solution of absolute value equation Question: If $x + |x| + y = 7$ and $x + |y| - y = 6$, then $x + y =?$
My solution:
I considered each cases of $x$ and $y$ i.e $x$ positive and $y$ positive, $x$ positive and $y$ negative, $x$ negative and $y$ positive, and finally $x$ negative and $y$ negative. After solving I found $x= 4$ and $y=-1$ is the solution. So, their sum is $4-1= 3$.  
Is there any smarter way to solve this type of equation?

You should consider one case:
If $y\ge0$ then $|y|=y$ , and
$$x+|y|-y=x=6$$
and
$$\underbrace{x+|x|}_{12}+y=7\implies y=-5$$
It is contradiction, hence $y<0.$ Similarly $x\le 0$ then $y=7$ whereas $x+\underbrace{|y|-y}_{0}=6$. So $x>0$ and $y<0$, we have
\begin{cases}
2x+y=7\\
x-2y=6
\end{cases}
