# Question 32919: Determinant of a Two-Variable Linear System [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-0019) 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. R0064-32919 · q-20261003-advance-20261004-r06-system-linear-equations-0019 ## Original question and givens Over the reals, the constraints (6)x+(1)y=1 and (1)x+(1)y=1 must both hold. Find the coefficient determinant D=ae−bd. - G001 — Over the reals, the constraints (6)x+(1)y=1 and (1)x+(1)y=1 must both hold. Find the coefficient determinant D=ae−bd. ## 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-determinant, determinant, reals, []] After derived = [[signed_determinant_integer_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 = [[[6, 1, 1], [1, 1, 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(6, 1) 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(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(6, 1) After System ORIGINAL signed determinant ae minus bd = 5 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 = [[[6, 1, 1], [1, 1, 1]]] After derived = [6, 1, 5] 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 ## Reverse checks - 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 5 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: c31797f4b40da978b22012737e0afca7f4bd417f1334e9fcfd315e68905187e8 - candidate record SHA-256: 983260de34ff67f6691a1224ed05890040add700bf1f1d4d4ecaa735ae4db437 - 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-0019.json)