@Carl Love Many thanks. 

@Kitonum Many thanks. IsolateNonlinearTerms does not seem to work when the coefficients are not numeric. (Curiosity since the polynomials in the question have terms with coefficients equal to 1).

Example:

w:=[[F*a, -a*x, -y, -z], [-b*x*z, x*y, G, -y], [b*x*y, x*z, -z]]

then when the function is applied the result is 

[F*a, -a*x, -b*x*z, x*y, b*x*y, x*z], [-y, -z, G, -y, -z]

@Carl Love Many thanks for helping me again. A comment on 3*x is given in another reply.

@dharr Many thanks.  In all my lists of polynomials from which the nonlinear terms have been extracted the coefficients of all terms are equal to one. That is the reason why 3*x is not an issue.   Anyways, the two solutions are far better than mine. I have a lot to learn. Thank you both ever so much.  

@acer I am not sure if I can give a good answer as requested.  I haven't said that the solution is optimal or best but simply it is one that fits my needs.  Yes, if just one variable is excluded, all the better.   I really don't know how to break ties.  

I don't know if the following definition will good enough for you but I will give a shot.  Definition: given a set of nonlinear equations, how to find the maximal set of unconflicted variables (and therefore a solution) and how to reduce the inconsistency to a minimal?  Sorry if I can do a better job.   Thanks for your patience.  

@acer Many thanks.   In the example given above, only alpha[2,3] and alpha [3,5] cannot be found (inconsistency). The remaining alphas can be found.  What I need is exactly that: find what is possible to find and show me the equations that cannot be solved.  (alpha[1,2]=-193/100,alpha[2.7]=-74/125,alpha[2,2]=477/1000,alpha[2,1]=139/100,alpha[1,4]=629/500,alpha[2,5]=1093/500).

@Carl Love Hi, Carl.  It would certainly.   That will make my life easier.  I have so many of these set of equations that anything to "shorten" them out would help. Many thanks.

@acer Many thanks.   Since I have lots of such systems, what I need is to find a way to automatically reduce the set of equations to 1) which variables can be found , 2) inconsistent equations (as you did in your example) and 3) is there a a[2,2] available?   

I am have a similar problem where the commands given below do not seem to work as expected (remove is one of them). 

Here is the example:

eqns := {alpha[1, 2] = -193/100, (-2*alpha[1, 4]-alpha[2, 5])*alpha[1, 2] = 453743/50000, alpha[1, 2]*alpha[1, 4]*alpha[2, 3]*alpha[3, 5] = -17388542089/25000000000, -2*alpha[2, 7] = 148/125, -alpha[1, 2]*alpha[2, 2] = 92061/100000, -alpha[1, 2]^2*alpha[2, 1] = -5177611/1000000, 2*alpha[1, 4]*alpha[2, 7] = -23273/15625, 4*alpha[1, 4]*alpha[2, 7] = -46546/15625, -4*alpha[1, 4]^2*alpha[2, 7] = 14638717/3906250, 3*alpha[1, 2]*alpha[1, 4]*alpha[2, 5] = -398060763/25000000, 2*alpha[1, 2]*alpha[1, 4]*alpha[2, 2]-alpha[1, 2]*alpha[2, 3]*alpha[3, 5] = -555270457/50000000}

fs := {alpha[1, 2], alpha[1, 4], alpha[2, 1], alpha[2, 2], alpha[2, 3], alpha[2, 5], alpha[2, 7], alpha[3, 5]}

Again, I need to extract those alphas that have a solution, eliminate the redundant equations and extract the inconsisting equations.

Many thanks again.

@vv Many thanks.   I was aware that the resulting rational number would have a huge denominator. How did you estimate such a number?   I was expecting that neither Maple nor Mathematica could come up with the sequence of rational numbers with huge numbers in the denominator, but Mathematica did (somehow I cannot reproduce that result anymore, but I have the notebook with it).  Regardless of the value of n, what is the guideline for changing the defaults to get more out of maple?  Many thanks. 

Thank you both for the nice solutions.   

@acer Many thanks.   As you said it is pretty straightforward but I still had to pass the result of map(int,eq,t) to Parts to get the result.  

@Christian Wolinski Would you know how to use Rule from Student[Calculus1] to do the same calculations as above? I could not figure out how to do it.  Many thanks.  

@Christian Wolinski Many thanks.  I did not know about the student[intparts].  

@acer Many thanks.  I am not sure what form I need. The only thing I have in mind is to get all indets that have solution sorted out and the rest of indets in a form that is as simple as possible.  Any special advice on that?   I know that I need to carefully go through the commands in the worksheets you sent in order to understand what they do.