An equation means that whatever is on the left of the equation sign, is the same as what is on the right of the sign, and we must always try to maintain that equality on both sides.

We can always write **6** times **3** put in our equation sign, is equal to **18**, or we could write **6** times **4** which is **24**, is equal to **21** plus **3** which is also **24**.

### Example 1

If I have **6** times **3** which is **18** is equal to ( = ) **14 plus 4**, that equation makes sense.

I can always manipulate my equation, as long as I do the same thing to both sides of the equation.

If I minus 3 from the left- hand side, the left hand side now becomes **15** and I cannot say that 15 is equal to 18, so what I need to do is the same operation, minus 3 on the right -hand side, now I can write **15 equals 15**.

The key here is to make sure that whatever you do on the left you do on the right or whatever you do on the right you do on the left, that way our equations are balanced.

Manipulating equations helps us to solve them when there is an unknown value in the equation.

### Example 2

We need to find a value of ‘x’ that will make that equation true. The key is to manipulate both sides so that we end up with only the “x” on one side.

**Solve the equation**

There is a plus 4 on the left – hand side and we want to get rid of it, so we minus 4 from the left – hand side, but I have to minus 4 from the right-hand side as well, in other words subtract 4 from both sides. The 4 has disappeared and I now have 7x is equal to 21.

I am looking at 7x on the left- hand side and the key is try and get rid of that 7. I am multiplying 7x’s, the inverse operation is going to be divide by 7, so if we divide both sides by 7 we get ‘x’ is equal to 3 and that is our answer.

### Example 3

Now we are going to try and deal with it when we have ‘x’s on both sides of the equation. Again, our goal is the same; end up with only “x” on one side. We have to remove either from the right -hand side or the left-hand side, all the ‘x’s that are on that side, and we do that by doing the opposite operation that is there with the ‘x’s.

**Solve the equation**

In this case we need to remove the 4x’s from both sides. So the equations becomes 6x**-4x** +7 =4x**-4x**+19

Simplified it becomes **2x +7 =19 **

Now we do an inverse operation, so we minus 7 from both sides.

**2x+ 7 -7 = 19 -7**

Simplified it becomes **2x =12**

We then divide both sides by 2 and we end up with **2x ÷2 =12 ÷2** which means x=6