An equation with two variables is unsolvable for either one, but how can I know if it's unsolvable as an expression for both? Weird title perhaps, so let me illustrate with the question that got me thinking about this problem:

You are buying a laptop and have two to choose from. What is the
  difference between the original prices of the two laptops?
What you know: After the laptops have come down in price 35 % and 45 % respectively, the difference in price between them is $50.
Can this answer be determined?

Intuitively, I would say no. If I write the problem algebraically I get:
0.65a - 0.55b = 50

It's obvious we can never work out a or b from this, but that's not what they're asking. They're asking what a - b (or rather |a - b|) is.
I understand that this answer still can't be determined, but if the percentages were 50 (or any same number) I could instead write:
0.5a - 0.5b = 50

Which simplifies to:
a - b = 100

I'm not quite understanding why I can't solve for a - b from any equation that includes a and b. What's the inutition to help me understand why this question is unsolvable in case of different percentages, but solvable in the case of the same?

Notice these are simultaneous equations.  You can break $a$ and $b$ out using a thing called parametric equations.
Suppose that $a=bc$ for some $c$.  There indeed must exist a $c$ for each $a,b$.
Then, we would have (original problem)
$$0.65bc-0.55b=50$$
$$b=\frac{50}{0.65c-0.55}$$
Similarly, if we multiply by $c$, we get $a=bc$, so
$$a=\frac{50c}{0.65c-0.55}$$
Now we can solve for $a-b$, which is
$$a-b=\frac{50(c-1)}{0.65c-0.55}$$
In the case that it was a 50/50 relationship, the denominator would have a factor of $c-1$, which would cancel and we'd have no $c$'s left over.  However, this is not the case, and so we are left with this, our solution in terms of $c$.
