Dimension of intersection of affine subspaces

Question

As an assignment we are to calculate the dimension of the intersection of two affine subspaces $A$,$B$ in $\mathbb{R}^6$ each defined by a system of linear equations of at most 6 variables and a constant.

How does one calculate that?

I tried to stack together the equations and simplify, then parametrize the solution as a $\vec v$ + {set of parametrized vectors}. Then the dimensionality of their intersection should be the size of the set. But when I carry out the calculations I get a contradiction.

Is my procedure valid? How else could I arrive at the solution? What are other methods of finding the intersection?

*Edit: Does contradiction indicate that the intersection is empty?

*Edit2: I add code I used to check if I made any numerical errors. After running the code, result is EmptySet()

The first 3 rows of $M$ is the augumented form of $A$.

The bottom 3 rows of $M$ is the augumented form of $B$

import sympy

M = Matrix([
[3, 4,  6,  0,  0,  0, 3],
[4, 0,  0,  1,  1,  0, 2],
[0, 1, -4,  0,  0,  0, 8],
[0, 1,  1, -1, -1,  2, 0],
[0, 0, -1,  1,  1, -2, 0],
[0, 1, -7,  0,  0,  0, 0]])

x1,x2,x3,x4,x5,x6 = sympy.symbols('x1,x2,x3,x4,x5,x6')
result = sympy.linsolve(M,(x1,x2,x3,x4,x5,x6))

Answer 1

Yes: the fact that you got the output EmptySet() indicates that the intersection of the two affine spaces is empty.

Using W|A confirms this result.