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Question 31562: Classifying a General Linear Equation

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0062-31562 · q-20260924-equation-linear-solution-class-0012

Original question and givens

Over the rational numbers, classify the solution set of 12x+(11)=12x+(12).

  • G001 — Over the rational numbers, classify the solution set of 12x+(11)=12x+(12).

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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: R62_EXACT_source_keys_domain_operation_form_AND_output_types · Uses: G001

S002 · implies

Before

source roles = [12, 11, 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 dependencies

Rule: Equation_BASE_integer_coefficients_declared_Z_OR_Q_NOT_blanket_SAFE · Uses: G001, S001

S003 · calculation

Before

Equation class ORIGINAL a minus c := sub(12, 12)

After

Equation class ORIGINAL a minus c = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Equation class ORIGINAL d minus b := sub(12, 11)

After

Equation class ORIGINAL d minus b = 1

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

source roles = [12, 11, 12, 12]

After

derived = [0, 1]

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 dependencies

Rule: Equation_CLASS_both_sides_move_terms_equivalent_Q · Uses: G001, S004

S006 · implies

Before

source roles = [0, 1]

After

derived = [no solution]

Conditions: none

At coefficient0,a zero right side gives0=0 and ALL Q solutions;a nonzero right side is contradictory with NO solution.Never divide by0 or use numerical0 as an empty solution set.

Step identifiers and dependencies

Rule: Equation_CLASS_ZERO_coefficient_all_Q_OR_contradiction_NOdivision · Uses: G001, S005

Reverse checks

  • S006 · 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.
  • 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

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: fa04cbed724f585759010ff76338dcf69e3af3fee48abcfe5b3c70a80339654c
  • candidate record SHA-256: 9157852d75a19f04c201cfbbac5d25932f105c1d53f30bd0fa1ebce7d54b1d06
  • semantic review receipt SHA-256: 963099ffc9f754069fdce6792949a85ed350472465eaceb3038fa8011cf672bd
  • question-first witness SHA-256: d7655324c5d11528e1af90b757384cebd3313e19dc5f6f5b9b8ea7624fb06f6f
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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