The Ultimate Guide To Algebra

Algebra, the area of Maths that uses:

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 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 external image gif.latex?%5Cinline%20x%5E%7B2%7D written down before but what does it mean? Well it means external image gif.latex?%5Cinline%20xx. external image gif.latex?%5Cinline%20%5E%7B2%7D is what's called an index (plural indices). external image gif.latex?%5Cinline%20%5E%7B3%7D(external image gif.latex?%5Cinline%20xxx) 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:
external image gif.latex?%5Cinline%20x%5E%7Ba%7Dx%5E%7Bb%7D=x%5E%7Ba+b%7D
external image gif.latex?%5Cinline%20%5Cfrac%7Bx%5E%7Ba%7D%7D%7Bx%5E%7Bb%7D%7D=x%5E%7Ba-b%7D
external image gif.latex?%5Cinline%20%5Cleft%20(%20x%5E%7Ba%7D%20%5Cright%20)%5E%7Bb%7D=x%5E%7Bab%7D
external image gif.latex?%5Cinline%20x%5E%7B-a%7D=%5Cfrac%7B1%7D%7Bx%5E%7Ba%7D%7D
external image gif.latex?%5Cinline%20x%5E0=1
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 external image gif.latex?%5Cinline%20x is if you're given the statement external image gif.latex?%5Cinline%202x=10? Well, you'd divide by 2 to get external image gif.latex?%5Cinline%20x=5.
How about external image gif.latex?%5Cinline%202x-3=7. Add 3 and you're back to external image gif.latex?%5Cinline%202x=10. Divide by 2 and you once again have external image gif.latex?%5Cinline%20x=5.
How about external image gif.latex?%5Cinline%20%5Cfrac%7B2x-3%7D%7B7%7D=1? Multiply by 7 and you're back at external image gif.latex?%5Cinline%202x-3=7. Etc. etc.

Simplifying Equations

external image gif.latex?%5Cinline%20x+x+2+4+y+3. Quite complicated, isn't it? But it doesn't have to be. You could add all the like terms together (external image gif.latex?%5Cinline%20xs, external image gif.latex?%5Cinline%20ys and numbers). This would give you
external image gif.latex?%5Cinline%202x+y+9. Remember though, you can't simplify something like external image gif.latex?%5Cinline%20x%5E%7B2%7D+x+2y+3 because none of them are like terms.

Intermediate Equations

What if you had external image gif.latex?%5Cinline%20x in both sides? external image gif.latex?%5Cinline%203x+2=x+6 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 external image gif.latex?%5Cinline%20x to get external image gif.latex?%5Cinline%202x+2=6, then subtract 2 to get external image gif.latex?%5Cinline%202x=4. From here, divide by 2 to get external image gif.latex?%5Cinline%20x=2.
But hold on. What if its something like external image gif.latex?%5Cinline%20-x+12=2-3x? Well then you subtract 2 (external image gif.latex?%5Cinline%20-x+10=-3x), add external image gif.latex?%5Cinline%20x (external image gif.latex?%5Cinline%2010=-2x) and divide by -2 to get external image gif.latex?%5Cinline%20-5=x. However, it is better to have external image gif.latex?%5Cinline%20x on the left, so you switch the sides to get external image gif.latex?%5Cinline%20x=-5. Problem solved.

Next, what if you have a fraction with external image gif.latex?%5Cinline%20x in both sides? What if you had something like external image gif.latex?%5Cinline%20%5Cfrac%7Bx%5E%7B2%7D%7D%7B2x%7D=3x+4? Well first you have to get rid of the fraction, so multiply by external image gif.latex?%5Cinline%202x and you'll have
external image gif.latex?%5Cinline%20x%5E%7B2%7D=6x%5E%7B2%7D+8x. Now, you would divide by external image gif.latex?%5Cinline%20x and have external image gif.latex?%5Cinline%20x=6x+8. Now you can subtract external image gif.latex?%5Cinline%206x and be left with external image gif.latex?%5Cinline%20-5x=8 divide by -5 and you'll have external image gif.latex?%5Cinline%20x=-%5Cfrac%7B8%7D%7B5%7D.


What is external image gif.latex?%5Cinline%202(x+3) without the brackets? external image gif.latex?%5Cinline%202x+3? No, it's external image gif.latex?%5Cinline%202x+6 because you have to multiply everything inside the brackets by the thing you're multiplying the brackets by. So how about external image gif.latex?%5Cinline%20x(x+y+3)? You'd get external image gif.latex?%5Cinline%20x%5E%7B2%7D+xy+3x. How about external image gif.latex?%5Cinline%204x(x%5E%7B2%7D+3y+%5Csqrt%7Bz%7D)? You'd get external image gif.latex?%5Cinline%204x%5E%7B3%7D+12xy+4x%5Csqrt%7Bz%7D.


Factorising is the opposite of expanding. When you factorise, you put the terms into brackets. So, imagine you had to factorise external image gif.latex?%5Cinline%202x%5E%7B2%7D+8xy+2x. 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 external image gif.latex?%5Cinline%202x. Now you divide all of the terms by that factor: external image gif.latex?%5Cinline%20x+4y+1. Now you put brackets around that and write next to it the factor: external image gif.latex?%5Cinline%202x(x+4y+1).

Basic Simultaneous Equations

external image gif.latex?%5Cinline%202x+3y=10
external image gif.latex?%5Cinline%20x+y=4
How would you work out what external image gif.latex?%5Cinline%20x and external image gif.latex?%5Cinline%20y 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 external image gif.latex?%5Cinline%20x=4-y. Now, we can substitute that into the first equation to get external image gif.latex?%5Cinline%202(4-y)+3y=10. Expand the brackets and you'll have external image gif.latex?%5Cinline%208-2y+3y=10. This simplifies down to external image gif.latex?%5Cinline%20y+8=10. From here you subtract 8 and you'll have external image gif.latex?%5Cinline%20y=2. Now, you can use that to work out external image gif.latex?%5Cinline%20x by replacing external image gif.latex?%5Cinline%20y in the first equation with 2: external image gif.latex?%5Cinline%202x+6=10. Surely, you can work that out: external image gif.latex?%5Cinline%20x=2. So now, you have:
external image gif.latex?%5Cinline%20x=2
external image gif.latex?%5Cinline%20y=2

Basic Graphs

external image gif.latex?%5Cinline%20y=2. 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 external image gif.latex?%5Cinline%20y=n is a horizontal line, just like external image gif.latex?%5Cinline%20x=n is a vertical line. But what about a diagonal line? Well that is external image gif.latex?%5Cinline%20y=mx+c. Here, external image gif.latex?%5Cinline%20m is the gradient of the line and external image gif.latex?%5Cinline%20c
is the point at which the line intersects the external image gif.latex?%5Cinline%20y axis. So external image gif.latex?%5Cinline%20y=2x+1 looks like this:


Intermediate Expansion

What if you had to do something like external image gif.latex?%5Cinline%20(x+2)(y+3)? 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 external image gif.latex?%5Cinline%20xy+3x+2y+6. Now, obviously, you can't simplify that, but often, you can simplify the result, such as external image gif.latex?(x+12)(x+3)=x%5E%7B2%7D+3x+12x+36=x%5E%7B2%7D+15x+36.

If you have three sets of brackets, e.g.external image gif.latex?(x%5E%7B2%7D+12)(3y+16)(%5Csqrt%7Bz%7D+20), you expand the first two sets of brackets first. This gives you external image gif.latex?3x%5E%7B2%7Dy+16x%5E%7B2%7D+36y+192. Now you put this in brackets and multiply it by the third set of brackets, so external image gif.latex?(3x%5E%7B2%7Dy+16x%5E%7B2%7D+36y+192)(%5Csqrt%7Bz%7D+20). 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:external image gif.latex?3x%5E%7B2%7Dy%5Csqrt%7Bz%7D+60x%5E%7B2%7Dy+16x%5E%7B2%7D%5Csqrt%7Bz%7D+320x%5E%7B2%7D+36y%5Csqrt%7Bz%7D+720y+192%5Csqrt%7Bz%7D+3840.

Intermediate Factorisation

external image gif.latex?4x%5E%7B2%7D+8x+4y. Obviously, this can be factorised or else it wouldn't be in this section. This factorises into external image gif.latex?4(x%5E%7B2%7D+2x+y). 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 getexternal image gif.latex?4(x(x+2)+y). Always watch out for things like that.

Advanced Equations

external image gif.latex?%5Cfrac%7B1%7D%7Bx%7D=%5Cfrac%7Bx%5E%7B2%7D%7D%7B5%7D. 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 external image gif.latex?5x and get external image gif.latex?x%5E%7B3%7D=5.Now cube root and get [[image:[3]%7B5%7D]].

Basic Rearranging Equations

What is you had external image gif.latex?v=u+at 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 external image gif.latex?vis the subject but in the example question you need to makeexternal image gif.latex?uthe subject. You need to change it to external image gif.latex?u=at-v so that the equation is equal.
Now, how about rearranging external image gif.latex?s=ut+%5Cfrac%7B1%7D%7B2%7Dat%5E%7B2%7Dto make external image gif.latex?athe subject. You would need to change it to external image gif.latex?a=%5Cfrac%7Bs-ut%7D%7B%5Cfrac%7B1%7D%7B2%7Dt%5E%7B2%7D%7D. Remember, the key to rearranging equations is to keep the equation equal.

Advanced Expansion

external image gif.latex?(x+y)%5E%7B2%7D. Expand. The answer is:
external image gif.latex?x%5E%7B2%7D+y%5E%7B2%7D. No. You need to think of it as
external image gif.latex?(x+y)(x+y). This means that using FOIL you can get
external image gif.latex?x%5E%7B2%7D+xy+xy+y%5E%7B2%7D which simplifies to become:
external image gif.latex?x%5E%7B2%7D+2xy+y%5E%7B2%7D.

´╗┐Advanced Factorisation