{
  "$schema":"../../../schemas/mko-v0.4.schema.json","schema_version":"mko-v0.4","id":"mko-linear-equation","semantic_family_id":"sf-linear-equation","type":"definition","version":"0.10.0","language":"zh-Hant",
  "titles":{"zh-Hant":"一元一次方程式","en":"Linear equation in one variable"},
  "summary":{"zh-Hant":"一元一次方程式可寫成 ax+b=0，且一次係數 a 必須非零。"},
  "statement":{"zh-Hant":"在明示的有理數或實數語境中，ax+b=0 且 a≠0 是一元一次方程式，唯一解為 x=-b/a；a=0 時是退化常數等式，不屬於本定義。"},
  "formula":{"tex":"x+1=0","mathml":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>=</mo><mn>0</mn></mrow></math>","semantic_ast":{"type":"equation","lhs":{"type":"addition","operands":[{"type":"symbol","name":"x","glyph":"x"},{"type":"number","value":1}]},"rhs":{"type":"number","value":0}},"compiler":{"id":"ocme-formula-core","version":"0.4.0","source_sha256":"112b539e76655c6fdcf821d796c23cdb8d698a1b0424f943ed6bfc24f41a6894"}},
  "assumptions":[{"id":"a1","zh-Hant":"變數 x 與係數 a、b 位於明示的有理數或實數語境。"},{"id":"a2","zh-Hant":"一次係數 a 不等於 0。"}],
  "symbols":[{"token":"x","role":"unknown_variable","role_zh":"唯一未知數","scope":"equation"},{"token":"a","role":"linear_coefficient","role_zh":"非零一次係數","scope":"general_statement"},{"token":"b","role":"constant_term","role_zh":"常數項","scope":"general_statement"}],
  "dependencies":[{"id":"mko-equation","title_zh":"方程式","reason_zh":"一元一次方程式是具有特定表達式形狀與解集條件的方程式。"}],
  "explanation":{"paragraphs_zh":["名稱中的『一次』表示未知數只以一階線性項出現；代表形式為 ax+b=0，且 a≠0。","在有理數或實數中，從 ax+b=0 得 ax=-b，再除以非零 a，得到唯一解 x=-b/a；求得後仍可代回原式核對。","若 a=0，方程退化成 b=0。此時 b=0 代表所有允許 x 都成立，b≠0 則無解；兩者都不能冒充具有唯一解的一次方程。"],"common_misconception_zh":"常見誤解是允許 a=0、把直線方程與任何含 x 的式子混同、除以未證明非零的係數，或只做變形而不代回檢查。"},
  "proofs":[],"computational_companions":[],
  "verification":{"computational_status":"not_evaluated_no_computational_claim","warning_zh":"小整數係數與精確有理數解可做有限重算，但不構成所有線性系統、線性函數或一般方程求解理論的形式證明。","evidence_refs":[],"producers":[{"id":"lean-mathlib","adapter":"formal_system","config_path":"formal/lean/lakefile.toml","status":"configured"}]},
  "formalization":{"status":"not_formalized","target_systems":["Lean 4","Mathlib"],"next_obligation_zh":"綁定 a≠0、ax+b=0、唯一解 -b/a 與退化 a=0 分類的精確聲明。"},
  "provenance":{"origin":"ocme_local_ai_materialization_candidate","note_zh":"依 research/atlas-linear-equation.md 整理；待人工與形式化審查。","content_license":"CC-BY-SA-4.0","code_license":"MIT","created_at":"2026-09-27"}
}