Numerical issues in solving linear systems
The $2\times 2$ matrix has determinant close to zero, and so its condition number is very large, causing numerical instability.

The explicit solution of your system is
$$
x=\frac{a}{a^2-b^2}, y=-\frac{b}{a^2-b^2}.
$$
Therefore
$$
x+y = \frac{a-b}{a^2-b^2} = \frac{1}{a+b},
$$
thus avoiding the numerical issues.