OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 32926: Solution Classification of a Linear System

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

Return to catalogue page 165

← Question derivation index · Original question batch

R0064-32926 · q-20261003-advance-20261004-r06-system-linear-equations-0026

Original question and givens

Over the reals, the constraints (0)x+(0)y=0 and (0)x+(0)y=0 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

  • G001 — Over the reals, the constraints (0)x+(0)y=0 and (0)x+(0)y=0 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

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.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-linear-system-solution-class, solution_class, reals, []]

After

derived = [[SOURCE_solution_class_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 dependencies

Rule: R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles · Uses: G001

S002 · implies

Before

source roles = [[[0, 0, 0], [0, 0, 0]]]

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 dependencies

Rule: System_RAW_exact2_augmented_rows3_ALL_SAFEincl_unused_rhs · Uses: G001, S001

S003 · calculation

Before

System ORIGINAL a times e main product := mul(0, 0)

After

System ORIGINAL a times e main product = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

System ORIGINAL b times d other product := mul(0, 0)

After

System ORIGINAL b times d other product = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

System ORIGINAL signed determinant ae minus bd := sub(0, 0)

After

System ORIGINAL signed determinant ae minus bd = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [[[0, 0, 0], [0, 0, 0]]]

After

derived = [0, 0, 0]

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 dependencies

Rule: System_DETERMINANT_coefficients_ONLY_raw_order_sign · Uses: G001, S005

S007 · implies

Before

source roles = [[[0, 0, 0], [0, 0, 0]], 0]

After

derived = [[]]

Conditions: none

At D0,FIRST detect any0x+0y=nonzero contradiction.Such a row is impossible;zero other minors do not establish consistency,and D must not be divided.

Step identifiers and dependencies

Rule: System_DZERO_zero_row_contradiction_guard_BEFORE_rank_or_minors · Uses: G001, S006

S008 · implies

Before

source roles = [[[0, 0, 0], [0, 0, 0]]]

After

derived = [all (x,y) in R^2]

Conditions: none

All coefficients are0 with no contradiction,so BOTH rows are0=0 and EVERY pair in R² solves:infinitely many,not just the zero solution or only rational-domain pairs.

Step identifiers and dependencies

Rule: System_RANKZERO_both_tautologies_ALL_R2_infinitely_many · Uses: G001, S007

Reverse checks

  • 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

infinitely many solutions

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: b3523c460716b9b2fb32088e37251f68eff1b2cc52cc60ac8831cc9f8af2af4e
  • candidate record SHA-256: 997378d0a06e415348bb355d5adc5b3cb4dc16d0b217bbb67c76846941e96e4c
  • semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b
  • question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON