{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261003-r05-linear-function-0049",
  "pilot_number": "R0001-01149",
  "number": 1149,
  "family_id": "qf-affine-two-point-slope",
  "title": {
    "zh-Hant": "第 1149 題：仿射圖形的兩點斜率",
    "en": "Question 1149: Two-Point Slope of an Affine Graph"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261003-r05-linear-function-0049",
    "batch_id": "batch-20261003-r05-linear-function-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-linear-function"
    ],
    "title_zh": "仿射圖形的兩點斜率",
    "stem_zh": "在實數論域，仿射函數圖形含兩點 (11, 1)、(5, 12)，兩橫座標不同。求最簡正分母斜率。",
    "answer": {
      "kind": "expression",
      "value": "-11/6",
      "explanation_zh": "斜率=-11/6。使用 (y₂−y₁)/(x₂−x₁)，分母非零；此式不適用垂直線。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 3,
      "skills": [
        "仿射圖形的兩點斜率"
      ]
    },
    "variant": {
      "family_id": "qf-affine-two-point-slope",
      "seed": 202610035049,
      "parameters": {
        "x1": 11,
        "y1": 1,
        "x2": 5,
        "y2": 12,
        "domain": "reals",
        "operation": "slope",
        "semantic_template": "qf-affine-two-point-slope/{\"domain\":\"reals\",\"operation\":\"slope\",\"x1\":5,\"x2\":11,\"y1\":12,\"y2\":1}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication"
      ],
      "warning_zh": "僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "Codex independent peer (inherited model; exact runtime model not exposed)",
      "generated_at": "2026-10-03",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261003-r05-linear-function-0049",
    "batch_id": "batch-20261003-r05-linear-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "4d29ff7bb0eb40ec19864bc22c765d22abc7cf5188b627338dcfb8240b9e4832",
      "family_id": "qf-affine-two-point-slope",
      "seed": 202610035049,
      "semantic_template": "qf-affine-two-point-slope/{\"domain\":\"reals\",\"operation\":\"slope\",\"x1\":5,\"x2\":11,\"y1\":12,\"y2\":1}"
    },
    "fields": {
      "title": "Two-Point Slope of an Affine Graph",
      "stem": "Over the reals, an affine-function graph contains (11, 1) and (5, 12), with distinct first coordinates. Find the slope as a reduced fraction with positive denominator.",
      "answer": {
        "kind": "expression",
        "value": "-11/6",
        "explanation": "Use (y_2−y_1)/(x_2−x_1) to obtain -11/6. The denominator is nonzero; this formula does not give the slope of a vertical line."
      },
      "skills": [
        "Two-Point Slope of an Affine Graph"
      ],
      "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": "在實數論域，仿射函數圖形含兩點 (11, 1)、(5, 12)，兩橫座標不同。求最簡正分母斜率。",
        "en": "Over the reals, an affine-function graph contains (11, 1) and (5, 12), with distinct first coordinates. Find the slope as a reduced fraction with positive denominator."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "f:R→R; f(x)=mx+b",
      "after": "m=0 gives a constant affine function",
      "relation": "implies",
      "rule_id": "affine_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "仿射函數允許m=0；不能把一次方程式a≠0的限制誤套到所有仿射函數。",
        "en": "An affine function allows m=0; do not impose the nondegenerate-equation condition on every affine function."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Δx := sub(5, 11)",
      "after": "Δx = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "分母是兩橫座標的差。",
        "en": "The denominator is the horizontal-coordinate difference."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Δx，不以小數近似替代。",
          "en": "Recompute Δx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5",
          "11"
        ],
        "output": "-6",
        "label": "Δx"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Δy := sub(12, 1)",
      "after": "Δy = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "分子也採相同點次序。",
        "en": "Use the same point order for the vertical difference."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Δy，不以小數近似替代。",
          "en": "Recompute Δy with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "12",
          "1"
        ],
        "output": "11",
        "label": "Δy"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "slope := div(11, -6)",
      "after": "slope = -11/6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已確認Δx非零，才可計算斜率。",
        "en": "Compute the slope only after checking Δx is nonzero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slope，不以小數近似替代。",
          "en": "Recompute slope with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "11",
          "-6"
        ],
        "output": "-11/6",
        "label": "slope"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "slope·Δx := mul(-11/6, -6)",
      "after": "slope·Δx = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "乘回分母必恢復原縱差。",
        "en": "Multiplication by the denominator must recover the original vertical difference."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slope·Δx，不以小數近似替代。",
          "en": "Recompute slope·Δx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-11/6",
          "-6"
        ],
        "output": "11",
        "label": "slope·Δx"
      }
    }
  ],
  "math_model": {
    "dx": "-6",
    "dy": "11",
    "slope": "-11/6"
  },
  "conclusion": {
    "canonical_value": "-11/6",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
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    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
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    "original_candidate_line_sha256": "07ae1c33f41909f85555d9cf88c3435e26b578c993517c2f45f0ea0d81156bc8"
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  "historical_source_state": {
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      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
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      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
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    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human 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-20261026-004-p2",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings 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.",
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}