# Question 6645: Original Denominator Guard versus Cross-Multiplication [Return to catalogue page 34](https://ocme.evemisslab.com/en/questions/page/34/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261010-advance-20261014-r03-equation-foundations/#q-20261010-advance-20261014-r03-equation-0070) 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. R0012-06645 · q-20261010-advance-20261014-r03-equation-0070 ## Original question and givens Over Q, the original equation is ((7)x+(20))/(x−(2))=0. 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 ((7)x+(20))/(x−(2))=0. 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](https://ocme.evemisslab.com/en/mko/mko-equation/) ## Step-by-step derivation S001 · impliesBefore common rational domain with original equation definitions After evaluate/classify the given model without enlarging its original domain Conditions: none Original definedness precedes substitution and transformations.Check the declared model,not arbitrary equation solving inferred from cross multiplication or one-point values. Step identifiers and dependenciesRule: original_rational_equation_scope · Uses: G001 S002 · calculationBefore query0 original denominator := sub(2, 2) After query0 original denominator = 0 Conditions: none Evaluate the original denominator x−h first;zero excludes this point from the original equation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore query0 numerator linear product := mul(7, 2) After query0 numerator linear product = 14 Conditions: none Substitute the specified x into this side's coefficient product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore query0 numerator affine value := sum(14, 20) After query0 numerator affine value = 34 Conditions: none Then add this side's own constant,not the other side's. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore query0 cross right side := mul(0, 0) After query0 cross right side = 0 Conditions: none The cross-multiplied right side is c(x−h),a polynomial value computable even at an excluded point. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore query0 cross residual := sub(34, 0) After query0 cross residual = 34 Conditions: none The cross residual is numerator minus c(x−h),reported separately from original definedness. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore x=2/1;x−h=0;cross residual=34 After original defined=No;original residual=undefined;original solution=No Conditions: none Cross multiplication is equivalent only when x≠h;even numerator0 and cross residual0 at an excluded point cannot restore an original solution. Step identifiers and dependenciesRule: original_denominator_exclusion_persists · Uses: G001, S006 S008 · calculationBefore query1 original denominator := sub(5/2, 2) After query1 original denominator = 1/2 Conditions: none Evaluate the original denominator x−h first;zero excludes this point from the original equation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore query1 numerator linear product := mul(7, 5/2) After query1 numerator linear product = 35/2 Conditions: none Substitute the specified x into this side's coefficient product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore query1 numerator affine value := sum(35/2, 20) After query1 numerator affine value = 75/2 Conditions: none Then add this side's own constant,not the other side's. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore query1 cross right side := mul(0, 1/2) After query1 cross right side = 0 Conditions: none The cross-multiplied right side is c(x−h),a polynomial value computable even at an excluded point. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore query1 cross residual := sub(75/2, 0) After query1 cross residual = 75/2 Conditions: none The cross residual is numerator minus c(x−h),reported separately from original definedness. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore query1 original quotient := div(75/2, 1/2) After query1 original quotient = 75 Conditions: none Evaluate the original quotient only with a nonzero original denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore query1 original residual := sub(75, 0) After query1 original residual = 75 Conditions: none Original quotient minus c is the original-equation residual. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore x=5/2;x−h=1/2;cross residual=75/2 After original defined=Yes;original residual=75/1;original solution=No Conditions: none Cross multiplication is equivalent only when x≠h;even numerator0 and cross residual0 at an excluded point cannot restore an original solution. Step identifiers and dependenciesRule: original_denominator_exclusion_persists · Uses: G001, S014 S016 · calculationBefore query2 original denominator := sub(4, 2) After query2 original denominator = 2 Conditions: none Evaluate the original denominator x−h first;zero excludes this point from the original equation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore query2 numerator linear product := mul(7, 4) After query2 numerator linear product = 28 Conditions: none Substitute the specified x into this side's coefficient product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore query2 numerator affine value := sum(28, 20) After query2 numerator affine value = 48 Conditions: none Then add this side's own constant,not the other side's. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore query2 cross right side := mul(0, 2) After query2 cross right side = 0 Conditions: none The cross-multiplied right side is c(x−h),a polynomial value computable even at an excluded point. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore query2 cross residual := sub(48, 0) After query2 cross residual = 48 Conditions: none The cross residual is numerator minus c(x−h),reported separately from original definedness. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore query2 original quotient := div(48, 2) After query2 original quotient = 24 Conditions: none Evaluate the original quotient only with a nonzero original denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore query2 original residual := sub(24, 0) After query2 original residual = 24 Conditions: none Original quotient minus c is the original-equation residual. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · impliesBefore x=4/1;x−h=2;cross residual=48 After original defined=Yes;original residual=24/1;original solution=No Conditions: none Cross multiplication is equivalent only when x≠h;even numerator0 and cross residual0 at an excluded point cannot restore an original solution. Step identifiers and dependenciesRule: original_denominator_exclusion_persists · Uses: G001, S022 ## 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, 75/1, 24/1], [34/1, 75/2, 48/1], [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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 459a12de6a3c0dc9b474a67a8ef9ac89c11c2de2f28db3b7b89157392ddeff71 - candidate record SHA-256: f5cc64256cdd5201aae8334ec8b8c995a6747a040f95dd5406804a96e221e180 - semantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00 - question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261010-advance-20261014-r03-equation-0070.json)