Question: Finding the values of coefficients in a set of linear equations

I have a set of around 60 linear equations with symbolic coefficients. ie


a*x1 + b*x2 + ... + c*x60 = 1

x1 + (c-a)*x2 + ... + d*x60 = 0


c*x1 + d*x2 + ... + b*x60 = a


The coefficients a,b,c,d are functions of x1...x60. I am trying to find the values of these coefficients. When I had a smaller set of equations I was solving them symbolically to find x1...x60 in terms of a,b,c,d and then using this solution to solve for a,b,c,d. I can no longer solve the set of equations symbolically as it is too large. How do I find the coefficients? I had some sort of optimization routine in mind.

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