why can you use common factor to simplify?

The question given could be analysed in general. But this
attempt is only  to the limitted sense of solving problems related with rational fractions and
rational algebraic expressions.

If a  rational fraction in the form
p/q is given, we simplify it  to the lowest posissible integers in both numerator and
denominator.  The purpose is that it easy for graping and also it is easy to deal with further
calcalations in the simpler form.

Example 1: WE assume  a  given
fraction  like 21/77 is equal to 7*3/7*11.Here both numerator and denominators  have the common
factor 7. So we can divide both numerator and denominator by 7 and the result is
3/11.

 Example 2: The given expression, (x^2-1)/(x+1)  is equal
to (x-1)(x+1)/(x-1) The common factor is x-1. So we simplify the given expression to x+1 of
course x= not equal to zero.