Which is the correct answer? (adding algebraic fractions)

Question

Which is the correct answer to this question: 2/y+1 + 3/y-1

 

There are 2 different websites which have different answers to this-

one answer is: -(y +5)/(y+1)(y-1)

the other 2 answers are: 5y+1/(y+1)(y-1)   and   5y+1/y²-1

 

I unerstand how each question arrived to each answer but which would be the most correct answer? and can you convert the numerator of the first fraction to the numerators of the other two fractions or are they different.

Sorry so simple im re-learning this stuff again. thanks

Answer 1

2/(y+1)+3/(y-1)

[2(y-1)+3(y+1)]/(y+1)(y-1)

(2y-2+3y+3)/(y+1)(y-1)

(5y+1)/(y+1)(y-1) or (5y+1)/(y²-1) because (y+1)(y-1)=y²-1

Answer 2

First, when you post a question like this online, you need an extra set of parenthesis around the whole denominator. The question should be posted like 2/(y+1) + 3/(y-1). The reason is when you write this online, the simple slash doesn't tell people where it covers, so the additional parenthesis is needed to avoid the ambiguity.

 

To solve this question, you identified that the denominators are different so you need to make them the same, the way to do this is multiplying each other, and multiply the numerator by the same thing to keep the whole fractional constant. Namely, you change 2/(y+1) to 2(y-1)/((y+1)(y-1)) by multiplying both the numerator and the denominator by (y-1), similarly, you multiply both the numerator and the denominator of 3/(y-1) by (y+1), and you get 3(y+1)/((y+1)(y-1)). Note that I added a new set of parenthesis to avoid ambiguity. Now the denominator are the same and it's a piece of cake for us to add them up, to (2(y-1)+3(y+1))/((y+1)(y-1)) = (5y+1)/((y+1)(y-1)), or it's (5y+1)/(y^2-1)