# Question 32957: Checking a Rational Candidate in Both Rows [Return to catalogue page 165](https://ocme.evemisslab.com/en/questions/page/165/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261003-advance-20261004-r06-system-linear-equations-foundations/#q-20261003-advance-20261004-r06-system-linear-equations-0057) 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. R0064-32957 · q-20261003-advance-20261004-r06-system-linear-equations-0057 ## Original question and givens Over the reals, the constraints (2)x+(3)y=-3 and (4)x+(-3)y=-15 must both hold. Does the reduced positive-denominator candidate x=-3/1, y=1/1 solve the entire system? - G001 — Over the reals, the constraints (2)x+(3)y=-3 and (4)x+(-3)y=-15 must both hold. Does the reduced positive-denominator candidate x=-3/1, y=1/1 solve the entire system? ## 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-system-linear-equations](https://ocme.evemisslab.com/en/mko/mko-system-linear-equations/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-linear-system-solution-verification, verify_solution, reals, []] After derived = [[ALL_rows_YesNo_STRING]] Conditions: none Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status. Step identifiers and dependenciesRule: R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles · Uses: G001 S002 · impliesBefore source roles = [[[2, 3, -3], [4, -3, -15]]] After derived = [[coefficient_x, coefficient_y, right_side]] Conditions: none Each ORIGINAL row[a,b,c]means ax+by=c;the THIRD column is RHS,not a coefficient.Type ALL SIX raw SAFE values,including unused RHS in a determinant task.Keep original row order. Step identifiers and dependenciesRule: System_RAW_exact2_augmented_rows3_ALL_SAFEincl_unused_rhs · Uses: G001, S001 S003 · calculationBefore System ORIGINAL a times e main product := mul(2, -3) After System ORIGINAL a times e main product = -6 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore System ORIGINAL b times d other product := mul(3, 4) After System ORIGINAL b times d other product = 12 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore System ORIGINAL signed determinant ae minus bd := sub(-6, 12) After System ORIGINAL signed determinant ae minus bd = -18 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [[[2, 3, -3], [4, -3, -15]]] After derived = [-6, 12, -18] Conditions: none D=ae−bd uses ONLY the first TWO columns,retaining ORIGINAL row order/sign.RHS is excluded;swapping rows changes the sign.Do not replace originals by semantic sorting. Step identifiers and dependenciesRule: System_DETERMINANT_coefficients_ONLY_raw_order_sign · Uses: G001, S005 S007 · impliesBefore source roles = [x candidate, [-3, 1]] After derived = [-3/1] Conditions: none The RAW candidate must ALREADY be reduced with positive nonzero denominator;zero is0/1.Do not repair an invalid candidate by normalization.Validate EVERY component,even in a zero-row system. Step identifiers and dependenciesRule: System_RAW_candidate_fraction_ALREADY_reduced_positive_den · Uses: G001, S006 S008 · impliesBefore source roles = [y candidate, [1, 1]] After derived = [1/1] Conditions: none The RAW candidate must ALREADY be reduced with positive nonzero denominator;zero is0/1.Do not repair an invalid candidate by normalization.Validate EVERY component,even in a zero-row system. Step identifiers and dependenciesRule: System_RAW_candidate_fraction_ALREADY_reduced_positive_den · Uses: G001, S007 S009 · calculationBefore System candidate ORIGINAL two positive denominators product := mul(1, 1) After System candidate ORIGINAL two positive denominators product = 1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore System cleared row0 ORIGINAL coefficient x times numerator := mul(2, -3) After System cleared row0 ORIGINAL coefficient x times numerator = -6 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore System cleared row0 ORIGINAL x term times y denominator := mul(-6, 1) After System cleared row0 ORIGINAL x term times y denominator = -6 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore System cleared row0 ORIGINAL coefficient y times numerator := mul(3, 1) After System cleared row0 ORIGINAL coefficient y times numerator = 3 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore System cleared row0 ORIGINAL y term times x denominator := mul(3, 1) After System cleared row0 ORIGINAL y term times x denominator = 3 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore System cleared row0 ORIGINAL rhs times both denominators := mul(-3, 1) After System cleared row0 ORIGINAL rhs times both denominators = -3 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore System cleared row0 ORIGINAL two left terms sum := sum(-6, 3) After System cleared row0 ORIGINAL two left terms sum = -3 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore System cleared row0 ORIGINAL residual := sub(-3, -3) After System cleared row0 ORIGINAL residual = 0 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore source roles = [0, 2, 3, -3, -3, 1, 1, 1] After derived = [-6, 3, -3, 0, Yes] Conditions: none Multiplying BOTH positive nonzero denominators preserves equality.Check EACH u*xn*yd+v*yn*xd−w*xd*yd=0.Do not test only one row or invent a D≠0 requirement. Step identifiers and dependenciesRule: System_CANDIDATE_EACH_row_clear_BOTH_nonzero_denominators · Uses: G001, S016 S018 · calculationBefore System cleared row1 ORIGINAL coefficient x times numerator := mul(4, -3) After System cleared row1 ORIGINAL coefficient x times numerator = -12 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore System cleared row1 ORIGINAL x term times y denominator := mul(-12, 1) After System cleared row1 ORIGINAL x term times y denominator = -12 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore System cleared row1 ORIGINAL coefficient y times numerator := mul(-3, 1) After System cleared row1 ORIGINAL coefficient y times numerator = -3 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore System cleared row1 ORIGINAL y term times x denominator := mul(-3, 1) After System cleared row1 ORIGINAL y term times x denominator = -3 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore System cleared row1 ORIGINAL rhs times both denominators := mul(-15, 1) After System cleared row1 ORIGINAL rhs times both denominators = -15 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore System cleared row1 ORIGINAL two left terms sum := sum(-12, -3) After System cleared row1 ORIGINAL two left terms sum = -15 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore System cleared row1 ORIGINAL residual := sub(-15, -15) After System cleared row1 ORIGINAL residual = 0 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · impliesBefore source roles = [1, 4, -3, -15, -3, 1, 1, 1] After derived = [-12, -3, -15, 0, Yes] Conditions: none Multiplying BOTH positive nonzero denominators preserves equality.Check EACH u*xn*yd+v*yn*xd−w*xd*yd=0.Do not test only one row or invent a D≠0 requirement. Step identifiers and dependenciesRule: System_CANDIDATE_EACH_row_clear_BOTH_nonzero_denominators · Uses: G001, S024 S026 · calculationBefore Candidate direct reverse ORIGINAL row0 x term := mul(2, -3) After Candidate direct reverse ORIGINAL row0 x term = -6 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore Candidate direct reverse ORIGINAL row0 y term := mul(3, 1) After Candidate direct reverse ORIGINAL row0 y term = 3 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Candidate direct reverse ORIGINAL row0 terms sum := sum(-6, 3) After Candidate direct reverse ORIGINAL row0 terms sum = -3 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore Candidate direct reverse ORIGINAL row0 residual minus rhs := sub(-3, -3) After Candidate direct reverse ORIGINAL row0 residual minus rhs = 0 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · impliesBefore source roles = [Candidate direct reverse, 0, 2, 3, -3, -3/1, 1/1] After derived = [-6, 3, -3, 0/1] Conditions: none Substitute into EACH ORIGINAL row with its ORIGINAL coefficients/RHS.ALL residuals must be0,not merely one transformed row. Step identifiers and dependenciesRule: System_EACH_original_row_reverse_exact_substitution · Uses: G001, S029 S031 · calculationBefore Candidate direct reverse ORIGINAL row1 x term := mul(4, -3) After Candidate direct reverse ORIGINAL row1 x term = -12 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore Candidate direct reverse ORIGINAL row1 y term := mul(-3, 1) After Candidate direct reverse ORIGINAL row1 y term = -3 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore Candidate direct reverse ORIGINAL row1 terms sum := sum(-12, -3) After Candidate direct reverse ORIGINAL row1 terms sum = -15 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore Candidate direct reverse ORIGINAL row1 residual minus rhs := sub(-15, -15) After Candidate direct reverse ORIGINAL row1 residual minus rhs = 0 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · impliesBefore source roles = [Candidate direct reverse, 1, 4, -3, -15, -3/1, 1/1] After derived = [-12, -3, -15, 0/1] Conditions: none Substitute into EACH ORIGINAL row with its ORIGINAL coefficients/RHS.ALL residuals must be0,not merely one transformed row. Step identifiers and dependenciesRule: System_EACH_original_row_reverse_exact_substitution · Uses: G001, S034 ## Reverse checks - S035 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S034 · premise_and_result_audit — Recompute Candidate direct reverse ORIGINAL row1 residual minus rhs with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Candidate direct reverse ORIGINAL row1 terms sum with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Candidate direct reverse ORIGINAL row1 y term with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Candidate direct reverse ORIGINAL row1 x term with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S029 · premise_and_result_audit — Recompute Candidate direct reverse ORIGINAL row0 residual minus rhs with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Candidate direct reverse ORIGINAL row0 terms sum with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Candidate direct reverse ORIGINAL row0 y term with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Candidate direct reverse ORIGINAL row0 x term with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S024 · premise_and_result_audit — Recompute System cleared row1 ORIGINAL residual with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute System cleared row1 ORIGINAL two left terms sum with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute System cleared row1 ORIGINAL rhs times both denominators with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute System cleared row1 ORIGINAL y term times x denominator with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute System cleared row1 ORIGINAL coefficient y times numerator with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute System cleared row1 ORIGINAL x term times y denominator with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute System cleared row1 ORIGINAL coefficient x times numerator with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S016 · premise_and_result_audit — Recompute System cleared row0 ORIGINAL residual with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute System cleared row0 ORIGINAL two left terms sum with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute System cleared row0 ORIGINAL rhs times both denominators with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute System cleared row0 ORIGINAL y term times x denominator with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute System cleared row0 ORIGINAL coefficient y times numerator with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute System cleared row0 ORIGINAL x term times y denominator with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute System cleared row0 ORIGINAL coefficient x times numerator with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute System candidate ORIGINAL two positive denominators product with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S007 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S006 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S005 · premise_and_result_audit — Recompute System ORIGINAL signed determinant ae minus bd with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute System ORIGINAL b times d other product with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute System ORIGINAL a times e main product with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. ## Result Yes 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: 22269db5c2cdb25395318a3b52ff6490d2b832df61fbae4fbe966e7fb56f9b10 - candidate record SHA-256: 491a860dcb6ece392eabbaaf2d6418dfe18ce8cce35dc019b6a4dd37dccc43a7 - semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b - question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-advance-20261004-r06-system-linear-equations-0057.json)