  We present an explicit expression for the $q$-difference system, which the
$BC_1$-type Jackson integral ($q$-series) satisfies, as first order
simultaneous $q$-difference equations with a concrete basis. As an application,
we give a simple proof for the hypergeometric summation formula introduced by
Gustafson and the product formula of the $q$-integral introduced by
Nassrallah-Rahman and Gustafson.
