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{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0003","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"一次資格與規一列","stem_zh":"Q上ax+b=0；非零a才是一次；候選a=2、b=-10。依序答 (a非零資格, [1/1,b/a]或未定義, 原解集class)。","answer":{"kind":"expression","value":"(是, [1/1, -5/1], 單解 5/1)","explanation_zh":"一次資格規一列與解集=(是, [1/1, -5/1], 單解 5/1)。先檢a，非零才除係數規一；零係數另按常數式分類。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["一次資格與規一列"]},"variant":{"family_id":"qf-density14-linear-normalized-row","seed":202610144003,"parameters":{"a":2,"b":-10,"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_normalized_row","semantic_template":"qf-density14-linear-normalized-row/\"單解 5/1\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0011","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"一次資格與規一列","stem_zh":"Q上ax+b=0；非零a才是一次；候選a=2、b=-2。依序答 (a非零資格, [1/1,b/a]或未定義, 原解集class)。","answer":{"kind":"expression","value":"(是, [1/1, -1/1], 單解 1/1)","explanation_zh":"一次資格規一列與解集=(是, [1/1, -1/1], 單解 1/1)。先檢a，非零才除係數規一；零係數另按常數式分類。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["一次資格與規一列"]},"variant":{"family_id":"qf-density14-linear-normalized-row","seed":202610144011,"parameters":{"a":2,"b":-2,"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_normalized_row","semantic_template":"qf-density14-linear-normalized-row/\"單解 1/1\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0023","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"一次資格與規一列","stem_zh":"Q上ax+b=0；非零a才是一次；候選a=2、b=10。依序答 (a非零資格, [1/1,b/a]或未定義, 原解集class)。","answer":{"kind":"expression","value":"(是, [1/1, 5/1], 單解 -5/1)","explanation_zh":"一次資格規一列與解集=(是, [1/1, 5/1], 單解 -5/1)。先檢a，非零才除係數規一；零係數另按常數式分類。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["一次資格與規一列"]},"variant":{"family_id":"qf-density14-linear-normalized-row","seed":202610144023,"parameters":{"a":2,"b":10,"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_normalized_row","semantic_template":"qf-density14-linear-normalized-row/\"單解 -5/1\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0024","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"一次資格與規一列","stem_zh":"Q上ax+b=0；非零a才是一次；候選a=-2、b=-11。依序答 (a非零資格, [1/1,b/a]或未定義, 原解集class)。","answer":{"kind":"expression","value":"(是, [1/1, 11/2], 單解 -11/2)","explanation_zh":"一次資格規一列與解集=(是, [1/1, 11/2], 單解 -11/2)。先檢a，非零才除係數規一；零係數另按常數式分類。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["一次資格與規一列"]},"variant":{"family_id":"qf-density14-linear-normalized-row","seed":202610144024,"parameters":{"a":-2,"b":-11,"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_normalized_row","semantic_template":"qf-density14-linear-normalized-row/\"單解 -11/2\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0025","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"一次資格與規一列","stem_zh":"Q上ax+b=0；非零a才是一次；候選a=2、b=12。依序答 (a非零資格, [1/1,b/a]或未定義, 原解集class)。","answer":{"kind":"expression","value":"(是, [1/1, 6/1], 單解 -6/1)","explanation_zh":"一次資格規一列與解集=(是, [1/1, 6/1], 單解 -6/1)。先檢a，非零才除係數規一；零係數另按常數式分類。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["一次資格與規一列"]},"variant":{"family_id":"qf-density14-linear-normalized-row","seed":202610144025,"parameters":{"a":2,"b":12,"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_normalized_row","semantic_template":"qf-density14-linear-normalized-row/\"單解 -6/1\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0026","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=0、b=0、ts=[0, 1, -1]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([0, 1, -1], [否, 是, 是], [全Q, 單解 0/1, 單解 0/1])","explanation_zh":"參數切片一次資格與解集=([0, 1, -1], [否, 是, 是], [全Q, 單解 0/1, 單解 0/1])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144026,"parameters":{"k":0,"b":0,"ts":[0,1,-1],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"0\",\"1\",\"-1\"],[\"全Q\",\"單解 0/1\",\"單解 0/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0027","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=2、b=1、ts=[2, 3, 1]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([0, 1, -1], [否, 是, 是], [無解, 單解 -1/1, 單解 1/1])","explanation_zh":"參數切片一次資格與解集=([0, 1, -1], [否, 是, 是], [無解, 單解 -1/1, 單解 1/1])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144027,"parameters":{"k":2,"b":1,"ts":[2,3,1],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"0\",\"1\",\"-1\"],[\"無解\",\"單解 -1/1\",\"單解 1/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0028","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=1、b=-10、ts=[-9, 4, 1]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([-10, 3, 0], [是, 是, 否], [單解 -1/1, 單解 10/3, 無解])","explanation_zh":"參數切片一次資格與解集=([-10, 3, 0], [是, 是, 否], [單解 -1/1, 單解 10/3, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144028,"parameters":{"k":1,"b":-10,"ts":[-9,4,1],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"-10\",\"3\",\"0\"],[\"單解 -1/1\",\"單解 10/3\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0029","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=2、b=-9、ts=[-7, 6, 2]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([-9, 4, 0], [是, 是, 否], [單解 -1/1, 單解 9/4, 無解])","explanation_zh":"參數切片一次資格與解集=([-9, 4, 0], [是, 是, 否], [單解 -1/1, 單解 9/4, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144029,"parameters":{"k":2,"b":-9,"ts":[-7,6,2],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"-9\",\"4\",\"0\"],[\"單解 -1/1\",\"單解 9/4\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0030","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=-1、b=-8、ts=[-9, 4, -1]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([-8, 5, 0], [是, 是, 否], [單解 -1/1, 單解 8/5, 無解])","explanation_zh":"參數切片一次資格與解集=([-8, 5, 0], [是, 是, 否], [單解 -1/1, 單解 8/5, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144030,"parameters":{"k":-1,"b":-8,"ts":[-9,4,-1],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"-8\",\"5\",\"0\"],[\"單解 -1/1\",\"單解 8/5\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0031","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=0、b=-7、ts=[-7, 6, 0]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([-7, 6, 0], [是, 是, 否], [單解 -1/1, 單解 7/6, 無解])","explanation_zh":"參數切片一次資格與解集=([-7, 6, 0], [是, 是, 否], [單解 -1/1, 單解 7/6, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144031,"parameters":{"k":0,"b":-7,"ts":[-7,6,0],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"-7\",\"6\",\"0\"],[\"單解 -1/1\",\"單解 7/6\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0032","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=1、b=-6、ts=[-5, 8, 1]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([-6, 7, 0], [是, 是, 否], [單解 -1/1, 單解 6/7, 無解])","explanation_zh":"參數切片一次資格與解集=([-6, 7, 0], [是, 是, 否], [單解 -1/1, 單解 6/7, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144032,"parameters":{"k":1,"b":-6,"ts":[-5,8,1],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"-6\",\"7\",\"0\"],[\"單解 -1/1\",\"單解 6/7\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0033","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=2、b=-5、ts=[-3, 10, 2]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([-5, 8, 0], [是, 是, 否], [單解 -1/1, 單解 5/8, 無解])","explanation_zh":"參數切片一次資格與解集=([-5, 8, 0], [是, 是, 否], [單解 -1/1, 單解 5/8, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144033,"parameters":{"k":2,"b":-5,"ts":[-3,10,2],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"-5\",\"8\",\"0\"],[\"單解 -1/1\",\"單解 5/8\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0034","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=-1、b=-4、ts=[-5, 8, -1]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([-4, 9, 0], [是, 是, 否], [單解 -1/1, 單解 4/9, 無解])","explanation_zh":"參數切片一次資格與解集=([-4, 9, 0], [是, 是, 否], [單解 -1/1, 單解 4/9, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144034,"parameters":{"k":-1,"b":-4,"ts":[-5,8,-1],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"-4\",\"9\",\"0\"],[\"單解 -1/1\",\"單解 4/9\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0035","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=0、b=-3、ts=[-3, 10, 0]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([-3, 10, 0], [是, 是, 否], [單解 -1/1, 單解 3/10, 無解])","explanation_zh":"參數切片一次資格與解集=([-3, 10, 0], [是, 是, 否], [單解 -1/1, 單解 3/10, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144035,"parameters":{"k":0,"b":-3,"ts":[-3,10,0],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"-3\",\"10\",\"0\"],[\"單解 -1/1\",\"單解 3/10\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0036","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=1、b=-2、ts=[-1, 12, 1]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([-2, 11, 0], [是, 是, 否], [單解 -1/1, 單解 2/11, 無解])","explanation_zh":"參數切片一次資格與解集=([-2, 11, 0], [是, 是, 否], [單解 -1/1, 單解 2/11, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144036,"parameters":{"k":1,"b":-2,"ts":[-1,12,1],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"-2\",\"11\",\"0\"],[\"單解 -1/1\",\"單解 2/11\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0037","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=2、b=-1、ts=[1, 14, 2]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([-1, 12, 0], [是, 是, 否], [單解 -1/1, 單解 1/12, 無解])","explanation_zh":"參數切片一次資格與解集=([-1, 12, 0], [是, 是, 否], [單解 -1/1, 單解 1/12, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144037,"parameters":{"k":2,"b":-1,"ts":[1,14,2],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"-1\",\"12\",\"0\"],[\"單解 -1/1\",\"單解 1/12\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0038","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=-1、b=0、ts=[-1, 12, -1]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([0, 13, 0], [否, 是, 否], [全Q, 單解 0/1, 全Q])","explanation_zh":"參數切片一次資格與解集=([0, 13, 0], [否, 是, 否], [全Q, 單解 0/1, 全Q])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144038,"parameters":{"k":-1,"b":0,"ts":[-1,12,-1],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"0\",\"13\",\"0\"],[\"全Q\",\"單解 0/1\",\"全Q\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0039","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=0、b=1、ts=[1, 14, 0]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([1, 14, 0], [是, 是, 否], [單解 -1/1, 單解 -1/14, 無解])","explanation_zh":"參數切片一次資格與解集=([1, 14, 0], [是, 是, 否], [單解 -1/1, 單解 -1/14, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144039,"parameters":{"k":0,"b":1,"ts":[1,14,0],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"1\",\"14\",\"0\"],[\"單解 -1/1\",\"單解 -1/14\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0040","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=1、b=2、ts=[3, 16, 1]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([2, 15, 0], [是, 是, 否], [單解 -1/1, 單解 -2/15, 無解])","explanation_zh":"參數切片一次資格與解集=([2, 15, 0], [是, 是, 否], [單解 -1/1, 單解 -2/15, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144040,"parameters":{"k":1,"b":2,"ts":[3,16,1],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"2\",\"15\",\"0\"],[\"單解 -1/1\",\"單解 -2/15\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0041","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=2、b=3、ts=[5, 18, 2]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([3, 16, 0], [是, 是, 否], [單解 -1/1, 單解 -3/16, 無解])","explanation_zh":"參數切片一次資格與解集=([3, 16, 0], [是, 是, 否], [單解 -1/1, 單解 -3/16, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144041,"parameters":{"k":2,"b":3,"ts":[5,18,2],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"3\",\"16\",\"0\"],[\"單解 -1/1\",\"單解 -3/16\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0042","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=-1、b=4、ts=[3, 16, -1]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([4, 17, 0], [是, 是, 否], [單解 -1/1, 單解 -4/17, 無解])","explanation_zh":"參數切片一次資格與解集=([4, 17, 0], [是, 是, 否], [單解 -1/1, 單解 -4/17, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144042,"parameters":{"k":-1,"b":4,"ts":[3,16,-1],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"4\",\"17\",\"0\"],[\"單解 -1/1\",\"單解 -4/17\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0043","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=0、b=5、ts=[5, 18, 0]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([5, 18, 0], [是, 是, 否], [單解 -1/1, 單解 -5/18, 無解])","explanation_zh":"參數切片一次資格與解集=([5, 18, 0], [是, 是, 否], [單解 -1/1, 單解 -5/18, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144043,"parameters":{"k":0,"b":5,"ts":[5,18,0],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"5\",\"18\",\"0\"],[\"單解 -1/1\",\"單解 -5/18\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0044","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=1、b=6、ts=[7, 20, 1]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([6, 19, 0], [是, 是, 否], [單解 -1/1, 單解 -6/19, 無解])","explanation_zh":"參數切片一次資格與解集=([6, 19, 0], [是, 是, 否], [單解 -1/1, 單解 -6/19, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144044,"parameters":{"k":1,"b":6,"ts":[7,20,1],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"6\",\"19\",\"0\"],[\"單解 -1/1\",\"單解 -6/19\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0045","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=2、b=7、ts=[9, 22, 2]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([7, 20, 0], [是, 是, 否], [單解 -1/1, 單解 -7/20, 無解])","explanation_zh":"參數切片一次資格與解集=([7, 20, 0], [是, 是, 否], [單解 -1/1, 單解 -7/20, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144045,"parameters":{"k":2,"b":7,"ts":[9,22,2],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"7\",\"20\",\"0\"],[\"單解 -1/1\",\"單解 -7/20\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0046","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=-1、b=8、ts=[7, 20, -1]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([8, 21, 0], [是, 是, 否], [單解 -1/1, 單解 -8/21, 無解])","explanation_zh":"參數切片一次資格與解集=([8, 21, 0], [是, 是, 否], [單解 -1/1, 單解 -8/21, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144046,"parameters":{"k":-1,"b":8,"ts":[7,20,-1],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"8\",\"21\",\"0\"],[\"單解 -1/1\",\"單解 -8/21\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0047","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=0、b=9、ts=[9, 22, 0]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([9, 22, 0], [是, 是, 否], [單解 -1/1, 單解 -9/22, 無解])","explanation_zh":"參數切片一次資格與解集=([9, 22, 0], [是, 是, 否], [單解 -1/1, 單解 -9/22, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144047,"parameters":{"k":0,"b":9,"ts":[9,22,0],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"9\",\"22\",\"0\"],[\"單解 -1/1\",\"單解 -9/22\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0048","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=1、b=10、ts=[11, 24, 1]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([10, 23, 0], [是, 是, 否], [單解 -1/1, 單解 -10/23, 無解])","explanation_zh":"參數切片一次資格與解集=([10, 23, 0], [是, 是, 否], [單解 -1/1, 單解 -10/23, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144048,"parameters":{"k":1,"b":10,"ts":[11,24,1],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"10\",\"23\",\"0\"],[\"單解 -1/1\",\"單解 -10/23\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0049","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=2、b=11、ts=[13, 26, 2]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([11, 24, 0], [是, 是, 否], [單解 -1/1, 單解 -11/24, 無解])","explanation_zh":"參數切片一次資格與解集=([11, 24, 0], [是, 是, 否], [單解 -1/1, 單解 -11/24, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144049,"parameters":{"k":2,"b":11,"ts":[13,26,2],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"11\",\"24\",\"0\"],[\"單解 -1/1\",\"單解 -11/24\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0050","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"三個參數切片的一次資格","stem_zh":"Q上ax+b=0；非零a才是一次；a_i=t_i−k，k=-1、b=12、ts=[11, 24, -1]。依序列 (三係數, 三個一次資格, 三個解集class)，不忽略a_i=0。","answer":{"kind":"expression","value":"([12, 25, 0], [是, 是, 否], [單解 -1/1, 單解 -12/25, 無解])","explanation_zh":"參數切片一次資格與解集=([12, 25, 0], [是, 是, 否], [單解 -1/1, 單解 -12/25, 無解])。逐個形成實際係數，分開非零根與退化常數式。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三個參數切片的一次資格"]},"variant":{"family_id":"qf-density14-linear-parameter-slice","seed":202610144050,"parameters":{"k":-1,"b":12,"ts":[11,24,-1],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_parameter_slice","semantic_template":"qf-density14-linear-parameter-slice/[[\"12\",\"25\",\"0\"],[\"單解 -1/1\",\"單解 -12/25\",\"無解\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0051","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=1≠0、b=0，offsets=[0/1, 0/2, 0/-1]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(0/1, [0/1, 0/1, 0/1], [0/1, 0/1, 0/1], [是, 是, 是])","explanation_zh":"根周圍候選與回代殘差=(0/1, [0/1, 0/1, 0/1], [0/1, 0/1, 0/1], [是, 是, 是])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144051,"parameters":{"a":1,"b":0,"offsets":[[0,1],[0,2],[0,-1]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"0/1\",[\"0/1\",\"0/1\",\"0/1\"],[\"0/1\",\"0/1\",\"0/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0052","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=-2≠0、b=1，offsets=[0/1, 1/2, -1/2]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(1/2, [1/2, 1/1, 0/1], [0/1, -1/1, 1/1], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(1/2, [1/2, 1/1, 0/1], [0/1, -1/1, 1/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144052,"parameters":{"a":-2,"b":1,"offsets":[[0,1],[1,2],[-1,2]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"1/2\",[\"1/2\",\"1/1\",\"0/1\"],[\"0/1\",\"-1/1\",\"1/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0053","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=3≠0、b=-10，offsets=[0/1, 3/2, -2/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(10/3, [10/3, 29/6, 8/3], [0/1, 9/2, -2/1], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(10/3, [10/3, 29/6, 8/3], [0/1, 9/2, -2/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144053,"parameters":{"a":3,"b":-10,"offsets":[[0,1],[3,2],[-2,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"10/3\",[\"10/3\",\"29/6\",\"8/3\"],[\"0/1\",\"9/2\",\"-2/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0054","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=-4≠0、b=-9，offsets=[0/1, 4/2, -3/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(-9/4, [-9/4, -1/4, -13/4], [0/1, -8/1, 4/1], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(-9/4, [-9/4, -1/4, -13/4], [0/1, -8/1, 4/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144054,"parameters":{"a":-4,"b":-9,"offsets":[[0,1],[4,2],[-3,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"-9/4\",[\"-9/4\",\"-1/4\",\"-13/4\"],[\"0/1\",\"-8/1\",\"4/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0055","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=5≠0、b=-8，offsets=[0/1, 5/2, -4/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(8/5, [8/5, 41/10, 4/15], [0/1, 25/2, -20/3], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(8/5, [8/5, 41/10, 4/15], [0/1, 25/2, -20/3], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144055,"parameters":{"a":5,"b":-8,"offsets":[[0,1],[5,2],[-4,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"8/5\",[\"8/5\",\"41/10\",\"4/15\"],[\"0/1\",\"25/2\",\"-20/3\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0056","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=-6≠0、b=-7，offsets=[0/1, 6/2, -5/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(-7/6, [-7/6, 11/6, -17/6], [0/1, -18/1, 10/1], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(-7/6, [-7/6, 11/6, -17/6], [0/1, -18/1, 10/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144056,"parameters":{"a":-6,"b":-7,"offsets":[[0,1],[6,2],[-5,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"-7/6\",[\"-7/6\",\"11/6\",\"-17/6\"],[\"0/1\",\"-18/1\",\"10/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0057","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=7≠0、b=-6，offsets=[0/1, 7/2, -6/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(6/7, [6/7, 61/14, -8/7], [0/1, 49/2, -14/1], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(6/7, [6/7, 61/14, -8/7], [0/1, 49/2, -14/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144057,"parameters":{"a":7,"b":-6,"offsets":[[0,1],[7,2],[-6,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"6/7\",[\"6/7\",\"61/14\",\"-8/7\"],[\"0/1\",\"49/2\",\"-14/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0058","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=-8≠0、b=-5，offsets=[0/1, 8/2, -7/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(-5/8, [-5/8, 27/8, -71/24], [0/1, -32/1, 56/3], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(-5/8, [-5/8, 27/8, -71/24], [0/1, -32/1, 56/3], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144058,"parameters":{"a":-8,"b":-5,"offsets":[[0,1],[8,2],[-7,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"-5/8\",[\"-5/8\",\"27/8\",\"-71/24\"],[\"0/1\",\"-32/1\",\"56/3\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0059","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=9≠0、b=-4，offsets=[0/1, 9/2, -8/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(4/9, [4/9, 89/18, -20/9], [0/1, 81/2, -24/1], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(4/9, [4/9, 89/18, -20/9], [0/1, 81/2, -24/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144059,"parameters":{"a":9,"b":-4,"offsets":[[0,1],[9,2],[-8,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"4/9\",[\"4/9\",\"89/18\",\"-20/9\"],[\"0/1\",\"81/2\",\"-24/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0060","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=-10≠0、b=-3，offsets=[0/1, 10/2, -9/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(-3/10, [-3/10, 47/10, -33/10], [0/1, -50/1, 30/1], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(-3/10, [-3/10, 47/10, -33/10], [0/1, -50/1, 30/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144060,"parameters":{"a":-10,"b":-3,"offsets":[[0,1],[10,2],[-9,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"-3/10\",[\"-3/10\",\"47/10\",\"-33/10\"],[\"0/1\",\"-50/1\",\"30/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0061","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=11≠0、b=-2，offsets=[0/1, 11/2, -10/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(2/11, [2/11, 125/22, -104/33], [0/1, 121/2, -110/3], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(2/11, [2/11, 125/22, -104/33], [0/1, 121/2, -110/3], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144061,"parameters":{"a":11,"b":-2,"offsets":[[0,1],[11,2],[-10,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"2/11\",[\"2/11\",\"125/22\",\"-104/33\"],[\"0/1\",\"121/2\",\"-110/3\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0062","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=-12≠0、b=-1，offsets=[0/1, 12/2, -11/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(-1/12, [-1/12, 71/12, -15/4], [0/1, -72/1, 44/1], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(-1/12, [-1/12, 71/12, -15/4], [0/1, -72/1, 44/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144062,"parameters":{"a":-12,"b":-1,"offsets":[[0,1],[12,2],[-11,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"-1/12\",[\"-1/12\",\"71/12\",\"-15/4\"],[\"0/1\",\"-72/1\",\"44/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0063","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=13≠0、b=0，offsets=[0/1, 13/2, -12/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(0/1, [0/1, 13/2, -4/1], [0/1, 169/2, -52/1], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(0/1, [0/1, 13/2, -4/1], [0/1, 169/2, -52/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144063,"parameters":{"a":13,"b":0,"offsets":[[0,1],[13,2],[-12,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"0/1\",[\"0/1\",\"13/2\",\"-4/1\"],[\"0/1\",\"169/2\",\"-52/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0064","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=-14≠0、b=1，offsets=[0/1, 14/2, -13/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(1/14, [1/14, 99/14, -179/42], [0/1, -98/1, 182/3], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(1/14, [1/14, 99/14, -179/42], [0/1, -98/1, 182/3], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144064,"parameters":{"a":-14,"b":1,"offsets":[[0,1],[14,2],[-13,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"1/14\",[\"1/14\",\"99/14\",\"-179/42\"],[\"0/1\",\"-98/1\",\"182/3\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0065","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=15≠0、b=2，offsets=[0/1, 15/2, -14/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(-2/15, [-2/15, 221/30, -24/5], [0/1, 225/2, -70/1], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(-2/15, [-2/15, 221/30, -24/5], [0/1, 225/2, -70/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144065,"parameters":{"a":15,"b":2,"offsets":[[0,1],[15,2],[-14,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"-2/15\",[\"-2/15\",\"221/30\",\"-24/5\"],[\"0/1\",\"225/2\",\"-70/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0066","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=-16≠0、b=3，offsets=[0/1, 16/2, -15/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(3/16, [3/16, 131/16, -77/16], [0/1, -128/1, 80/1], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(3/16, [3/16, 131/16, -77/16], [0/1, -128/1, 80/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144066,"parameters":{"a":-16,"b":3,"offsets":[[0,1],[16,2],[-15,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"3/16\",[\"3/16\",\"131/16\",\"-77/16\"],[\"0/1\",\"-128/1\",\"80/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0067","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=17≠0、b=4，offsets=[0/1, 17/2, -16/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(-4/17, [-4/17, 281/34, -284/51], [0/1, 289/2, -272/3], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(-4/17, [-4/17, 281/34, -284/51], [0/1, 289/2, -272/3], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144067,"parameters":{"a":17,"b":4,"offsets":[[0,1],[17,2],[-16,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"-4/17\",[\"-4/17\",\"281/34\",\"-284/51\"],[\"0/1\",\"289/2\",\"-272/3\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0068","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=-18≠0、b=5，offsets=[0/1, 18/2, -17/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(5/18, [5/18, 167/18, -97/18], [0/1, -162/1, 102/1], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(5/18, [5/18, 167/18, -97/18], [0/1, -162/1, 102/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144068,"parameters":{"a":-18,"b":5,"offsets":[[0,1],[18,2],[-17,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"5/18\",[\"5/18\",\"167/18\",\"-97/18\"],[\"0/1\",\"-162/1\",\"102/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0069","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=19≠0、b=6，offsets=[0/1, 19/2, -18/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(-6/19, [-6/19, 349/38, -120/19], [0/1, 361/2, -114/1], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(-6/19, [-6/19, 349/38, -120/19], [0/1, 361/2, -114/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144069,"parameters":{"a":19,"b":6,"offsets":[[0,1],[19,2],[-18,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"-6/19\",[\"-6/19\",\"349/38\",\"-120/19\"],[\"0/1\",\"361/2\",\"-114/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0070","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=-20≠0、b=7，offsets=[0/1, 20/2, -19/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(7/20, [7/20, 207/20, -359/60], [0/1, -200/1, 380/3], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(7/20, [7/20, 207/20, -359/60], [0/1, -200/1, 380/3], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144070,"parameters":{"a":-20,"b":7,"offsets":[[0,1],[20,2],[-19,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"7/20\",[\"7/20\",\"207/20\",\"-359/60\"],[\"0/1\",\"-200/1\",\"380/3\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0071","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=21≠0、b=8，offsets=[0/1, 21/2, -20/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(-8/21, [-8/21, 425/42, -148/21], [0/1, 441/2, -140/1], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(-8/21, [-8/21, 425/42, -148/21], [0/1, 441/2, -140/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144071,"parameters":{"a":21,"b":8,"offsets":[[0,1],[21,2],[-20,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"-8/21\",[\"-8/21\",\"425/42\",\"-148/21\"],[\"0/1\",\"441/2\",\"-140/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0072","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=-22≠0、b=9，offsets=[0/1, 22/2, -21/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(9/22, [9/22, 251/22, -145/22], [0/1, -242/1, 154/1], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(9/22, [9/22, 251/22, -145/22], [0/1, -242/1, 154/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144072,"parameters":{"a":-22,"b":9,"offsets":[[0,1],[22,2],[-21,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"9/22\",[\"9/22\",\"251/22\",\"-145/22\"],[\"0/1\",\"-242/1\",\"154/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0073","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=23≠0、b=10，offsets=[0/1, 23/2, -22/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(-10/23, [-10/23, 509/46, -536/69], [0/1, 529/2, -506/3], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(-10/23, [-10/23, 509/46, -536/69], [0/1, 529/2, -506/3], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144073,"parameters":{"a":23,"b":10,"offsets":[[0,1],[23,2],[-22,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"-10/23\",[\"-10/23\",\"509/46\",\"-536/69\"],[\"0/1\",\"529/2\",\"-506/3\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0074","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=-24≠0、b=11，offsets=[0/1, 24/2, -23/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(11/24, [11/24, 299/24, -173/24], [0/1, -288/1, 184/1], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(11/24, [11/24, 299/24, -173/24], [0/1, -288/1, 184/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144074,"parameters":{"a":-24,"b":11,"offsets":[[0,1],[24,2],[-23,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"11/24\",[\"11/24\",\"299/24\",\"-173/24\"],[\"0/1\",\"-288/1\",\"184/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0075","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"根周圍的候選與回代","stem_zh":"Q上ax+b=0；非零a才是一次；a=25≠0、b=12，offsets=[0/1, 25/2, -24/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。","answer":{"kind":"expression","value":"(-12/25, [-12/25, 601/50, -212/25], [0/1, 625/2, -200/1], [是, 否, 否])","explanation_zh":"根周圍候選與回代殘差=(-12/25, [-12/25, 601/50, -212/25], [0/1, 625/2, -200/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["根周圍的候選與回代"]},"variant":{"family_id":"qf-density14-linear-root-substitution-grid","seed":202610144075,"parameters":{"a":25,"b":12,"offsets":[[0,1],[25,2],[-24,3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_root_substitution_grid","semantic_template":"qf-density14-linear-root-substitution-grid/[\"-12/25\",[\"-12/25\",\"601/50\",\"-212/25\"],[\"0/1\",\"625/2\",\"-200/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0076","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[1, 0], [2, 0]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(0/1, 0/1, 是, 0/1)","explanation_zh":"兩個一次方程的根與共同解=(0/1, 0/1, 是, 0/1)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144076,"parameters":{"rows":[[1,0],[2,0]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"0/1\",\"0/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0077","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[1, -1], [-2, 2]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(1/1, 1/1, 是, 1/1)","explanation_zh":"兩個一次方程的根與共同解=(1/1, 1/1, 是, 1/1)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144077,"parameters":{"rows":[[1,-1],[-2,2]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"1/1\",\"1/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0078","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -2], [-3, -1]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(1/1, -1/3, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(1/1, -1/3, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144078,"parameters":{"rows":[[2,-2],[-3,-1]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"1/1\",\"-1/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0079","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -3], [-3, -2]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(3/2, -2/3, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(3/2, -2/3, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144079,"parameters":{"rows":[[2,-3],[-3,-2]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"3/2\",\"-2/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0080","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -4], [-3, -3]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(2/1, -1/1, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(2/1, -1/1, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144080,"parameters":{"rows":[[2,-4],[-3,-3]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"2/1\",\"-1/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0081","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -5], [-3, -4]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(5/2, -4/3, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(5/2, -4/3, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144081,"parameters":{"rows":[[2,-5],[-3,-4]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"5/2\",\"-4/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0082","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -6], [-3, -5]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(3/1, -5/3, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(3/1, -5/3, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144082,"parameters":{"rows":[[2,-6],[-3,-5]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"3/1\",\"-5/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0083","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -7], [-3, -6]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(7/2, -2/1, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(7/2, -2/1, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144083,"parameters":{"rows":[[2,-7],[-3,-6]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"7/2\",\"-2/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0084","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -8], [-3, -7]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(4/1, -7/3, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(4/1, -7/3, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144084,"parameters":{"rows":[[2,-8],[-3,-7]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"4/1\",\"-7/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0085","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -9], [-3, -8]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(9/2, -8/3, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(9/2, -8/3, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144085,"parameters":{"rows":[[2,-9],[-3,-8]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"9/2\",\"-8/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0086","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -10], [-3, -9]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(5/1, -3/1, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(5/1, -3/1, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144086,"parameters":{"rows":[[2,-10],[-3,-9]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"5/1\",\"-3/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0087","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -11], [-3, -10]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(11/2, -10/3, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(11/2, -10/3, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144087,"parameters":{"rows":[[2,-11],[-3,-10]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"11/2\",\"-10/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0088","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -12], [-3, -11]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(6/1, -11/3, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(6/1, -11/3, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144088,"parameters":{"rows":[[2,-12],[-3,-11]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"6/1\",\"-11/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0089","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -13], [-3, -12]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(13/2, -4/1, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(13/2, -4/1, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144089,"parameters":{"rows":[[2,-13],[-3,-12]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"13/2\",\"-4/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0090","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -14], [-3, -13]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(7/1, -13/3, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(7/1, -13/3, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144090,"parameters":{"rows":[[2,-14],[-3,-13]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"7/1\",\"-13/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0091","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -15], [-3, -14]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(15/2, -14/3, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(15/2, -14/3, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144091,"parameters":{"rows":[[2,-15],[-3,-14]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"15/2\",\"-14/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0092","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -16], [-3, -15]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(8/1, -5/1, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(8/1, -5/1, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144092,"parameters":{"rows":[[2,-16],[-3,-15]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"8/1\",\"-5/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0093","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -17], [-3, -16]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(17/2, -16/3, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(17/2, -16/3, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144093,"parameters":{"rows":[[2,-17],[-3,-16]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"17/2\",\"-16/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0094","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -18], [-3, -17]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(9/1, -17/3, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(9/1, -17/3, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144094,"parameters":{"rows":[[2,-18],[-3,-17]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"9/1\",\"-17/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0095","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -19], [-3, -18]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(19/2, -6/1, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(19/2, -6/1, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144095,"parameters":{"rows":[[2,-19],[-3,-18]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"19/2\",\"-6/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0096","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -20], [-3, -19]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(10/1, -19/3, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(10/1, -19/3, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144096,"parameters":{"rows":[[2,-20],[-3,-19]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"10/1\",\"-19/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0097","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -21], [-3, -20]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(21/2, -20/3, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(21/2, -20/3, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144097,"parameters":{"rows":[[2,-21],[-3,-20]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"21/2\",\"-20/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0098","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -22], [-3, -21]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(11/1, -7/1, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(11/1, -7/1, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144098,"parameters":{"rows":[[2,-22],[-3,-21]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"11/1\",\"-7/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0099","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -23], [-3, -22]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(23/2, -22/3, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(23/2, -22/3, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144099,"parameters":{"rows":[[2,-23],[-3,-22]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"23/2\",\"-22/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r04-linear-equation-0100","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-equation"],"title_zh":"兩行一次式的共同根","stem_zh":"Q上ax+b=0；非零a才是一次；有序rows=[[2, -24], [-3, -23]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。","answer":{"kind":"expression","value":"(12/1, -23/3, 否, 無)","explanation_zh":"兩個一次方程的根與共同解=(12/1, -23/3, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩行一次式的共同根"]},"variant":{"family_id":"qf-density14-linear-two-row-coincidence","seed":202610144100,"parameters":{"rows":[[2,-24],[-3,-23]],"domain":"finite_Q_affine_row_protocol","operation":"density14_linear_two_row_coincidence","semantic_template":"qf-density14-linear-two-row-coincidence/[\"12/1\",\"-23/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}