# Question 7807: Two Rational Substitution Residuals for a System

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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.

R0014-07807 · q-20261012-advance-20261016-r05-system-linear-equations-0007



## Original question and givens

Over R, rows=[[1, 0, -6], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 0/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).


- G001 — Over R, rows=[[1, 0, -6], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 0/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).





## 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

rows=[[1, 0, -6], [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.

Step identifiers and dependenciesRule: two_equations_real_system_scope · Uses: G001





S002 · impliesBefore

raw rational pairs=[[0, 1], [0, 1]]

After

candidate=[0/1, 0/1]

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.

Step identifiers and dependenciesRule: candidate_rational_denominator_guard · Uses: G001, S001





S003 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

candidate row0 by := mul(0, 0)

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

candidate row0 lhs := sum(0, 0)

After

candidate row0 lhs = 0

Conditions: none

Add the already aligned component or position,without mixing coordinates.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

candidate row0 residual := sub(0, -6)

After

candidate row0 residual = 6

Conditions: none

Subtract the second operand from the first in the requested order,retaining the residual sign.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

candidate row1 by := mul(1, 0)

After

candidate row1 by = 0

Conditions: none

Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

candidate row1 lhs := sum(0, 0)

After

candidate row1 lhs = 0

Conditions: none

Add the already aligned component or position,without mixing coordinates.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · calculationBefore

candidate row1 residual := sub(0, 0)

After

candidate row1 residual = 0

Conditions: none

Subtract the second operand from the first in the requested order,retaining the residual sign.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · impliesBefore

residuals=[6/1, 0/1]

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.

Step identifiers and dependenciesRule: system_candidate_both_rows · Uses: G001, S010









## Reverse checks


- 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.





## Result

([6/1, 0/1], No)

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: b2c5ddca84a111f9ae43c3bf2d24bd9c397fc1453b39922dfb69db210089ee3d
- candidate record SHA-256: 6c512ffc6a117248ebcdf8955dc19984678ac964369ea459b1fe1f4006ffebf3
- semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77
- question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261016-r05-system-linear-equations-0007.json)