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P1000-0263 · q-20260927-linear-equation-form-validity-0013
Over the rational numbers, consider -1x+(0)=0. Under this MKO, is it a linear equation in one variable?
G001 — Over the rational numbers, consider -1x+(0)=0. Under this MKO, is it a linear equation in one variable?Use the rational domain. A linear equation requires a≠0; a=0 is a separate degenerate constant equation.
Before
a=-1, b=0, x∈Q
After
equation: -1x+(0)=0
Conditions: none
Read the linear coefficient, constant and rational domain explicitly; a name alone does not establish nondegeneracy.
Rule: read_coefficients · Uses: G001
Before
a=-1
After
a≠0: Yes
Conditions: none
The definition of a nondegenerate linear equation requires a nonzero leading coefficient.
Rule: nondegenerate_definition · Uses: G001, S001
Before
-1x+(0)=0
After
linear form qualified: Yes
Conditions: none
This checks the definition, not merely whether the equation has a solution.
Rule: definition_application · Uses: G001, S002
Before
if a=0 then ax+b=b
After
constant equality is a separate boundary
Conditions: none
When a=0, b=0 gives every rational solution and b≠0 gives none; neither is nondegenerate linear form.
Rule: degenerate_boundary · Uses: G001, S003
S004 · premise_and_result_audit — Substitute into the original equation; retain every nonzero condition before division.S003 · premise_and_result_audit — Substitute into the original equation; retain every nonzero condition before division.S002 · premise_and_result_audit — Substitute into the original equation; retain every nonzero condition before division.S001 · premise_and_result_audit — Substitute into the original equation; retain every nonzero condition before division.Yes
The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.
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.
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