Question 2698: The Nonvertical Affine Rule through Two Distinct Inputs

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], [2, 6]]. 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], [2, 6]]. 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

Step-by-step derivation

declared symbolic rule or complete finite table

keep the stated domain and representation distinct

x₂−x₁ := sub(2, 0)

x₂−x₁ = 2

y₂−y₁ := sub(6, 10)

y₂−y₁ = -4

slope := div(-4, 2)

slope = -2

slope times x₁ := mul(-2, 0)

slope times x₁ = 0

intercept := sub(10, 0)

intercept = 10

reverse x=0 ax := mul(-2, 0)

reverse x=0 ax = 0

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

reverse x=0 ax+b = 10

predicted=10; supplied=10

both values agree

reverse x=2 ax := mul(-2, 2)

reverse x=2 ax = -4

reverse x=2 ax+b := sum(-4, 10)

reverse x=2 ax+b = 6

predicted=6; supplied=6

both values agree

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

unique affine coefficients=(-2,10)

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=2 ax+b with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute reverse x=2 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], [2, 6]], -2, 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.