Generating regular elements

Let R be a prime right Goldie ring. A useful fact is that, if $a,b\in R$ are such that $aR+bR$ contains a regular element, then there exists $\lambda \in R$ such that $a+b\lambda$ is regular. We show that the analogous result holds for $n?1$ pairs of elements: if R contains a field of cardinality at least $n+1$, and if $a_i,b_i\in R$ are such that $a_{i}R+b_{i}R$ contains a regular element for $1?i?n$, then there exists a single element $\lambda \in R$ such that $a_i+b_i\lambda$ is regular for each i.