X:Y pair can be (+,+), (-,-), (+,-) or (-,+)

This part is correct.
From here we need to try each case and see which one avoids contradictory results.

Example, case 1 (+,+):
If y is positive then x + |y| – y = 6 simplifies to be x=6 (since |y|-y = 0 for all positive values of y.
When we plug x=6 into the equation x + |x| + y = 7, we get y=-5
But this can't be since this case assumes that y is positive.
Contradictory result.

Each case except "x is positive and y is negative" will yield a contradictory result (or solution that does not work for both equations).

The answer is C