Variable-rate linear network codes are investigated in this paper, which are referred to as linear network codes that can support a demanded range of transmission rates on a common netowrk. A new kind of linear network code, called as strict linear broadcast, is defined. Compared with general linear broadcast, it imposes more rigid constraints on the global encoding kernels, but does not require larger finite field size for construction. Then, an efficient scheme is proposed to construct variable-rate linear broadcasts based on the strict linear broadcast. Instead of construcing a fix-rate linear broadcast for each demanded transmission rate, this scheme implements variable-rate linear broadcasts with a single-rate strict linear broadcast. Every node in the network, including the source node, needs to store only one local encoding kernel. When transmission rate varies, the coding operations performed on every network node remain unchanged. Thus, small storage space and no kernel-swithching operations are required on any network code. Furthermore, by combining the strict linear broadcast with a special source-data packetization strategy, a hierarchical broadcast scheme is proposed. With this scheme, multi-rate service can be provided by a single-rate strict linear broadcast to heterogeneous receivers, even at variable transmission rate. Thus, the variable-rate linear broadcasts constructed in this paper are also applicable to the network with heterogeneous receivers.