<p>This paper develops a comprehensive Galerkin approximation theory for quasilinear elliptic systems of the form: a positive constant times u minus the divergence of a matrix-valued coefficient a(x,u) times the gradient of u plus a lower-order term b(x,u,grad u) equals a given right-hand side f. The coefficient a(x,u) is assumed to be uniformly elliptic, and the nonlinearity b(x,u,grad u) satisfies form-bounded growth conditions. We construct a nonlinear operator from the Sobolev space of functions with one derivative in Lp into its dual and establish its key properties: coercivity, strict monotonicity in Lp, and hemicontinuity. Using these properties, we prove the convergence of Galerkin approximations and establish existence, uniqueness, and regularity results for weak solutions. The framework accommodates vector-valued problems and provides explicit convergence rates under appropriate spectral gap conditions. We further extend the theory to measurable coefficients via mollification techniques, establish higher regularity under strengthened assumptions, and develop a parallel theory for variable exponent Sobolev spaces.</p>