# Question 30620: Two Affine Roots and a Common Root [Return to catalogue page 154](https://ocme.evemisslab.com/en/questions/page/154/) [← 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-0095) 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. R0060-30620 · q-20261010-advance-20261014-r04-linear-equation-0095 ## Original question and givens Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -19], [-3, -18]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers. - G001 — Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -19], [-3, -18]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers. ## 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-two-row-coincidence, density14_linear_two_row_coincidence, finite_Q_affine_row_protocol] After derived = [[root1, root2, equal, common_root_or_none]] 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 original row0 ORIGINAL negated constant := sub(0, -19) After Linear original row0 ORIGINAL negated constant = 19 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 original row0 ORIGINAL negative b over a := div(19, 2) After Linear original row0 ORIGINAL negative b over a = 19/2 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 original row0 reverse ORIGINAL a times root := mul(2, 19/2) After Linear original row0 reverse ORIGINAL a times root = 19 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 original row0 reverse ORIGINAL residual plus b := sum(19, -19) After Linear original row0 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, S004 S006 · impliesBefore source roles = [Linear original row0, 2, -19] After derived = [19/2, 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, S005 S007 · calculationBefore Linear original row1 ORIGINAL negated constant := sub(0, -18) After Linear original row1 ORIGINAL negated constant = 18 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, S006 S008 · calculationBefore Linear original row1 ORIGINAL negative b over a := div(18, -3) After Linear original row1 ORIGINAL negative b over a = -6 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 original row1 reverse ORIGINAL a times root := mul(-3, -6) After Linear original row1 reverse ORIGINAL a times root = 18 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 original row1 reverse ORIGINAL residual plus b := sum(18, -18) After Linear original row1 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, S009 S011 · impliesBefore source roles = [Linear original row1, -3, -18] After derived = [-6/1, 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, S010 S012 · impliesBefore source roles = [[[2, -19], [-3, -18]]] After derived = [[19/2, -6/1], No, none] Conditions: none Require nonzero a in BOTH rows,solve each root independently,then compare exactly.Different coefficients can share a root.Without a common root return none,not numerical0 or approximate equality. Step identifiers and dependenciesRule: Linear_TWO_independent_nonzero_rows_root_coincidence_NOT_coefficients · Uses: G001, S011 ## Reverse checks - S012 · 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. - S011 · 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. - S010 · premise_and_result_audit — Recompute Linear original row1 reverse ORIGINAL residual plus b with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Linear original row1 reverse ORIGINAL a times root with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Linear original row1 ORIGINAL negative b over a with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Linear original row1 ORIGINAL negated constant with exact rational arithmetic, not rounded decimals. - S006 · 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. - S005 · premise_and_result_audit — Recompute Linear original row0 reverse ORIGINAL residual plus b with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Linear original row0 reverse ORIGINAL a times root with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Linear original row0 ORIGINAL negative b over a with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Linear original row0 ORIGINAL negated constant 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 (19/2, -6/1, No, none) 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: 7f49ced02eb38d921809109d0f6b9308bbf71d0cc6baaab36299f5adaa3ed0ff - candidate record SHA-256: 15847a166db0f9c3536f622ddcad1f5aded61ed443e5079c2a06e461a4be921d - 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-0095.json)