A general linear technique is discussed which constructs unitary scattering amplitudes without expanding in partial waves and in the presence of inelastic channels. Two- and three-particle intermediate states are discussed explicitly, but the method can be extended directly to any finite number of particles. A new approximation technique suggested by this formalism is applied to electroproduction of pions from pions and pion-K-meson scattering. A form of the impulse approximation is derived for both the coupled form factor and the coupled scattering amplitude problems. The nucleon and deuteron form factor system is briefly discussed. Finally, a model field theory which contains three-particle intermediate states is formulated and solved by the linear technique for purely pedagogical reasons.