Step by step
The question
Question: make x the subject of ax + 3 = 2x + c.
The subject is the letter on its own: x on one side, everything else on the other.
Here x is in two places, so we must bring them together.
1 Get the x terms on one side
2x is on the right.
To remove it, we take away 2x.
We do it on both sides, so they stay equal.
Careful!
Taking away 2x is written − 2x, on both sides.
2x and −2x cancel each other.
They make zero.
Now we have ax + 3 − 2x = c.
Is x alone now?
Not yet: + 3 is still with x.
How do we remove it?
3 has no x.
To remove it from the left, we take away 3.
We do it on both sides, so they stay equal.
3 and −3 cancel each other.
They make zero.
Now both x terms are on the left: ax − 2x = c − 3.
So after step 1: we took away 2x and took away 3, and now we have ax − 2x = c − 3.
2 Take x out
x is in every part.
So we take it out once, and write what is left in a bracket.
It is opening a bracket, backwards.
From ax, a is left.
From −2x, −2 is left.
So x times (a − 2) is c − 3.
3 Remove the (a − 2)
x(a − 2) means x times (a − 2).
We want just x.
To remove times (a − 2), we divide by (a − 2).
We do it on both sides.
(a − 2) divided by (a − 2) is 1.
They cancel each other.
So just x is left.
So x = (c − 3)/(a − 2).
4 Check
Let's check with numbers.
We pick small numbers that give a whole answer: a = 3, c = 4.
Our answer gives x = 1.
Now we put all the numbers into the question.
First the left side.
Now the right side.
Both sides are 6.
So our answer is right!
Let's look back.
First: get the x terms on one side.
Then: take x out.
Then: remove the (a − 2).
Last: check.