There is a general method to "unsolve" a system of linear parametric equations, that is, to go from this system to a system of linear equations whose solution set is the affine manifold which is expressed by the original system. I found this method by myself, but it is likely that somebody else came up with the same method before me.
Suppose we have an affine manifold in R^n given in linear parametric form by
x=b+At,
where x is an n-column vector which gives a point on the manifold, b a constant n-column vector, A an n x m -matrix, and t an m-column-vector of parameters.
Now, write down the n x (m+1) - matrix (b A). On this matrix we perform a sequence of elementary column operations. Such an operation is like an elementary row operation but using columns instead of rows. But we exclude all such operations which involve the first column (originally b), except that we allow adding a multiple of another column to the first column (but not the other way round). One can verify that such an operation does not change the manifold given by the corresponding system of linear parametric equations.
We choose this sequence of column operations in a way that corresponds to Gauss-Jordan elimination, but using columns instead of rows (and vice versa) and starting in the bottom right corner (instead of the upper left one) and going left and upwards.
Let us write the resulting matrix as
(b' A'),
where b' is its first column.
The submatrix A' will now be in what we may call upside down reduced column echelon form, which is as reduced row echelon form with columns instead of rows (and vice versa) and bottom/right instead of upper/left (and vice versa).
Also, if a row in A' is a unit row (one element is 0 and the other 0:s), the element in b' at this row is also 0.
Next, if there is a set of 0-columns collected at the left of A', we remove these, so that we obtain new matrix
(b' A'')
of type n x (k+1), with k\le\max(m,n).
With t' as a k-column vectors of parameters, the system of linear parametric equations
x'=b'+A''t' (*)
gives the same manifold as the original one. m-k (which may be 0) parameters turned out to be superfluous. The dimension of the manifold is k.
This new system (*) of linear parametric equations has precisely the standard form which is obtained if we start from a system of linear equations, solve it by Gauss-Jordan elimiation and write the solution in parametric form in the standard way.
Now, we can, from (*), go "backwards" and (re)construct an (n-k) x (n+1) - matrix in reduced row echelon form, which is the augmented matrix of a system of linear equations, whose solution set is the given manifold.
Ok, I know I should give an example to illistrate this, but it is so cumbersome doing it in TeX-notation here... Maybe some day...
