In this paper, we propose and study a new joint load balancing (LB) and power allocation (PA) scheme for a hybrid visible light communication (VLC) and radio frequency (RF) system consisting of one RF access point (AP) and multiple VLC APs. An iterative algorithm is proposed to distribute users on APs and distribute the powers of the APs on their users. In the PA subproblem, an optimization problem is formulated to allocate the power of each AP to the connected users for total achievable data rate maximization. In this subproblem, we propose a new efficient algorithm that finds optimal dual variables after formulating them in terms of each other. This new algorithm provides faster convergence and better performance than the traditional subgradient method. In addition, it does not depend on the step size or the initial values of the variables, which we look for, as the subgradient does. Then, we start with the user of the minimum data rate seeking another AP that offers a higher data rate for that user. Users with lower data rates continue reconnecting from one AP to another to balance the load only if this travel increases the summation of the achievable data rates and enhances the system fairness. Two approaches are proposed to have the joint PA and LB performed: a main approach that considers the exact interference information for all users, and a suboptimal approach that aims to decrease the complexity of the first approach by considering only the approximate interference information of users. The numerical results demonstrate that the proposed algorithms improve the system capacity and system fairness with fast convergence.