# Question 32938: Solution Classification of a 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-0038) 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. R0064-32938 · q-20261003-advance-20261004-r06-system-linear-equations-0038 ## Original question and givens Over the reals, the constraints (1)x+(6)y=8 and (2)x+(12)y=16 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 (1)x+(6)y=8 and (2)x+(12)y=16 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. - [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-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 dependenciesRule: R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles · Uses: G001 S002 · impliesBefore source roles = [[[1, 6, 8], [2, 12, 16]]] 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, 12) After System ORIGINAL a times e main 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, S002 S004 · calculationBefore System ORIGINAL b times d other product := mul(6, 2) 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(12, 12) 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [[[1, 6, 8], [2, 12, 16]]] After derived = [12, 12, 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 dependenciesRule: System_DETERMINANT_coefficients_ONLY_raw_order_sign · Uses: G001, S005 S007 · impliesBefore source roles = [[[1, 6, 8], [2, 12, 16]], 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 dependenciesRule: System_DZERO_zero_row_contradiction_guard_BEFORE_rank_or_minors · Uses: G001, S006 S008 · calculationBefore System rank1 ORIGINAL a times f augmented minor := mul(1, 16) After System rank1 ORIGINAL a times f augmented minor = 16 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 System rank1 ORIGINAL c times d augmented minor := mul(8, 2) After System rank1 ORIGINAL c times d augmented minor = 16 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 rank1 ORIGINAL augmented x minor := sub(16, 16) After System rank1 ORIGINAL augmented x minor = 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, S009 S011 · calculationBefore System rank1 ORIGINAL b times f augmented minor := mul(6, 16) After System rank1 ORIGINAL b times f augmented minor = 96 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 rank1 ORIGINAL c times e augmented minor := mul(8, 12) After System rank1 ORIGINAL c times e augmented minor = 96 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 rank1 ORIGINAL augmented y minor := sub(96, 96) After System rank1 ORIGINAL augmented y minor = 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, S012 S014 · impliesBefore source roles = [[[1, 6, 8], [2, 12, 16]]] After derived = [0, 0, Yes] Conditions: none AFTER excluding rank0/zero-row contradictions,the D0 coefficient rank is1.BOTH augmented minors must vanish for consistency;any nonzero minor yields an elimination contradiction.One zero minor is insufficient. Step identifiers and dependenciesRule: System_RANKONE_BOTH_augmented_minors_consistency · Uses: G001, S013 S015 · calculationBefore System free parameter ORIGINAL rhs over x pivot := div(8, 1) After System free parameter ORIGINAL rhs over x pivot = 8 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 free parameter ORIGINAL negative y coefficient := sub(0, 6) After System free parameter ORIGINAL negative y coefficient = -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, S015 S017 · calculationBefore System free parameter ORIGINAL x slope in free y := div(-6, 1) After System free parameter ORIGINAL x slope in free y = -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, S016 S018 · calculationBefore System free parameter reverse OTHER x coefficient times constant := mul(2, 8) After System free parameter reverse OTHER x coefficient times constant = 16 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 free parameter reverse OTHER x coefficient times slope := mul(2, -6) After System free parameter reverse OTHER x coefficient times slope = -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 free parameter reverse OTHER add y coefficient := sum(-12, 12) After System free parameter reverse OTHER add y coefficient = 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, S019 S021 · impliesBefore source roles = [[[1, 6, 8], [2, 12, 16]], 0] After derived = [[x = 8/1 + (-6/1)*t, y = t], 16/1, 0/1, t arbitrary in R] Conditions: none Choose an ACTUAL nonzero pivot and exhibit a free REAL parameter t.Substitution into the OTHER row gives its exact RHS and zero t-coefficient,so the WHOLE family solves.Different free-coordinate t values give infinitely many distinct solutions. Step identifiers and dependenciesRule: System_RANKONE_explicit_free_REAL_parameter_and_ALL_row_identity · Uses: G001, S020 ## Reverse checks - S021 · 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. - S020 · premise_and_result_audit — Recompute System free parameter reverse OTHER add y coefficient with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute System free parameter reverse OTHER x coefficient times slope with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute System free parameter reverse OTHER x coefficient times constant with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute System free parameter ORIGINAL x slope in free y with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute System free parameter ORIGINAL negative y coefficient with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute System free parameter ORIGINAL rhs over x pivot with exact rational arithmetic, not rounded decimals. - S014 · 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. - S013 · premise_and_result_audit — Recompute System rank1 ORIGINAL augmented y minor with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute System rank1 ORIGINAL c times e augmented minor with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute System rank1 ORIGINAL b times f augmented minor with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute System rank1 ORIGINAL augmented x minor with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute System rank1 ORIGINAL c times d augmented minor with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute System rank1 ORIGINAL a times f augmented minor with exact rational arithmetic, not rounded decimals. - 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: 8cb72a25f1c01e2e8139c22031ddb9ced74a842db5fd836b936eab48584931c6 - candidate record SHA-256: 921dea97a90bc7c89b1adffa5cc98eb70c4b1ed5a36abefb927e5840b11b69b2 - 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-0038.json)