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?

Substitute $y=7-x-|x|$, giving an equation in a single unknown, $$x+|7-x-|x||-(7-x-|x|)=6.$$
Then for $x\le0$, $x=6$, which is impossible and for $x\ge0$, $x+|7-2x|-(7-2x)=6$.
You still  have to distinguish $2x\le7$, giving $x=6$, which is impossible, and for $2x\ge7$, $x-14+4x=6$ or $$\color{green}{x=4,y=-1}.$$
Not much simpler.
