.

15 = 4x 3x + 6 Cutting the 3x from the left and pasting it to the right (weveinserted it just after the 4x because these are like terms andcan be simplified)

Now get rid of the 6 so its with its mate, 15, on the left. Tidy the 4x 3x at thesame time. 4x 3x = x.

15 6 = x 21 = x

x = 21 ANSWER!!



51/66


Solve each of these equations.

Try to use a method each time, even though it might be easier just to guessthe answer!

1 x + 1 = 5 11 7 y = 8

2 a 3 = 6 12 1 p = 2

3 b + 7 = 3 13 3 a = 3

4 y + 2 = 6 14 2x + 1 = x + 6

5 4 + a = 9 15 5x 2 = 4x + 3

6 7 + a = 3 16 3x 1 = 2 + 4x

7 4 = b + 1 17 x 7 = 3 x*careful! see Example 5

8 12 = x 8 18 2(x 3) = x + 1

9 7 = 3 x 19 3x 2 = 2(2x 5)

10 9 = 5 x 20 2(x + 3) = 3(4 + x)




52/66

(d) MULTIPLY, DIVIDE to REWRITE EQUATIONS

RULE 2

An equation remains true if the same number multiplies both sides, ordivides both sides.

Example 1Begin with 12 8 = 4 (True?)

Divide both sides by 2 6 4 = 2 (Still true!)

Example 2Begin with 3 + 5 = 6 + 2 (True?)

Multiply both sides by 7 21 + 35 = 42 + 14 (True!)

Now divide both sides by 2 10.5 + 17.5 = 21 + 7 (Still true!)

Example 3Begin with 10 3 = 5 + 2 (True?)

Pick any number you like and multiply right through by it!(Ill pick 4!)

Multiply each term by 4: 40 12 = 20 + 8 (True! 28 = 28)

Now, try multiplying by 5 (more care needed here!)

Starting again with 10 3 = 5 + 2

Well multiply each of the four numbers by 5. Watch what happens.

Left side-5 10 = -50-5 -3 = +15

Right side-5 5 = -25-5 +2 = -10

10 3 = 5 + 2

50 +15 = 25 10



53/66

35 = 35 STILL TRUE!!!!

Example 4

In primary school you learned (hopefully!) that

5 53

= 3 4 41

= 172

7 = 2

etc. When a fraction is multiplied by itsdenominator (bottom) the denominatorvanishes and youre left with the top! Here

420

4 = 20. The quick way is to just delete

the 4s and write down the top (20).

5 = 420

True?

Now multiply both sides by 4:

5 4 =4

20 4

20 = 20. Still True!

Example 5

315 = 5


315 3 = 5 3

Could you see quickly that3

15 3 = 15?

This is just our rule from the last example.

15 = 15 True !



54/66

Example 6

25

10=


510 15 = 2 15

What happens on the left side is important to understand!

510

15 can be worked out simply by cancelling the 5

and 15 as follows:

510

15 =10

15

= 10 3= 30

3

5 1

30 = 30 True!

Q: What have we learned here?A: That if we begin with a true statement, then as long as we multiply

or divide both sides by the same number, it will remain a truestatement. Examples 4 and 5 especially show you how easy it is toget rid of a fraction (a skill youll need with algebra!)



55/66

(e) SOLVING EQUATIONS using MULTIPLY, DIVIDE

Now lets move into algebra.

Example 1

Solve 3 x = 21

Remember our aim is to get the answer looking like x =

As the question really means 3 x = 21, to remove the 3 we cant just cut n paste as wewould do if it were 3 + x = 21, but rather we have to divide by 3, so the 3 and themultiplication are cancelled. So.

3 x = 21

Dividing both sides by 3

321

33

= x

321

33

= x

Now canceling the 3s on the left side, and simplifying321

on the right side,

x = 7 . ANSWER!

Example 2

Solve 5 x = 32

Again recognizing that this really means 5 x = 32, to remove the 5 x bit, wehave to divide both sides by 5:

532

55 = x

Simplifying,

x =52

65

32= .. ANSWER!!



56/66


57/66


58/66


59/66

Example 7

Solve53

x 72=

Here we have two fractions ! Arrrrgh! Lets get rid of them first.

We can get rid of both of them in the one step if we multiply both sides by 15 .Where did we get 15 from? 15 is the smallest number that both 3 and 5 divideinto! (In mathematical jargon, 15 is the lowest common multiple (LCM) orlowest common denominator (LCD) of 3 and 5).

32 x 15 =

57 15

Now do you remember what happens in situations like this? See Demo and takecareful note of whats written in the box!

On the left side, the 3 and 15 cancel to leave a 5On the right side, the 5 and 15 cancel to leave a 3

32 x 15 =

57 15

35

2 x 5 = 7 3

10 x = 21

Remember what to do now? (See earlier examples )Divide both sides by 10 !!

1010 x

=1021

Cancelling the 10s on the left, we get

x =1021

or 2101

or 2.1 . ANSWER!!



60/66

(f) MORE ADVANCED EQUATIONS WITH ALLFOUR OPERATIONS

Example 8:

Solve 3 x 5 = 22 6 x

There arent any fractions to begin with. What a relief!First we need to move the xs to the side where most of them are.

As 3 is bigger than 6, and 3x is on the left, we move xs to the left.This means non xs (the 22 and 5) will move to the right.

So cut n paste the 6x to the left, and the 5 to the right.Note that the 6x becomes +6x and the 5 becomes +5. (You should be familiarwith this by now!!)

3 x + 6 x = 22 + 5Tidy up both sides

9 x = 27

Divide both sides by 9

x = 3 ANSWER!!

Example 9

Solve 5 x 3( x 4) = 7 x + 17

No fractions to get rid of, so get rid of the brackets by expanding. Rememberwhat to do? What does the 3 in front do to the sign inside the brackets?

5 x 3x + 12 = 7 x + 17Did you remember to change the sign to a + ? because 3 4 = +12

Now tidy up

2 x + 12 = 7 x + 17

Now its time for cut n paste, as we did in Example 8 above. xs go to the right.Why? (Because 7 is bigger than 2)

12 17 = 7 x 2 x 5 = 5 x

Divide both sides by 5

1 = x , or x = -1 (ANSWER!!)



61/66


62/66

Example 11 :

Solve 34

)2(35

)3(2=

x x

This looks nasty, because its got the lot! Fractions, brackets, you name it!

Go back to the box on the previous page, and well follow through the steps.

Step 1 . Get rid of fractions. We multiply by 20. Remember to multiply everyterm.

5)3(2 x 20

4)2(3 x 20 = 3 20

Now cancel the 5 and 20, then the 4 and 20.

5)3(2 x 20

4)2(3 x 20 = 3 20

1

45

1

Clean up.

2( x 3) 4 3(2 x) 5 = 60

Now clean up further by multiplying the 2 by 4, and the 3 by 5:

8( x 3) 15 (2 x) = 60

Remove brackets by expandingStep 2 .

8x 24 30 + 15 x = 60

Step 3 . Tidy up on each side, collecting like terms

23 x 54 = 60

Move xs to one side, non-x-terms to the other (this is cut n paste)Step 4 .

23 x = 60 + 54Tidy up Step 5 .

23 x = 114

Divide both sides by 23Step 6 .

x =23

114 = 4.96 ANSWER!



63/66

Example 12:

Solve2

1231

21

512 +

+=

+ x x x

**An important step to begin with:wherever you have numerators (tops of fractions) with morethan 1 term as we have in the first, second and fourth terms),its a good idea to put brackets in. This helps when we get rid offractions and tidy up later on .

2)12(

31

2)1(

5)12( +

+=

+ x x x

Step 1 Get rid of fractions. What number do you think we should multiply by?If you think 30, then youd be correct! 60 would also be OK, but the smaller the better!

Remember to multiply all 4 fractions by 30:

5)12( + x 30

2)1( x

30 = 31

30 + 2

)12( + x 30

Cancel & clean up

6(2x + 1) 15 (x 1 ) = 10 + 15(2x + 1)Can you see why we put the brackets in at the start?

Step 2 Get rid of brackets

12x + 6 15x + 15 = 10 + 30x + 15

Step 3 Tidy Up

3x + 21 = 30x + 25

Step 4 Cut n paste (xs to right side as 30 is bigger than 3)

21 25 = 30x + 3x

Step 5 Tidy Up

4 = 33x

Step 6 Divide both sides by 33

334

= x ANSWER!!!!



64/66

Practice Exercises 11

Solve the following equations

1 3a 5 = a + 7 11 123

=+ x x

x

2 4 3t = 2t 6 1231

43

= x

x

3 5(a 3) = 2(4 a) 1332

452

= x x

4 2 x 7 = 3 (4 x) 14 x x x 53

132

1=

+

+

5 5a 4 = 7 2(3a 5) 15 24

5

3

2=

x x

6 3p + 5 = p + 3(p 1) 1636

12432 x x x

=

7 4(a + 3) = 3a 2(5 a)8 5

732

= x

9 735

=+ x x

1046

25 x x=




65/66

ANSWERS Ex 1 :

(1) 28 (2) 10 (3) 12 (4) 4 (5) 48 (6) 8 (7) 63 (8) 7 (9) 60 (10) 7 (11) 4 (12) 81 (13) 5(14) 24 (15) -11 (16) 60 (17) 5 (18) 35 (19) 2 (20) 8

ANSWERS Ex 2 :

(1) 11 (2) 18 (3) 7 (4) 25 (5) 14 (6) 4 (7) 9 (8) 10 (9) 5 (10) 5(11) 12 (12) 18 (13) 13 (14) 11 (15) 6 (16) 12 (17) 10 (18) -1 (19) 10 (20) 18

ANSWERS Ex 3:

(1) 10a (2) 7a (3) 9a (4) -3t (5) 13ab (6) 7ay or 7ya (7) same (8) same (9) same (10)10ab or 10ba (11) 1wx or wx or 1xw or xw (12) same (13) a 2 (14) 2ab (15) 0 (16) 3xy or 3yx (17) same (18) same (19) same (20) same (21) same (22) -2rs or 2sr (23) same (24)same (25) same (26) same (27) same (28) 6ab (29) 7y (30) 13abc (31) -3abc (32) same(33) 9a- 9b (34) 2p+ 11 (35) 3a 15 (36) 11a 2 (37) 3a 3 (38) same (39) 7a 4 (40) same

ANSWERS Ex 4 (Remember if there are 2 terms or more in the answer then there is morethan one way of writing the answer). In the first 5 answers below, both possibilities areshown.

(1) 7a + 8b or 8b + 7a (2) 4a + b or b + 4a (3) -2x + 7y or 7y 2x (4) 7a 7b or 7b + 7a(5) 4a + 8b or 8b 4a (6) 9 2a (7) a 14b (8) 8ab + 6a (9) x (10) -2ab (11) 10 + 3a(12) -2w 4t (13) 2y 4yz (14) 5a 9 (15) 2ab + 4b (16) 4a 2 4a (17) 3 2a a 2 (18) 3ab + 3a 2 (19) 3a + 6ab + 3b (20) 2k 2 3k 3 Note: In Q9, you might get 0xy. Remember this is just 0! This also happens in Q10 and 15.

ANSWERS Ex 5: (1) xyz (2) 6a (3) 15b (4) 8ac (5) 6abc (6) 8ab (7) 20x 2 (8) 2ab (9)12a 2 (10) 6a 2b (11) 10ab (12) 42ab (13) 24ab (14) 8a 2b2 (15) 30a 3 (16) 18a 3 (17)12a 3b2 (18) -24a 2b2c 2 (19) a 3b 3c3 (20) 30a 3b2c2

ANSWERS Ex 6:(1) 5ab 5ab = 25a 2b2 (2) 3ab 3ab 3ab = 27a 3b3 (3) 4a 4a = 16a 2 (4) 2ab 2ab 2ab = 8a 3b3 (5) 25a 2 + 16a 2 = 41a 2 (6) 3a 3a 2b 2b = 36a 2b2 (7) 3a 3a 5a 5a = 225a 4 (8) 2ab 2ab 2ab 2ab = 16a 4b4 (9) 5abc 5abc 5abc = 125a 3b3c3 (10) ab ab ab ab ab = a 5b5

ANSWERS Ex 7:

(1) 6a 15 (2) 12ay 8py (3) 3ab 3a (4) 6a 2 14a (5) 10a 2b + 15ab 2 (6) 4a + 4b or 4b 4a (7) 6a 3 or 3 6a (8) x 2 5x (9) 2a 2b + 6ab 2 (10) 4py + 4p 2 20pt

ANSWERS Ex 8: (1) a(c y) (2) b(t + a) (3) a(b + c 2) (4) 3(b + 2c) (5) 3a(b 2t)(6) 5ab(1 2c) (7) 12abc(1 2d) (8) 3a(a 3) (9) 6a(3a 2bc) (10) 10abc(3ab 2c)

ANSWERS Ex 9: (1) ab + 3a + 2b + 6 (2) a 2 + 11a + 30 (3) ab a 3b + 1 (4) 2a 2 + 13a +20 (5) 6a 2 + 23a + 20 (6) a 2 36 (How come this has only 2 terms? How is the questiondifferent?) (7) 4y 2 + 4y 15 (8) y 2- 36 (another one like Q6) (9) 6a 2 7a 20(10) a 2 + 10a + 25 (11) 4a 2 + 28a + 49 (12) 9a 2 24a + 16



66/66

ANSWERS Ex 10 : (1) 4 (2) 9 (3) 4 (4) 4 (5) 5 (6) 4 (7) 3 (8) 20 (9) 4 (10) 14 (11) 1 (12) 1 (13) 0 (14) 5 (15) 5 (16) 3 (17) 5 (18) 7 (19) 8 (20) 6

ANSWERS EX 11: (1) a = 6 (2) t = 2 (3) a = 23/7 (4) x = 5 (5) a = 21/11 (6) p = 8(7) a = - 2/9 (8) x = 19 (9) x = 105/8 (10) x= 10/7 (11) x = 1 (12) x = 4/3(13) x = 40/9 (14) x = 1/21 (15) x = 39/11 (16) x = 8/17

Documents

Algebra Toolbox Part 1