# Question 31578: Multiplicative Equation [Return to catalogue page 158](https://ocme.evemisslab.com/en/questions/page/158/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260924-equation-foundations/#q-20260924-equation-multiplicative-solve-0003) 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. R0062-31578 · q-20260924-equation-multiplicative-solve-0003 ## Original question and givens Solve 3×x=-9 over the integers; the coefficient 3 is explicitly nonzero. - G001 — Solve 3×x=-9 over the integers; the coefficient 3 is explicitly nonzero. ## 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-equation](https://ocme.evemisslab.com/en/mko/mko-equation/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-equation-multiplicative-solve, integers, no-operation-field, a*x=b] After derived = [[integer_solution_STRING]] Conditions: none Check THIS bank interface:BASE equations use form/domain;BASE events/complement have no operation;factorials have no domain.ext18 uses local hyphens,other legacy operations underscores.Do not force every answer into a tuple or fraction. Step identifiers and dependenciesRule: R62_EXACT_source_keys_domain_operation_form_AND_output_types · Uses: G001 S002 · impliesBefore source roles = [3, -9, a*x=b, integers] After derived = [Yes] Conditions: none The BASE bank checks integers,not an added SAFE limit.This candidate checks the frozen exact integer slice,not arbitrary unsafe JavaScript-number behavior.Distinguish solutions over Z and Q. Step identifiers and dependenciesRule: Equation_BASE_integer_coefficients_declared_Z_OR_Q_NOT_blanket_SAFE · Uses: G001, S001 S003 · calculationBefore Equation multiplicative ORIGINAL b over nonzero a := div(-9, 3) After Equation multiplicative ORIGINAL b over nonzero a = -3 Conditions: none Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Equation multiplicative reverse ORIGINAL a times solution := mul(3, -3) After Equation multiplicative reverse ORIGINAL a times solution = -9 Conditions: none Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore source roles = [3, -9] After derived = [0, -3, -9] Conditions: none FIRST require a≠0 AND a divides b,then solve x=b/a IN Z.Negative a is legal.Nondivisibility fails this source family qualification;do not present a rational answer as an integer solution. Step identifiers and dependenciesRule: Equation_MULT_Z_nonzero_AND_divisibility_BEFORE_division · Uses: G001, S004 ## Reverse checks - S005 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate. - S004 · premise_and_result_audit — Recompute Equation multiplicative reverse ORIGINAL a times solution with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Equation multiplicative ORIGINAL b over nonzero a with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate. - S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate. ## Result -3 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: ccad2bd97e8778cea6236f7f9330b7f0d904326558fc0719465d96885260c8b3 - candidate record SHA-256: 9b972db0f2a563811a51e25803914c233e5324bbcaef8938f550bc0380c1a7b9 - semantic review receipt SHA-256: 963099ffc9f754069fdce6792949a85ed350472465eaceb3038fa8011cf672bd - question-first witness SHA-256: d7655324c5d11528e1af90b757384cebd3313e19dc5f6f5b9b8ea7624fb06f6f - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260924-equation-multiplicative-solve-0003.json)