Say, in what domain are we solving this? If it's the reals, than this assumption doesn't hold. Take for example the way less general (a=1, b=1):

So your assumption would imply that c+d=2, but one can find other c and d that solve the above system. Let u=c+d, so from the second:

Adding twice to the first equation

As, expected, one solution is u=c+d=1+1=2. But the other roots

are the roots of the quadratic:

And those are .

So, c+d can be equal to those numbers, and not just to 2=1+1=a+b.

For example:

Where

one of the roots of the above quadratic.