Mr Barton’s Maths Notes

Algebra

1. Rules of Algebra

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1. The Rules of Algebra

Let’s just get one thing clear before we start…Algebra really isn’t anything to be afraid of, I promiseIf anything, dealing with letters is a lot easier than just dealing with numbers.Why?... because, as you will see, letters are always cancelling each other out, meaning the questions get easier and easier the more you get into them,And… quite often you can know for sure if your answer is correct, or not!

So, take a deep breath, think positive thoughts, and let’s give this Algebra thing a go… What is Algebra and Why do we need it?• On a simple level, Algebra is just maths with letters… but it is a lot more than that!

• By bringing in letters as well as numbers we can work out things that numbers alone would not allow us to.

• In Algebra, letters are called “Unknowns”. Basically, we stick a letter in to stand for something when we don’t know it’s true value.

• Now, this could be anything from the price of Nintendo Wii, the number of hours you spend watching TV in a week, or the speed you walk to school in the morning. • If we don’t know what it is, call it a letter – any letter you like – and let’s let algebra figure everything out for us.

And the whole of Algebra – right up to A Level and beyond – is built around 3 rules…

The Lingo You Need:

Term – this is basically any part of an expression or equation that involves a letter

e.g. 4m -2r and p are all terms

Expression – this is kind of like a collection of terms, maybe with a few numbers chucked in e.g. 4m + 2r and 8z – 5p + 6q2 – 7 are all expressions

Equation – this is just the same as an expression, but with an equals sign

e.g. 4m + 2r = 7 and 8z – 5p + 6q2 – 7 = a

Rule 1: You can add or subtract LIKE TERMS but you cannot add or subtract DIFFERENT TERMS.

Okay, so by a LIKE TERM I mean a term that contains the exact same letter (or letters) as another term

e.g. + 2= m m m 3 + 2 5 = p p p 2 2216 - 4 2 = 1t t t 10 - 7 = 3pq pq pq

BUT…

3 lots of something, plus 2 lots of something, gives you 5 lots of something

16 lots of something, minus 4 lots of something, gives you 12 lots of something

+ m p Does Not = mp 3 + 2r t Does Not = 5rt

Because the terms are different!

Simplifying Expressions

Now, once you have got to grips with Rule 1, it allows you to simplify nasty looking expressions into nice simple ones… which is called, believe it or not… simplifying.To Simplify and Expression: Draw boxes around all the LIKE TERMS and deal with each set of like terms on their own.

Okay, let’s draw boxes around all the LIKE TERMS

Remember: Draw around the sign in front on the term as well!

Simplify:

Example 1

4 + 2 6m p m p

4 + 2 6m p m p

So, let’s see what we’ve got:

4 = 3m m m

2 + 6 8 = p p p

Which gives us our answer of:3 + 8m pNote: if you cannot see a sign in front of a

term, then just assume it is a PLUS

Okay, let’s draw boxes around all the LIKE TERMS

Remember: t and t2 are DIFFERENT!

Simplify:

Example 2 – Tricky!2 24 5 2 3t t t t

So, let’s see what we’ve got:

2 2 24 3 = t t t

5 2 = 7 tt t

Which gives us our answer of: 2 7tt

2 24 5 2 3t t t t

Note: write this instead of 21 t

Note: see how important it is you remember how to work with NEGATIVE NUMBERS!

Rule 2: When Multiplying with Algebra, we need to remember the following things:

1. We CAN multiply different terms and like terms together

2. Always multiply the numbers together first

3. Put the letters in alphabetical order

4. Leave out the Multiplication Sign

1. Okay, each of the three terms is different, but we are multiplying, so it’s not a problem!

Simplify:

Example 1

5 2 3b c a

2. Let’s multiply the numbers together first:

5 2 303 =

3. Now let’s deal with the letters, remembering to write them in alphabetical order and leave out the multiplication sign = = b c a bca abc

4. Putting them together, and again leaving out the multiplication sign, gives us our answer:

30abc

1. Again, no problem with the different terms

Simplify:

Example 2

4 3 3 r p r q

2. Let’s multiply the numbers together first, being very careful with our negatives!:

4 3 3 1 = 3 6

3. Now let’s deal with the letters:

2 = = r p r q pqrr pqr

4. Which together gives us:236pqr

Note: there was no number in front of the q, which means it is just a 1!

Remember: if you multiply something by itself, it just means you are squaring it!

Rule 3: When Dividing with Algebra, the rules are just the same as when multiplying, but instead of a division sign like this ÷ we tend to write divisions as fractions!

Crucial: When dividing, watch for things cancelling out and disappearing!

1. Okay, just like when multiplying, different terms are no problem!

Simplify:

Example 12

3

5

35

a b

ab

2. Let’s divide the numbers first:

20 4 5 = 3. Now let’s deal with the letters:

= xyz z xy

4. So, our answer is:5xy

What happened there? well, when you divide the z on the top by the z on the bottom you are left with 1 (just like if you divide anything by itself you get 1). But multiplying or dividing by 1 does not make a difference to our answer, so we can say that the z cancelled out!

Example 2 – Nightmare!

Simplify:20

4

xyz

z1. Different terms, no problem.

2. Dividing the numbers first:

55 35 = =

35 7

1

Note: when you don’t get a nice answer like in Example 1, you need to use Fractions!

3. Now let’s deal with the letters (this requires a bit of knowledge about INDICES):

the a on the bottom wipes out one a on top, but still leaves an a behind on the top

the b3 on the bottom wipes out the b on the top, and still leaves a b2 behind on the bottom.

4. So, our answer is: 27

a

b

Forming your own Expressions

Now, once you have got to grips with Rules 1 - 3, you should be able to have a go at forming your very own algebraic expressions.

Have a look at the following diagram and see if you can figure out where I have got the expressions below from. I am sure you could make up some much better ones.

b

y y

r

rr

bb

y

gb g

y y y y y y y yy

rr rr

2b r 4b y 2 3g r 6b r y 3 3r y g

Now sometimes you are told a story and asked to form an expression using the information you are given. No problem, so long as you understood Rules 1 - 3

Once upon a time, in the Land of Algebra

Mr Barton has been sent has been sent on a shopping trip by his girlfriend and he is trying to figure out how much money he needs to bring.

A glance down the list reveals he needs 5 pears, 2 tins of beans, and a box of chocolates.

What is the total cost of these items?

Well, until I get there I don’t know how much each item will cost, so I’ll need to make up some letters… hmm… how about p for the price of pears, b for the price of beans, and g for the price of the box of chocolates (because I know she likes Galaxy!). It does not matter at all which you choose!

So what’s the total cost so far?…

Well, 5 pears and each one costs p

2 tins of beans each costing b

and a box of chocolates costing g

5p

2b

g

So, Total Cost = 5p + 2b + g

Then I get a text saying we need two more tins of beans and one less pear. Now how much will it cost?

So, Total Cost = 5p + 2b + g + 2b - p = 4p + 4b + g

Then she announces that her parents are coming around, so we need twice as much of everything! Cost?

So, Total Cost = 2 x (4p + 4b + g) = 8p + 8b + 2g

Note: the terms are DIFFERENT so don’t try and add them together!

Substitution

One other thing the Rules of Algebra allow you to do is to substitute numbers into expressions. For this, you need to remember our friends BODMAS and Negative Numbers!

If: a = 2 b = 5 c = -3 d = -10

Work out the values of the following expressions:

3abWell this means:

3 a b Using our values:

3 2 5 = 30

6cb

6 c b

6 3 5 = 90

Well this means:

Using our values:

2 ac acdOkay, so we have to do our multiplications first

2 2 2 12 3 ac 2 3 610 0acd

So together we have: 12 60 2 7

25adOkay, we must sort out our power first:

2 10 10 100 d Now we can multiply together

5 2 100 1000

Good luck with your revision!