# Question 1136: Two-Point Slope of an Affine Graph

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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.

R0001-01136 · q-20261003-r05-linear-function-0036



## Original question and givens

Over the reals, an affine-function graph contains (-2, -2) and (3, -4), with distinct first coordinates. Find the slope as a reduced fraction with positive denominator.


- G001 — Over the reals, an affine-function graph contains (-2, -2) and (3, -4), with distinct first coordinates. Find the slope as a reduced fraction with positive denominator.





## 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-linear-function](https://ocme.evemisslab.com/en/mko/mko-linear-function/)





## Step-by-step derivation

S001 · impliesBefore

f:R→R; f(x)=mx+b

After

m=0 gives a constant affine function

Conditions: none

An affine function allows m=0; do not impose the nondegenerate-equation condition on every affine function.

Step identifiers and dependenciesRule: affine_domain · Uses: G001





S002 · calculationBefore

Δx := sub(3, -2)

After

Δx = 5

Conditions: none

The denominator is the horizontal-coordinate difference.

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





S003 · calculationBefore

Δy := sub(-4, -2)

After

Δy = -2

Conditions: none

Use the same point order for the vertical difference.

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





S004 · calculationBefore

slope := div(-2, 5)

After

slope = -2/5

Conditions: none

Compute the slope only after checking Δx is nonzero.

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





S005 · calculationBefore

slope·Δx := mul(-2/5, 5)

After

slope·Δx = -2

Conditions: none

Multiplication by the denominator must recover the original vertical difference.

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









## Reverse checks


- S005 · premise_and_result_audit — Recompute slope·Δx 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 Δy with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute Δx 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

-2/5

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: 374638d8cd14303f6b0d352b9cef39ed81ae48c70d242be7b7ac6a498114bf25
- candidate record SHA-256: a956a8c19942db6d2b3a29d6417dadac914e18f014a4a729aa55209cfe96a358
- semantic review receipt SHA-256: c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496cc
- question-first witness SHA-256: 3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7ad
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-r05-linear-function-0036.json)