# Question 32997: Exact Unique Solution with a Nonzero Determinant [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-0097) 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. R0064-32997 · q-20261003-advance-20261004-r06-system-linear-equations-0097 ## Original question and givens Over the reals, the constraints (1)x+(1)y=9 and (1)x+(-2)y=-1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator. - G001 — Over the reals, the constraints (1)x+(1)y=9 and (1)x+(-2)y=-1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator. ## 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-unique-solve, unique_solve, reals, []] After derived = [[solution_pair_fractions_RETAIN1]] 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 = [[[1, 1, 9], [1, -2, -1]]] 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(1, -2) After System ORIGINAL a times e main product = -2 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(1, 1) After System ORIGINAL b times d other 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, S003 S005 · calculationBefore System ORIGINAL signed determinant ae minus bd := sub(-2, 1) After System ORIGINAL signed determinant ae minus bd = -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, S004 S006 · impliesBefore source roles = [[[1, 1, 9], [1, -2, -1]]] After derived = [-2, 1, -3] 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 · calculationBefore Cramer ORIGINAL c times e := mul(9, -2) After Cramer ORIGINAL c times e = -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, S006 S008 · calculationBefore Cramer ORIGINAL b times f := mul(1, -1) After Cramer ORIGINAL b times f = -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, S007 S009 · calculationBefore Cramer ORIGINAL numerator x ce minus bf := sub(-18, -1) After Cramer ORIGINAL numerator x ce minus bf = -17 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 Cramer ORIGINAL a times f := mul(1, -1) After Cramer ORIGINAL a times f = -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, S009 S011 · calculationBefore Cramer ORIGINAL c times d := mul(9, 1) After Cramer ORIGINAL c times d = 9 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 Cramer ORIGINAL numerator y af minus cd := sub(-1, 9) After Cramer ORIGINAL numerator y af minus cd = -10 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 Cramer ORIGINAL x numerator over NONZERO D := div(-17, -3) After Cramer ORIGINAL x numerator over NONZERO D = 17/3 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 Cramer ORIGINAL y numerator over NONZERO D := div(-10, -3) After Cramer ORIGINAL y numerator over NONZERO D = 10/3 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 Cramer reverse ORIGINAL row0 x term := mul(1, 17/3) After Cramer reverse ORIGINAL row0 x term = 17/3 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 Cramer reverse ORIGINAL row0 y term := mul(1, 10/3) After Cramer reverse ORIGINAL row0 y term = 10/3 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, S015 S017 · calculationBefore Cramer reverse ORIGINAL row0 terms sum := sum(17/3, 10/3) After Cramer reverse ORIGINAL row0 terms sum = 9 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, S016 S018 · calculationBefore Cramer reverse ORIGINAL row0 residual minus rhs := sub(9, 9) After Cramer 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, S017 S019 · impliesBefore source roles = [Cramer reverse, 0, 1, 1, 9, 17/3, 10/3] After derived = [17/3, 10/3, 9, 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, S018 S020 · calculationBefore Cramer reverse ORIGINAL row1 x term := mul(1, 17/3) After Cramer reverse ORIGINAL row1 x term = 17/3 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 Cramer reverse ORIGINAL row1 y term := mul(-2, 10/3) After Cramer reverse ORIGINAL row1 y term = -20/3 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 Cramer reverse ORIGINAL row1 terms sum := sum(17/3, -20/3) After Cramer reverse ORIGINAL row1 terms sum = -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, S021 S023 · calculationBefore Cramer reverse ORIGINAL row1 residual minus rhs := sub(-1, -1) After Cramer 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, S022 S024 · impliesBefore source roles = [Cramer reverse, 1, 1, -2, -1, 17/3, 10/3] After derived = [17/3, -20/3, -1, 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, S023 S025 · impliesBefore source roles = [[[1, 1, 9], [1, -2, -1]], -3] After derived = [-17, -10, 17/3, 10/3] Conditions: none Divide ONLY for D≠0:x=(ce−bf)/D,y=(af−cd)/D.Reduce positive denominators,retain/1,and verify BOTH original rows.The difference of two solutions obeys the homogeneous equations;elimination gives DΔx=DΔy=0,hence uniqueness over R,not an integer-unit criterion. Step identifiers and dependenciesRule: System_CRAMER_NONZERO_D_BOTH_fractions_and_uniqueness · Uses: G001, S024 ## Reverse checks - 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 — 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. - S023 · premise_and_result_audit — Recompute Cramer reverse ORIGINAL row1 residual minus rhs with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Cramer reverse ORIGINAL row1 terms sum with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Cramer reverse ORIGINAL row1 y term with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Cramer reverse ORIGINAL row1 x term with exact rational arithmetic, not rounded decimals. - S019 · 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. - S018 · premise_and_result_audit — Recompute Cramer reverse ORIGINAL row0 residual minus rhs with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Cramer reverse ORIGINAL row0 terms sum with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Cramer reverse ORIGINAL row0 y term with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Cramer reverse ORIGINAL row0 x term with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Cramer ORIGINAL y numerator over NONZERO D with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Cramer ORIGINAL x numerator over NONZERO D with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Cramer ORIGINAL numerator y af minus cd with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Cramer ORIGINAL c times d with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Cramer ORIGINAL a times f with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Cramer ORIGINAL numerator x ce minus bf with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Cramer ORIGINAL b times f with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Cramer ORIGINAL c times e with exact rational arithmetic, not rounded decimals. - 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 (17/3, 10/3) 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: 115a0c9ddbae3d53c36733c88da12fd6f675795adda7b822f22d263a00be34f8 - candidate record SHA-256: e0610a495df3af3a396fc1be79782630f59ae03d725fbf46c6c926b5110cc92e - 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-0097.json)