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R0012-06695 · q-20261010-advance-20261014-r03-equation-0045
Over Q, the original equation is (2)x+(-16)=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 (2)x+(-16)=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 negative constant := sub(0, -16)
After
original negative constant = 16
Conditions: none
Rearrange Ax+B=0 into Ax=−B.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
original unique root := div(16, 2)
After
original unique root = 8
Conditions: none
Divide only because A is nonzero,to obtain the unique rational root.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
original reverse substitution linear product := mul(2, 8)
After
original reverse substitution linear product = 16
Conditions: none
Substitute the specified x into this side's coefficient product.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
original reverse substitution affine value := sum(16, -16)
After
original reverse substitution affine value = 0
Conditions: none
Then add this side's own constant,not the other side's.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
original:A=2≠0;root=8/1
After
Ax+B=0
Conditions: none
Rearrangement and division by a nonzero number are reversible,so uniqueness does not come merely from testing one candidate.
Rule: nonzero_affine_equation_unique_root · Uses: G001, S005
Before
scaled coefficient := mul(0, 2)
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, S006
Before
scaled constant := mul(0, -16)
After
scaled constant = 0
Conditions: none
Multiply the constant term by k as well,not just one coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S007
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, S008
Before
k=0;before=single solution 8/1;after=all Q
After
same solution class=No
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, S009
S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Recompute scaled constant with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute scaled coefficient with exact rational arithmetic, not rounded decimals.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 — Recompute original reverse substitution affine value with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute original reverse substitution linear product with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute original unique root with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute original negative constant 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.(single-solution 8/1, all Q, 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: 4617a5065257c889f3bbbf367e97e3cb8a42ed478083f75ff7e21484eb296417candidate record SHA-256: 06ade7f35c66d2eb9ea8ffd7b77291c1319452c00fc3f5afaf70d9b5c15d7cc5semantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9Lean: NOT_RUN · human review: NOT_PERFORMED