# Question 4783: Affine Expression in Two Environments

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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.

R0008-04783 · q-20261008-advance-20261011-r03-variable-expression-0008



## Original question and givens

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-7)x+(0) is an expression, not an equation. Ordered x environments=[7/3, -8/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.


- G001 — All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-7)x+(0) is an expression, not an equation. Ordered x environments=[7/3, -8/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-variable-expression](https://ocme.evemisslab.com/en/mko/mko-variable-expression/)





## Step-by-step derivation

S001 · impliesBefore

typed rational expression and explicitly given environment

After

evaluation,not solving an equation

Conditions: none

Evaluating an expression is not solving an equation;substitute only the supplied environment without inventing roots or changing bindings.

Step identifiers and dependenciesRule: expression_environment_scope · Uses: G001





S002 · calculationBefore

environment0 ax := mul(-7, 7/3)

After

environment0 ax = -49/3

Conditions: none

Substitute this environment's x into the product term.

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





S003 · calculationBefore

environment0 ax+b := sum(-49/3, 0)

After

environment0 ax+b = -49/3

Conditions: none

Then add b from the same expression.

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





S004 · calculationBefore

environment1 ax := mul(-7, -4)

After

environment1 ax = 28

Conditions: none

Substitute this environment's x into the product term.

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





S005 · calculationBefore

environment1 ax+b := sum(28, 0)

After

environment1 ax+b = 28

Conditions: none

Then add b from the same expression.

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





S006 · impliesBefore

[-49/3, 28/1]

After

ordered independent evaluations of the same expression

Conditions: none

Read both environments from the original data;the second evaluation does not inherit a mutated first substitution.When a=0 the constant expression is independent of x.

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









## Reverse checks


- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Recompute environment1 ax+b with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute environment1 ax with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute environment0 ax+b with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute environment0 ax with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(-49/3, 28/1)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## 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: 4b309fc913807298647e63a85107de6b4882070a8b94dd4f848f007163e37272
- candidate record SHA-256: 3e3a3f0adc5d6a45f08c3b9132660c6524a4bec7472c30e958257ea8861159a6
- semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289
- question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67
- Lean: NOT_RUN · human review: NOT_PERFORMED

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