Expert: Richard J. Raridon - 7/3/2010

Question
QUESTION: Here are the three fractions that I want to combine:
(s/(s+t))-(t/(t^2-s^2))+(1/2)
The question that I have is why is it necessary to multiply the middle fraction by (-1/-1) before solving?  This will produce a LCD of 2(s+t)(s-t).  However, with the problem stated above, I believe the LCD is 2(t+s)(t-s).  As you can see, the results are different when multiplying the middle term by (-1/-1).  Can you explain the rules regarding this so I won't make the same mistakes?

ANSWER: It's equivalent either way.  If you use LCD = 2(t+s)(t-s) you get
[t^2-3s^2-2t+2st]/[2(t+s)(t-s)]
If you use LCD = 2(s+t)(s-t) you get [3s^2-t^2-2st+2t]/[2(s+t)(s-t)]
Those are the same.

---------- FOLLOW-UP ----------

QUESTION: According to an online algebra solver, the original problem [ (s/(s+t)) – (t/((t^2-s^2)) + (1/2) ] has a -2t in the numerator.  However, when changing the sign of the middle term, I believe the problem becomes [ (s/(s+t)) + (t/((s^2-t^2)) + (1/2) ] and the algebra solver’s result has a +2t in the numerator.  Therefore these two problems are not the same.  How do you change the sign of the middle term without changing the problem?

ANSWER: You can't change -2t/(t^2-s^2)  to t/(s^2-t^2).  If the original problem had -2t, then that's a different problem.  
Changing the sign of the NUMERATOR in the middle term does not change the problem and won't give you a different answer, as I pointed out previously.  

---------- FOLLOW-UP ----------

QUESTION: It is obvious that I am the one who is confused, and I apologize for that.  The middle term of the original problem is: [ - ((t) / (t^2 – s^2)) ].  I believe that this middle term is equivalent to: [ + ((t) / (s^2 – t^2)) ] by multiplying this fraction by -1/-1.  Apparently I am mistaken because this changes the answer.  The only differences between these two answers is the sign of the 2t term in the numerator, otherwise the answers would be identical.  So, the question is: How do you change the middle term (of the original problem) denominator from (t^2 – s^2) to (s^2 – t^2) without changing the problem?

Answer
Yes, those two middle terms are equivalent, and no, it can't change the answer.  I gave you the two answers and you can go from one to the other by multiplying by -1/-1.  All of the signs in the numerators of the two answers are opposite, not just 2t.  I hope you can eventually get it straight.