Question 6637: Original Denominator Guard versus Cross-Multiplication

Original question and givens

Over Q, the original equation is ((-1)x+(12))/(x−(-1))=1. Exclude x=h=-1; ordered queries=[-1/1, -1/2, 1/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

• G001 — Over Q, the original equation is ((-1)x+(12))/(x−(-1))=1. Exclude x=h=-1; ordered queries=[-1/1, -1/2, 1/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

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-equation

Step-by-step derivation

common rational domain with original equation definitions

evaluate/classify the given model without enlarging its original domain

query0 original denominator := sub(-1, -1)

query0 original denominator = 0

query0 numerator linear product := mul(-1, -1)

query0 numerator linear product = 1

query0 numerator affine value := sum(1, 12)

query0 numerator affine value = 13

query0 cross right side := mul(1, 0)

query0 cross right side = 0

query0 cross residual := sub(13, 0)

query0 cross residual = 13

x=-1/1;x−h=0;cross residual=13

original defined=No;original residual=undefined;original solution=No

query1 original denominator := sub(-1/2, -1)

query1 original denominator = 1/2

query1 numerator linear product := mul(-1, -1/2)

query1 numerator linear product = 1/2

query1 numerator affine value := sum(1/2, 12)

query1 numerator affine value = 25/2

query1 cross right side := mul(1, 1/2)

query1 cross right side = 1/2

query1 cross residual := sub(25/2, 1/2)

query1 cross residual = 12

query1 original quotient := div(25/2, 1/2)

query1 original quotient = 25

query1 original residual := sub(25, 1)

query1 original residual = 24

x=-1/2;x−h=1/2;cross residual=12

original defined=Yes;original residual=24/1;original solution=No

query2 original denominator := sub(1, -1)

query2 original denominator = 2

query2 numerator linear product := mul(-1, 1)

query2 numerator linear product = -1

query2 numerator affine value := sum(-1, 12)

query2 numerator affine value = 11

query2 cross right side := mul(1, 2)

query2 cross right side = 2

query2 cross residual := sub(11, 2)

query2 cross residual = 9

query2 original quotient := div(11, 2)

query2 original quotient = 11/2

query2 original residual := sub(11/2, 1)

query2 original residual = 9/2

x=1/1;x−h=2;cross residual=9

original defined=Yes;original residual=9/2;original solution=No

Reverse checks

• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Recompute query2 original residual with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute query2 original quotient with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute query2 cross residual with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute query2 cross right side with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute query2 numerator affine value with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute query2 numerator linear product with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute query2 original denominator with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute query1 original residual with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute query1 original quotient with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute query1 cross residual with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute query1 cross right side with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute query1 numerator affine value with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute query1 numerator linear product with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute query1 original denominator with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute query0 cross residual with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute query0 cross right side with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute query0 numerator affine value with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute query0 numerator linear product with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute query0 original denominator 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

([No, Yes, Yes], [undefined, 24/1, 9/2], [13/1, 12/1, 9/1], [No, No, No])

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.