@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.   

@acer Thank you ever so much. I will have to go through the worksheets step by step to understand what you did.  Meanwhile, would you mind looking at a much more complicated example, please?  example1_eqns.mw

@acer Many thanks.  I have uploaded a worksheet as you requested.
 

Download example_eqns.mw
 

@Christian Wolinski thanks for pointing out eliminate and a couple of new alternatives that I wasn't aware of.  The downside of the solution is that you entered with the values of theta[2,1] and theta[3,0],  

@acer Many thanks

eqns := {-theta[1, 2]*theta[2, 6]-theta[1, 3]*theta[3, 6]+1 = 0, theta[1, 2]*theta[2, 6]+theta[1, 3]*theta[3, 6]-1 = 0, -theta[1, 2]*theta[2, 6]*theta[3, 6]-theta[1, 3]*theta[3, 6]^2+1 = 0, -theta[1, 1]*theta[1, 2]*theta[2, 2]*theta[2, 6]*theta[3, 6]-theta[1, 1]*theta[1, 3]*theta[2, 2]*theta[3, 6]^2+1 = 0, -theta[1, 3]*theta[2, 2]+theta[1, 3]*theta[3, 3]+alpha+rho = 0, theta[1, 0]*theta[1, 3]*theta[2, 2]^2*theta[3, 3]-theta[1, 0]*theta[1, 3]*theta[2, 2]*theta[3, 3]^2-alpha^2*beta-alpha*beta*rho = 0, theta[1, 1]*theta[1, 2]*theta[2, 6]+theta[1, 1]*theta[1, 3]*theta[3, 6]+theta[1, 2]*theta[2, 2]*theta[2, 6]+theta[1, 3]*theta[2, 2]*theta[3, 6]-alpha = 0, theta[1, 1]*theta[1, 2]*theta[2, 6]*theta[3, 6]+theta[1, 1]*theta[1, 3]*theta[3, 6]^2+theta[1, 2]*theta[2, 2]*theta[2, 6]*theta[3, 6]+theta[1, 3]*theta[2, 2]*theta[3, 6]^2-alpha = 0, theta[1, 1]*theta[1, 3]*theta[2, 2]-theta[1, 1]*theta[1, 3]*theta[3, 3]+theta[1, 3]*theta[2, 2]^2-theta[1, 3]*theta[3, 3]^2-alpha^2+rho^2 = 0, -theta[1, 1]*theta[1, 2]*theta[2, 6]-theta[1, 1]*theta[1, 3]*theta[3, 6]-theta[1, 2]*theta[2, 2]*theta[2, 6]-theta[1, 2]*theta[2, 6]*theta[3, 3]-2*theta[1, 3]*theta[3, 3]*theta[3, 6]-2*rho = 0, -theta[1, 0]*theta[1, 2]*theta[2, 2]*theta[2, 6]*theta[3, 6]-theta[1, 0]*theta[1, 3]*theta[2, 2]*theta[3, 6]^2-theta[1, 1]*theta[1, 2]*theta[2, 2]*theta[2, 6]*theta[3, 3]+theta[1, 1]*theta[1, 3]*theta[2, 2]^2*theta[3, 6]-2*theta[1, 1]*theta[1, 3]*theta[2, 2]*theta[3, 3]*theta[3, 6]-alpha-2*rho = 0, theta[1, 1]*theta[1, 2]*theta[2, 6]*theta[3, 3]-theta[1, 1]*theta[1, 3]*theta[2, 2]*theta[3, 6]+2*theta[1, 1]*theta[1, 3]*theta[3, 3]*theta[3, 6]+theta[1, 2]*theta[2, 2]*theta[2, 6]*theta[3, 3]-theta[1, 3]*theta[2, 2]^2*theta[3, 6]+2*theta[1, 3]*theta[2, 2]*theta[3, 3]*theta[3, 6]+alpha^2+2*alpha*rho = 0, -theta[1, 0]*theta[1, 2]*theta[2, 2]*theta[2, 6]*theta[3, 3]+theta[1, 0]*theta[1, 3]*theta[2, 2]^2*theta[3, 6]-2*theta[1, 0]*theta[1, 3]*theta[2, 2]*theta[3, 3]*theta[3, 6]+theta[1, 1]*theta[1, 3]*theta[2, 2]^2*theta[3, 3]-theta[1, 1]*theta[1, 3]*theta[2, 2]*theta[3, 3]^2+alpha*beta+alpha*rho+rho^2 = 0, -theta[1, 0]*theta[1, 2]*theta[2, 2]*theta[2, 6]-theta[1, 0]*theta[1, 3]*theta[2, 2]*theta[3, 6]-theta[1, 1]*theta[1, 3]*theta[2, 2]^2+theta[1, 1]*theta[1, 3]*theta[3, 3]^2-theta[1, 3]*theta[2, 2]^2*theta[3, 3]+theta[1, 3]*theta[2, 2]*theta[3, 3]^2-alpha^2*rho-alpha*rho^2+alpha+beta+rho = 0}

and

fc := {theta[1, 0], theta[1, 1], theta[1, 2], theta[1, 3], theta[2, 2], theta[2, 6], theta[3, 3], theta[3, 6]}

I found out how to add the second module. Save allows me to add more modules.

@acer Many thanks. It worked as you said (including loading automatically).   I have followed all the steps in the help file but to no avail. 

As I said I have two modules and if I got it right Save only saves one module.  How can I append the second module to the same mla file?

Yes, it is OSX. The files were created using the line codes of the example in the help files.  Maple put .mapleinit in /Users/eduardo . What bugs me is that Maple help uses around 15 lines of code to do that same as you did in 5 lines and does not work.  

Regarding adding help files,  what you recommend?  

Many thanks.

@Christian Wolinski  Many thanks.   Now it returns 14 lines.  I will apply the function to other examples to see how it goes.