Basically if you have two things that are equal, they will still be equal if you subtract the same amount from both sides.
You see it all the time in algebra.
4x + 8 = 10
Now subtract 8 from both sides:
4x + 8 - 8 = 10 - 8
4x = 2
The rule you are using in this step is the "subtraction property of equality"
I don't think you need more examples to understand it. Just if you ever find yourself subtracting the same amount from both sides... you've just invoked the property.