Why does combining these two linear diophantine equations yield solutions, when each individually has none? The two following linear diophantine equations have no solutions:
$$412x + 18y = 49$$
$$33x + 99y = 15$$
We can however combine them as $412x + 18y - 49=0, 33x + 49y - 15 = 0$ gives
$412x+33x+18y+49y-49-15=0 \to $
$$445x + 117y = 64$$
This has solutions $x=-3584 + 117k, y=-13632-445k$. Why? Doesn't $A+C=B+D$ imply $A=B, C=D$, and if not, why does it work fine when we combine equations so many other times in math? 

You lost some information.  You have $a=0$ and $b=0$ and (essentially) conclude that $a=b$, which is correct.  But you've lost the information that each side equals $0$.  You'd get the same conclusion if $a=5$ and $b=5$.  Concluding $a=b$ is correct, but you've lost the $5$-ness.
Since you lost information, you have fewer restrictions on your variables, and so the new equation allows more solutions.
