# Question 31557: Classifying a General Linear 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-linear-solution-class-0007) 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. R0062-31557 · q-20260924-equation-linear-solution-class-0007 ## Original question and givens Over the rational numbers, classify the solution set of 8x+(6)=11x+(0). - G001 — Over the rational numbers, classify the solution set of 8x+(6)=11x+(0). ## 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-linear-solution-class, rationals, no-operation-field, a*x+b=c*x+d] After derived = [[all_Q_no_solution_or_unique_compact_rational_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 = [8, 6, a*x+b=c*x+d, rationals] 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 class ORIGINAL a minus c := sub(8, 11) After Equation class ORIGINAL a minus c = -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 class ORIGINAL d minus b := sub(0, 6) After Equation class ORIGINAL d minus b = -6 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 = [8, 6, 11, 0] After derived = [-3, -6] Conditions: none Subtract cx and b from BOTH sides to obtain the equivalent(a−c)x=d−b.IMPORTANT:do not divide by a−c until degeneracy is classified. Step identifiers and dependenciesRule: Equation_CLASS_both_sides_move_terms_equivalent_Q · Uses: G001, S004 S006 · calculationBefore Equation class ORIGINAL constant over nonzero coefficient := div(-6, -3) After Equation class ORIGINAL constant over nonzero coefficient = 2 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, S005 S007 · calculationBefore Equation class reverse ORIGINAL a times x := mul(8, 2) After Equation class reverse ORIGINAL a times x = 16 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, S006 S008 · calculationBefore Equation class reverse ORIGINAL ax plus b := sum(16, 6) After Equation class reverse ORIGINAL ax plus b = 22 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, S007 S009 · calculationBefore Equation class reverse ORIGINAL c times x := mul(11, 2) After Equation class reverse ORIGINAL c times x = 22 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, S008 S010 · calculationBefore Equation class reverse ORIGINAL cx plus d := sum(22, 0) After Equation class reverse ORIGINAL cx plus d = 22 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, S009 S011 · calculationBefore Equation class reverse ORIGINAL sides difference := sub(22, 22) After Equation class reverse ORIGINAL sides difference = 0 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, S010 S012 · impliesBefore source roles = [8, 6, 11, 0, -3, -6] After derived = [2/1, 22, 22, 0, unique solution x=2] Conditions: none ONLY the nonzero coefficient permits division,giving a UNIQUE Q solution.Reduce to a positive denominator and OMIT/1 for integer display.Substitute termwise into BOTH ORIGINAL sides,not only the rearranged equation. Step identifiers and dependenciesRule: Equation_CLASS_NONZERO_unique_Q_root_positive_den_OMIT1 · Uses: G001, S011 ## Reverse checks - S012 · 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. - S011 · premise_and_result_audit — Recompute Equation class reverse ORIGINAL sides difference with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Equation class reverse ORIGINAL cx plus d with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Equation class reverse ORIGINAL c times x with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Equation class reverse ORIGINAL ax plus b with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Equation class reverse ORIGINAL a times x with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Equation class ORIGINAL constant over nonzero coefficient with exact rational arithmetic, not rounded decimals. - 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 class ORIGINAL d minus b with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Equation class ORIGINAL a minus c 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 unique solution x=2 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: bde08f9650cca97fcf85a9f6c2dde9c7655f95623ce406eb1d48ba83e345a7a3 - candidate record SHA-256: cce2ee25abd8a54a59717042a6e8f6237e89766af4e84742a4787d55b67113f0 - 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-linear-solution-class-0007.json)