Question 16460: Membership in a Homogeneous Real Line Is an Exact Equation Check

Original question and givens

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[10, -5]. Selected point=[-3, -1]. Give (point,normal,equation_value,in_line?).

• G001 — Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[10, -5]. Selected point=[-3, -1]. Give (point,normal,equation_value,in_line?).

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-vector-subspace

Step-by-step derivation

homogeneous selected first coordinate term := mul(10, -3)

homogeneous selected first coordinate term = -30

homogeneous selected second coordinate term := mul(-5, -1)

homogeneous selected second coordinate term = 5

homogeneous selected equation value := sum(-30, 5)

homogeneous selected equation value = -25

same homogeneous line normal=[10, -5];selected point=[-3, -1];equation value=-25

selected point belongs to line=No;one membership check is not universal subspace closure

Reverse checks

• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Recompute homogeneous selected equation value with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute homogeneous selected second coordinate term with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Recompute homogeneous selected first coordinate term with exact rational arithmetic, not rounded decimals.

Result

([-3, -1], [10, -5], -25, 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.