If S and T are non-zero numbers and

Question

If S and T are non-zero numbers and  \frac{1}{S} + \frac{1}{T} = S + T , which of the following must be true?

A.  ST = 1 
B.  S + T = 1 
C.  \frac{1}{S} = T 
D.  \frac{S}{T} = 1 
E. None of the above

M18-19

Explanation provided:

1/S + 1/T = S+T;

T+S/ST = S+T→;

Cross-multiply: S+T=(S+T)∗ST;

(S+T)(ST−1)=0. Either S+T=0 or ST=1. Now, notice that if S+T=0 is true then none of the options must be true.

The correct answer is E

Question: 
I understand the way of the equation, however, what I would have done is interfere at the following step:
S+T = (S+T)*ST 
-> (S+T)/(S+T)=ST
-> ST = 1

Is there some rule which forbids me to take this step? Or is the only option to realize so, that you perform the above given equation as well and realise that "S+T = 0" negates all other options than E.... ??

Thanks in advance, 
best regards

P.S. Sry if the format is terrible, this is the first question I am copying out of somewhere.

Bunuel · Accepted Answer

If S and T are non-zero numbers and \frac{1}{S} + \frac{1}{T} = S + T , which of the following must be true? A. ST = 1 B. S + T = 1 C. \frac{1}{S} = T D. \frac{S}{T} = 1 E. None of the above OE: \frac{1}{S} + \frac{1}{T} = S + T --> \frac{T+S}{ST}=S+T --> cross-multiply: S+T=(S+T)*ST --> (S+T)(ST-1)=0 --> either S+T=0 or ST=1 . So, if S+T=0 is true then none of the options must be true. Answer: E. P.S. Please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Please pay attention to the rule #3. Thank you.

Vips0000 · Answer

Yes, this is not correct way of cancelling. I'll show you one example.
5* 0 = 3*0
if we cancel 0 on both side, we get 5=3. is it correct? No.

The crux (and the rule) is, we can cancel out a term only when we know its not 0. So the way it is done in explanation is absolutely correct and the right method.

Hope it helps

MacFauz · Answer

There was this fun derivation that my math teacher showed us in school. Just to demonstrate how cancelling of 0 could yield wrong results. He claimed that he could prove that 1=2 and hence all numbers are equal.

It goes as below:

Let,
 a=b

Multiplying both sides by a, We get

a^2 = ab

Subtracting  b^2  from both sides

a^2 - b^2 = ab - b^2

(a+b)(a-b) = b(a-b)

Cancelling  (a-b)  on both sides,

a+b = b

Since  a=b

a+a = a

2a = a

2=1

SirGMAT · Answer

Thanks - lol I am an idiot - ...! I even thought of the zero number things, but mistakenly memorized the prompt as telling me that the respective equation could not be "0", though it only said that each number alone is non-zero.... Thanks guys!

kartik222 · Answer

Hi bunuel,

In this step:
s+t = (s+t)st can't we just cancel (s+t) and get ---> st =1?

thanks,
-K

Marcab · Answer

Bunuel, 
in your solution I need to ask one thing.
Inequalities, there is a rule that if you dont know the sign of denominator, then dont cross multiply. In your solution, how can you be so sure of the sign of ST. Please let me know if i am missing something

Bunuel · Answer

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We can not divide by zero.

So, if you divide (reduce) s+t = (s+t)st by (s+t), you assume, with no ground for it, that (s+t) does not equal to zero thus exclude a possible solution (notice that both st=1 AND (s+t)=0 satisfy the equation).

Hope it's clear.

Bunuel · Answer

We are concerned with the sign when we cross-multiply an inequality because this operation might affect (flip) its sign (> to <, for example) but it's always safe to cross-multiply an equation.

Sachin9 · Answer

Bunuel,
Since S+T=0   OR   ST=1 and the question asks what   must   be true, the answer is E ? Another way to answer the question. . Is my reasoning right?

Bunuel · Answer

Yes, if for example s=1 and t=-1 (s+t=1-1=0), then none of the options is true (none of the options MUST be true).

Sachin9 · Answer

I understand what you are trying to say bunuel..
my question is that since the equation results in 2 soln and we have a OR .. . that is soln 1 OR soln 2 and the question asks for MUST be true..

So based on this reasoning, can we say the answer is E..?

For something  must be true  , we cannot have soln 1  OR  soln 2.. we need to have 1 soln / soln1 AND soln2..

Hope you are getting what I am trying to ask..

Vips0000 · Answer

I would say that don't generalize this point. You know that because there are two solutions, any of the given options need not be a MUST. But if u really have an option that says (s+t)(st-1)=0 then that MUST be true.

Sachin9 · Answer

But if u really have an option that says (s+t)(st-1)=0 then that MUST be true.

Ididn;t understand this.. if (s+t)(st-1)=0 then either s+t=0 or st-1=0.. we still have a OR here

Vips0000 · Answer

If the question were to be this:
If S and T are non-zero numbers and  \frac{1}{S} + \frac{1}{T} = S + T , which of the following must be true?

A.  ST = 1 
B.  S + T = 1 
C.  \frac{1}{S} = T 
D.  \frac{S}{T} = 1 
E. None of the above
F. (s+t)(st-1)=0

Then your generalization will go wrong as you have an ans choice F that must hold true.

kuttingchai · Answer

Hello Bunuel,

How did u get S+T=(S+T)*ST --> (S+T)(ST-1)=0 ??

What did i miss in he below equation?? how come you got (S+T)(ST-1)=0 ?? Ca you please explain
1/S+1/T = S+T
(S+T) = (S+T) (ST) 
divide both side by (S+T) we get
1=1(ST)
therefore ST=1

Thank you.

Bunuel · Answer

Check here: if-s-and-t-are-non-zero-numbers-and-141887.html#p1140729 Hope it helps.

kuttingchai · Answer

I think I got the answer

from your previous post I got "
Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We can not divide by zero.

So, if you divide (reduce) s+t = (s+t)st by (s+t), you assume, with no ground for it, that (s+t) does not equal to zero thus exclude a possible solution (notice that both st=1 AND (s+t)=0 satisfy the equation).
"

therefore
1/S+1/T = S+T
(S+T) = (S+T) (ST) 
0= (S+T) (ST) - (S+T) 
0 = (S+T) (ST-1)

(S+T)=0 or ST = 1

Thank you

ArunpriyanJ · Answer

Bunuel,

Not able to understand the highlighted part. Kindly explain in detail...

Thx,
Arun

Bunuel · Answer

S+T=(S+T)*ST

(S+T)*ST-(S+T)=0

Factor s+t:  (S+T)(ST-1)=0

Hope it's clear.

If S and T are non-zero numbers and

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If S and T are non-zero numbers and

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