Solving Equations

Today we had a look at how we solve equations. By solving equations, what I really mean is we want to find a value for our pro-numeral (letter). So we want to manipulate our equation in such a way that we have our pro-numeral on one side of the equals sign, and a number on the other side of our equals sign.

To do this, we want to use a few techniques to "free up" the pro-numeral. Basically, what we want to do is move the numbers away from the pro-numeral by doing the opposite to what the number is actually doing. Hopefully after a few examples,  this will become clearer.

So for example, if we have the expression x + 5. We want to somehow get rid of the 5 so we are just left with x. To get rid of the 5, we must do the opposite of what 5 is currently doing. In the expression, we are adding 5 to x. So to get rid of the 5, we must subtract 5 from the expression, since the opposite of adding is subtracting.
So, we then get: x + 5 - 5
We then see that + 5 - 5 is equal to zero, so we are just left with x.

Similarly, if we have 4x, we want to get rid of the 4 so we are just left with x. In this expression, the 4 is multiplied by x, so to get rid of the 4, we want to do the opposite of multiplying by 4, which is dividing by 4.
So we get 4x ÷ 4  
This then simply becomes x

This is quite easy for simple expressions like these, however we will usually have to solve a lot more complex equations. I will go through a few examples of these, hopefully demonstrating what we need to do.

Example: x + 6 = 8

We want to get rid of the 6, so we are just left with x on one side of the equals sign, and numbers on the other side.
To get rid of the 6, we need to do the opposite of what it is currently doing. In this equation, we have "plus 6" so to get rid of it we must "minus 6", since this is the opposite.
We also need to remember that whatever we do on one side of our equals sign, we must do it on the other side as well. In this equation we need to subtract 6 from the left hand side, to get rid of the 6 that is already there. This means we must subtract 6 from the right hand side too.
So we are left with:
x + 6 - 6 = 8 - 6
If we work these sums out, we get
x = 2
So we have now solved the equation x + 6 = 8, and found that x = 2.

The do get more complicated than this. For example, we may have:

4x - 7 = 13

We want to get rid of the 7 and the 4, so this one will require 2 steps. First we will look at getting rid of the 7.
In this equation we have "minus 7", so to get rid of this, we need to "plus 7", remembering to do this to both sides. We then get:
4x - 7 + 7 = 13 + 7
If we do these sums we get:
4x = 20

Now, we want to get rid of the 4, so we will divide both sides by 4. This gives us:
4x ÷ 4 = 20 ÷ 4
Again, working out these sums we get:
x = 5

So we have now solved the equation 4x - 7 = 13, and found that x = 5. 

For a little more information on this, see this website: http://www.mathsisfun.com/algebra/equations-solving.html 

There is a new worksheet on this in the "Worksheets" tab at the top of the page. Work through questions 1 - 3 on this by Friday. 

As always, if you have any questions feel free to comment on this tab, or email or chat to me in person.