For two lines to intersect, don't they have to share a point? If so, then:
. . . . .1 + 2t = 1 + 3s
. . . . .-1 - t = 2 + 2s
. . . . .3t = 3 + 4s
Pick any two of the equations, and solve the resulting system of equations for the values of "s" and "t".
Whatever the result, plug that into the remaining original equation, and see if it works. If it does, then I believe you have an intersection point. Otherwise, not.