  With the use of the $(f,g)$-matrix inversion under specializations that
$f=1-xy,g=y-x$, we establish an $(1-xy,y-x)$-expansion formula. When
specialized to basic hypergeometric series, this $(1-xy,y-x)$-expansion formula
leads us to some expansion formulas expressing any ${}_{r}\phi_{s}$ series in
variable $x~t$ in terms of a linear combination of ${}_{r+2}\phi_{s+1}$ series
in $t$, as well as various specifications. All these results can be regarded as
common generalizations of many konwn expansion formulas in the setting of
$q$-series. As specific applications, some new transformation formulas of
$q$-series including new approach to the Askey-Wilson polynomials, the
Rogers-Fine identity, Andrews' four-parametric reciprocity theorem and
Ramanujan's ${}_1\psi_1$ summation formula, as well as a transformation for
certain well-poised Bailey pairs, are presented.
