The x is a placeholder for some currently unknown number. Let's say that number was 10. So we replace x with 10 to get 10-3 = 12. That statement is false because 10-3 is actually 7. What you can do is keep guessing until you get the right answer.
A much better approach is to undo what is happening to x. We have x and we subtract off 3 to get 12. Following this in reverse, we need to add 3 to both sides so that x is isolated.
x-3 = 12
x-3+3 = 12+3
x = 15
Let's replace x with 15 and see if the equation is true
x - 3 = 12
15 - 3 = 12
12 = 12
Sure enough we get the right answer.
The idea of adding 3 to both sides is possible through the additive property of equality (if a = b, then a+c = b+c)
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In summary, I added 3 to both sides using the additive property of equality. Doing this isolated x to get x = 15.