Gröbner Basis and linear basis

Question

Let $I$ be an ideal of a polynomial algebra $A$ with a Gröbner basis $G$. Suppose we know how to describe the leading terms of all elements in $G$, denoted by $\{i_1,\dots,i_k\}$, so that we can give an explicit basis for the vector space $A/I$, denoted by $\{b_1,\dots,b_n\}$. Now, suppose we have a principal ideal $J=\langle g\rangle$. How to construct a Gröbner basis for $I+J$ and an explicit basis for $A/(I+J)$ with the data above?