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As stated, the nine equations of r1 (with implicit zero right-had side) are mutually incompatible.

A brute-force search of the subsets of mutually incompatible equations can be done as follows :

# Search for all incompatible sets of equations in r1
DIS={}
for k in (1..len(r1)-1):
    IS=[]
    for s in Combinations(r1,k):
        v=[w for w in va if w in list(set(flatten([u.variables() for u in s])))]
        S=solve(s, v, solution_dict=True)
        if len(S)==0 : IS.append(s)
    if len(IS)>0: DIS.update({k:IS})
# End of search

Running this code shows that these incompatibilities may be quite specific :

sage: {u:len(DIS[u]) for u in DIS.keys()}
{4: 1, 5: 7, 6: 18, 7: 20, 8: 9}

For example, the four equations :

sage: DIS[4]
[[-y_1_2 + x_2_2/x_2_3,
  -y_2_1 + x_1_2/x_1_3,
  x_1_2*x_2_2*x_3_2 - y_2_3,
  x_1_3*x_2_3*x_3_2 - y_3_3]]

are mutually incompatible.

HTH,

As stated, the nine equations of r1 (with implicit zero right-had side) are mutually incompatible.

A brute-force search of the subsets of mutually incompatible equations can be done as follows :

# Search for all incompatible sets of equations in r1
DIS={}
for k in (1..len(r1)-1):
    IS=[]
    for s in Combinations(r1,k):
        v=[w for w in va if w in list(set(flatten([u.variables() for u in s])))]
        S=solve(s, v, solution_dict=True)
        if len(S)==0 : IS.append(s)
    if len(IS)>0: DIS.update({k:IS})
# End of search

Running this code shows that these incompatibilities may be quite specific :

sage: {u:len(DIS[u]) {u:(len(DIS[u]),binomial(9,u)) for u in DIS.keys()}
{4: 1, (1, 126), 5: 7, (7, 126), 6: 18, (18, 84), 7: 20, (20, 36), 8: 9}
(9, 9)}

For example, the four equations :

sage: DIS[4]
[[-y_1_2 + x_2_2/x_2_3,
  -y_2_1 + x_1_2/x_1_3,
  x_1_2*x_2_2*x_3_2 - y_2_3,
  x_1_3*x_2_3*x_3_2 - y_3_3]]

are mutually incompatible.

HTH,