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R0012-06676 · q-20261010-advance-20261014-r03-equation-0026
Over Q, the original equation is (0)x+(0)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.
G001 — Over Q, the original equation is (0)x+(0)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.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
common rational domain with original equation definitions
After
evaluate/classify the given model without enlarging its original domain
Conditions: none
Original definedness precedes substitution and transformations.Check the declared model,not arbitrary equation solving inferred from cross multiplication or one-point values.
Rule: original_rational_equation_scope · Uses: G001
Before
original:0·x+(0)=0
After
all Q
Conditions: none
With coefficient0,this is constant:constant0 is true for all Q and a nonzero constant is inconsistent.Do not force a degenerate equation into a linear-equation premise.
Rule: constant_equation_solution_class · Uses: G001, S001
Before
scaled coefficient := mul(0, 0)
After
scaled coefficient = 0
Conditions: none
Multiplying both sides by k also multiplies the x-coefficient by that same k.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
scaled constant := mul(0, 0)
After
scaled constant = 0
Conditions: none
Multiply the constant term by k as well,not just one coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
scaled:0·x+(0)=0
After
all Q
Conditions: none
With coefficient0,this is constant:constant0 is true for all Q and a nonzero constant is inconsistent.Do not force a degenerate equation into a linear-equation premise.
Rule: constant_equation_solution_class · Uses: G001, S004
Before
k=0;before=all Q;after=all Q
After
same solution class=Yes
Conditions: none
A nonzero k can be divided back,retaining solutions;k=0 turns the equation into0=0 and can change no-solution or singleton into all Q.If the original was already all Q,this does not change its class.
Rule: zero_scaling_solution_equivalence_guard · Uses: G001, S005
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Recompute scaled constant with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute scaled coefficient 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.(all Q, all Q, Yes)
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: f3fa846b9dd10f03afe20c5481fb492d7da68eb47f3fdc95b08139e0307fafbecandidate record SHA-256: aa139184feabd594b8eef09f2d2ba0c06d0267dfec7dd2a8a0c58ac9e2120dc8semantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9Lean: NOT_RUN · human review: NOT_PERFORMED