A lattice QCD computation of the coupled channel $\pi\Sigma−\bar{K}N$ scattering amplitudes in the $\Lambda(1405)$ region is detailed. Results are obtained using a single ensemble of gauge field configurations with $N_f=2+1$ dynamical quark flavors and $m_\pi\approx200$ MeV and $m_K\approx487$ MeV. Hermitian correlation matrices using both single baryon and meson-baryon interpolating operators for a variety of different total momenta and irreducible representations are used. Several parametrizations of the two-channel scattering K-matrix are utilized to obtain the scattering amplitudes from the finite-volume spectrum. The amplitudes, continued to the complex energy plane, exhibit a virtual bound state below the $\pi\Sigma$ threshold and a resonance pole just below the $\bar{K}N$ threshold.