# Question 258: Validity of a Linear-Equation Form

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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.

P1000-0258 · q-20260927-linear-equation-form-validity-0008



## Original question and givens

Over the rational numbers, consider 3x+(-5)=0. Under this MKO, is it a linear equation in one variable?


- G001 — Over the rational numbers, consider 3x+(-5)=0. Under this MKO, is it a linear equation in one variable?





## Definition and scope

Use the rational domain. A linear equation requires a≠0; a=0 is a separate degenerate constant equation.


- [mko-linear-equation](https://ocme.evemisslab.com/en/mko/mko-linear-equation/)





## Step-by-step derivation

S001 · impliesBefore

a=3, b=-5, x∈Q

After

equation: 3x+(-5)=0

Conditions: none

Read the linear coefficient, constant and rational domain explicitly; a name alone does not establish nondegeneracy.

Step identifiers and dependenciesRule: read_coefficients · Uses: G001





S002 · impliesBefore

a=3

After

a≠0: Yes

Conditions: none

The definition of a nondegenerate linear equation requires a nonzero leading coefficient.

Step identifiers and dependenciesRule: nondegenerate_definition · Uses: G001, S001





S003 · impliesBefore

3x+(-5)=0

After

linear form qualified: Yes

Conditions: none

This checks the definition, not merely whether the equation has a solution.

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





S004 · impliesBefore

if a=0 then ax+b=b

After

constant equality is a separate boundary

Conditions: none

When a=0, b=0 gives every rational solution and b≠0 gives none; neither is nondegenerate linear form.

Step identifiers and dependenciesRule: degenerate_boundary · Uses: G001, S003









## Reverse checks


- S004 · premise_and_result_audit — Substitute into the original equation; retain every nonzero condition before division.
- S003 · premise_and_result_audit — Substitute into the original equation; retain every nonzero condition before division.
- S002 · premise_and_result_audit — Substitute into the original equation; retain every nonzero condition before division.
- S001 · premise_and_result_audit — Substitute into the original equation; retain every nonzero condition before division.





## Result

Yes

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.





## 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: a777b98b6b4207137ce3f8fbd388db09f831ad6c0c91d63d01f2e1df127eeea4
- candidate record SHA-256: b882e293e042d5624ab9fcbe5445607b1ae7b77d54fe36f2e171761e05c2d643
- semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
- question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260927-linear-equation-form-validity-0008.json)