Why can you not divide expressions by common factors?

With equations, e.g. 4x = 12, the 4x variable has a given value. This means you can apply the same changes to both sides (at the same time). For example, both sides can be divided by 4 to give x=3
However expressions, e.g. 4x + 12, do not have an equal sign. This means the value of x is unknown and is simply a variable and the expression cannot be changed. Instead the expression can be simplified, for example by factorisation 4(x+3).