# Question 301: Cross-Multiplication Check of a Rational Solution

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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.

P1000-0301 · q-20260927-linear-equation-solution-verification-0001



## Original question and givens

Over the rational numbers, consider -4x+(-14)=0 and the candidate x=-7/2. Is the candidate a solution?


- G001 — Over the rational numbers, consider -4x+(-14)=0 and the candidate x=-7/2. Is the candidate a solution?





## Definition and scope

Use the rational domain. A candidate p/q needs q≠0; a zero residual is equivalent to the cross-multiplied identity.


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





## Step-by-step derivation

S001 · impliesBefore

a=-4, b=-14, x∈Q

After

equation: -4x+(-14)=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 · calculationBefore

candidate x := div(-7, 2)

After

candidate x = -7/2

Conditions: none

Read the candidate as an exact rational; its denominator must be nonzero.

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





S003 · calculationBefore

a·x := mul(-4, -7/2)

After

a·x = 14

Conditions: none

Use the original coefficient in the linear term.

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





S004 · calculationBefore

a·x+b := sum(14, -14)

After

a·x+b = 0

Conditions: none

The candidate solves the original equation exactly when the residual is0.

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





S005 · calculationBefore

a·p := mul(-4, -7)

After

a·p = 28

Conditions: none

Independently form the first cross-multiplied integer term.

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





S006 · calculationBefore

b·q := mul(-14, 2)

After

b·q = -28

Conditions: none

Multiply the constant by the same nonzero denominator.

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





S007 · calculationBefore

a·p+b·q := sum(28, -28)

After

a·p+b·q = 0

Conditions: none

Nonzero q makes this equality equivalent to a zero original residual.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · impliesBefore

residual=0; cross=0

After

candidate solves equation: Yes

Conditions: none

The two exact checks agree on the candidate's validity.

Step identifiers and dependenciesRule: substitution_verdict · Uses: G001, S007









## Reverse checks


- S008 · premise_and_result_audit — Substitute into the original equation; retain every nonzero condition before division.
- S007 · premise_and_result_audit — Recompute a·p+b·q with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute b·q with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute a·p with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute a·x+b with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute a·x with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute candidate x with exact rational arithmetic, not rounded decimals.
- 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: efdce7dd45caf29b5ee862159044605dbefe9569b3add1133d91ff7a65c7f030
- candidate record SHA-256: ade9bb19403897d0934e7f588c6105061ea570d12e68970cc51f0a8014db576f
- 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-solution-verification-0001.json)