Recently a few people have asked about how to solve equations with an x on both sides.
When I did my GCSEs many, many years ago dad suggested that I imagined a set of scales. Whatever I did to one side, I had to ensure I did to the other side as well to make sure the scales stayed balanced.
These are very much like solving normal equations except there may be an extra step or two.
Imagine we are given the equation 3x+2 = 6x-4
The way that dad explained this to me (and made instant sense) was to imagine a set of scales that I need to keep balanced.
Place everything to the left of the equals sign into the left-hand side of the scales and everything to the right into the right-hand side.
Our goal is to have just the x’s in one side and the numbers in the other.
Right now, I have x’s in both sides so I’m going to remove the 3x from the left-hand side.
This has made the right-hand side heavier, so I need to remove 3x from there as well.
This has left me with 2 on the left, and 3x-4 on the right.
2=3x-4
I have -4 on the right which I don’t want as that side is going to contain just x’s.
So, if I add 4 to the right it will remove the -4. (-4+4=0)
Again, to keep both sides of the scales balanced I will need to add 4 to the left-hand side as well.
6=3x
I have now worked my way down to a fairly simple, 1 step equation.
3x something =6
6÷3=2
So, x=2
I can double check that by substituting it back into the original equation to make sure it works.
3x2+2 = 6+2=8
6x2-4=12-4=8
Both sides equal 8, so, we’re right.
I’ll give you a second example
4x+3 = 3x-6
5
This one looks slightly harder. With the 5 underneath the 4x+3 it means that is all being divided by 5.
To get rid of a divide by 5 I have to do the opposite so I will multiply both sides by 5 to keep them balanced. (If I just multiply the left-hand side, it will instantly become heavier – both sides need to be treated the same).
4x+3 = 5(3x-6)
5(3x-6) simply means I have 5 lots of everything inside the brackets.
My next step will be to expand the brackets:
5 times 3x= 15x
5 times -6 = -30
So now my scales will read:
4x+3 =15x-30
We’ll keep the x’s in the right-hand side as we have more in there to get us started.
So, step one will be to take the 4x from the left-hand side, and then also remove 4x from the right.
This leaves me with 3=11x-30
I next need to remove the -30 from the right-hand side so that I only have x’s on that side. To do this I will counteract -30 by adding 30 in.
To keep the scales balanced I will add 30 to the left as well.
I now have 33=11x
I am now back to a 1-step equation.
11x? = 33
My x (which is simply a question mark or “what” or “something” is now 3, because 11x3=33)
Therefore, x=3.
Once again if I check:
4x+3 = 4x3+3 = 12+3 = 15 = 3
5 5 5 5
3x-6 = 3x3 -6 =9-6=3
I hope this makes sense any questions, please do ask
(I will add this alongside a worksheet and other resources to help you to revise in the membership group. You can find the details here: The Clara James Approach
