As high-performance computing (HPC) scales to tackle increasingly complex problems, balancing computational efficiency and precision has become a challenge. Reduced-precision floating-point formats, such as FP16 and FP32, offer significant speedups but require careful strategies to maintain numerical stability. This talk explores how mixed-precision approaches can enhance the performance of dense linear algebra operations and solve linear systems efficiently. Highlighting techniques implemented on modern GPU architectures, we discuss their applicability to scientific computing workloads, including LU factorization and beyond. The discussion also addresses challenges in optimizing sequential portions of computation, which often leave GPU resources underutilized, and presents strategies for improving both parallel and sequential execution.