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? 
 A: 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.
A: $A+C$=$B+D$ doesn't imply $A=B, C=D$. Why should it? $1+(-1)=(-1)+1$, but $1\neq -1$. 
The converse: $A=B, C=D\Rightarrow A+C=B+D$ is always true, this is where I think you're getting confused. 
A: $A+C=B+D$ does not imply $A=B \land C= D$. Consider for instance $A=2,C=2,B=1,D=3$. In fact, there is an infinite amount of real number quadruples that are counter examples to your conjecture.
A: You modify what your answer has to fit. $$33x+99y=15$$ forces it to be a multiple of 5.  Rewriting it as :$$(6\cdot 5+3)x+(19\cdot 5+4)y$$ distributing, not regrouping, subtracting away the multiples of 5 to the right hand side, and factoring out the factor 5, we get:$$3x+4y$$ needs to be a multiple of 5. This is possible, but it changes what multiple of 5 we need to get to, too much to get to it. At least in integer solutions. 
As others have laboured, $$A+C=B+D$$ doesn't mean any of the variables are equal. Goldbach's Conjecture, could be settled if this were the case. You can actually find out, that without loss of much generality :$$C-B=D-A$$ As long as the variables came from opposite sides, you can make a statement of this sort. 
