Ian's post is very informative. We can summarize the exceptions as follows:
if you have n distinct linear equations and n variables, you can always solve the entire system.
Most of the "exceptions" that appear on the GMAT aren't actually exceptions to the rule; in the cases in which it turns out that it looks like we have enough information to solve but we don't, it's not because the rule doesn't apply: it's because our equations either aren't linear or aren't distinct. Ian does a great job of explaining those terms, so I won't bother!
The more common situation we find ourselves in, especially in DS, is when we don't actually need n distinct linear equations to solve the problem.
We only need those n distinct linear equations for sure if we want to solve the entire system. When we're solving for part of the system, or for a relationship among the variables rather than the variables themselves, we can often solve with as few as 1 equation.
Here are a couple of examples:
Q: what's the value of 3x - 4y?
(1) 9x - 12y = 21
(2) 9x + 12y = 33
Our first thought may be: hey, 2 equations for 2 unknowns, the answer is "C".
However, if we examine (1) more closely, we see that we can divide both sides by 3 to get:
3x - 4y = 7
which certainly answers the question "what's the value of 3x - 4y?".
So, since (1) gives us the exact relationship for which we're looking, it's good enough all by itself.
Q: what's the value of c?
(1) a - 2b - 2c = 14
(2) 12 - c = 2b - a
At first glance, it looks as though we have 2 equations but 3 variables, so there's no way to solve. However, if we rearrange the second equation and line it up under the first, we have:
a - 2b - 2c = 14
a - 2b - c = -12
and now when we subtract the second equation from the first, we get:
-c = 26
which certainly allows us to solve for c.
So, even though we only have 2 equations, combined they allow us to solve for the 1 variable about which the question asks: together sufficient.
Here's a list of questions to ask yourself in these situations:
1) do I need to solve the entire system?
if yes, then I need 1 distinct linear equation for each variable; if no, then I should look for special cases.
2) are the equations distinct and linear?
3) am I being asked to solve for the values of the variables or for a relationship among variables?
And here's the big thing to watch out for:
1) has the testmaker made it difficult to compare the statements? (e.g. are the variables on different sides of the equal sign?)
If so, that's because the testmaker doesn't want you to compare them; when in doubt, rearrange them to make them easier to compare.
