Find the Value of $a+b$ Given that $a + |a| + b= 7  $ and $a + |b| -b =6  $
The above equation  can be solved  by checking different value of $a$ and $b$. But how to do this  by solving the above two equations?
 A: There are $4$ cases:


*

*$a>0,b>0$


$$2a+b=7$$
$$a=6$$


*$a>0,b<0$


$$2a+b=7$$
$$a-2b=6$$


*$a<0,b>0$


$$b=7$$
$$a=6$$


*$a<0,b<0$


$$b=7$$
$$a-2b=6$$
Hence the solutions for $(a,b)$ are $(6,-5),(4,-1),(7,6),(20,7)$ for respective cases. 
Considering the respective domains, the only solution is $(4,-1)$.
A: Take the first equation, $a +|a|+b=7$.  If $a \lt 0$, this becomes $b=7$, which you can plug into the second equation.  If $a \ge 0$, this becomes $2a+b=7, b=7-2a$, which you can plug into the second.
A: Case 1: 
  a
  >
  0
  ,
  b
  >
  0
:
2a+b=7 and a+b-b=6 i.e. a=6, b=-5 , but this is not the case as we have taken b to be positive but value of b is coming out to be -5.
Case 2: 
  a
  <
  0
  ,
  b
  <
  0
:
a-a+b=7 and a-2b=6 i.e.b=7,a=20, this is also rejected for the similar resason as mentioned in case1.
Case 3: 
  a
  >
  0
  ,
  b
  <
  0
:
2a+b=7, a-2b=6 i.e. b=-1,a=4
Case 4 : 
  a
  <
  0
  ,
  b
  <
  0
:
a-a+b=7 ,a+b-b=6 i.e. b=7, a=6 ,similarly this is not possible.
so our final answer is a=4 and b=-1.
