# Question 6662: Original Rational Equation Sides and Defect [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-0012) 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. R0012-06662 · q-20261010-advance-20261014-r03-equation-0012 ## Original question and givens Use the original equation’s common rational domain. L(x)=((-1)x+(12))/(-2), R(x)=(1)x+(-11), q≠0, x=12/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers. - G001 — Use the original equation’s common rational domain. L(x)=((-1)x+(12))/(-2), R(x)=(1)x+(-11), q≠0, x=12/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers. ## 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 original left numerator linear product := mul(-1, 6) After original left numerator linear product = -6 Conditions: none Substitute the specified x into this side's coefficient product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore original left numerator affine value := sum(-6, 12) After original left numerator affine value = 6 Conditions: none Then add this side's own constant,not the other side's. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore original right linear product := mul(1, 6) After original right linear product = 6 Conditions: none Substitute the specified x into this side's coefficient product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore original right affine value := sum(6, -11) After original right affine value = -5 Conditions: none Then add this side's own constant,not the other side's. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore original left quotient := div(6, -2) After original left quotient = -3 Conditions: none The original constant denominator q is nonzero;negative q is also handled exactly. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore original two-side residual := sub(-3, -5) After original two-side residual = 2 Conditions: none Subtract the original right side from the original left;residual0 means the equation holds at this x. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore q=-2≠0;x=6/1;L−R=2 After original equation at this x=No Conditions: none This audits the legal original equation at the specified point;one-point agreement is not an identity for every x. Step identifiers and dependenciesRule: original_fraction_equation_point_check · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute original two-side residual with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute original left quotient with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute original right affine value with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute original right linear product with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute original left numerator affine value with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute original left numerator linear product 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 (-3/1, -5/1, 2/1, 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: e4c31f18124649e6718a483d9693e124c4f34b00bc7cf1847a3155062e1f3f0f - candidate record SHA-256: 72b7b756c0d316dfa6ef9a96e5e3db4f0543e54edea6938b8a544d303157caa5 - 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-0012.json)