Silicon carbide (SiC) crystal growth from ternary solutions Si-C-X where X is a transition metal was studied. In order to select the desirable transition element and to determine the solution composition, we have conducted the calculations of ternary phase diagrams by means of CALPHAD (CALculation of PHase Diagrams) method. Preliminary growth experiments without a seed crystal were also performed. Among various Si-based solutions, Si-C-Ti was one of the most effective solutions to increase crystal growth rate compared with Si-C. Optical microscopic observation of the obtained SiC etched by molten KOH showed no micropipe defects in the crystals. We have also performed the growth experiments with 6H-SiC seed crystal under temperature gradient. As a result, we have successfully obtained a 12mm X 12mm self standing SiC crystal.