Explore related knowledge and questions →
← Question derivation index · Original question batch
R0014-07810 · q-20261012-advance-20261016-r05-system-linear-equations-0010
Over R, rows=[[1, 0, -3], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 1/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).
G001 — Over R, rows=[[1, 0, -3], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 1/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).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.
Before
rows=[[1, 0, -3], [0, 1, 0]]
After
each row [a,b,c] means a*x+b*y=c over R
Conditions: none
Both equations must hold simultaneously.The right-hand constant is not a third unknown.Retain degenerate zero-coefficient rows rather than presuming a nonzero determinant.
Rule: two_equations_real_system_scope · Uses: G001
Before
raw rational pairs=[[0, 1], [1, 1]]
After
candidate=[0/1, 1/1]
Conditions: none
Both original candidate denominators must be nonzero.Negative denominators are legal;normalize sign and common factors without replacing the candidate by a solved answer.
Rule: candidate_rational_denominator_guard · Uses: G001, S001
Before
candidate row0 ax := mul(1, 0)
After
candidate row0 ax = 0
Conditions: none
Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
candidate row0 by := mul(0, 1)
After
candidate row0 by = 0
Conditions: none
Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
candidate row0 lhs := sum(0, 0)
After
candidate row0 lhs = 0
Conditions: none
Add the already aligned component or position,without mixing coordinates.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
candidate row0 residual := sub(0, -3)
After
candidate row0 residual = 3
Conditions: none
Subtract the second operand from the first in the requested order,retaining the residual sign.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
candidate row1 ax := mul(0, 0)
After
candidate row1 ax = 0
Conditions: none
Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
candidate row1 by := mul(1, 1)
After
candidate row1 by = 1
Conditions: none
Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
candidate row1 lhs := sum(0, 1)
After
candidate row1 lhs = 1
Conditions: none
Add the already aligned component or position,without mixing coordinates.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
candidate row1 residual := sub(1, 0)
After
candidate row1 residual = 1
Conditions: none
Subtract the second operand from the first in the requested order,retaining the residual sign.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
residuals=[3/1, 1/1]
After
simultaneous solution=No
Conditions: none
Both rational residuals must individually be zero.Passing one equation,or summing two opposite residuals to zero,is insufficient.
Rule: system_candidate_both_rows · Uses: G001, S010
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Recompute candidate row1 residual with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute candidate row1 lhs with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute candidate row1 by with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute candidate row1 ax with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute candidate row0 residual with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute candidate row0 lhs with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute candidate row0 by with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute candidate row0 ax with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.([3/1, 1/1], No)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 96f499ed7c670f15ef4567fc2b1e6ef791bc59d320e8ad7dd3b69bac012b7b9acandidate record SHA-256: 8092295e821d29c0069b84a242d912a2e3437a2ca9b45e1175a9f6a15f43c222semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705Lean: NOT_RUN · human review: NOT_PERFORMED