Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=-10. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
G001 — Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=-10. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
Definition and scope
Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status.
Linear row ORIGINAL b over nonzero a := div(-10, 2)
After
Linear row ORIGINAL b over nonzero a = -5
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Linear normalized row ORIGINAL negated constant := sub(0, -10)
After
Linear normalized row ORIGINAL negated constant = 10
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Linear normalized row ORIGINAL negative b over a := div(10, 2)
After
Linear normalized row ORIGINAL negative b over a = 5
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Linear normalized row reverse ORIGINAL a times root := mul(2, 5)
After
Linear normalized row reverse ORIGINAL a times root = 10
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Linear normalized row reverse ORIGINAL residual plus b := sum(10, -10)
After
Linear normalized row reverse ORIGINAL residual plus b = 0
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
ONLY nonzero a permits division:ax+b=0 IFF x=-b/a.Substitute into the ORIGINAL coefficients to verify zero residual.Normalize a negative a to a positive denominator;derived roots are not SAFE-capped or approximated.
Eligibility,the row after division by a,and the original solution set are SEPARATE fields.a=0 is a legal No/undefined row while the constant equation still has a solution class;all Q does not make it linear.
S008 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
S007 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
S006 · premise_and_result_audit — Recompute Linear normalized row reverse ORIGINAL residual plus b with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute Linear normalized row reverse ORIGINAL a times root with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute Linear normalized row ORIGINAL negative b over a with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute Linear normalized row ORIGINAL negated constant with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute Linear row ORIGINAL b over nonzero a with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
Result
(Yes, [1/1, -5/1], single-solution 5/1)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
This question review and source binding
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.