Over the reals, the constraints (4)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 (4)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.
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.
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.
System ORIGINAL a times e main product := mul(4, -1)
After
System ORIGINAL a times e main product = -4
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.
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.
System ORIGINAL signed determinant ae minus bd := sub(-4, 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.
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.
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.