# Question 292: Exact Solution of a Linear Equation

[Return to catalogue page 2](https://ocme.evemisslab.com/en/questions/page/2/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260927-linear-equation-foundations/#q-20260927-linear-equation-exact-solve-0017)

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.

P1000-0292 · q-20260927-linear-equation-exact-solve-0017



## Original question and givens

Solve 1x+(4)=0 exactly over the rational numbers; a=1 is explicitly nonzero.


- G001 — Solve 1x+(4)=0 exactly over the rational numbers; a=1 is explicitly nonzero.





## Definition and scope

Use the rational domain with a≠0. Move the constant, divide by nonzero a and give the reduced fraction with positive denominator.


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





## Step-by-step derivation

S001 · impliesBefore

a=1, b=4, x∈Q

After

equation: 1x+(4)=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 · equivalentBefore

1x+(4)=0

After

1x=-4

Conditions: none

Subtract b from both sides, preserving the solution set.

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





S003 · calculationBefore

x := div(-4, 1)

After

x = -4

Conditions: none

Division by a is allowed because 1≠0.

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





S004 · calculationBefore

a·x := mul(1, -4)

After

a·x = -4

Conditions: none

Substitute into the linear term.

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





S005 · calculationBefore

a·x+b := sum(-4, 4)

After

a·x+b = 0

Conditions: none

Adding the constant must give0.

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





S006 · impliesBefore

a(x₁−x₂)=0 and a≠0

After

x₁=x₂

Conditions: none

Subtract equations for any two solutions and cancel nonzero a to prove uniqueness.

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









## Reverse checks


- S006 · premise_and_result_audit — Substitute into the original equation; retain every nonzero condition before division.
- 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 — 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

-4

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: 552e0665efb99936067c3e2169800183468579c598b64810a55b4c33c09b0a64
- candidate record SHA-256: d7d3438981161997f003f4cefc71ce94593995547b45bb14882070c319562fd5
- 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-exact-solve-0017.json)