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 resultingsystem of equationsfor 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.