Magnetohydrodynamic (MHD) tokamak equilibria are found with values of β up to 20% and realistic MHD safety factor values (e.g. q(axis) = 1 and q(edge) = 4.8) for tokamaks with aspect ratio A = 4 and D-shaped cross-section. If such equilibria can be attained experimentally, they will be very attractive for decreasing the projected costs of tokamak power reactors. In the flux-conserving tokamak (FCT) model, where rapid heating is applied to an already relatively hot plasma, these high-β equilibria are achievable. The quasi-static evolution of FCT equilibria as β increases is studied. An operating window is found in the pressure profile width wp at the half-height: for high β, the values of wp lie between 0.40 and 0.55 times the minor plasma diameter. Within this window, plasma current and poloidal β increase with β. For fixed plasma boundary, significant poloidal surface currents are induced, but these can be eliminated by small increases in the plasma minor radius, the pressure profile width, or the vacuum toroidal field.