OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 269: Validity of a Linear-Equation Form

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

Return to catalogue page 2

← Question derivation index · Original question batch

P1000-0269 · q-20260927-linear-equation-form-validity-0019

Original question and givens

Over the rational numbers, consider -4x+(6)=0. Under this MKO, is it a linear equation in one variable?

  • G001 — Over the rational numbers, consider -4x+(6)=0. Under this MKO, is it a linear equation in one variable?

Definition and scope

Use the rational domain. A linear equation requires a≠0; a=0 is a separate degenerate constant equation.

Step-by-step derivation

S001 · implies

Before

a=-4, b=6, x∈Q

After

equation: -4x+(6)=0

Conditions: none

Read the linear coefficient, constant and rational domain explicitly; a name alone does not establish nondegeneracy.

Step identifiers and dependencies

Rule: read_coefficients · Uses: G001

S002 · implies

Before

a=-4

After

a≠0: Yes

Conditions: none

The definition of a nondegenerate linear equation requires a nonzero leading coefficient.

Step identifiers and dependencies

Rule: nondegenerate_definition · Uses: G001, S001

S003 · implies

Before

-4x+(6)=0

After

linear form qualified: Yes

Conditions: none

This checks the definition, not merely whether the equation has a solution.

Step identifiers and dependencies

Rule: definition_application · Uses: G001, S002

S004 · implies

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.

Step identifiers and dependencies

Rule: degenerate_boundary · Uses: G001, S003

Reverse checks

  • 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.

Result

Yes

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.

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: a1279617a0db69ef88744fb20b086091808b9fed2454b533086c8e81a04da7cc
  • candidate record SHA-256: eec9ea19343aaa39216383a1ef5fdad73f5f143c25ec998ea98e9e8f00ef2c82
  • semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
  • question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON