I wish to solve for $x$ in

$$ (A+B)x=y $$

given square symmetric matrices $A$ and $B$. For certain reasons I have already computed the Cholesky decompositions for A and B:

$$ A = L^T L $$ $$ B = M^T M $$

Can I use these solve for $x$ more efficiently than by naively computing the Cholesky decomposition of $A+B$?