Write the system first in matrix form:
((1,2,-3),(2,4,-5),(3,-1,1))*(x,y,z)=(4,12,3)
Next, check the determinant of the matrix. If it is zero, the system has no solution (or, if the right side was zero, infinitely many solutions).
In this case, the determinant is non-zero:
det ((1,2,-3),(2,4,-5),(3,-1,1))=7,
which means it the system has a unique solution. You find this solution by computing the inverse of the matrix,
((1,2,-3),(2,4,-5),(3,-1,1))^{-1}=1/7*((-1,1,2),(-17,10,-1),(-14,7,0))
and multiplying it by (4,12,3). You get
(x,y,z) = (2,7,4).