why its paradoxical to try make $a$ the subject of $(a-b)/(b-a) = c?$

Question

so one day i just made a random formula to try and make $a$ the subject and i made $a-b/b-a = c$ ok first you need to times both side by $b-a$ to get $a-b=cb-ca$ now you need to $b-ca$ on both sides to get $a+ca=b+cb$ now factorise to get $a(c+1)=b(c+1)$ divide by $c+1$ then cancel the c+1 to get a=b which means that if you was to plug this in to the original formula you would get $a-a/a-a=c$ in other words $0/0=c$ and since you can't do the times by 0 at the start so it really confused me what do you think of this?

Answer 1

The problem is:

• Either $a=b$ and you made a mistake by multiplying by $b-a$
• Or $a\ne b$ and then $c=-1$ and you made a mistake when you cancelled $c+1$.

To do it properly, you would need to always remember under which conditions you did the operation you did (e.g. multiplying by $b-a$ is not giving you an equivalent statement, unless $a\ne b$), and then do a separate analysis for the opposite case. This may lead you to distinguish quite a number of cases in some problems. Be systematic.