# Question 33: Solution Classification of a Linear Equation

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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.

P250-0033 · q-20260927-linear-equation-solution-class-0008



## Original question and givens

Over the rational numbers, classify the solution set of 0x+(7)=0.


- G001 — Over the rational numbers, classify the solution set of 0x+(7)=0.





## Definition and scope

Use the rational domain. Treat constant equations separately when a=0; divide by a only when a is nonzero.


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





## Step-by-step derivation

S001 · impliesBefore

0x+(7)=0, x∈Q

After

a=0; b=7

Conditions: none

Read the rational solution domain and check the leading coefficient before dividing by a.

Step identifiers and dependenciesRule: equation_domain · Uses: G001





S002 · impliesBefore

a=0

After

0x+(7)=7

Conditions: a=0

Zero times any rational number is zero, so the unknown term disappears.

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





S003 · impliesBefore

7=0

After

false for every x∈Q

Conditions: none

The remaining constant equality is independent of x; check its truth directly.

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





S004 · impliesBefore

constant equality has fixed truth value

After

Solution set = ∅

Conditions: none

If the equality is always true every allowed value is a solution; if always false there is no solution.

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





S005 · impliesBefore

a=0 and b=0 or b≠0

After

no solution

Conditions: none

These two constant branches exhaust the case a=0.

Step identifiers and dependenciesRule: exhaustive_cases · Uses: G001, S004









## Reverse checks


- S005 · premise_and_result_audit — In the original equation the truth of 0x+b=0 is unchanged for every rational x.
- S004 · premise_and_result_audit — All solutions and no solutions are both distinct from a unique solution.
- S003 · premise_and_result_audit — The classification depends on b=0, not on sampling a few values of x.
- S002 · premise_and_result_audit — No reciprocal of a or division by zero is introduced.
- S001 · premise_and_result_audit — When a=0 this is a degenerate constant equation, not a nondegenerate linear equation.





## Result

no solution

There is no solution.





## This question review and source binding

Review date: 2026-10-24

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: 1a9f35ae455bb4c91d1cb1064d39784445e82097b4e34951e9acf4e3efe14b75
- candidate record SHA-256: 336c5fe30bbce01ad12bfbc86e120a609f6eb65f6ef5d42798cb3b1bd2a31b11
- semantic review receipt SHA-256: 4727d1bd957a9b7f6cadb7972ddb5ab80a31f5259dc5394c1dff2b71944ed36b
- question-first witness SHA-256: 51e18649133238ac992229f6e16542a5f3af53ca73b2262ab0524508ea937dee
- Lean: NOT_RUN · human review: NOT_PERFORMED

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