# Question 6695: 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-0045) 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. R0012-06695 · q-20261010-advance-20261014-r03-equation-0045 ## Original question and givens 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. ## 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 · calculationBefore original negative constant := sub(0, -16) After original negative constant = 16 Conditions: none Rearrange Ax+B=0 into Ax=−B. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore 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. Step identifiers and dependenciesRule: nonzero_affine_equation_unique_root · Uses: G001, S005 S007 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore scaled constant := mul(0, -16) After scaled constant = 0 Conditions: none Multiply the constant term by k as well,not just one coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore 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. Step identifiers and dependenciesRule: constant_equation_solution_class · Uses: G001, S008 S010 · impliesBefore 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. Step identifiers and dependenciesRule: zero_scaling_solution_equivalence_guard · Uses: G001, S009 ## Reverse checks - 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. ## Result (single-solution 8/1, all Q, 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: 4617a5065257c889f3bbbf367e97e3cb8a42ed478083f75ff7e21484eb296417 - candidate record SHA-256: 06ade7f35c66d2eb9ea8ffd7b77291c1319452c00fc3f5afaf70d9b5c15d7cc5 - 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-0045.json)