The dynamics of an incompressible, triple-diffusive medium flowing across a linearly stretched surface are examined in this study, taking into account the effects of radiation, viscous dissipation, magnetohydrodynamics (MHD), and the Boussinesq approximation. The work uses simulations to investigate how the combined effects of Soret (thermal-diffusion) and Dufour (diffusion-thermo) phenomena interact with MHD restrictions. Using the Lie-group transformation approach, the goal is to find symmetry reductions in the governing partial differential equations. The method of Runge-Kutta shooting is used to solve the modified system of equations. To show the effects of key factors, such as the buoyancy ratio and magnetic field parameters, a visual analysis is performed. The results show that for both fluid flow scenarios, a rise in the magnetic field parameter causes a drop in the velocity profile, with velocity gradients decreasing by a good percentage. Similar to this, changes in the buoyancy ratio factors cause notable adjustments to the distributions of salt concentrations. When the magnetic parameter varies, the Nusselt number decreases approximately 14.29 %. While interesting behavior is depicted in triple diffusion scenarios, the Nusselt number increase approximately 9.52%. when values of soret effect Sr varies, the percentage of Sherwood number Shr in double diffusion decreases by 40% whereas in triple diffusion the Sherwood number increase interestingly about 33.33%. The precision and dependability of the conclusions are verified by contrasting the simulation results with previously published research.