  We extend the quark Born diagram formalism for hadron-hadron scattering to
processes with valence $q\bar q$ annihilation, specifically $I=0$ and $I=1$
$\pi\pi$ and $I=1/2$ $K\pi$ elastic scattering. This involves the $s$-channel
hybrid annihilation process $q^2\bar q^2 \to q\bar q g \to q^2\bar q^2$ and
conventional $s$-channel $q\bar q$ resonances, in addition to the $t$-channel
gluon exchange treated previously using this formalism. The strength of the
$t$-channel gluon amplitude is fixed by previous studies of $I=2$ $\pi\pi$ and
$I=3/2$ $K\pi$ scattering. The $s$-channel resonances $\rho(770)$, $K^*(892)$,
$f_0(1400)$ and $K_0^*(1430)$ are incorporated as relativized Breit-Wigner
amplitudes, with masses and energy-dependent widths fitted to experimental
phase shifts. The strength of the $s$-channel gluon ``hybrid" annihilation
diagrams is problematical since the perturbative massless-gluon energy
denominator must be modified to account for the effect of confinement on the
energy of the virtual hybrid state. Our naive expectation is that near
threshold the hybrid diagrams are comparable in magnitude but opposite in sign
to the contribution predicted using massless perturbative gluons. Fitting the
strength of this amplitude to $I=0$ $\pi\pi$ and $I=1/2$ $K\pi$ S-wave data
gives a result consistent with this expectation. We find good agreement with
experimental phase shifts from threshold to 0.9 GeV in $\pi\pi$ and 1.6 GeV in
$K\pi$ using this approach. We conclude that the most important contribution to
low energy S-wave scattering in these channels arises from the nonresonant
quark Born diagrams, but that the low-energy wings of the broad $s$-channel
resonances $f_0(1400)$ and $K^*_0(1430)$ also give important contributions near
threshold. The nonresonant contributions are found to be much smaller in $L\;
