python: less ugly solution to the intersection of 3 lines in 3 dimensional space
Instead of having variables firstX, secondX, thirdX, replace them with a list X
Don't put input and calculations in the same function, separate them
You do:

if D == 0:
     some stuff
 if D != 0:
     other stuff


use else instead


Your three lines calculating D, DX, DY, DZ are very similar. Make it a function.
The terms being subtractd parallel the ones being added. Extract the pair into a function.


Here is my result:

DEPENDENT = 0
INCONSISTENT = 1
GOOD = 2

def determinant(X, Y, Z):
    def term(x, y, z):
        return X[x]*Y[y]*Z[z] - X[z]*Y[y]*Z[z]
    return term(0, 1, 2) + term(2, 0, 1) + term(1,2,0)

def trinomial(X, Y, Z, A):

    D  = determinant(X, Y, Z)
    DX = determinant(A, Y, Z)
    DY = determinant(X, A, Z)
    DZ = determinant(X, Y, A)
#the next two if statements evaluate the D's, or determinants, to see if it's an inconsistant or identical systerm
#if it's not, then it does the math to solve the equation
    if D == 0:
       if DX == 0 and DY == 0 and DZ == 0:
           return DEPENDENT, None
       else:
           return INCONSISTENT, None
    else:
       return GOOD, [DX/D, DY/D, DZ/D]


def main():
    print "Trinomials require 3 equations with three variables each."
    X.append( input ("Please enter the coeficcient of X in the first equation: ") )
    Y.append( input ("Please enter the coefficient of Y in the first equation: ") )
    Z.append( input ("Please enter the coefficient of Z in the first equation: ") )
    A.append( = input ("and what's after the = sign?") )

    X.append( input ("Please enter the coeficcient of X in the second equation: ") )
    Y.append( input ("Please enter the coefficient of Y in the second equation: ") )
    Z.append( input ("Please enter the coefficient of Z in the second equation: ") )
    A.append( = input ("and what's after the = sign?") )

    X.append( input ("Please enter the coeficcient of X in the third equation: ") )
    Y.append( input ("Please enter the coefficient of Y in the third equation: ") )
    Z.append( input ("Please enter the coefficient of Z in the third equation: ") )
    A.append( = input ("and what's after the = sign?") )

    result, answer = trinomial(X, Y, Z, A)
    if result == DEPENDENT:
        print "System is dependant, there are infinite solutions"
        print
    elif result == INCONSISTENT:
        print "System is inconsistant, there are no solutions"
        print
    else:
        print "the solution to the set of equations is ("+ ",".join(map(str, answer)) + ")."
        print


But if you do any amount of python programming with this stuff you'll want to learn how to use numpy, which make your code:

def trinomial(X, Y, Z, A):
    system = numpy.array([X, Y, Z]).T
    unknowns = numpy.array([A])

    return numpy.linalg.solve(system, unknowns)