ok, let me see if i can unravel this for you. our system of equations is:
Mt = 0 (your equations seem to be missing something (i don't see a "z term")).
as it stands, this system is "under-determined" we have 10 variables to solve for, and only 6 equations.
so what we are going to do is create a bunch of 6x6 matrices from M, and calculate their determinants. the first matrix, will just be the first 6 columns of M. this represents our "baseline" (pretending x,y and z aren't a part of our system).
to calculate you need to know how to form the 6x6 matrix . basically this is a choice of "which 6 columns (of M) to use".
according to your description, for we would have:
(we use column 7 of M in place of column 1 of M...the "i" tells us which column of to omit, and the "j" tells us which one we're going to replace it with: the j+6-th column of M).
as another example, we have:
(we use column 9 of M in place of column 2 of M).
we have 6 possible columns of to replace (one of the first 6 columns of M), and 4 possible columns to use instead (columns 7 through 10 of the original M), giving us 24 determinants to calculate.
this is going to give a system of 6 equations in 4 unknowns ("over-determined") which (if consistent, and it should be, if the original system is) is now certainly solvable.