If a is not equal to b, what is the value of a-b

[dk08]

I didn't understand the reasoning behind the simplification.

If a is not equal to b, what is the value of a-b
_____

a^2 - b^2


(1) a + b = 8
(2) a - b = 6

The question can be factored to:

a - b
___________
(a - b)(a + b)


Here is my problem, I thought the (a-b)'s would cancel and you would be left with just (a+b). Is that incorrect? But the solution says that its 1/ (a+b). Usually, you cancel the two like terms. I though so anyway. Could someone please explain why that's not true here.

which can be further simplified to :

1
_____
a + b



[/b]

[dbernst]

dk,

When reducing a fraction, you cannot simply eliminate the numerator (top) when you "cancel" like terms. For example, the fraction 2/12 can be rewritten as 2/(6)(2). Then, the 2's "cancel" and the fraction in its reduced form is 1/6.

The same is true in this problem. When the (a-b) terms are "canceled," they are actually reduced to 1/1. Thus, the reduced fraction is 1/(a+b).

-dan

[HanJ791]

Sorry for bringing up old posts.

Not sure if this thread is still active. I had a question on this one. If we cancel out the (a-b), aren't we necessarily not finding all the values available? So technically we only half answered this question?

The math is straightforward where we can factor the a^2 - b^2, but we eliminate the ability to find both values when we select 1.

[Sage Pearce-Higgins]

It's fine to ask questions about an old thread. I'm having trouble locating the source of this problem (and the initial post doesn't make a lot of sense to me); please could you tell me where it's from, or post the whole problem?

It may be that cancelling the (a-b) doesn't exclude any possible solutions because we know that (a-b) is not equal to zero (we have to be careful when dividing by a variable if it could be zero).

[HanJ791]

It was one of the CAT questions. I took a screenshot and uploaded it to imgur.
https://imgur.com/a/eEqpf

[Sage Pearce-Higgins]

Thanks, it makes more sense now. Cancelling the (a-b) is no problem here as we have an expression, not an equation. We're not trying to solve an equation, but trying to find the value of an expression of a and b. This is a classic "combo" problem, in that we might think "We need to know what a and b are individually, so the answer must be C." Of course, there's a cunning way to get the value with just statement (1).

To make things clearer, try actually finding the value of the expression, both using statement (1) alone, and using statement (2). You could then try out some different values for 'a' and 'b' that work for statement (2) alone, and what results they give you when you plug them into the expression.