# Question 2678: The Nonvertical Affine Rule through Two Distinct Inputs

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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.

R0004-02678 · q-20261013-advance-20261021-r01-mathematical-representation-0028



## Original question and givens

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[0, -10], [3, -13]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.


- G001 — Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[0, -10], [3, -13]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.





## 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-mathematical-representation](https://ocme.evemisslab.com/en/mko/mko-mathematical-representation/)





## Step-by-step derivation

S001 · impliesBefore

declared symbolic rule or complete finite table

After

keep the stated domain and representation distinct

Conditions: none

Use only the declared rule or table;agreement on a finite table alone does not establish function equality on an unlisted domain.

Step identifiers and dependenciesRule: representation_scope · Uses: G001





S002 · calculationBefore

x₂−x₁ := sub(3, 0)

After

x₂−x₁ = 3

Conditions: none

Check that the two x-coordinates differ before division.

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





S003 · calculationBefore

y₂−y₁ := sub(-13, -10)

After

y₂−y₁ = -3

Conditions: none

Use the same point order in the y-difference;do not reverse just one difference.

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





S004 · calculationBefore

slope := div(-3, 3)

After

slope = -1

Conditions: none

Divide y₂−y₁=a(x₂−x₁) by the nonzero x-difference to obtain the affine slope.

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





S005 · calculationBefore

slope times x₁ := mul(-1, 0)

After

slope times x₁ = 0

Conditions: none

Substitute the slope at the first point.

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





S006 · calculationBefore

intercept := sub(-10, 0)

After

intercept = -10

Conditions: none

Rearrange y₁=ax₁+b to obtain the intercept.

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





S007 · calculationBefore

reverse x=0 ax := mul(-1, 0)

After

reverse x=0 ax = 0

Conditions: none

Evaluate a·x first;do not replace the rule by a(x+b).

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





S008 · calculationBefore

reverse x=0 ax+b := sum(0, -10)

After

reverse x=0 ax+b = -10

Conditions: none

Then add the intercept b,fully substituting this input.

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





S009 · impliesBefore

predicted=-10; supplied=-10

After

both values agree

Conditions: none

Substitute both points back;the unique affine interpolant is not the unique unrestricted curve through the points.

Step identifiers and dependenciesRule: two_point_substitution_audit · Uses: G001, S008





S010 · calculationBefore

reverse x=3 ax := mul(-1, 3)

After

reverse x=3 ax = -3

Conditions: none

Evaluate a·x first;do not replace the rule by a(x+b).

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · calculationBefore

reverse x=3 ax+b := sum(-3, -10)

After

reverse x=3 ax+b = -13

Conditions: none

Then add the intercept b,fully substituting this input.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · impliesBefore

predicted=-13; supplied=-13

After

both values agree

Conditions: none

Substitute both points back;the unique affine interpolant is not the unique unrestricted curve through the points.

Step identifiers and dependenciesRule: two_point_substitution_audit · Uses: G001, S011





S013 · impliesBefore

two distinct x-coordinates and an affine rule ax+b

After

unique affine coefficients=(-1,-10)

Conditions: none

Subtracting two affine fits gives(a−a′)(x₂−x₁)=0,so a=a′ and b=b′;without the affine restriction,other interpolants are possible.

Step identifiers and dependenciesRule: affine_model_uniqueness · Uses: G001, S012









## Reverse checks


- S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S011 · premise_and_result_audit — Recompute reverse x=3 ax+b with exact rational arithmetic, not rounded decimals.
- S010 · premise_and_result_audit — Recompute reverse x=3 ax with exact rational arithmetic, not rounded decimals.
- S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S008 · premise_and_result_audit — Recompute reverse x=0 ax+b with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute reverse x=0 ax with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute intercept with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute slope times x₁ with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute slope with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute y₂−y₁ with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute x₂−x₁ 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

([[0, -10], [3, -13]], -1, -10)

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: 9b42bfdde5c6938d7f5fac03b12eb22fe16fe77c71ab71ca3150f83b046ba5b7
- candidate record SHA-256: c9f1dfd2c952e4975a3817cf10210b69663fff29328aeebeafb569dace53c1ae
- semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56
- question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261021-r01-mathematical-representation-0028.json)