# Question 30527: Linear Eligibility, Normalized Row, and Solution Class

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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.

R0060-30527 · q-20261010-advance-20261014-r04-linear-equation-0002



## Original question and givens

Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=0, b=1. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.


- G001 — Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=0, b=1. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.





## 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-normalized-row, density14_linear_normalized_row, finite_Q_affine_row_protocol]

After

derived = [[linear, normalized_row_or_undefined, solution_class]]

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 · impliesBefore

source roles = [Linear normalized row, 0, 1]

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, S001





S003 · impliesBefore

source roles = [0, 1]

After

derived = [No, undefined, no solution]

Conditions: none

Eligibility,the row after division by a,and the original solution set are SEPARATE fields.a=0 is a legal No/undefined row while the constant equation still has a solution class;all Q does not make it linear.

Step identifiers and dependenciesRule: Linear_ELIGIBILITY_a0_LEGAL_no_normalized_row_not_division · Uses: G001, S002









## Reverse checks


- S003 · 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.
- S002 · 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.
- 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

(No, undefined, 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: 86c1b5c112cde58f90be1b06f27bd023a001dd93ff5dd5068d352bb0077453ae
- candidate record SHA-256: 3537a453eaddb0ec51daa7085e4963dd8dd991980c231aab16fe741655912cb9
- 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-0002.json)