From the two equations, we get \(x=6\) and \(y=1\), so \(x+y=7\), which is not among the answer choices you listed.



Here's the quick reasoning:

1. From \(x+|y|-y=6\) :

- If \(y \geq 0\), then \( |y|=y\), so the equation becomes \(x=6\).
- If \(y<0\), then \(|y|=-y\), so the equation becomes \( x-2 y=6\).

2. Use the first equation x+|x|+y=7 :

- If \(x \geq 0\), then \(|x|=x, so 2 x+y=7\).
- If \(x<0\), then \(|x|=-x, so y=7\).

Now check consistent sign cases:

- Case 1: \(x \geq 0, y \geq 0\)

From above, \(x=6\).

Plug into \(2 x+y=7: 12+y=7 \Rightarrow y=-5\), which contradicts \(y \geq 0\). So no solution here.

- Case 2: \(x \geq 0, y<0\)

From |y| equation: \(x-2 y=6\).
From |x| equation: \(2 x+y=7\).
Solve:

$$
\(2 x+y=7 \Rightarrow y=7-2 x\)
$$


Substitute into \(x-2 y=6 \):

$$
\(x-2(7-2 x)=6 \Rightarrow x-14+4 x=6 \Rightarrow 5 x=20 \Rightarrow x=4\)
$$


Then \(y=7-2(4)=-1\), consistent with y<0 and \(x \geq 0\).
- Case 3: \(x<0\) leads to contradictions when checked similarly.

So the only valid solution is x=4, y=-1, giving x+y=3.



Thus the correct answer is C. 3.