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R0014-07882 · q-20261012-advance-20261016-r05-system-linear-equations-0082
Over R, rows=[[1, 0, -6], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.
G001 — Over R, rows=[[1, 0, -6], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.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, -6], [0, 1, 2]]
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
before ae := mul(1, 1)
After
before ae = 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, S001
Before
before bd := mul(0, 0)
After
before bd = 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
before D := sub(1, 0)
After
before D = 1
Conditions: none
Subtract the second operand from the first in the requested order,retaining the residual sign.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
before af := mul(1, 2)
After
before af = 2
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, S004
Before
before cd := mul(-6, 0)
After
before cd = 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, S005
Before
before minor_x_rhs := sub(2, 0)
After
before minor_x_rhs = 2
Conditions: none
Subtract the second operand from the first in the requested order,retaining the residual sign.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
before bf := mul(0, 2)
After
before bf = 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, S007
Before
before ce := mul(-6, 1)
After
before ce = -6
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, S008
Before
before minor_y_rhs := sub(0, -6)
After
before minor_y_rhs = 6
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
phase=before;rows=[[1, 0, -6], [0, 1, 2]];2x2 minors=[1, 2, 6]
After
rankA=2;rankAb=2
Conditions: none
A nonzero coefficient 2×2 minor gives rank2;otherwise a nonzero coefficient gives rank1.The augmented matrix has three 2×2 minors;if all vanish,check its entries,including the right-hand constants.
Rule: system_rank_from_complete_minors · Uses: G001, S010
Before
before x numerator := sub(0, 6)
After
before x numerator = -6
Conditions: none
Subtract the second operand from the first in the requested order,retaining the residual sign.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
before x solution := div(-6, 1)
After
before x solution = -6
Conditions: none
The denominator has separately been checked nonzero;divide exactly as a rational number.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
before y solution := div(2, 1)
After
before y solution = 2
Conditions: none
The denominator has separately been checked nonzero;divide exactly as a rational number.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
before solution row0 ax := mul(1, -6)
After
before solution row0 ax = -6
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, S014
Before
before solution row0 by := mul(0, 2)
After
before solution 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, S015
Before
before solution row0 lhs := sum(-6, 0)
After
before solution row0 lhs = -6
Conditions: none
Add the already aligned component or position,without mixing coordinates.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
before solution row0 residual := sub(-6, -6)
After
before solution row0 residual = 0
Conditions: none
Subtract the second operand from the first in the requested order,retaining the residual sign.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
before solution row1 ax := mul(0, -6)
After
before solution 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, S018
Before
before solution row1 by := mul(1, 2)
After
before solution row1 by = 2
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, S019
Before
before solution row1 lhs := sum(0, 2)
After
before solution row1 lhs = 2
Conditions: none
Add the already aligned component or position,without mixing coordinates.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
before solution row1 residual := sub(2, 2)
After
before solution row1 residual = 0
Conditions: none
Subtract the second operand from the first in the requested order,retaining the residual sign.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
phase=before;D=1;D*x=-6;D*y=2
After
solution=[-6/1, 2/1];row residuals=[0, 0]
Conditions: none
Nonzero D permits division after Cramer elimination:D·x=ce−bf and D·y=af−cd.Substitute the rational pair into both original rows;both residuals are zero.Nonzero D also ensures uniqueness.
Rule: system_unique_solution_substitution · Uses: G001, S022
Before
phase=before;rankA=2;rankAb=2;unknowns=2
After
class=unique solution;solution/status=[-6/1, 2/1]
Conditions: none
Unequal ranks mean an inconsistent row and no solution.Equal rank below2 leaves a free variable and infinitely many real solutions.Only rank2 gives a unique solution.Never divide by D when D=0.
Rule: system_consistency_and_free_variables · Uses: G001, S023
Before
scaled row1[0] := mul(1, 1)
After
scaled row1[0] = 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, S024
Before
scaled row1[1] := mul(1, 0)
After
scaled row1[1] = 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, S025
Before
scaled row1[2] := mul(1, -6)
After
scaled row1[2] = -6
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, S026
Before
operation=system-row-scaling-solution;t=1;old rows=[[1, 0, -6], [0, 1, 2]]
After
new rows=[[1, 0, -6], [0, 1, 2]];invertible=Yes;inverse=row1 -> row1/(1)
Conditions: none
A row operation includes the right-hand constant.Adding any t times the other row is undone by adding−t times it.Scaling can be divided back only for t≠0.At t=0 a redundant system may coincidentally keep its solution set,but the operation is still not invertible.
Rule: row_operation_inverse_guard · Uses: G001, S027
Before
after ae := mul(1, 1)
After
after ae = 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, S028
Before
after bd := mul(0, 0)
After
after bd = 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, S029
Before
after D := sub(1, 0)
After
after D = 1
Conditions: none
Subtract the second operand from the first in the requested order,retaining the residual sign.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
after af := mul(1, 2)
After
after af = 2
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, S031
Before
after cd := mul(-6, 0)
After
after cd = 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, S032
Before
after minor_x_rhs := sub(2, 0)
After
after minor_x_rhs = 2
Conditions: none
Subtract the second operand from the first in the requested order,retaining the residual sign.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
after bf := mul(0, 2)
After
after bf = 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, S034
Before
after ce := mul(-6, 1)
After
after ce = -6
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, S035
Before
after minor_y_rhs := sub(0, -6)
After
after minor_y_rhs = 6
Conditions: none
Subtract the second operand from the first in the requested order,retaining the residual sign.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
phase=after;rows=[[1, 0, -6], [0, 1, 2]];2x2 minors=[1, 2, 6]
After
rankA=2;rankAb=2
Conditions: none
A nonzero coefficient 2×2 minor gives rank2;otherwise a nonzero coefficient gives rank1.The augmented matrix has three 2×2 minors;if all vanish,check its entries,including the right-hand constants.
Rule: system_rank_from_complete_minors · Uses: G001, S037
Before
after x numerator := sub(0, 6)
After
after x numerator = -6
Conditions: none
Subtract the second operand from the first in the requested order,retaining the residual sign.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
after x solution := div(-6, 1)
After
after x solution = -6
Conditions: none
The denominator has separately been checked nonzero;divide exactly as a rational number.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
after y solution := div(2, 1)
After
after y solution = 2
Conditions: none
The denominator has separately been checked nonzero;divide exactly as a rational number.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
after solution row0 ax := mul(1, -6)
After
after solution row0 ax = -6
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, S041
Before
after solution row0 by := mul(0, 2)
After
after solution 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, S042
Before
after solution row0 lhs := sum(-6, 0)
After
after solution row0 lhs = -6
Conditions: none
Add the already aligned component or position,without mixing coordinates.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
after solution row0 residual := sub(-6, -6)
After
after solution row0 residual = 0
Conditions: none
Subtract the second operand from the first in the requested order,retaining the residual sign.
Rule: exact_rational_arithmetic · Uses: G001, S044
Before
after solution row1 ax := mul(0, -6)
After
after solution 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, S045
Before
after solution row1 by := mul(1, 2)
After
after solution row1 by = 2
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, S046
Before
after solution row1 lhs := sum(0, 2)
After
after solution row1 lhs = 2
Conditions: none
Add the already aligned component or position,without mixing coordinates.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
after solution row1 residual := sub(2, 2)
After
after solution row1 residual = 0
Conditions: none
Subtract the second operand from the first in the requested order,retaining the residual sign.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
phase=after;D=1;D*x=-6;D*y=2
After
solution=[-6/1, 2/1];row residuals=[0, 0]
Conditions: none
Nonzero D permits division after Cramer elimination:D·x=ce−bf and D·y=af−cd.Substitute the rational pair into both original rows;both residuals are zero.Nonzero D also ensures uniqueness.
Rule: system_unique_solution_substitution · Uses: G001, S049
Before
phase=after;rankA=2;rankAb=2;unknowns=2
After
class=unique solution;solution/status=[-6/1, 2/1]
Conditions: none
Unequal ranks mean an inconsistent row and no solution.Equal rank below2 leaves a free variable and infinitely many real solutions.Only rank2 gives a unique solution.Never divide by D when D=0.
Rule: system_consistency_and_free_variables · Uses: G001, S050
S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S049 · premise_and_result_audit — Recompute after solution row1 residual with exact rational arithmetic, not rounded decimals.S048 · premise_and_result_audit — Recompute after solution row1 lhs with exact rational arithmetic, not rounded decimals.S047 · premise_and_result_audit — Recompute after solution row1 by with exact rational arithmetic, not rounded decimals.S046 · premise_and_result_audit — Recompute after solution row1 ax with exact rational arithmetic, not rounded decimals.S045 · premise_and_result_audit — Recompute after solution row0 residual with exact rational arithmetic, not rounded decimals.S044 · premise_and_result_audit — Recompute after solution row0 lhs with exact rational arithmetic, not rounded decimals.S043 · premise_and_result_audit — Recompute after solution row0 by with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Recompute after solution row0 ax with exact rational arithmetic, not rounded decimals.S041 · premise_and_result_audit — Recompute after y solution with exact rational arithmetic, not rounded decimals.S040 · premise_and_result_audit — Recompute after x solution with exact rational arithmetic, not rounded decimals.S039 · premise_and_result_audit — Recompute after x numerator with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S037 · premise_and_result_audit — Recompute after minor_y_rhs with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Recompute after ce with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Recompute after bf with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute after minor_x_rhs with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Recompute after cd with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Recompute after af with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute after D with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Recompute after bd with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Recompute after ae with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S027 · premise_and_result_audit — Recompute scaled row1[2] with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute scaled row1[1] with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute scaled row1[0] with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S022 · premise_and_result_audit — Recompute before solution row1 residual with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute before solution row1 lhs with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute before solution row1 by with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute before solution row1 ax with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute before solution row0 residual with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute before solution row0 lhs with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute before solution row0 by with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute before solution row0 ax with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute before y solution with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute before x solution with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute before x numerator with exact rational arithmetic, not rounded decimals.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 before minor_y_rhs with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute before ce with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute before bf with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute before minor_x_rhs with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute before cd with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute before af with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute before D with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute before bd with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute before ae with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.([[1, 0, -6], [0, 1, 2]], [1, 2, 2, unique-solution, [-6/1, 2/1]], [1, 2, 2, unique-solution, [-6/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)
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: 5c651fdd4c17a8a23e67d5e95f5e9af265045f7de6e47fa2adafefd34a288d92candidate record SHA-256: 308e4afb65f952d912082d8dff4d335c329822ffe96023003e5a0178b788f31esemantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705Lean: NOT_RUN · human review: NOT_PERFORMED