Equations Part 1

Algebra is about using letters as well as numbers. A lot of maths uses equations as you probably know from school. Equations are a mix of letters and numbers. At school, you are probably given equations to "solve" and they sometimes scare people as they don't know what to do with them. I would like to start with the very simplest equations so that you understand them and can feel confident with them.


Here is a very simple equation:-

x = 3

We are giving x the value 3.

So two lots of x must be 6. We write this as :-

2x  = 6

(This is the same as   x  + x  = 6)

And three lots of x must be 9. (I hope you know your times tables!)

3x = 9

(This is the same as   x  + x  + x  = 9)

In algebra, we are allowed to miss out the times, X, sign. So instead of saying 2 X x = 6   and 3 X x = 9

we just write  2x = 6 and 3x = 9.

We can keep going through the 3 times table, making more equations, like this:


4
x = 12,   5x = 15,   6x = 18,   7x = 21,   8x = 24,  

9x = 27,   10x = 30,   11x = 33,   12x = 36.

All these equations rely on x being equal to 3.

Let's take one of these equations and see what else we can do. We will choose   7x = 21

Now let's add a 1 to the 7x :    7x + 1

If we do this, we must add 1 to the other side too so that it balances:-

7x + 1 = 21 + 1

And we simplify this to 7x + 1 = 22.

If a teacher gave you  7x + 1 = 22   to "solve", you might wonder what to do. But you have seen how easy it is by the work we have just done.

So, we subtract the "+1" from both sides:-

7x + 1 -1 = 22 - 1

     7x   =  21

To "solve" the equation just means to find the value of x . We already know it is 3, but if you didn't know that you would do the following:-

7x = 21

x  = 21/7             or         x = 21 รท 7 

(We divide by the 7 so that we can find what one x is equal to.)

As long as you know your tables, you can then write

x  = 3

And now you have solved the equation 7x + 1 = 22,

since (7 X 3) + 1 = 22.    Easy, wasn't it!


Next we will use  x  = 5

See if you can work out what has happened in these two equations:-

x + 1 = 6     and     x + 4 = 9

The first one has had 1 added to the x and to the 5 and the second one has had 4 added to the x and to the 5.

If x = 5 then we can also say that  2x = 10 and we can add first 1 and then 4 to give the following equations:-

2x + 1 = 11    and    2x + 4 = 14

It is true that  4x = 20  if  x = 5 and, again, we can add first 1 and then 4 and we get the following:-

4x + 1 = 21     and    4x + 4 =24

Let's add some other numbers to these equations to get some variety (always remember that here  x = 5) :-

x + 6 = 11

x + 10 = 15

x + 16 = 21

x + 18 = 23

2x + 5 = 15

2x + 9 = 19

2x + 12 = 22

2x + 13 = 23

4x + 6 = 26

4x + 8 = 28

4x + 11 = 31

4x + 19 = 39

So far we have been adding to the equation  x = 5 so now let's subtract:-

x - 1 = 4

x - 3 = 2

2x - 1 = 9

2x - 3 = 7

4x - 1 = 19

4x - 3 = 17

x - 4 = 1

x - 5 = 0

2x - 5 = 5

2x - 7 = 3

4x - 8 = 12

4x - 17 = 3

Now let's take one of these equations and see what we need to do to get back to   x = 5 :-

4x - 1 = 19

Add 1 to both sides:-

4x - 1 + 1 = 20

4x = 20

x = 20/4 

x = 5

I hope that you have followed this introduction to equations. If there is something you don't understand then please email us on randmgibson@aol.com