# Question 30574: Parameter-Dependent Linear and Constant Cases [Return to catalogue page 153](https://ocme.evemisslab.com/en/questions/page/153/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261010-advance-20261014-r04-linear-equation-foundations/#q-20261010-advance-20261014-r04-linear-equation-0049) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0060-30574 · q-20261010-advance-20261014-r04-linear-equation-0049 ## Original question and givens Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=2, b=11, ordered ts=[13, 26, 2]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases. - G001 — Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=2, b=11, ordered ts=[13, 26, 2]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases. ## 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. - [mko-linear-equation](https://ocme.evemisslab.com/en/mko/mko-linear-equation/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-density14-linear-parameter-slice, density14_linear_parameter_slice, finite_Q_affine_row_protocol] After derived = [[ALL_derived_coefficients, ALL_linear_flags, ALL_solution_classes]] Conditions: none 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. Step identifiers and dependenciesRule: R60_EXACT_perbank_operation_domain_FRAME_and_output_roles · Uses: G001 S002 · calculationBefore Linear parameter ORIGINAL t0 minus k := sub(13, 2) After Linear parameter ORIGINAL t0 minus k = 11 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Linear parameter original slot0 ORIGINAL negated constant := sub(0, 11) After Linear parameter original slot0 ORIGINAL negated constant = -11 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Linear parameter original slot0 ORIGINAL negative b over a := div(-11, 11) After Linear parameter original slot0 ORIGINAL negative b over a = -1 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Linear parameter original slot0 reverse ORIGINAL a times root := mul(11, -1) After Linear parameter original slot0 reverse ORIGINAL a times root = -11 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Linear parameter original slot0 reverse ORIGINAL residual plus b := sum(-11, 11) After Linear parameter original slot0 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [Linear parameter original slot0, 11, 11] After derived = [-1/1, 0/1] Conditions: none 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. Step identifiers and dependenciesRule: Linear_NONZERO_a_exact_Q_root_AND_original_substitution · Uses: G001, S006 S008 · calculationBefore Linear parameter ORIGINAL t1 minus k := sub(26, 2) After Linear parameter ORIGINAL t1 minus k = 24 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Linear parameter original slot1 ORIGINAL negated constant := sub(0, 11) After Linear parameter original slot1 ORIGINAL negated constant = -11 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Linear parameter original slot1 ORIGINAL negative b over a := div(-11, 24) After Linear parameter original slot1 ORIGINAL negative b over a = -11/24 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Linear parameter original slot1 reverse ORIGINAL a times root := mul(24, -11/24) After Linear parameter original slot1 reverse ORIGINAL a times root = -11 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Linear parameter original slot1 reverse ORIGINAL residual plus b := sum(-11, 11) After Linear parameter original slot1 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore source roles = [Linear parameter original slot1, 24, 11] After derived = [-11/24, 0/1] Conditions: none 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. Step identifiers and dependenciesRule: Linear_NONZERO_a_exact_Q_root_AND_original_substitution · Uses: G001, S012 S014 · calculationBefore Linear parameter ORIGINAL t2 minus k := sub(2, 2) After Linear parameter ORIGINAL t2 minus k = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore source roles = [Linear parameter original slot2, 0, 11] After derived = [No, no solution] Conditions: none At a=0 this is NOT linear:b=0 gives0=0 and all Q solutions;nonzero b is a contradictory constant equation with no solution.Do not divide by0. Step identifiers and dependenciesRule: Linear_a0_CONSTANT_equation_separate_ALLQ_vs_NOsolution · Uses: G001, S014 S016 · impliesBefore source roles = [2, 11, [13, 26, 2]] After derived = [[11, 24, 0], [Yes, Yes, No], [single solution -1/1, single solution -11/24, no solution]] Conditions: none First compute a_i=t_i-k at EACH original position;derived coefficients may exceed SAFE.Then distinguish nonzero linear and zero-coefficient constant equations.Retain all three flags/classes without sorting/deduplicating parameters. Step identifiers and dependenciesRule: Linear_PARAMETER_derived_a_not_revalidated_RAW_and_ALL_classes · Uses: G001, S015 ## Reverse checks - S016 · 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. - S015 · 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. - S014 · premise_and_result_audit — Recompute Linear parameter ORIGINAL t2 minus k with exact rational arithmetic, not rounded decimals. - S013 · 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. - S012 · premise_and_result_audit — Recompute Linear parameter original slot1 reverse ORIGINAL residual plus b with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Linear parameter original slot1 reverse ORIGINAL a times root with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Linear parameter original slot1 ORIGINAL negative b over a with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Linear parameter original slot1 ORIGINAL negated constant with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Linear parameter ORIGINAL t1 minus k with exact rational arithmetic, not rounded decimals. - 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 parameter original slot0 reverse ORIGINAL residual plus b with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Linear parameter original slot0 reverse ORIGINAL a times root with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Linear parameter original slot0 ORIGINAL negative b over a with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Linear parameter original slot0 ORIGINAL negated constant with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Linear parameter ORIGINAL t0 minus k 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 ([11, 24, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -11/24, no-solution]) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 8eeef04b51200b099702cb6f7513f6cd677413c1a69ae8cdfd458fdf98754749 - candidate record SHA-256: 9b954af529f43b6a3dda291cde5b0a2dfa6622216b3307f51e3cd7ab182c9122 - semantic review receipt SHA-256: d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bf - question-first witness SHA-256: f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261010-advance-20261014-r04-linear-equation-0049.json)