We propose a linear electrolyzer model for steady-state load flow analysis of multi-carrier energy networks, where the electrolyzer is capable of producing hydrogen gas and heat. For our electrolyzer model, we show that there are boundary conditions that lead to a well-posed problem. We derive these conditions for two cases, namely with a known and unknown heat efficiency parameter. Furthermore, the derived conditions are validated numerically. Moreover, we investigate the extensibility of our model by including nonlinear models from electricity, gas, and heat. In this setting, we derived boundary conditions based on our previous findings. Due to the involvement of nonlinearity, it is a challenge to prove that the boundary conditions lead to a well-posed problem. Therefore, we simulated the electrolyzer connected with an electricity, gas, and heat system. Additionally, we considered a known and unknown heat efficiency parameter. The numerical results support that the linear electrolyzer model is solvable in a multi-carrier energy network.