Understanding the Foundations of Algebra:

This can look traumatising, but please don’t worry.

Substitution is simply giving the letter a value.

So, if you are told that:

a = 3

b=2

c=1

d=7

you would solve the following questions accordingly.

a+b

We know that the a is worth 3 and the b is worth 2, so, 3+2 =5. So, a+b=5

a+b+c

Again, we know that a = 3, b=2, and c=1.

3+2 +1 = 6,

Therefore a +b +c = 6

2 + b

Again, b =2, so 2+2 =4. 2+b=4

bd

this is how b x d would be written. In algebra, the multiplication sign generally isn’t written so that it can’t be mistaken as another value.

So, once again we know that b= 2 and d=4. Therefore, 2x4 = 8 and bd also equals 8.

Finally:

a+d

b

This means that everything above the line is divided by b.

So, a = 3, d =7 and b=2.

a+d = 3+7 =10

and 10 divided by b(2) = 5

Therefore:

a + d = 5

b

Collecting like terms – simplifying expressions

These are normally 1- or 2-mark questions found at the beginning of the paper. Once you know what to do, I suspect that you will find them quite easy.

Example: a+a+a

Here I have one a, then another a, and finally a third a. Altogether I have 3 a’s. So, my answer is 3a.

Example: a+ a +b

Again, I have an a, another a, and a b. So I have 2 a’s and a b, making my answer, my r 2a+b.

Example: a + a + b + a + b – a + 2a +b -3a

You will see I have colour coded each letter so that it stands out.

There’s an a + a + a– a + 2a -3a

a + a + b + a + b – a + 2a +b -3a

We can simplify the a’s 3a – a – 2a -3a

3a -a =2a

2a -2a = 0

0-3a = -3a

Now we can do the same with the b’s

So, we have a b +b + b

a + a + b + a + b – a + 2a +b -3a

Once simplified, this gives us 3b.

When we combine our answers we will have -3a +3b (or 3b-3a)

In a slightly different example, we could take:

a x a

This would give us the answer: a2

When we multiply the letters, we write a power indicating how many of the same letter we multiplied together. So,

a x a x a would give us a3

Example:

If our example had read: a x b x a x a

It would give us an answer of a3b (as a general rule of them, when simplifying don’t use a multiplication sign in case it is mistaken as an ‘x’ or another value.

A slightly different example to consider is:

a2 + a2 + a +b

Our answer here would be 2a2+a+b

Imagine the a2 is your custard and you’re a is the gravy. You just wouldn’t mix them; it would taste awful!

So, keep the a2 separate to your a’s, b’s and other letters.

The same applies if your example is

3 +2a +4a

Our 3 would remain separate from everything else. Therefore, our answer would be 3 +6a.

Solving 1 step equations

Sometimes in life you need the help of someone who has the patience of a saint.

For learning how to solve equations I relied on that very person; my dad.

We had studied this at school and it just never made sense.

My dad explained it to me using the following technique and from there it just seemed to be perfectly logical.

I have used his method to explain how to solve equations over the years and it seems to have worked for so many people.

So for this blog I want to say thank you to my dad

For the purpose of this blog we’ll start with simple 1 step equations.

We’ll move on to 2 step equations in a later blog.

Your question to solve is:

3 + a = 11

Imagine you have a set of scales.

In one side you have 3 + a. In the other side you have the number 11.

You want to work out the value of a so in order to do this you need to end up with just 1 letter a in one side and numbers in the other.

At present, in the left hand side you have a 3 along with the a.

Because we only want a’s in that side we will need to take the 3 out.

This leaves us with an a in one side and 11 in the other.

Now the right hand side (the side with the 11 in) is heavier because we only took the 3 out of one side.

Our next step is to take 3 away from the 11 as well to balance the scales again.

11 – 3 =8

So our scales now have 8 in one side and a in the other:

8 = a or a = 8

Another example of an equation we might be asked to solve is

x- 5 =1

Once again imagine your set of scales.

This time you have x- 5 in one side and 1 in the other.

The 5 in the left hand side is 5 less than 0.

To bring it up to zero I would need to add 5.

X – 5 + 5 =x (Contents of the left hand side).

Once again the left hand side is now heavier than the right so to keep them both balanced I need to add 5 to the right hand side as well.

1 +5 =6

I am left with x on the left hand side and 6 on the right hand side.

Therefore: x = 6

Another example of a one-step equation you might be offered is:

6a = 12

This means that at the moment I have 6 lots of a on the left hand side.

I only want 1 as I am trying to find the value of a (1a).

Because I have 6 lots of a, this is the same as 6 times a, so in order to get back down to one I have to do the opposite of multiply and divide by 6.

6a ÷ 6 = a

Once again, having divided the left hand side of the scales by 6 I need to do the same to the right hand side.

12 ÷ 6 =2.

I now know that a = 2.

The final type of one step equation to solve would be a question such as: x divided by 4 = 8

This could be written as:

X =8

4

At the moment by x is being divided by 4.

In order to counteract this and leave just the x in the left hand side, I need multiply the x by 4.

This once again makes the left hand side heavier. I need to make both sides equal which means multiplying the right hand side (the number 8) by 4 as well.

8 x 4 = 32

Now we are left with x= 32