{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20260927-linear-equation-exact-solve-0020",
  "pilot_number": "P1000-0295",
  "number": 295,
  "family_id": "qf-linear-equation-exact-solve",
  "title": {
    "zh-Hant": "第 295 題：一次方程的精確解",
    "en": "Question 295: Exact Solution of a Linear Equation"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20260927-linear-equation-exact-solve-0020",
    "batch_id": "batch-20260927-linear-equation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-linear-equation"
    ],
    "title_zh": "一次方程的精確解",
    "stem_zh": "在有理數論域精確解 3x+(-6)=0；a=3 已明示非零。",
    "answer": {
      "kind": "short_text",
      "value": "2",
      "explanation_zh": "x=−(-6)/3=2；以 gcd 約成正分母 canonical rational。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 3,
      "skills": [
        "有理數",
        "canonical fraction"
      ]
    },
    "variant": {
      "family_id": "qf-linear-equation-exact-solve",
      "seed": 20261047,
      "parameters": {
        "a": 3,
        "b": -6,
        "domain": "rationals",
        "operation": "exact_solve",
        "semantic_template": "qf-linear-equation-exact-solve/{\"a\":3,\"b\":-6,\"domain\":\"rationals\",\"operation\":\"exact_solve\"}"
      }
    },
    "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"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20260927-linear-equation-exact-solve-0020",
    "batch_id": "batch-20260927-linear-equation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "b72c580be7c9c86299f860f0fc54eeb58b86ae1c8ad4d6b67ecb734cb6b55c19",
      "family_id": "qf-linear-equation-exact-solve",
      "seed": 20261047,
      "semantic_template": "qf-linear-equation-exact-solve/{\"a\":3,\"b\":-6,\"domain\":\"rationals\",\"operation\":\"exact_solve\"}"
    },
    "fields": {
      "title": "Exact Solution of a Linear Equation",
      "stem": "Solve 3x+(-6)=0 exactly over the rational numbers; a=3 is explicitly nonzero.",
      "answer": {
        "kind": "short_text",
        "value": "2",
        "explanation": "x=−(-6)/3=2; the result is reduced by the gcd to a canonical rational with a positive denominator."
      },
      "skills": [
        "rational numbers",
        "canonical fraction"
      ],
      "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": "在有理數論域精確解 3x+(-6)=0；a=3 已明示非零。",
        "en": "Solve 3x+(-6)=0 exactly over the rational numbers; a=3 is explicitly nonzero."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "a=3, b=-6, x∈Q",
      "after": "equation: 3x+(-6)=0",
      "relation": "implies",
      "rule_id": "read_coefficients",
      "conditions": [],
      "explanation": {
        "zh-Hant": "明確讀取一次項、常數項與有理數域，不從式子表面名稱推斷非退化。",
        "en": "Read the linear coefficient, constant and rational domain explicitly; a name alone does not establish nondegeneracy."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "代回原式核對，除法前的非零條件不可省略。",
          "en": "Substitute into the original equation; retain every nonzero condition before division."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "3x+(-6)=0",
      "after": "3x=6",
      "relation": "equivalent",
      "rule_id": "subtract_constant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩邊同減常數 b，等價保留原解集。",
        "en": "Subtract b from both sides, preserving the solution set."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "代回原式核對，除法前的非零條件不可省略。",
          "en": "Substitute into the original equation; retain every nonzero condition before division."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "x := div(6, 3)",
      "after": "x = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "因 3≠0，才可同除以 a。",
        "en": "Division by a is allowed because 3≠0."
      },
      "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",
          "3"
        ],
        "output": "2",
        "label": "x"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "a·x := mul(3, 2)",
      "after": "a·x = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先代回一次項。",
        "en": "Substitute into 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": [
          "3",
          "2"
        ],
        "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": "加回常數須得到0。",
        "en": "Adding the constant must give0."
      },
      "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−x₂)=0 and a≠0",
      "after": "x₁=x₂",
      "relation": "implies",
      "rule_id": "unique_solution",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若另有解，兩解方程相減並約去非零 a，必相等。",
        "en": "Subtract equations for any two solutions and cancel nonzero a to prove uniqueness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "代回原式核對，除法前的非零條件不可省略。",
          "en": "Substitute into the original equation; retain every nonzero condition before division."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "2"
  },
  "conclusion": {
    "canonical_value": "2",
    "explanation": {
      "zh-Hant": "上述完整代值與條件核對得到原題答案；欄位次序及真假標記保持原始合約。",
      "en": "The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings."
    }
  },
  "definition_note": {
    "zh-Hant": "有理數論域且 a≠0；先移項再除以非零 a，答案為正分母的既約分數。",
    "en": "Use the rational domain with a≠0. Move the constant, divide by nonzero a and give the reduced fraction with positive denominator."
  },
  "source": {
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      "question_compact_sha256": "b72c580be7c9c86299f860f0fc54eeb58b86ae1c8ad4d6b67ecb734cb6b55c19",
      "english_manifest_path": "public/data/i18n/editions/batch-20260927-linear-equation-foundations.en.json",
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    "original_candidate_line_sha256": "43bfc36d0d8789df56fd6114fd9c714a22dc2c34b5a9786c63347121747e8809"
  },
  "historical_source_state": {
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      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定的教學推導候選；精確算術及有限列舉不是獨立AI審查或一般形式化證明。原題、答案、MKO、公式、AST及歷史保持不變。",
      "en": "Source-bound pedagogical derivation candidate: exact arithmetic and finite enumeration are not independent AI review or general formal proof. Original questions, answers, MKOs, formulas, ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-003-01",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "Thirty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct sentence shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full answer tuples, model results and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
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    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 6,
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}