# Question 6678: Solution Classes Before and After Scaling [Return to catalogue page 34](https://ocme.evemisslab.com/en/questions/page/34/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261010-advance-20261014-r03-equation-foundations/#q-20261010-advance-20261014-r03-equation-0028) 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. R0012-06678 · q-20261010-advance-20261014-r03-equation-0028 ## Original question and givens Over Q, the original equation is (0)x+(1)=0; multiply both sides by k=2. 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+(1)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d. ## 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-equation](https://ocme.evemisslab.com/en/mko/mko-equation/) ## Step-by-step derivation S001 · impliesBefore 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. Step identifiers and dependenciesRule: original_rational_equation_scope · Uses: G001 S002 · impliesBefore original:0·x+(1)=0 After no solution 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. Step identifiers and dependenciesRule: constant_equation_solution_class · Uses: G001, S001 S003 · calculationBefore scaled coefficient := mul(2, 0) After scaled coefficient = 0 Conditions: none Multiplying both sides by k also multiplies the x-coefficient by that same k. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore scaled constant := mul(2, 1) After scaled constant = 2 Conditions: none Multiply the constant term by k as well,not just one coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore scaled:0·x+(2)=0 After no solution 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. Step identifiers and dependenciesRule: constant_equation_solution_class · Uses: G001, S004 S006 · impliesBefore k=2;before=no solution;after=no solution 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. Step identifiers and dependenciesRule: zero_scaling_solution_equivalence_guard · Uses: G001, S005 ## Reverse checks - 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. ## Result (no-solution, no-solution, Yes) 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: e6b0474384d8cc819456ab0a8a21a43f009b2d8b704ad950220069aaf24a85f0 - candidate record SHA-256: afb724db8354484e94b026f322cba0b55c55afa2698f3c483d479594cbf9a003 - semantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00 - question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261010-advance-20261014-r03-equation-0028.json)