The paper presents two approximate methods for solving the band-structure problem in a physically transparent and efficient manner. Both methods use the variational principle for the one-electron Hamiltonian, with trial functions being linear combinations of energy-independent augmented plane waves (APWs) and muffin-tin orbitals (MTOs). The secular equations are eigenvalue equations linear in energy. The trial functions are defined with respect to a muffin-tin potential, and the energy bands depend on the potential parameters, which describe the energy dependence of the logarithmic derivatives. Inside the spheres, the energy-independent APW is a linear combination of an exact solution at a fixed energy and its energy derivative, matching continuously and differentiably onto the plane-wave part in the interstitial region. The errors in the energies obtained with the linear-APW method for the muffin-tin potential are of order $(E - E_0)^4$. Similarly, the energy-independent MTO is a linear combination that matches onto the solution of the Laplace equation in the interstitial region, which is regular at infinity. The errors in the energies obtained with the linear-MTO method are additional errors of order $(E - V_{\text{mt}})^2$ arising from the interstitial region where the potential is $V_{\text{mt}}$. The linear-APW (LAPW) method combines desirable features of the APW and OPW methods, treating $d$ bands and having a linear energy dependence of its pseudopotential. The linear-MTO (LMTO) method is particularly suited for closely packed structures, combining features of Korringa-Kohn-Rostoker, linear-combination-of-atomic-orbitals, and cellular methods. The secular matrix is linear in energy, and the overlap integrals factorize as potential parameters and structure constants, which are canonical and do not depend on energy or cell volume. The method is well-suited for self-consistent calculations. The empty-lattice test is applied to the linear-MTO method, accurately reproducing the free-electron energy bands. The paper also discusses how relativistic effects can be included in both the LAPW and LMTO methods.The paper presents two approximate methods for solving the band-structure problem in a physically transparent and efficient manner. Both methods use the variational principle for the one-electron Hamiltonian, with trial functions being linear combinations of energy-independent augmented plane waves (APWs) and muffin-tin orbitals (MTOs). The secular equations are eigenvalue equations linear in energy. The trial functions are defined with respect to a muffin-tin potential, and the energy bands depend on the potential parameters, which describe the energy dependence of the logarithmic derivatives. Inside the spheres, the energy-independent APW is a linear combination of an exact solution at a fixed energy and its energy derivative, matching continuously and differentiably onto the plane-wave part in the interstitial region. The errors in the energies obtained with the linear-APW method for the muffin-tin potential are of order $(E - E_0)^4$. Similarly, the energy-independent MTO is a linear combination that matches onto the solution of the Laplace equation in the interstitial region, which is regular at infinity. The errors in the energies obtained with the linear-MTO method are additional errors of order $(E - V_{\text{mt}})^2$ arising from the interstitial region where the potential is $V_{\text{mt}}$. The linear-APW (LAPW) method combines desirable features of the APW and OPW methods, treating $d$ bands and having a linear energy dependence of its pseudopotential. The linear-MTO (LMTO) method is particularly suited for closely packed structures, combining features of Korringa-Kohn-Rostoker, linear-combination-of-atomic-orbitals, and cellular methods. The secular matrix is linear in energy, and the overlap integrals factorize as potential parameters and structure constants, which are canonical and do not depend on energy or cell volume. The method is well-suited for self-consistent calculations. The empty-lattice test is applied to the linear-MTO method, accurately reproducing the free-electron energy bands. The paper also discusses how relativistic effects can be included in both the LAPW and LMTO methods.