What can be substituted in this system of equations with absolute values?

Question

I am trying to figure out what I can substitute in this system of equations that involve absolute values - $$|x+1|+|y|=1$$ $$|x+y|=2$$ My method is to do every combination and check with my cases to see what answers are correct. But they used a substitution method, it looks like they substituted $x$, but why do that in the first place?
If anyone can point me as to what I could use as a substitute I would be grateful. The answers they got are this $(t,-2-t)$ and $t∈[-2,-1]$ and checking my answers I got the correct ones $(-2,0)$ and $(-1,-1)$ my only two answers from 8, but I would like to learn what they did too.

Answer 1

$|x+y|=2 \implies (x+y=2)$ or $(x+y=-2)$. In the first case: $$|x+1|+|y|=1 \implies |x+1|+|2-x|=1$$ Check the cases: If both are positive, we may remove the absolute value signs. If either is not positive, we must multiply the inside by negative one....

Otherwise: $$|x+1|+|y|=1 \implies |x+1|+|-2-x|=1$$ And do the same here.