Question 6690: Solution Classes Before and After Scaling

Original question and givens

Over Q, the original equation is (2)x+(-11)=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 (2)x+(-11)=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

Step-by-step derivation

common rational domain with original equation definitions

evaluate/classify the given model without enlarging its original domain

original negative constant := sub(0, -11)

original negative constant = 11

original unique root := div(11, 2)

original unique root = 11/2

original reverse substitution linear product := mul(2, 11/2)

original reverse substitution linear product = 11

original reverse substitution affine value := sum(11, -11)

original reverse substitution affine value = 0

original:A=2≠0;root=11/2

Ax+B=0

scaled coefficient := mul(2, 2)

scaled coefficient = 4

scaled constant := mul(2, -11)

scaled constant = -22

scaled negative constant := sub(0, -22)

scaled negative constant = 22

scaled unique root := div(22, 4)

scaled unique root = 11/2

scaled reverse substitution linear product := mul(4, 11/2)

scaled reverse substitution linear product = 22

scaled reverse substitution affine value := sum(22, -22)

scaled reverse substitution affine value = 0

scaled:A=4≠0;root=11/2

Ax+B=0

k=2;before=single solution 11/2;after=single solution 11/2

same solution class=Yes

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Recompute scaled reverse substitution affine value with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute scaled reverse substitution linear product with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute scaled unique root with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute scaled negative constant with exact rational arithmetic, not rounded decimals.
• 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 11/2, single-solution 11/2, 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.