jmc

Taking the conjugate of both sides, we get $$\overline{a+\overline{b}}=\overline{2-7i}=2+7i.$$But $\overline{a+\overline{b}}=\overline{a}+\overline{\overline{b}}=\overline{a}+b,$ so $$\overline{a}+b=\boxed{2+7i}.$$