This article presents a numerical study to investigate boundary-layer heat transfer fluid associated with a moving flat body in cooperation of variable plate and streaming-free velocity along the boundary surface in the laminar flow. The thermal conductivity is supposed to vary linearly with temperature. Similarity transformations are applied to render the governing partial differential equations for mass, momentum and energy into a system of ordinary differential equations to reveal the possible existence of dual solutions. MATLAB package has been used to solve the boundary value problem numerically. We present the effects of various parameters such as velocity ratio, thermal conductivity and variable viscosity on velocity and temperature distribution. The analysis of the results concerning Skin friction and Nusselt number near the wall is also presented. It is focused on the detection and description of the dual solutions. The study reveals that the undertaken problem admits dual solutions in particular range of values of different physical parameters. It can be seen that for the first branch solution, the fluid velocity decreases near the sheet, but it increases far away from the sheet for velocity ratio parameter, whereas the opposite effect is induced for second branch solution. Skin friction coefficient and rate of heat transfer increase due to increase in thermal conductivity parameter.