But how does maths work with no numbers? Well, in place of numbers, it uses letters. x and y instead of 1 and 2. So how does one go about this? Well, first you have to know various rules...

BIDMAS

BIDMAS is the order that operations are done: Brackets, Indices, Division, Multiplication, Addition, Subtraction. Operations inside brackets also happen in the order of BIDMAS. Writing Algebraic Terms

It is important to remember that x is often used so writing x in order to mean "multiplied by" would be very confusing so axb is written as ab.

The Laws Of Indices

You've probably seen written down before but what does it mean? Well it means . is what's called an index (plural indices). () is also an index but they aren't the only indices. Any number or letter can be an index. A number like 2 always has index of 1. However there are certain rules that control the use of indices:

So, what do you think of the laws of indices? Well, they're going to be pretty useful later on.

Basic Equations

Before working out any equations, you have to know that WHATEVER YOU DO TO ONE SIDE, YOU HAVE TO DO TO THE OTHER SIDE AS WELL!
How would you work out what is if you're given the statement ? Well, you'd divide by 2 to get .
How about . Add 3 and you're back to . Divide by 2 and you once again have .
How about ? Multiply by 7 and you're back at . Etc. etc.

Simplifying Equations

. Quite complicated, isn't it? But it doesn't have to be. You could add all the like terms together (s, s and numbers). This would give you . Remember though, you can't simplify something like because none of them are like terms.

Intermediate Equations

What if you had in both sides? for example. What you have to do is get all the letters on one side and the numbers on the other. So what you'd do is subtract to get , then subtract 2 to get . From here, divide by 2 to get .
But hold on. What if its something like ? Well then you subtract 2 (), add () and divide by -2 to get . However, it is better to have on the left, so you switch the sides to get . Problem solved.

Next, what if you have a fraction with in both sides? What if you had something like ? Well first you have to get rid of the fraction, so multiply by and you'll have
. Now, you would divide by and have . Now you can subtract and be left with divide by -5 and you'll have .

Expansion

What is without the brackets? ? No, it's because you have to multiply everything inside the brackets by the thing you're multiplying the brackets by. So how about ? You'd get . How about ? You'd get .

Factorisation

Factorising is the opposite of expanding. When you factorise, you put the terms into brackets. So, imagine you had to factorise . How would you do it? Well, first, you have to find the highest common factor of all of those terms, so for this it is . Now you divide all of the terms by that factor: . Now you put brackets around that and write next to it the factor: .

Basic Simultaneous Equations

How would you work out what and are? First, you have to work out what one letter is in terms of the other one. So, in this case we can rearrange the second equation to get . Now, we can substitute that into the first equation to get . Expand the brackets and you'll have . This simplifies down to . From here you subtract 8 and you'll have . Now, you can use that to work out by replacing in the first equation with 2: . Surely, you can work that out: . So now, you have:

Basic Graphs

. Fairly simple: it's an equation. But it can also be used to plot a line on a graph. This line, for example is like so:

Anything that is is a horizontal line, just like is a vertical line. But what about a diagonal line? Well that is . Here, is the gradient of the line and
is the point at which the line intersects the axis. So looks like this:

Intermediate Expansion

What if you had to do something like ? For this, you have to use FOIL (First, Outside, Inside, Last) . This means that you multiply the first term in each bracket, the two outside terms, the two inside terms and then the last term in each bracket. This would give you . Now, obviously, you can't simplify that, but often, you can simplify the result, such as .

If you have three sets of brackets, e.g., you expand the first two sets of brackets first. This gives you . Now you put this in brackets and multiply it by the third set of brackets, so . But wait, you can't use FOIL here. Just multiply each term in the first set of brackets by both terms in the second set of brackets:.

Intermediate Factorisation

. Obviously, this can be factorised or else it wouldn't be in this section. This factorises into . But do you see the problem here? It can be factorised again. Look at what's inside the brackets. You can't factorise all of it, but you can get. Always watch out for things like that.

Advanced Equations

. Oh no! Fractions in both sides: what do I do?! Cross-multiply. What's that? It's when you multiply the denominators together and the multiply both sides by the result, so in this case you'd multiply by and get .Now cube root and get[[image:http://latex.codecogs.com/gif.latex?x=%5Csqrt[3]%7B5%7D]].

Basic Rearranging Equations

What is you had and were told to rearrange it? What is rearranging an equation? Rearranging an equation is when you change the subject of an equation. So, in this case is the subject but in the example question you need to makethe subject. You need to change it to so that the equation is equal. Now, how about rearranging to make the subject. You would need to change it to . Remember, the key to rearranging equations is to keep the equation equal.

Advanced Expansion

. Expand. The answer is:
. No. You need to think of it as
. This means that using FOIL you can get
which simplifies to become:
.

## The Ultimate Guide To Algebra

Algebra, the area of Maths that uses:

NO NUMBERS!

But how does maths work with no numbers? Well, in place of numbers, it uses letters. x and y instead of 1 and 2. So how does one go about this? Well, first you have to know various rules...

## BIDMAS

BIDMAS is the order that operations are done: Brackets, Indices, Division, Multiplication, Addition, Subtraction. Operations inside brackets also happen in the order of BIDMAS.

Writing Algebraic Terms

It is important to remember that x is often used so writing x in order to mean "multiplied by" would be very confusing so axb is written as ab.

## The Laws Of Indices

You've probably seen written down before but what does it mean? Well it means . is what's called an index (plural indices). () is also an index but they aren't the only indices. Any number or letter can be an index. A number like 2 always has index of 1. However there are certain rules that control the use of indices:

[[image:http://latex.codecogs.com/gif.latex?%5Cinline%20x%5E%7B%5Cfrac%7B1%7D%7Ba%7D%7D=%5Csqrt[a]%7Bx%7D]]

So, what do you think of the laws of indices? Well, they're going to be pretty useful later on.

## Basic Equations

Before working out any equations, you have to know that WHATEVER YOU DO TO ONE SIDE, YOU HAVE TO DO TO THE OTHER SIDE AS WELL!

How would you work out what is if you're given the statement ? Well, you'd divide by 2 to get .

How about . Add 3 and you're back to . Divide by 2 and you once again have .

How about ? Multiply by 7 and you're back at . Etc. etc.

Simplifying Equations. Quite complicated, isn't it? But it doesn't have to be. You could add all the like terms together (s, s and numbers). This would give you

. Remember though, you can't simplify something like because none of them are like terms.

## Intermediate Equations

What if you had in both sides? for example. What you have to do is get all the letters on one side and the numbers on the other. So what you'd do is subtract to get , then subtract 2 to get . From here, divide by 2 to get .

But hold on. What if its something like ? Well then you subtract 2 (), add () and divide by -2 to get . However, it is better to have on the left, so you switch the sides to get . Problem solved.

Next, what if you have a fraction with in both sides? What if you had something like ? Well first you have to get rid of the fraction, so multiply by and you'll have

. Now, you would divide by and have . Now you can subtract and be left with divide by -5 and you'll have .

## Expansion

What is without the brackets? ? No, it's because you have to multiply everything inside the brackets by the thing you're multiplying the brackets by. So how about ? You'd get . How about ? You'd get .

## Factorisation

Factorising is the opposite of expanding. When you factorise, you put the terms into brackets. So, imagine you had to factorise . How would you do it? Well, first, you have to find the highest common factor of all of those terms, so for this it is . Now you divide all of the terms by that factor: . Now you put brackets around that and write next to it the factor: .

## Basic Simultaneous Equations

How would you work out what and are? First, you have to work out what one letter is in terms of the other one. So, in this case we can rearrange the second equation to get . Now, we can substitute that into the first equation to get . Expand the brackets and you'll have . This simplifies down to . From here you subtract 8 and you'll have . Now, you can use that to work out by replacing in the first equation with 2: . Surely, you can work that out: . So now, you have:

## Basic Graphs

. Fairly simple: it's an equation. But it can also be used to plot a line on a graph. This line, for example is like so:

Anything that is is a horizontal line, just like is a vertical line. But what about a diagonal line? Well that is . Here, is the gradient of the line and

is the point at which the line intersects the axis. So looks like this:

## Intermediate Expansion

What if you had to do something like ? For this, you have to use FOIL (First, Outside, Inside, Last) . This means that you multiply the first term in each bracket, the two outside terms, the two inside terms and then the last term in each bracket. This would give you . Now, obviously, you can't simplify that, but often, you can simplify the result, such as .

If you have three sets of brackets, e.g., you expand the first two sets of brackets first. This gives you . Now you put this in brackets and multiply it by the third set of brackets, so . But wait, you can't use FOIL here. Just multiply each term in the first set of brackets by both terms in the second set of brackets:.

## Intermediate Factorisation

. Obviously, this can be factorised or else it wouldn't be in this section. This factorises into . But do you see the problem here? It can be factorised again. Look at what's inside the brackets. You can't factorise all of it, but you can get. Always watch out for things like that.

## Advanced Equations

. Oh no! Fractions in both sides: what do I do?! Cross-multiply. What's that? It's when you multiply the denominators together and the multiply both sides by the result, so in this case you'd multiply by and get .Now cube root and get [[image:http://latex.codecogs.com/gif.latex?x=%5Csqrt[3]%7B5%7D]].

## Basic Rearranging Equations

What is you had and were told to rearrange it? What is rearranging an equation? Rearranging an equation is when you change the subject of an equation. So, in this case is the subject but in the example question you need to makethe subject. You need to change it to so that the equation is equal.

Now, how about rearranging to make the subject. You would need to change it to . Remember, the key to rearranging equations is to keep the equation equal.

## Advanced Expansion

. Expand. The answer is:

. No. You need to think of it as

. This means that using FOIL you can get

which simplifies to become:

.

## Advanced Factorisation