Two variables - two equations trap

Dear iliavko,

I'm happy to respond.

I will distinguish three cases:
Case I: one equation enough to solve
Case II: two equations needed to solve
Case III: two equations not enough to solve for two variables
I point out that Case II is statistically the most common case, and most often tested. That's the rule, but there are exceptions.

Case I only happens when the question is asking not for the value of x or the value of y, but simply for the value of some expression. Here's a very simple example. Suppose the question is "What is the value of y - x?" Suppose we are given the single equation 2y - x = 28 + x. We could rearrange that equation to solve for the desired expression, which equals 14, despite the fact that we cannot find the individual values of the variables.

Case II happens when we have two different equations that represent two lines with different slope. Any two lines in the x-y plane that have different slopes will intersect at exactly one and only one (x, y) point. That determines a unique and unambiguous solution.

Case III happens when the two lines have the same slope. For example:
Equation #1: x - 2y = 5
Equation #2: 4y + 6 = 2x
It's not necessarily obvious at first glance, but if you were to re-write these both in y = mx + b form, you would see that they have the same slope. If you try to solve them, you will get mathematical nonsense. In fact, they are parallel lines, and they never intersect, so there's no solution.
A variant on this case:
Equation #1: 3x + 2y = 5
Equation #2: 4x = 10 - 6y
This what mathematicians call a "degenerate system," because the two equations are really one equation in two different forms. Here, we can't solve, because absolutely every value that works in equation #1 will work in equation #2 as well, because they're the same equation! This has a continuous infinity of possible solutions, which in practice, is equivalent to having no solution.

Does all this make sense?
Mike