equality of equations

i once believed that simplifying an equation yields an equation that is the same as the original. However, an equation such as y=(x-3)/(x-3) is undefined at 3 where as y=1 is not. why?

zero divisors, buddy.

y = (x-3)/(x-3)
=> y(x-3)=(x-3)
=> y(x-3)=1(x-3)
cancellation laws only hold for ab=ac when you know a is not 0
x-3 is not zero for everything except 3.  when you multiply both sides of the equation y = 1 by (x-3) youre potentially multiplying by zero, so everything you get from that equation depends on whether or not what you multiplied by is equal to zero or not.

adding on in case you wanted a shorter response

x/x doesnt necessarily equal 1

Failed trolling.  Try harder.

>>1
y = (x-3)/(x-3) is non-continuous for x = 3 where it's (3-3)/(3-3) = 0/0 = √-1 .