We report the structural and magnetic properties of a quantum magnet PbCuTeO5. The triclinic structure of PbCuTeO5 comprises of alternating layers (ab-planes), in which one layer is composed of S = 1/2 dimer chains and another layer is composed of S = 1/2 trimer chains formed by corner-shared CuO4 units. Magnetic susceptibility data show a Curie-Weiss behavior with an antiferromagnetic Curie-Weiss temperature (Theta_CW) of -165 K. At low temperature, both the heat capacity Cp(T) and susceptibility data show an anomaly at Tc = 6 K with a weak ferromagnetic (WFM) moment suggesting the appearance of long-range order. The magnetization vs. field M(H) data at 2 K also provide evidence for WFM behavior. Magnetic frustration with a frustration parameter f = Theta_CW/Tc of about 27 is observed. Magnetic specific heat data suggest the presence of a large entropy in the paramagnetic region, well above Tc, suggesting the presence of short-range spin correlations. The observed results might originate from the frustrated network of S = 1/2 distorted checkerboard lattice formed due to the coupling of the spin chains via TeO6 octahedral units in the ab-plane.