The map
$$X \mapsto N(X) := \#(\text{connected components of }X)$$
extends to a ring homomorphism $K_0(Var/k) \to \mathbf{Z}$. So $[X] + [Y] = 0$ implies that $N(X) = N(Y) = 0$, hence $X$ and $Y$ are both empty.
HYL
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