I'm finding out that this is due to a problem in algebra called "extraneous solutions".
Why does this problem arise? I'm having problems finding it... Is algebra flawed?
(x+1)/(x-2)=3/(x-2)+5 - Wolfram|Alpha
Why on earth can't I just multiply by (t-2) and get the solution t=2 ?!
To make it absolutely clear that I should be able to just multiply by (t-2), I can rewrite this equation as:
I am VERY confused..
I do realize that t can't be 2 because of the denominator becoming 0, but why does it give me the algebraic solution t=2 and then it doesn't work?
The reason that extraneous solutions sometimes occur is because if you were to raise both sides of the equation to a higher power such as cubed, you would increase the number of the solution. This new number night not satisfy the original equation.
Well it wouldn't give you the solution t=2 if you followed the rules of algebra and did not take (which is what you are doing when you cancel to 1.
Simply put this equation has no solution because it is wrong to begin with, you can see this more clearly if you rearrange it.
Nothing divided by itself can equal 0 so you can see that the equation is false to begin with.
It is not true for all t. When t=2 it is not equal to 1. So when you are trying to find a solution to the equation when you make the step to turn (t-2)/(t-2) into 1 you must make a note that any conclusions you reach are under the assumption that t is not equal to 2. And when you continued to find a solution you reached the conclusion that t=2. As I said "any conclusions you reach are under the assumption that t is not equal to 2." so your conclusion that t=2 is under the assumption that t is not equal to 2.