{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20260927-linear-equation-solution-class-0019",
  "pilot_number": "P250-0044",
  "number": 44,
  "family_id": "qf-linear-equation-solution-class",
  "title": {
    "zh-Hant": "第 44 題：一次方程的解分類",
    "en": "Question 44: Solution Classification of a Linear Equation"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20260927-linear-equation-solution-class-0019",
    "batch_id": "batch-20260927-linear-equation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-linear-equation"
    ],
    "title_zh": "一次方程的解分類",
    "stem_zh": "在有理數論域，分類 5x+(6)=0 的解集。",
    "answer": {
      "kind": "short_text",
      "value": "唯一解 x=-6/5",
      "explanation_zh": "a=5≠0，所以唯一解為 −b/a=-6/5。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "解分類",
        "退化常數等式"
      ]
    },
    "variant": {
      "family_id": "qf-linear-equation-solution-class",
      "seed": 20261246,
      "parameters": {
        "a": 5,
        "b": 6,
        "domain": "rationals",
        "operation": "solution_class",
        "semantic_template": "qf-linear-equation-solution-class/{\"a\":5,\"b\":6,\"domain\":\"rationals\",\"operation\":\"solution_class\"}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "schema_pending",
        "independent_recomputation_pending"
      ],
      "warning_zh": "本題為 candidate；整數與有理數的有限精確重算不是形式證明或人工審查。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "GPT-5.6 Terra",
      "generated_at": "2026-09-27",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
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    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20260927-linear-equation-solution-class-0019",
    "batch_id": "batch-20260927-linear-equation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2a9ab2047bfa8f4a848e0c29bf1e474b55921cef6e352fdcd254ddd45c8196b1",
      "family_id": "qf-linear-equation-solution-class",
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      "semantic_template": "qf-linear-equation-solution-class/{\"a\":5,\"b\":6,\"domain\":\"rationals\",\"operation\":\"solution_class\"}"
    },
    "fields": {
      "title": "Solution Classification of a Linear Equation",
      "stem": "Over the rational numbers, classify the solution set of 5x+(6)=0.",
      "answer": {
        "kind": "short_text",
        "value": "unique solution x=-6/5",
        "explanation": "Because a=5≠0, the equation has the unique solution x=-6/5."
      },
      "skills": [
        "solution classification",
        "degenerate constant equality"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "在有理數論域，分類 5x+(6)=0 的解集。",
        "en": "Over the rational numbers, classify the solution set of 5x+(6)=0."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "5x+(6)=0, x∈Q",
      "after": "a=5; b=6",
      "relation": "implies",
      "rule_id": "equation_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先讀取有理數解域，並判斷一次係數是否為零；尚不可直接除以 a。",
        "en": "Read the rational solution domain and check the leading coefficient before dividing by a."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "a=0 是退化常數等式，不能冒充非退化一元一次方程式。",
          "en": "When a=0 this is a degenerate constant equation, not a nondegenerate linear equation."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "5x+(6)=0",
      "after": "5x=-6",
      "relation": "equivalent",
      "rule_id": "subtract_same_constant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "等式兩邊同減 6，因此此步等價。",
        "en": "Subtract 6 from both sides, preserving equivalence."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "兩邊同加 b 可回到原式。",
          "en": "Adding b back to both sides recovers the original equation."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "x := div(-6, 5)",
      "after": "x = -6/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已知 a=5 非零，兩邊才可同除以它。",
        "en": "Now a=5 is known to be nonzero, so division by it is valid."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x，不以小數近似替代。",
          "en": "Recompute x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-6",
          "5"
        ],
        "output": "-6/5",
        "label": "x"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "a·x := mul(5, -6/5)",
      "after": "a·x = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先代回計算一次項。",
        "en": "Substitute the candidate and calculate the linear term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a·x，不以小數近似替代。",
          "en": "Recompute a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "-6/5"
        ],
        "output": "-6",
        "label": "a·x"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "a·x+b := sum(-6, 6)",
      "after": "a·x+b = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把常數項加回，得到 0 才符合原方程式。",
        "en": "Add the constant back; a result of 0 verifies the original equation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a·x+b，不以小數近似替代。",
          "en": "Recompute a·x+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-6",
          "6"
        ],
        "output": "0",
        "label": "a·x+b"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "a x₁+b=0 and a x₂+b=0",
      "after": "a(x₁−x₂)=0 ⇒ x₁=x₂",
      "relation": "implies",
      "rule_id": "nonzero_cancellation",
      "conditions": [
        "a≠0"
      ],
      "explanation": {
        "zh-Hant": "兩個假設解相減，非零 a 可約掉，所以不存在另一個不同解。",
        "en": "Subtract two proposed solution equations and cancel the nonzero a; no distinct second solution exists."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "這個唯一性論證的 a≠0 前提不可省略。",
          "en": "The hypothesis a≠0 is essential to this uniqueness argument."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "branch": "unique",
    "a": 5,
    "b": 6,
    "solution": "-6/5"
  },
  "conclusion": {
    "canonical_value": "唯一解 x=-6/5",
    "explanation": {
      "zh-Hant": "唯一解 x=-6/5",
      "en": "The unique rational solution is x=-6/5."
    }
  },
  "definition_note": {
    "zh-Hant": "有理數論域；a=0 分開處理常數等式，a≠0 才除以 a。",
    "en": "Use the rational domain. Treat constant equations separately when a=0; divide by a only when a is nonzero."
  },
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    "original_candidate_line_sha256": "402cf887123b4069d1bd195692884b3dc306e6c18b2aeb9d675cd664182dddc8"
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  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
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    "scope_note": {
      "zh-Hant": "本稿是來源綁定的教學推導候選；精確重算不等於獨立數學審查或形式化證明。原題、答案、公式、AST 與歷史未被修改。",
      "en": "This is a source-bound pedagogical derivation candidate. Exact recomputation is not independent mathematical review or formal proof. Original questions, answers, formulas, ASTs and history are unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
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