How can I verify that I have all solutions of a system of equations with absolute values

Question

The problem is proposed as follows:
For two real numbers $a$ and $b$ with $|a| \neq |b| $ find all solutions to the following system of equations:
$x + |y| = a$ and $|x| + y = b$, where $x$ and $y$ are real numbers.
I approached this problem by first solving for $y$ and then substituting it into the second equation:
$y = a - x \vee y = x - a$, so for $x$ the solutions are $x = \frac{a + b}{2} \wedge \frac{a - b}{2}$ . Substituting $x$ into the first equation again then gives $y = a - \frac{a-b}{2} \vee y = -a + \frac{a-b}{2} \vee y = a - \frac{a + b}{2} \vee y = -a + \frac{a + b}{2}$. But by "symmetry" of the problem the solutions for $x$ and $y$ can be interchanged if you switch all the $a$s and $b$s. I am not sure if those are all the solutions, because I didn't really use the condition that $|a| \neq |b$. Considering this equation, you have to put some restrictions on $x$ and $y$, for example that both of them can't be positive or negative at the same time, which on the other hand puts restrictions on $a$ and $b$, since the system of equation was solved relative to them, wouldn't it? So, how should I proceed in verifying the solutions of the problem? Excuse me, if I missused some terms, I am not a native speaker and essentially have to translate the terms and the problems, so I might have made a mistake there somewhere.