# Question 41: 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.

P250-0041 · q-20260927-linear-equation-solution-class-0016



## Original question and givens

Over the rational numbers, classify the solution set of 2x+(3)=0.


- G001 — Over the rational numbers, classify the solution set of 2x+(3)=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

2x+(3)=0, x∈Q

After

a=2; b=3

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 · equivalentBefore

2x+(3)=0

After

2x=-3

Conditions: none

Subtract 3 from both sides, preserving equivalence.

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





S003 · calculationBefore

x := div(-3, 2)

After

x = -3/2

Conditions: none

Now a=2 is known to be nonzero, so division by it is valid.

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





S004 · calculationBefore

a·x := mul(2, -3/2)

After

a·x = -3

Conditions: none

Substitute the candidate and calculate the linear term.

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





S005 · calculationBefore

a·x+b := sum(-3, 3)

After

a·x+b = 0

Conditions: none

Add the constant back; a result of 0 verifies the original equation.

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





S006 · impliesBefore

a x₁+b=0 and a x₂+b=0

After

a(x₁−x₂)=0 ⇒ x₁=x₂

Conditions: a≠0

Subtract two proposed solution equations and cancel the nonzero a; no distinct second solution exists.

Step identifiers and dependenciesRule: nonzero_cancellation · Uses: G001, S005









## Reverse checks


- S006 · premise_and_result_audit — The hypothesis a≠0 is essential to this uniqueness argument.
- S005 · premise_and_result_audit — Recompute a·x+b with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute a·x with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute x with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Adding b back to both sides recovers the original equation.
- S001 · premise_and_result_audit — When a=0 this is a degenerate constant equation, not a nondegenerate linear equation.





## Result

unique solution x=-3/2

The unique rational solution is x=-3/2.





## 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: 6dba6030b10377067a1d5c3770c03034c5db81c745c1f96b0bbe4e1869ca4811
- candidate record SHA-256: 3869be07ef1ee420a4884eb19ec30d4824ee838ebd65fef569b109b4a7fb5b0d
- semantic review receipt SHA-256: 4727d1bd957a9b7f6cadb7972ddb5ab80a31f5259dc5394c1dff2b71944ed36b
- question-first witness SHA-256: 51e18649133238ac992229f6e16542a5f3af53ca73b2262ab0524508ea937dee
- Lean: NOT_RUN · human review: NOT_PERFORMED

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