How do you solve this equation? 10 pts. (a-5)/b = (a+5)/b + (b+5)/a?

Question

For some reason I keep on getting different answers to this. Sorry, the answer isn't in the back of the book so I don't know how to get there. 10 pts. to someone who can explain with the clearest method and direction. Thank you so much!

Solve for a:

(a-5)/b = (a+5)/b + (b+5)/a

I always end up with 5a=ab, but then it screws up after that.

Thanks.

Answer 1

If you end up with 5a = ab, then 5a - ab = 0

a(5 - b) = 0, so either a = 0, or b = 5

Now to see if that's what you should end up with!

(a - 5) / b = (a + 5) / b + (b + 5) / a

multiply through by ab

a(a - 5) = a(a + 5) + b(b + 5)

a^2 - 5a = a^2 + 5a + b^2 + 5b

-10a = b^2 + 5b

a = -(b^2 + 5b) / 10 <==== answer

edit: if the last fraction is supposed to be (b - 5) / a, then that only changes one thing:

-10a = b^2 - 5b

(EDIT for clarification)

a = (b^2 - 5b) / (-10) = -(b^2 - 5b) / 10 = (-b^2 + 5b) / 10

a = (5b - b^2) / 10