{
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  "question_id": "q-20260927-linear-equation-solution-class-0007",
  "pilot_number": "P250-0032",
  "number": 32,
  "family_id": "qf-linear-equation-solution-class",
  "title": {
    "zh-Hant": "第 32 題：一次方程的解分類",
    "en": "Question 32: Solution Classification of a Linear Equation"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20260927-linear-equation-solution-class-0007",
    "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 0x+(6)=0 的解集。",
    "answer": {
      "kind": "short_text",
      "value": "無解",
      "explanation_zh": "a=0 且 b=6≠0，等式為 6=0，所以無解。"
    },
    "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": 20261234,
      "parameters": {
        "a": 0,
        "b": 6,
        "domain": "rationals",
        "operation": "solution_class",
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      }
    },
    "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",
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    }
  },
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    "question_id": "q-20260927-linear-equation-solution-class-0007",
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    },
    "fields": {
      "title": "Solution Classification of a Linear Equation",
      "stem": "Over the rational numbers, classify the solution set of 0x+(6)=0.",
      "answer": {
        "kind": "short_text",
        "value": "no solution",
        "explanation": "With a=0 and b=6≠0, the equation is 6=0, so it has no solution."
      },
      "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 0x+(6)=0 的解集。",
        "en": "Over the rational numbers, classify the solution set of 0x+(6)=0."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "0x+(6)=0, x∈Q",
      "after": "a=0; 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": "a=0",
      "after": "0x+(6)=6",
      "relation": "implies",
      "rule_id": "zero_coefficient",
      "conditions": [
        "a=0"
      ],
      "explanation": {
        "zh-Hant": "零乘任何有理數都為零，所以未知數項完全消失。",
        "en": "Zero times any rational number is zero, so the unknown term disappears."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "此處不建立 1/a，也不執行除以零。",
          "en": "No reciprocal of a or division by zero is introduced."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "6=0",
      "after": "false for every x∈Q",
      "relation": "implies",
      "rule_id": "constant_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "剩下的是與 x 無關的常數等式，直接檢查其真假。",
        "en": "The remaining constant equality is independent of x; check its truth directly."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "分類取決於 b 是否為零，不依抽樣試代幾個 x。",
          "en": "The classification depends on b=0, not on sampling a few values of x."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "constant equality has fixed truth value",
      "after": "Solution set = ∅",
      "relation": "implies",
      "rule_id": "solution_set_definition",
      "conditions": [],
      "explanation": {
        "zh-Hant": "恆真時每個允許值皆為解；恆假時不存在任何解。",
        "en": "If the equality is always true every allowed value is a solution; if always false there is no solution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "所有解與沒有解都不同於唯一解。",
          "en": "All solutions and no solutions are both distinct from a unique solution."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "a=0 and b=0 or b≠0",
      "after": "無解",
      "relation": "implies",
      "rule_id": "exhaustive_cases",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這兩種常數分支已窮盡 a=0 的情況。",
        "en": "These two constant branches exhaust the case a=0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原式，0x+b 的真值對所有有理數保持不變。",
          "en": "In the original equation the truth of 0x+b=0 is unchanged for every rational x."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "branch": "none",
    "a": 0,
    "b": 6,
    "solution": null
  },
  "conclusion": {
    "canonical_value": "無解",
    "explanation": {
      "zh-Hant": "無解",
      "en": "There is no solution."
    }
  },
  "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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