Question 16417: Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

Original question and givens

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[1, 3, 4]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

• G001 — In R² with usual operations consider a*x+b*y=c with [a,b,c]=[1, 3, 4]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

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

affine nonempty particular x coordinate := div(4, 1)

affine nonempty particular x coordinate = 4

affine particular first term := mul(1, 4)

affine particular first term = 4

affine particular second term := mul(3, 0)

affine particular second term = 0

affine particular equation value := sum(4, 0)

affine particular equation value = 4

nonzero original normal=[1, 3];nonzero right side=4;particular real solution=[4, 0];checked equation=4

nonempty affine line;zero gives0 rather than nonzero right side;affine_line_not_through_zero;subspace=No

Reverse checks

• 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 affine particular equation value with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute affine particular second term with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute affine particular first term with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Recompute affine nonempty particular x coordinate with exact rational arithmetic, not rounded decimals.

Result

([1, 3, 4], affine_line_not_through_zero, No, affine_line_not_through_zero)

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.