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?

I believe its because the difference between $a$ and $b$ is dependent on their values in every case and it just so happens the case where the difference did not depend on them is when they both drop by the same percentage. Let me illustrate using the example you've given.
$$0.65a - 0.55b = 50$$
$$0.55a - 0.55b = 50 - 0.10a$$
$$a-b = \frac{50-0.10a}{0.55}$$
As you can see the difference now depends on $a$ and this is generally what happens when the percentage drops are not equal. Intuitively think about it this way, if $a$ and $b$ both drop by a particular same percentage the difference will aswell, but if they don't the difference depends on how much of an impact the dicount on $a$ does with respect to $b$ which depends on their values.
