Is a ring R, modulo an ideal I (generated by x), then modulo an ideal J (generated by n) the same as R modulo the ideal generated by (n,x)?

Question

Is the following statement true?

$$ R/(x,n) = \left[ R/(x) \right] / (n) $$

My thinking behind it was as follows:

\begin{array}{ccc} \left[ R/(x) \right] / (n) & = & \{ r+(n) : r \in R/(x) \} \\ & = & \{ \left[ r + (x) \right] + (n) : r \in R \}\\ & = & \{ r + ax + bn : r,a,b \in R \} \\ & = & \{ r + (n,x) : r \in R \} = R/(n,x) \end{array}

Answer 1

This is one of the isomorphism theorems:

If $I \subset J$, we have a canonical isomorphism $(R/I)/(J/I) \cong R/J$.

Take $I=(x), J=(x,n)$.