Question 6640: Original Denominator Guard versus Cross-Multiplication

Original question and givens

Over Q, the original equation is ((2)x+(15))/(x−(2))=1. Exclude x=h=2; ordered queries=[2/1, 5/2, 4/1]. 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 ((2)x+(15))/(x−(2))=1. Exclude x=h=2; ordered queries=[2/1, 5/2, 4/1]. 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(2, 2)

query0 original denominator = 0

query0 numerator linear product := mul(2, 2)

query0 numerator linear product = 4

query0 numerator affine value := sum(4, 15)

query0 numerator affine value = 19

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

query0 cross right side = 0

query0 cross residual := sub(19, 0)

query0 cross residual = 19

x=2/1;x−h=0;cross residual=19

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

query1 original denominator := sub(5/2, 2)

query1 original denominator = 1/2

query1 numerator linear product := mul(2, 5/2)

query1 numerator linear product = 5

query1 numerator affine value := sum(5, 15)

query1 numerator affine value = 20

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

query1 cross right side = 1/2

query1 cross residual := sub(20, 1/2)

query1 cross residual = 39/2

query1 original quotient := div(20, 1/2)

query1 original quotient = 40

query1 original residual := sub(40, 1)

query1 original residual = 39

x=5/2;x−h=1/2;cross residual=39/2

original defined=Yes;original residual=39/1;original solution=No

query2 original denominator := sub(4, 2)

query2 original denominator = 2

query2 numerator linear product := mul(2, 4)

query2 numerator linear product = 8

query2 numerator affine value := sum(8, 15)

query2 numerator affine value = 23

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

query2 cross right side = 2

query2 cross residual := sub(23, 2)

query2 cross residual = 21

query2 original quotient := div(23, 2)

query2 original quotient = 23/2

query2 original residual := sub(23/2, 1)

query2 original residual = 21/2

x=4/1;x−h=2;cross residual=21

original defined=Yes;original residual=21/2;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, 39/1, 21/2], [19/1, 39/2, 21/1], [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.