Recently, reconfigurable computing systems have been built which employ Field-Programmable Gate Arrays (FPGAs) as hardware accelerators for general-purpose processors. These systems provide new opportunities for highperformance computing. In this paper, we investigate hybrid designs that effectively utilize both the FPGAs and processors in the reconfigurable computing systems. Based on a high-level computational model, we propose designs for floating-point matrix multiplication and block LU decomposition. In our designs, the workload of an application is partitioned between the FPGAs and processors in a balanced way; the FPGAs and processors work cooperatively without data hazards or memory access conflicts. Experimental results on Cray XD1 show that with one Xilinx XC2VP50 FPGA (a relatively small device available in XD1) and an AMD 2.2 GHz processor, our designs achieve up to 1.4X/2X speedup over the design that employs AMD processors/FPGAs only. The performance of our designs scales with the number of nodes. Moreover, our designs achieve higher performance when improved floating-point units or larger devices are used.