Algebra

Algebra

Using one dictionary’s explanation, algebra is:

“A branch of mathematics in which symbols, usually letters of the alphabet, represent numbers or members of a specified set and are used to represent quantities and to express general relationships that hold for all members of the set.”

Now that was the easiest explanation out of several that I read.

Basically we are going to use letters to help us solve problems where we are not really sure of the actual numerical value. Here is an example of what we are trying to aim for as a worked example, look here now

Adding and subtracting with letters
Multiplying with letters
Dividing with letters
Factorising         Common Factors
                                                 Quadratics
                                                 Difference of two squares

Adding and subtracting with letters

Look at this problem: 2 + 3 = 5

Now you may have used your fingers or beads, when you first did a question like this but now you probably did it without really thinking too hard.

Using the same numbers, look at this problem

2a + 3a = 5a

Hey what does 2a mean? Exactly that 2a means 2 of the quantity “a”. “ a” could be any number.
We do this type of questions in lots of different places for instance;
When we are buying the groceries, if we needed to check if we had enough money to buy 2 bottles of milk. We would simply multiply the price by two.

For Example: A bottle of milk costs $2.59 so two bottles would cost (2 x $2.59) and that would be $5.18 . Here in Australia we do not have 1 or 2 cents anymore so it is rounded to the nearest 5 cents and hence it becomes $5.20.

The thought processes that we went through to work out the total price of the milk are the same ones that we do in algebra except it appears more difficult as it is abstract.

So all we do is add and subtract terms that look exactly the same, here are a few to try and then you can jump to the answer.

Set One

Simplify the following expressions:

35abc + 3cad - 20bca
70xy + 5xy - 37gh + 45gh Answer
3a - 7a + 20a - 6a Answer

I will start you off by answering question 1

Horror upon horrors we have more than one letter, there is no need to hide as we still can only add and subtract terms that look exactly the same. Firstly let’s put the letters in the same order usually alphabetical.

35abc + 3acd - 20 abc

Now put the ones that look the same together with the sign before it. The first term only ever has a sign in front, if it is a negative(minus) sign. See how the negative(minus) sign stays in front of the 20abc. So it would look like:

35abc- 20 abc + 3acd

Now we have two distincts parts, the number and the letters, so we can do adding and subtracting with the numbers and then just write the letters down after the number.

So 35 - 20 is 15, and then write down the letters, 15 abc. The other term,3acd, we have nothing that looks exactly the same to add or subtract with.

So the answer is 15abc +3acd

Try the questions two and three now, and there will be more next time.

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Multiplying and dividing with letters

This is a little more difficult than adding or subtracting but we will take it very slowly.

Multiplying with letters

We will have to learn some new terms such as powers but we will have a look with numbers first and then apply the same principles to the letters.

The number 24 can be written as 2 times 3 times 4
now we could write 24 as letters such as xyz where x = 2, y =3, and z =4.

Just a small point here 2 times 3 times 4 is the same as 3 times 2 times 4 is the same as 4 times 3 times 2 and so on. When we multiply the order is not that important, the same is true when we add numbers such as 3 + 5 is the same as 5 + 3.

When we added and subtracted before, we had terms such as 35abc, and 45gh, what are these letters representing?

35 times a times b times c and we write it 35abc, where a, b and c can stand for any numbers.

Now 4 times 4 is 4²so replacing 4 with a,
we would get a times a which would be a^2.

When we multiply letters together we need a new way to write this so it is not confused with multiplying by a number.
Now a² and a³, we say “a squared” and “a cubed”, the rest we use the words ”to the power of ...”. For example: a⁵is “a to the power of five”.

What does a⁵( a to the power of five) represent?

a x a x a x a x a = a⁵We have multiply a by itself 5 times.

Count the number of a’s we have multiplied together, this then becomes the power.

We can check that a²is not the same as 2a.

a²= a times a whereas 2a = 2 times a, check by substituting any number for a,

lets choose the value of a to be 3

so 3 times 3 = 9 2 times 3 = 6

Before you look at the next set of examples, try these few questions to check that you understand how to write and interpret letters raised to a power.

Set Two

Write using powers

a x a x a x a x a x a x a x a = a ⁸
count the number of a’s, and there are eight.

w x w x w x w x w x w x w answer
b x b x b x b x b x b x h x h x h x h x h answer

Let’s look at a question and solve it slowly

Example One

4mn x 5a

Step One

Collect the numbers together and collect all the same letters together next.

4 x 5 x m x n x a

Remember 4mn can be written as 4 x m x n and we can change the order without affecting the answer.

For example:

2 x 3 = 3 x 2 = 6

Step Two

Now multiply the numbers and same letters together

4 x 5 x m x n x a = 20 x m x n x a

Step Three

Now write numbers and letters without the multiplication signs and put in alphabetical order.

20 x m x n x a = 20 amn

This is the answer

Example Two

3ab ² x 20a ⁵

Step One

Collect the numbers together and collect all the same letters together next.

3 x 20 x a x a ⁵ x b ²

Step Two

Now multiply the numbers and same letters together

3 x 20 x a x a x a x a x a x a x b x b

= 60 x a ⁶ x b ²= 60a ⁶b ²

Step Three

Now write numbers and letters without the multiplication signs and put in alphabetical order.

60 x a ⁶ x b ²= 60a ⁶b ²

This is the answer

Set Three Now try these questions and see how you get on.

5n ⁴y x 2any Answer
2a ⁷by x a ⁴b ⁴y Answer
2p ¹¹x ⁵ x p ⁶x ³ Answer

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Dividing with letters

This is not too difficult as long as you can think of the following:

3 7 10 a a ²3 7 10 a a ²

These all equal one, anything divided by itself is one. Just for the smart people 0/0 is a special case.

We do have to remember all about indices for the this section as well. It is a simple case of cancelling like terms.

Let’s simplify the following expression 3 a b ² 6 a ²b

We will write the expression out in full just like we did for multiplying letters.

3 x a x x b x b
6 x a x a x b

We will rewrite 6 as 3 x 2

3 x x a x b x b
3 x 2 x a x a x b

Now we will cancel all like terms.
As we said before 3/3, a/a, and b/b are all equal to one so our expression becomes

1 x x 1 x 1 x b
x 2 x x a

We will just write the expression now without the multiplication signs and the number ones.

b
2a

This is the answer.

The expression is now in the simplest form that it can be.

You do not have to write it out in full each time if you can see what is happening and you are confident. I suggest that you write it out in full until you are sure of the process. It is more usual to say for the numbers 3 into 3 goes once and 3 into 6 goes twice rather than writing 6 as three times two

Example Three

27 p ⁶ l ²
9 p ⁴l ⁴

Step One

Write out in full if needed

27 x p x p x p x p x p x p x l x l
9 x p x p x p x p x l x l x l x l

Step Two

Now rewrite 27 as 9 x 3 if needed or you can say 9 into 9 goes once and 9 into 27 goes 3 times.

9 x 3 x p x p x p x p x p x p x l x l
9 x p x p x p x p x x l x l x l x l

Step Three

Now we will cancel all like terms. 9/9, and p/p to 1.

1 x 3 x 1 x 1 x 1 x 1 x p x p x 1 x 1
x x x x x x x x l x l

Step Four