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If you want to take two polynomials and match coefficients of like terms, you could do something like

from collections import defaultdict
def solve_matched_polynomials(eq, coeff_vars):
    # get the non-coefficient variables
    vv = set(eq.variables()).difference(coeff_vars)
    d = defaultdict(Integer)
    # loop over each term in the equation
    for term in (eq.lhs()-eq.rhs()).operands():
        # get the degrees of the variables
        vsig = tuple(term.degree(v) for v in vv)
        # find the non-variable part
        d[vsig] += term/(prod(v**vs for v,vs in zip(vv, vsig)))
    # get solutions with free variables
    return solns = solve(d.values(), coeff_vars)

which will return the standard free variables form

sage: solve_matched_polynomials(5*a*x^2*y+b*x+y == 4*c*x**2*y-d*y, [a,b,c,d])
[[a == 4/5*r66, b == 0, c == r66, d == -1]]

and then if you really wanted to eliminate the free variables, you could.

If you want to take two polynomials and match coefficients of like terms, you could do something like

from collections import defaultdict
def solve_matched_polynomials(eq, coeff_vars):
    # get the non-coefficient variables
    vv = set(eq.variables()).difference(coeff_vars)
sorted(set(eq.variables()).difference(coeff_vars))
    d = defaultdict(Integer)
    # loop over each term in the equation
    for term in (eq.lhs()-eq.rhs()).operands():
        # get the degrees of the variables
        vsig = tuple(term.degree(v) for v in vv)
        # find the non-variable part
        d[vsig] += term/(prod(v**vs for v,vs in zip(vv, vsig)))
    # get solutions with free variables
    return solns = solve(d.values(), coeff_vars)

which will return the standard free variables form

sage: solve_matched_polynomials(5*a*x^2*y+b*x+y == 4*c*x**2*y-d*y, [a,b,c,d])
[[a == 4/5*r66, b == 0, c == r66, d == -1]]

and then if you really wanted to eliminate the free variables, you could.

If you want to take two polynomials and match coefficients of like terms, you could do something like

from collections import defaultdict
def solve_matched_polynomials(eq, coeff_vars):
    # get the non-coefficient variables
    vv = sorted(set(eq.variables()).difference(coeff_vars))
    d = defaultdict(Integer)
    # loop over each term in the equation
    for term in (eq.lhs()-eq.rhs()).operands():
        # get the degrees of the variables
        vsig = tuple(term.degree(v) for v in vv)
        # find the non-variable part
        d[vsig] += term/(prod(v**vs for v,vs in zip(vv, vsig)))
    # get solutions with free variables
    return solns = solve(d.values(), coeff_vars)

which will return the standard free variables form

sage: solve_matched_polynomials(5*a*x^2*y+b*x+y == 4*c*x**2*y-d*y, [a,b,c,d])
[[a == 4/5*r66, b == 0, c == r66, d == -1]]

and then if you really wanted to eliminate the free variables, you could.

If you want to match coefficients of like terms, you could do something like

from collections import defaultdict
def solve_matched_polynomials(eq, coeff_vars):
    # get the non-coefficient variables
    vv = sorted(set(eq.variables()).difference(coeff_vars))
    d = defaultdict(Integer)
    # loop over each term in the equation
    for term in (eq.lhs()-eq.rhs()).operands():
(eq.lhs().operands())+(-(eq.rhs())).operands():
        # get the degrees of the variables
        vsig = tuple(term.degree(v) for v in vv)
        # find the non-variable part
        d[vsig] += term/(prod(v**vs for v,vs in zip(vv, vsig)))
    # get solutions with free variables
    return solns = solve(d.values(), coeff_vars)

which will return the standard free variables form

sage: solve_matched_polynomials(5*a*x^2*y+b*x+y == 4*c*x**2*y-d*y, [a,b,c,d])
[[a == 4/5*r66, b == 0, c == r66, d == -1]]

and then if you really wanted to eliminate the free variables, you could.

If you want to match coefficients of like terms, you could do something like

from collections import defaultdict
def solve_matched_polynomials(eq, coeff_vars):
    # get the non-coefficient variables
    vv = sorted(set(eq.variables()).difference(coeff_vars))
    # make sure they're polynomials
    assert all(eq.lhs().is_polynomial(v) and eq.rhs().is_polynomial(v) for v in vv)
    d = defaultdict(Integer)
    # loop over each term in the equation
    for term in (eq.lhs().operands())+(-(eq.rhs())).operands():
        # get the degrees of the variables
        vsig = tuple(term.degree(v) for v in vv)
        # find the non-variable part
        d[vsig] += term/(prod(v**vs for v,vs in zip(vv, vsig)))
    # get solutions with free variables
    return solns = solve(d.values(), coeff_vars)

which will return the standard free variables form

sage: solve_matched_polynomials(5*a*x^2*y+b*x+y == 4*c*x**2*y-d*y, [a,b,c,d])
[[a == 4/5*r66, b == 0, c == r66, d == -1]]

and then if you really wanted to eliminate the free variables, you could.