{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0001","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d71b06077b96639dcd1ceb5221a6e9a266523bf8ef3bb8a840bb08d45b878b5d","family_id":"qf-density14-linear-normalized-row","seed":202610144001,"semantic_template":"qf-density14-linear-normalized-row/\"全Q\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=0, b=0. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(No, undefined, all Q)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (No, undefined, all Q)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0002","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86c1b5c112cde58f90be1b06f27bd023a001dd93ff5dd5068d352bb0077453ae","family_id":"qf-density14-linear-normalized-row","seed":202610144002,"semantic_template":"qf-density14-linear-normalized-row/\"無解\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=0, b=1. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(No, undefined, no-solution)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (No, undefined, no-solution)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0003","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d0debfdea1fe35ba3cb5ce10c577f3a8cac840faa45726b3deb8266000a5210","family_id":"qf-density14-linear-normalized-row","seed":202610144003,"semantic_template":"qf-density14-linear-normalized-row/\"單解 5/1\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=-10. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, -5/1], single-solution 5/1)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -5/1], single-solution 5/1)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0004","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e4f634adf82cc72171d7f8a81a68fbf0f4762c45183ad3b76f29bc9d3285214","family_id":"qf-density14-linear-normalized-row","seed":202610144004,"semantic_template":"qf-density14-linear-normalized-row/\"單解 9/2\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=9. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, -9/2], single-solution 9/2)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -9/2], single-solution 9/2)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0005","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b74df61afbd4b568b191be1ff20d43a282ba3fb1717deba662c87ba3f0015a4","family_id":"qf-density14-linear-normalized-row","seed":202610144005,"semantic_template":"qf-density14-linear-normalized-row/\"單解 4/1\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=-8. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, -4/1], single-solution 4/1)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -4/1], single-solution 4/1)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0006","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d502634251fb3651a52a50b2bd652c990bfbe5bcf473325d6240ccd56efe4aca","family_id":"qf-density14-linear-normalized-row","seed":202610144006,"semantic_template":"qf-density14-linear-normalized-row/\"單解 7/2\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=7. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, -7/2], single-solution 7/2)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -7/2], single-solution 7/2)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0007","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"119116e3c1669451a19021361b6bc87108ecadd9954a38dbc0a750d17f85defd","family_id":"qf-density14-linear-normalized-row","seed":202610144007,"semantic_template":"qf-density14-linear-normalized-row/\"單解 3/1\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=-6. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, -3/1], single-solution 3/1)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -3/1], single-solution 3/1)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0008","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef6b8e0aae72ff3979c07b8c66ae00df7fe151df184f56a5406b165b36826c48","family_id":"qf-density14-linear-normalized-row","seed":202610144008,"semantic_template":"qf-density14-linear-normalized-row/\"單解 5/2\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=5. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, -5/2], single-solution 5/2)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -5/2], single-solution 5/2)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0009","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f852568e5e13cce4e986f3abc22c1d4c5126e2fa4a5ea10b99ef86709268a3f","family_id":"qf-density14-linear-normalized-row","seed":202610144009,"semantic_template":"qf-density14-linear-normalized-row/\"單解 2/1\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=-4. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, -2/1], single-solution 2/1)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -2/1], single-solution 2/1)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0010","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01de9832fffa2b0e56ab87d2eef334eb4607815feadab7f3f0c004d046b87b9c","family_id":"qf-density14-linear-normalized-row","seed":202610144010,"semantic_template":"qf-density14-linear-normalized-row/\"單解 3/2\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=3. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, -3/2], single-solution 3/2)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -3/2], single-solution 3/2)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0011","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a9a3e274d271834203005b8d3af1193f7355a13ec543219a8fbb4c3f16603d7","family_id":"qf-density14-linear-normalized-row","seed":202610144011,"semantic_template":"qf-density14-linear-normalized-row/\"單解 1/1\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=-2. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, -1/1], single-solution 1/1)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -1/1], single-solution 1/1)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0012","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d570e9ded25d8bb7c0c0cdc5035fee27b323c1e672cc94a7558223c594d7de2c","family_id":"qf-density14-linear-normalized-row","seed":202610144012,"semantic_template":"qf-density14-linear-normalized-row/\"單解 1/2\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=1. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, -1/2], single-solution 1/2)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -1/2], single-solution 1/2)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0013","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6faff8b22dcb0262d5e2a9a2f91ed402dc334614d825a53d1defebc1c3603776","family_id":"qf-density14-linear-normalized-row","seed":202610144013,"semantic_template":"qf-density14-linear-normalized-row/\"單解 0/1\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=0. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, 0/1], single-solution 0/1)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 0/1], single-solution 0/1)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0014","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb42f7a0306d0d537666ac18d7f8e1934df39b36e9184510f1c9ad9e7c7860ea","family_id":"qf-density14-linear-normalized-row","seed":202610144014,"semantic_template":"qf-density14-linear-normalized-row/\"單解 -1/2\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=-1. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, 1/2], single-solution -1/2)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 1/2], single-solution -1/2)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0015","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa207a413704e0ebd4e848f153a5925007d5dc6d42635c05c1575be86f2a8a52","family_id":"qf-density14-linear-normalized-row","seed":202610144015,"semantic_template":"qf-density14-linear-normalized-row/\"單解 -1/1\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=2. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, 1/1], single-solution -1/1)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 1/1], single-solution -1/1)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0016","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67ceb7838590c2f1a2e41831c75df7a0ca380084ab99f1b50d41ab5911a851b2","family_id":"qf-density14-linear-normalized-row","seed":202610144016,"semantic_template":"qf-density14-linear-normalized-row/\"單解 -3/2\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=-3. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, 3/2], single-solution -3/2)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 3/2], single-solution -3/2)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0017","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f33506297db8f9d9cafd105f96ca2e9caff3bb848667f4785af7ae85374e6aa6","family_id":"qf-density14-linear-normalized-row","seed":202610144017,"semantic_template":"qf-density14-linear-normalized-row/\"單解 -2/1\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=4. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, 2/1], single-solution -2/1)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 2/1], single-solution -2/1)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0018","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ae757c748249194036473f670ae9ef7f1c94a6f3e645be2fc8e0df5d4434f2e","family_id":"qf-density14-linear-normalized-row","seed":202610144018,"semantic_template":"qf-density14-linear-normalized-row/\"單解 -5/2\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=-5. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, 5/2], single-solution -5/2)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 5/2], single-solution -5/2)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0019","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34425bf7e7d8572ce9a722a2c8e673b672353025ef18a92c690120eb204b9d4c","family_id":"qf-density14-linear-normalized-row","seed":202610144019,"semantic_template":"qf-density14-linear-normalized-row/\"單解 -3/1\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=6. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, 3/1], single-solution -3/1)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 3/1], single-solution -3/1)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0020","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba67fce31980d9d27b46934d7ce11db12d6c101b455fc19c775ac34250e06c5d","family_id":"qf-density14-linear-normalized-row","seed":202610144020,"semantic_template":"qf-density14-linear-normalized-row/\"單解 -7/2\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=-7. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, 7/2], single-solution -7/2)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 7/2], single-solution -7/2)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0021","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d0add29199c734ce7c82e579f64e05e30bd750e8102608cfb2fd3e4a1bc54a2","family_id":"qf-density14-linear-normalized-row","seed":202610144021,"semantic_template":"qf-density14-linear-normalized-row/\"單解 -4/1\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=8. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, 4/1], single-solution -4/1)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 4/1], single-solution -4/1)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0022","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13dd962c6ad43015d801dd350be2070356d698670771041c8c3d7c4187fd6ef0","family_id":"qf-density14-linear-normalized-row","seed":202610144022,"semantic_template":"qf-density14-linear-normalized-row/\"單解 -9/2\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=-9. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, 9/2], single-solution -9/2)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 9/2], single-solution -9/2)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0023","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d77b11c2eb337c5a609258b66110413446b5d51dab7b669c1e855e06b5b71b7","family_id":"qf-density14-linear-normalized-row","seed":202610144023,"semantic_template":"qf-density14-linear-normalized-row/\"單解 -5/1\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=10. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, 5/1], single-solution -5/1)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 5/1], single-solution -5/1)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0024","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4de0efe61fac28fd6a205068ceb044a46a97945adf4382e4690cbb8da7e5ef8","family_id":"qf-density14-linear-normalized-row","seed":202610144024,"semantic_template":"qf-density14-linear-normalized-row/\"單解 -11/2\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=-11. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, 11/2], single-solution -11/2)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 11/2], single-solution -11/2)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0025","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81e1a141265a19edb246b0afb71592a65fd64aa715712acaafdbb186414babcd","family_id":"qf-density14-linear-normalized-row","seed":202610144025,"semantic_template":"qf-density14-linear-normalized-row/\"單解 -6/1\""},"fields":{"title":"Linear Eligibility, Normalized Row, and Solution Class","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=12. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(Yes, [1/1, 6/1], single-solution -6/1)","explanation":"Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 6/1], single-solution -6/1)."},"skills":["Linear Eligibility, Normalized Row, and Solution Class"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0026","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"607699b752bada33b7182feae6f41ceae3a618a470a32f76885d151e43d8a875","family_id":"qf-density14-linear-parameter-slice","seed":202610144026,"semantic_template":"qf-density14-linear-parameter-slice/[[\"0\",\"1\",\"-1\"],[\"全Q\",\"單解 0/1\",\"單解 0/1\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=0, b=0, ordered ts=[0, 1, -1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([0, 1, -1], [No, Yes, Yes], [all Q, single-solution 0/1, single-solution 0/1])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([0, 1, -1], [No, Yes, Yes], [all Q, single-solution 0/1, single-solution 0/1])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0027","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e6687712ba2973161f62f0cdce503e432255704e351b353d897934da00fdcac","family_id":"qf-density14-linear-parameter-slice","seed":202610144027,"semantic_template":"qf-density14-linear-parameter-slice/[[\"0\",\"1\",\"-1\"],[\"無解\",\"單解 -1/1\",\"單解 1/1\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=2, b=1, ordered ts=[2, 3, 1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([0, 1, -1], [No, Yes, Yes], [no-solution, single-solution -1/1, single-solution 1/1])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([0, 1, -1], [No, Yes, Yes], [no-solution, single-solution -1/1, single-solution 1/1])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0028","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73711bc7a5fb62b42baec79bdaa4a19ec3e5f4a7b1ca7ea4f3e835973c8feb94","family_id":"qf-density14-linear-parameter-slice","seed":202610144028,"semantic_template":"qf-density14-linear-parameter-slice/[[\"-10\",\"3\",\"0\"],[\"單解 -1/1\",\"單解 10/3\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=1, b=-10, ordered ts=[-9, 4, 1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([-10, 3, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 10/3, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-10, 3, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 10/3, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0029","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d32038cc721a13fed5489ed90093067182579f6cb868f99fbb429754a7384a7","family_id":"qf-density14-linear-parameter-slice","seed":202610144029,"semantic_template":"qf-density14-linear-parameter-slice/[[\"-9\",\"4\",\"0\"],[\"單解 -1/1\",\"單解 9/4\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=2, b=-9, ordered ts=[-7, 6, 2]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([-9, 4, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 9/4, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-9, 4, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 9/4, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0030","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81decfa7302a5b553da866af5834060de880ca69290606bf92a736ab75ec1db9","family_id":"qf-density14-linear-parameter-slice","seed":202610144030,"semantic_template":"qf-density14-linear-parameter-slice/[[\"-8\",\"5\",\"0\"],[\"單解 -1/1\",\"單解 8/5\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=-1, b=-8, ordered ts=[-9, 4, -1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([-8, 5, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 8/5, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-8, 5, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 8/5, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0031","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d716741d57ed5603d0a26b542d5b9b5bf7f472cb6e5bd0d7779f975da4110982","family_id":"qf-density14-linear-parameter-slice","seed":202610144031,"semantic_template":"qf-density14-linear-parameter-slice/[[\"-7\",\"6\",\"0\"],[\"單解 -1/1\",\"單解 7/6\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=0, b=-7, ordered ts=[-7, 6, 0]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([-7, 6, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 7/6, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-7, 6, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 7/6, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0032","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7f125fc28321944fe23df06c25eb4bea5b92dede5e70281652db293ad8ac2dc","family_id":"qf-density14-linear-parameter-slice","seed":202610144032,"semantic_template":"qf-density14-linear-parameter-slice/[[\"-6\",\"7\",\"0\"],[\"單解 -1/1\",\"單解 6/7\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=1, b=-6, ordered ts=[-5, 8, 1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([-6, 7, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 6/7, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-6, 7, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 6/7, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0033","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9e3e2bd6ab4c42b15414a1bc2711164e50682996146df7e72561f6f7a786560","family_id":"qf-density14-linear-parameter-slice","seed":202610144033,"semantic_template":"qf-density14-linear-parameter-slice/[[\"-5\",\"8\",\"0\"],[\"單解 -1/1\",\"單解 5/8\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=2, b=-5, ordered ts=[-3, 10, 2]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([-5, 8, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 5/8, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-5, 8, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 5/8, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0034","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31773f62e05dfbfca585d2b1df6598a543218cdca63e47743084b549ec530fdb","family_id":"qf-density14-linear-parameter-slice","seed":202610144034,"semantic_template":"qf-density14-linear-parameter-slice/[[\"-4\",\"9\",\"0\"],[\"單解 -1/1\",\"單解 4/9\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=-1, b=-4, ordered ts=[-5, 8, -1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([-4, 9, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 4/9, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-4, 9, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 4/9, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0035","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34c40c64d0d28f3a0632273db617ba71aa46a7bd2bc052eb307fd7beab3a147c","family_id":"qf-density14-linear-parameter-slice","seed":202610144035,"semantic_template":"qf-density14-linear-parameter-slice/[[\"-3\",\"10\",\"0\"],[\"單解 -1/1\",\"單解 3/10\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=0, b=-3, ordered ts=[-3, 10, 0]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([-3, 10, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 3/10, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-3, 10, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 3/10, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0036","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d74cb5e51ee8e731a518b93086b2f55e2177bf873ee5e2d3321017c550afb3b","family_id":"qf-density14-linear-parameter-slice","seed":202610144036,"semantic_template":"qf-density14-linear-parameter-slice/[[\"-2\",\"11\",\"0\"],[\"單解 -1/1\",\"單解 2/11\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=1, b=-2, ordered ts=[-1, 12, 1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([-2, 11, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 2/11, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-2, 11, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 2/11, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0037","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"608b199b3b644e426f5e239e36095a644445a22a9efc7ebf64bd323fca9e4e1c","family_id":"qf-density14-linear-parameter-slice","seed":202610144037,"semantic_template":"qf-density14-linear-parameter-slice/[[\"-1\",\"12\",\"0\"],[\"單解 -1/1\",\"單解 1/12\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=2, b=-1, ordered ts=[1, 14, 2]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([-1, 12, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 1/12, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-1, 12, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 1/12, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0038","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f79f08955342f801d6a3b4cb65d99c72ebbdd35927781d7801aa2e09ed26eff3","family_id":"qf-density14-linear-parameter-slice","seed":202610144038,"semantic_template":"qf-density14-linear-parameter-slice/[[\"0\",\"13\",\"0\"],[\"全Q\",\"單解 0/1\",\"全Q\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=-1, b=0, ordered ts=[-1, 12, -1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([0, 13, 0], [No, Yes, No], [all Q, single-solution 0/1, all Q])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([0, 13, 0], [No, Yes, No], [all Q, single-solution 0/1, all Q])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0039","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bfbb9f863220154ff02ad81af906a208bc016ac0c3954f815853671afb0dafd","family_id":"qf-density14-linear-parameter-slice","seed":202610144039,"semantic_template":"qf-density14-linear-parameter-slice/[[\"1\",\"14\",\"0\"],[\"單解 -1/1\",\"單解 -1/14\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=0, b=1, ordered ts=[1, 14, 0]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([1, 14, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -1/14, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([1, 14, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -1/14, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0040","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c3b8f8d7e66d5ab4a8f1e4367ef352d731a432b4f05914a15ba847348ed42ae","family_id":"qf-density14-linear-parameter-slice","seed":202610144040,"semantic_template":"qf-density14-linear-parameter-slice/[[\"2\",\"15\",\"0\"],[\"單解 -1/1\",\"單解 -2/15\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=1, b=2, ordered ts=[3, 16, 1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([2, 15, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -2/15, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([2, 15, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -2/15, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0041","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18967a96fe104b647fd525a46953d5dbe278a880a18151c4ecb31515edb91383","family_id":"qf-density14-linear-parameter-slice","seed":202610144041,"semantic_template":"qf-density14-linear-parameter-slice/[[\"3\",\"16\",\"0\"],[\"單解 -1/1\",\"單解 -3/16\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=2, b=3, ordered ts=[5, 18, 2]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([3, 16, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -3/16, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([3, 16, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -3/16, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0042","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"486c419f24475c0d0a6639ddc4ad7a0b1b0f87e9ce20d61f57e91341c97c019c","family_id":"qf-density14-linear-parameter-slice","seed":202610144042,"semantic_template":"qf-density14-linear-parameter-slice/[[\"4\",\"17\",\"0\"],[\"單解 -1/1\",\"單解 -4/17\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=-1, b=4, ordered ts=[3, 16, -1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([4, 17, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -4/17, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([4, 17, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -4/17, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0043","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4f61538ec15a4badb77d23bd71a27b7dc0c4e5790ea5c84df11aab43073b3fa","family_id":"qf-density14-linear-parameter-slice","seed":202610144043,"semantic_template":"qf-density14-linear-parameter-slice/[[\"5\",\"18\",\"0\"],[\"單解 -1/1\",\"單解 -5/18\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=0, b=5, ordered ts=[5, 18, 0]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([5, 18, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -5/18, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([5, 18, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -5/18, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0044","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3bbf68a8e7cbaa01c9edaee7eb41a80d783b1780c775f9d875726cdecdd7143","family_id":"qf-density14-linear-parameter-slice","seed":202610144044,"semantic_template":"qf-density14-linear-parameter-slice/[[\"6\",\"19\",\"0\"],[\"單解 -1/1\",\"單解 -6/19\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=1, b=6, ordered ts=[7, 20, 1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([6, 19, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -6/19, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([6, 19, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -6/19, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0045","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6279038880079c9180c5516f138f840b0d6e9f989a5344b2a04b28f3ca4ed79d","family_id":"qf-density14-linear-parameter-slice","seed":202610144045,"semantic_template":"qf-density14-linear-parameter-slice/[[\"7\",\"20\",\"0\"],[\"單解 -1/1\",\"單解 -7/20\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=2, b=7, ordered ts=[9, 22, 2]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([7, 20, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -7/20, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([7, 20, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -7/20, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0046","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"625534ede8cf8a9fb865ea7507d7a613d528b0f92bfcdc67605a43dcc2cdb630","family_id":"qf-density14-linear-parameter-slice","seed":202610144046,"semantic_template":"qf-density14-linear-parameter-slice/[[\"8\",\"21\",\"0\"],[\"單解 -1/1\",\"單解 -8/21\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=-1, b=8, ordered ts=[7, 20, -1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([8, 21, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -8/21, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([8, 21, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -8/21, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0047","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92feb752821d3f1dbdf7681409c5a58f3b74dabbabc037c0d2c99a5c10d011c7","family_id":"qf-density14-linear-parameter-slice","seed":202610144047,"semantic_template":"qf-density14-linear-parameter-slice/[[\"9\",\"22\",\"0\"],[\"單解 -1/1\",\"單解 -9/22\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=0, b=9, ordered ts=[9, 22, 0]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([9, 22, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -9/22, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([9, 22, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -9/22, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0048","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e00fecc465772f70137ad1473799d5290835d7e84601f6fff439b5e6e04560c","family_id":"qf-density14-linear-parameter-slice","seed":202610144048,"semantic_template":"qf-density14-linear-parameter-slice/[[\"10\",\"23\",\"0\"],[\"單解 -1/1\",\"單解 -10/23\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=1, b=10, ordered ts=[11, 24, 1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([10, 23, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -10/23, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([10, 23, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -10/23, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0049","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8eeef04b51200b099702cb6f7513f6cd677413c1a69ae8cdfd458fdf98754749","family_id":"qf-density14-linear-parameter-slice","seed":202610144049,"semantic_template":"qf-density14-linear-parameter-slice/[[\"11\",\"24\",\"0\"],[\"單解 -1/1\",\"單解 -11/24\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=2, b=11, ordered ts=[13, 26, 2]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([11, 24, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -11/24, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([11, 24, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -11/24, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0050","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c206e701beaf067c4f052db4473f98122a7e3a9d7c5c2ec198e3556b9e6591e6","family_id":"qf-density14-linear-parameter-slice","seed":202610144050,"semantic_template":"qf-density14-linear-parameter-slice/[[\"12\",\"25\",\"0\"],[\"單解 -1/1\",\"單解 -12/25\",\"無解\"]]"},"fields":{"title":"Parameter-Dependent Linear and Constant Cases","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=-1, b=12, ordered ts=[11, 24, -1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.","answer":{"kind":"expression","value":"([12, 25, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -12/25, no-solution])","explanation":"Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([12, 25, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -12/25, no-solution])."},"skills":["Parameter-Dependent Linear and Constant Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0051","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46aca1d230e5048460a8c8c08fbbce71ac26b95394836a7309b476ae5aa78b6e","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144051,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"0/1\",[\"0/1\",\"0/1\",\"0/1\"],[\"0/1\",\"0/1\",\"0/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=1≠0, b=0, ordered offsets=[0/1, 0/2, 0/-1]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, [0/1, 0/1, 0/1], [0/1, 0/1, 0/1], [Yes, Yes, Yes])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (0/1, [0/1, 0/1, 0/1], [0/1, 0/1, 0/1], [Yes, Yes, Yes])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0052","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d62d02343b84af4fd452c43f97be870370d017a867faa7a71aa1d0169a3a0c5d","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144052,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"1/2\",[\"1/2\",\"1/1\",\"0/1\"],[\"0/1\",\"-1/1\",\"1/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-2≠0, b=1, ordered offsets=[0/1, 1/2, -1/2]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, [1/2, 1/1, 0/1], [0/1, -1/1, 1/1], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (1/2, [1/2, 1/1, 0/1], [0/1, -1/1, 1/1], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0053","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"520a9eb284ebc0ab2b87f72b984fe2e3622f63bb2bbc810f51fc6df0f1b5cfc1","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144053,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"10/3\",[\"10/3\",\"29/6\",\"8/3\"],[\"0/1\",\"9/2\",\"-2/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=3≠0, b=-10, ordered offsets=[0/1, 3/2, -2/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(10/3, [10/3, 29/6, 8/3], [0/1, 9/2, -2/1], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (10/3, [10/3, 29/6, 8/3], [0/1, 9/2, -2/1], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0054","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72b703c4b609af0097a2f6fae63d9d386c61e158e9fd904692b348a0917a4706","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144054,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"-9/4\",[\"-9/4\",\"-1/4\",\"-13/4\"],[\"0/1\",\"-8/1\",\"4/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-4≠0, b=-9, ordered offsets=[0/1, 4/2, -3/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-9/4, [-9/4, -1/4, -13/4], [0/1, -8/1, 4/1], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-9/4, [-9/4, -1/4, -13/4], [0/1, -8/1, 4/1], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0055","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81a263709160fcac9abc8721982e895ccd7476097520839e6920dc866abf1638","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144055,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"8/5\",[\"8/5\",\"41/10\",\"4/15\"],[\"0/1\",\"25/2\",\"-20/3\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=5≠0, b=-8, ordered offsets=[0/1, 5/2, -4/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(8/5, [8/5, 41/10, 4/15], [0/1, 25/2, -20/3], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (8/5, [8/5, 41/10, 4/15], [0/1, 25/2, -20/3], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0056","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35ddd7a4db2bcec1530356f01eabbdd3be8a1324f452445d25972990a127aed2","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144056,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"-7/6\",[\"-7/6\",\"11/6\",\"-17/6\"],[\"0/1\",\"-18/1\",\"10/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-6≠0, b=-7, ordered offsets=[0/1, 6/2, -5/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-7/6, [-7/6, 11/6, -17/6], [0/1, -18/1, 10/1], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-7/6, [-7/6, 11/6, -17/6], [0/1, -18/1, 10/1], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0057","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af39a6bf377703e7c34bbb4249045db11f42c269eaf765547761543ef8f809d1","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144057,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"6/7\",[\"6/7\",\"61/14\",\"-8/7\"],[\"0/1\",\"49/2\",\"-14/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=7≠0, b=-6, ordered offsets=[0/1, 7/2, -6/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(6/7, [6/7, 61/14, -8/7], [0/1, 49/2, -14/1], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (6/7, [6/7, 61/14, -8/7], [0/1, 49/2, -14/1], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0058","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"686c588188fa336146a8b2559ac24fa5752eae6e7970563f500d498f791ff416","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144058,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"-5/8\",[\"-5/8\",\"27/8\",\"-71/24\"],[\"0/1\",\"-32/1\",\"56/3\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-8≠0, b=-5, ordered offsets=[0/1, 8/2, -7/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-5/8, [-5/8, 27/8, -71/24], [0/1, -32/1, 56/3], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-5/8, [-5/8, 27/8, -71/24], [0/1, -32/1, 56/3], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0059","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b38e287e87317ca22971db1a552da8ddaae1ee3d28b779efa4c2fc4651e55e89","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144059,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"4/9\",[\"4/9\",\"89/18\",\"-20/9\"],[\"0/1\",\"81/2\",\"-24/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=9≠0, b=-4, ordered offsets=[0/1, 9/2, -8/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(4/9, [4/9, 89/18, -20/9], [0/1, 81/2, -24/1], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (4/9, [4/9, 89/18, -20/9], [0/1, 81/2, -24/1], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0060","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af4f0015c998a3552d8308d8abefb5e593176c5c1ff37ea2c043334c850c16b8","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144060,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"-3/10\",[\"-3/10\",\"47/10\",\"-33/10\"],[\"0/1\",\"-50/1\",\"30/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-10≠0, b=-3, ordered offsets=[0/1, 10/2, -9/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-3/10, [-3/10, 47/10, -33/10], [0/1, -50/1, 30/1], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-3/10, [-3/10, 47/10, -33/10], [0/1, -50/1, 30/1], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0061","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b3abd2b12eb50fcaff44602a8f92e893d1891e5cc54046161318df2b8059882","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144061,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"2/11\",[\"2/11\",\"125/22\",\"-104/33\"],[\"0/1\",\"121/2\",\"-110/3\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=11≠0, b=-2, ordered offsets=[0/1, 11/2, -10/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(2/11, [2/11, 125/22, -104/33], [0/1, 121/2, -110/3], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (2/11, [2/11, 125/22, -104/33], [0/1, 121/2, -110/3], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0062","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4536560795607490101bfa1fceef8ed4c638152a69544be81ac5a798cc357e0c","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144062,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"-1/12\",[\"-1/12\",\"71/12\",\"-15/4\"],[\"0/1\",\"-72/1\",\"44/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-12≠0, b=-1, ordered offsets=[0/1, 12/2, -11/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/12, [-1/12, 71/12, -15/4], [0/1, -72/1, 44/1], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-1/12, [-1/12, 71/12, -15/4], [0/1, -72/1, 44/1], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0063","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4c75b016faad02d8dcfe3b807aac0ec32889686d62c81315a0a875a92e80390","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144063,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"0/1\",[\"0/1\",\"13/2\",\"-4/1\"],[\"0/1\",\"169/2\",\"-52/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=13≠0, b=0, ordered offsets=[0/1, 13/2, -12/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, [0/1, 13/2, -4/1], [0/1, 169/2, -52/1], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (0/1, [0/1, 13/2, -4/1], [0/1, 169/2, -52/1], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0064","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"966167941a4342e3d03aa73cd8e0bd0788ec4ea52b94c2ecb245dc8c38f4b10b","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144064,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"1/14\",[\"1/14\",\"99/14\",\"-179/42\"],[\"0/1\",\"-98/1\",\"182/3\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-14≠0, b=1, ordered offsets=[0/1, 14/2, -13/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/14, [1/14, 99/14, -179/42], [0/1, -98/1, 182/3], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (1/14, [1/14, 99/14, -179/42], [0/1, -98/1, 182/3], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0065","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22cddaa975570aca2a314d3959c329217bac940fcfdf845d592b303ccd6c930a","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144065,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"-2/15\",[\"-2/15\",\"221/30\",\"-24/5\"],[\"0/1\",\"225/2\",\"-70/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=15≠0, b=2, ordered offsets=[0/1, 15/2, -14/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-2/15, [-2/15, 221/30, -24/5], [0/1, 225/2, -70/1], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-2/15, [-2/15, 221/30, -24/5], [0/1, 225/2, -70/1], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0066","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf87745888f80f31ff4b8f97906d1188c83c396edbf3c89bcd1d1b11dddd1b51","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144066,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"3/16\",[\"3/16\",\"131/16\",\"-77/16\"],[\"0/1\",\"-128/1\",\"80/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-16≠0, b=3, ordered offsets=[0/1, 16/2, -15/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(3/16, [3/16, 131/16, -77/16], [0/1, -128/1, 80/1], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (3/16, [3/16, 131/16, -77/16], [0/1, -128/1, 80/1], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0067","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf2fdacb17e4fe89e0e851a0bbda3e64d534f9aa9b85618ad2c4892f8a6e338a","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144067,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"-4/17\",[\"-4/17\",\"281/34\",\"-284/51\"],[\"0/1\",\"289/2\",\"-272/3\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=17≠0, b=4, ordered offsets=[0/1, 17/2, -16/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-4/17, [-4/17, 281/34, -284/51], [0/1, 289/2, -272/3], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-4/17, [-4/17, 281/34, -284/51], [0/1, 289/2, -272/3], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0068","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8def8ec22ee2c9f0b1581c29ee299a5b48b75beee6756e7a7056cf43d1a613df","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144068,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"5/18\",[\"5/18\",\"167/18\",\"-97/18\"],[\"0/1\",\"-162/1\",\"102/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-18≠0, b=5, ordered offsets=[0/1, 18/2, -17/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(5/18, [5/18, 167/18, -97/18], [0/1, -162/1, 102/1], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (5/18, [5/18, 167/18, -97/18], [0/1, -162/1, 102/1], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0069","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"369dc0911f16218e3107579632acf4687972ab2f4b32c3fcc3744c52338ebb7a","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144069,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"-6/19\",[\"-6/19\",\"349/38\",\"-120/19\"],[\"0/1\",\"361/2\",\"-114/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=19≠0, b=6, ordered offsets=[0/1, 19/2, -18/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-6/19, [-6/19, 349/38, -120/19], [0/1, 361/2, -114/1], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-6/19, [-6/19, 349/38, -120/19], [0/1, 361/2, -114/1], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0070","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4292966be01a464d4701e4c1b8f324f159f9d127bff492fe546daae0e1b5ed72","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144070,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"7/20\",[\"7/20\",\"207/20\",\"-359/60\"],[\"0/1\",\"-200/1\",\"380/3\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-20≠0, b=7, ordered offsets=[0/1, 20/2, -19/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(7/20, [7/20, 207/20, -359/60], [0/1, -200/1, 380/3], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (7/20, [7/20, 207/20, -359/60], [0/1, -200/1, 380/3], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0071","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4f9d7504f514b9f64290e9eb1f079b23b4e2ddd72f84a9bb5163b81d5b6b6af","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144071,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"-8/21\",[\"-8/21\",\"425/42\",\"-148/21\"],[\"0/1\",\"441/2\",\"-140/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=21≠0, b=8, ordered offsets=[0/1, 21/2, -20/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-8/21, [-8/21, 425/42, -148/21], [0/1, 441/2, -140/1], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-8/21, [-8/21, 425/42, -148/21], [0/1, 441/2, -140/1], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0072","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a5a9894dc21a7d88fc42a6e2e257e52b054dc51c5514026430790b5b36e3b7d","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144072,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"9/22\",[\"9/22\",\"251/22\",\"-145/22\"],[\"0/1\",\"-242/1\",\"154/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-22≠0, b=9, ordered offsets=[0/1, 22/2, -21/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(9/22, [9/22, 251/22, -145/22], [0/1, -242/1, 154/1], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (9/22, [9/22, 251/22, -145/22], [0/1, -242/1, 154/1], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0073","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48ed551819fd52197fb47886704f15dc581f9d29593b619f00b8c3a1aef888a0","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144073,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"-10/23\",[\"-10/23\",\"509/46\",\"-536/69\"],[\"0/1\",\"529/2\",\"-506/3\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=23≠0, b=10, ordered offsets=[0/1, 23/2, -22/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-10/23, [-10/23, 509/46, -536/69], [0/1, 529/2, -506/3], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-10/23, [-10/23, 509/46, -536/69], [0/1, 529/2, -506/3], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0074","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9376250210c51e956a6b0ec22ebf4d3d908e21aa3afad9cd66a8ff48e880ed8f","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144074,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"11/24\",[\"11/24\",\"299/24\",\"-173/24\"],[\"0/1\",\"-288/1\",\"184/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-24≠0, b=11, ordered offsets=[0/1, 24/2, -23/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(11/24, [11/24, 299/24, -173/24], [0/1, -288/1, 184/1], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (11/24, [11/24, 299/24, -173/24], [0/1, -288/1, 184/1], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0075","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34a3ff81d10baf3e195997410fc6d5e6de4f907e0f0b6ee6f62a1d60e7d4fe3f","family_id":"qf-density14-linear-root-substitution-grid","seed":202610144075,"semantic_template":"qf-density14-linear-root-substitution-grid/[\"-12/25\",[\"-12/25\",\"601/50\",\"-212/25\"],[\"0/1\",\"625/2\",\"-200/1\"]]"},"fields":{"title":"Exact Root Offsets and Original Substitution","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=25≠0, b=12, ordered offsets=[0/1, 25/2, -24/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-12/25, [-12/25, 601/50, -212/25], [0/1, 625/2, -200/1], [Yes, No, No])","explanation":"Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-12/25, [-12/25, 601/50, -212/25], [0/1, 625/2, -200/1], [Yes, No, No])."},"skills":["Exact Root Offsets and Original Substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0076","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e14015c35c620ae99f0aa2ba0b84f75008dd13003f64985f1da72a9d7297c063","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144076,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"0/1\",\"0/1\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[1, 0], [2, 0]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, 0/1, Yes, 0/1)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (0/1, 0/1, Yes, 0/1)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0077","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33429ff55b0e44c022d29a62f861221ba08a8f5cc716d9ec91a1b7c44e475a2e","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144077,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"1/1\",\"1/1\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[1, -1], [-2, 2]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/1, 1/1, Yes, 1/1)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (1/1, 1/1, Yes, 1/1)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0078","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"046a17776dd8c8286eb519b74d0e3b83dc22cc017aae04ed66f3ae8ed8ec2565","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144078,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"1/1\",\"-1/3\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -2], [-3, -1]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/1, -1/3, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (1/1, -1/3, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0079","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98b1e61a00d1b9408f9ce32d335b4fc04ccb068f78323abcd11bec87e9889362","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144079,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"3/2\",\"-2/3\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -3], [-3, -2]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(3/2, -2/3, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (3/2, -2/3, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0080","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3deff00dec686721a0c17eae21bf4525d6492838f2e1881b925bebdecd8ed1b","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144080,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"2/1\",\"-1/1\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -4], [-3, -3]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(2/1, -1/1, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (2/1, -1/1, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0081","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fec38904023772d933e48d5e258c5e929b7b2ba7ef8c98b3b3ee836cd3fad857","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144081,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"5/2\",\"-4/3\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -5], [-3, -4]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(5/2, -4/3, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (5/2, -4/3, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0082","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee1a860b042e4c7edb38f3664419790f73b0969495060246be9afe8cbc02e0bf","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144082,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"3/1\",\"-5/3\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -6], [-3, -5]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(3/1, -5/3, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (3/1, -5/3, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0083","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0a37769aa5f33f59c2bdfe9cfb82799f0fd9e2f7ceef2ba5d56957a610a6aa0","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144083,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"7/2\",\"-2/1\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -7], [-3, -6]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(7/2, -2/1, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (7/2, -2/1, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0084","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71d4f1e90a2f8ce536465fd1e0d7dae9e3a1027ed62dad65299b000175f853ff","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144084,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"4/1\",\"-7/3\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -8], [-3, -7]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(4/1, -7/3, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (4/1, -7/3, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0085","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b9e35c5cffc9eb8b2adfeccab03d2546e1b456c9635c899bd106d63c082ed3a","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144085,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"9/2\",\"-8/3\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -9], [-3, -8]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(9/2, -8/3, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (9/2, -8/3, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0086","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfc5575d78aca0c0978ab8ae134ca079142fc6ebddf0523a6df7c83c49862db9","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144086,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"5/1\",\"-3/1\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -10], [-3, -9]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(5/1, -3/1, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (5/1, -3/1, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0087","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1126940767dfa081cc39a3580cc7b4142441cd49072d7bcbc5e63890b83100ab","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144087,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"11/2\",\"-10/3\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -11], [-3, -10]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(11/2, -10/3, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (11/2, -10/3, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0088","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67ff4466226d8883775ad27086659363ebdbdbb038ba33fe99a104bf62262cd5","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144088,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"6/1\",\"-11/3\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -12], [-3, -11]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(6/1, -11/3, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (6/1, -11/3, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0089","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d30e03e699b9c286906e61ee852bf09ba2985e89ed3eb139dab6597503bfef2e","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144089,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"13/2\",\"-4/1\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -13], [-3, -12]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(13/2, -4/1, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (13/2, -4/1, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0090","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1da576313f91701a2e02850ccdd2cdec13dd990c0073220052a87d3977dd35d7","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144090,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"7/1\",\"-13/3\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -14], [-3, -13]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(7/1, -13/3, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (7/1, -13/3, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0091","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87f82c625d2f5b49cc4fef45fde2a46cb20b9ee1632a92e4cac76474d646a03e","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144091,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"15/2\",\"-14/3\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -15], [-3, -14]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(15/2, -14/3, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (15/2, -14/3, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0092","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4d747f80286e5443e3d943bf1aac4a79ecfea820f89e1bf493c30c1a7993d60","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144092,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"8/1\",\"-5/1\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -16], [-3, -15]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(8/1, -5/1, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (8/1, -5/1, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0093","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1be919e2fdf4ac606d0c9d1a8de65f93e777c34893780fa5505157e8a3c135e0","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144093,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"17/2\",\"-16/3\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -17], [-3, -16]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(17/2, -16/3, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (17/2, -16/3, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0094","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f26f4b1d349c42be7ba69157ebae2deae7e702f1f3cc2d3c093e7ae696d61ca6","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144094,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"9/1\",\"-17/3\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -18], [-3, -17]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(9/1, -17/3, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (9/1, -17/3, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0095","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f49ced02eb38d921809109d0f6b9308bbf71d0cc6baaab36299f5adaa3ed0ff","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144095,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"19/2\",\"-6/1\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -19], [-3, -18]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(19/2, -6/1, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (19/2, -6/1, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0096","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60c01f28215d7f14b3d5369414be90555fbfbf8fed4996d4bd5bde5ff2edaf56","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144096,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"10/1\",\"-19/3\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -20], [-3, -19]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(10/1, -19/3, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (10/1, -19/3, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0097","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fdb3fe880fbe8051a1c811b63de8b6dca81eab38ebf7d33ecacede296a06113","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144097,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"21/2\",\"-20/3\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -21], [-3, -20]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(21/2, -20/3, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (21/2, -20/3, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0098","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f3f729c54fd53f1f58fcf5062c52d21a7ea5bca75cba0a8916f400b14540ec1","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144098,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"11/1\",\"-7/1\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -22], [-3, -21]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(11/1, -7/1, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (11/1, -7/1, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0099","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2150cb2fb8a0aed66563dd0e8a49374c664523e58d62d9ae4b2f7b2e8aa3c1ee","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144099,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"23/2\",\"-22/3\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -23], [-3, -22]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(23/2, -22/3, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (23/2, -22/3, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r04-linear-equation-0100","batch_id":"batch-20261010-advance-20261014-r04-linear-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30ccdfd36f42b2e7e6dffefef112c5d5cf919d744cc19f50fa9571ea108183bc","family_id":"qf-density14-linear-two-row-coincidence","seed":202610144100,"semantic_template":"qf-density14-linear-two-row-coincidence/[\"12/1\",\"-23/3\"]"},"fields":{"title":"Two Affine Roots and a Common Root","stem":"Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -24], [-3, -23]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(12/1, -23/3, No, none)","explanation":"Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (12/1, -23/3, No, none)."},"skills":["Two Affine Roots and a Common Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}