Hello to everyone.

What the question means is that different ways of expressing the same relation between the data and unknown variables produce really weird fit results:

The problem: I have the unknown variables $x_1,x_2,x_3,x_4,x_5,x_6$ and the observations $\vec{q}_i=\Big(q_i(1),q_i(2),q_i(3)\Big), i=1\ldots N$
($N$ observations of real valued 3-vectors)

The relation between the unknowns and the observations is:

$\vec{q}_i\cdot(x_1,x_3,x_5)^T=-1$ (1st Set of $N$ equations)
$\vec{q}_i\cdot(x_2,x_4,x_6)^T=3$ (2nd Set of $N$ equations)

If i solve each set of equations as a separate overdetermined linear system the solution error is very bad.

But if i exploit that $3=(-3)*(-1)$ and combine the 2 previous Equations into one equation

$\vec{q}_i\cdot(x_2,x_4,x_6)^T=-3\ \vec{q}_i\cdot(x_1,x_3,x_5)^T$

This is a relation between the observations and the unknowns that results in the following overdetermined homogeneous system:

$3q_i(1)x_1+q_i(1)x_2+3q_i(2)x_3+q_i(2)x_4+3q_i(3)x_5+q_i(3)x_6=0,\ \ \ \ \ i=1\ldots N$

I use MATLAB for calculations.
Solving this for $x_1,x_2,x_3,x_4,x_5,x_6$ using SVD gives excellent fitting results and extremely low error.
What is so bad about solving separately the two overdetermined linear systems?
Are my variables correlated?
What are some guidelines to be followed when formulating overdetermined systems of equations?

I have been frantically looking for answer on this one for days.
Some article-book references would be really valuable.