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?
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1You need some parens. – user4894 Feb 10 '18 at 22:38
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Because $a-b/b-a=a-a-b/b=0-1=-1$. It happens to also be true that $(a-b)/(b-a)=-1$ – Ross Millikan Feb 10 '18 at 22:41
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Is "make $a$ the subject of" Newspeak for "solve for $a$"? – bof Feb 10 '18 at 22:57
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Here's a simpler one: Solve $x-x=3$ for $x.$ – bof Feb 10 '18 at 22:58
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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.