{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r06-system-linear-equations-0001","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf9e4a76c211e43c635d34966aa8e4adaa2ccf106c4523dd2323b3dce5d0f3a5","family_id":"qf-linear-system-determinant","seed":202610046001,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"0\",\"0\"],[\"0\",\"0\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (0)x+(0)y=0 and (0)x+(0)y=0 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"0","explanation":"The coefficient determinant is D=0, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0002","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abe7922192291f1e9f833f18c7903ad527e71c1d43bb8f280281b6432f0cd376","family_id":"qf-linear-system-determinant","seed":202610046002,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-11\",\"1\"],[\"1\",\"-1\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (-11)x+(1)y=-1 and (1)x+(-1)y=-2 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"10","explanation":"The coefficient determinant is D=10, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0003","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9908280faf19ff5d5427940ebd0948b10925ef377feddfc3713b140d59680ae","family_id":"qf-linear-system-determinant","seed":202610046003,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-10\",\"1\"],[\"1\",\"0\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (-10)x+(1)y=0 and (1)x+(0)y=-1 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"-1","explanation":"The coefficient determinant is D=-1, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0004","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a31a9378d65e83cc0c01e12409e1010984a1448457390a05f4b696fefa6e449","family_id":"qf-linear-system-determinant","seed":202610046004,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-9\",\"1\"],[\"1\",\"1\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (-9)x+(1)y=1 and (1)x+(1)y=0 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"-10","explanation":"The coefficient determinant is D=-10, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0005","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc1f91df259751113c3aff40e6d47509940cefe3660ab98ca96dc70f624ca087","family_id":"qf-linear-system-determinant","seed":202610046005,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-8\",\"1\"],[\"1\",\"2\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (-8)x+(1)y=2 and (1)x+(2)y=1 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"-17","explanation":"The coefficient determinant is D=-17, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0006","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fa21d2913748738f39f885ce3b9e848c192cc0de5382652c25b165578ca006b","family_id":"qf-linear-system-determinant","seed":202610046006,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-7\",\"1\"],[\"1\",\"-2\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (-7)x+(1)y=-2 and (1)x+(-2)y=2 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"13","explanation":"The coefficient determinant is D=13, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0007","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11b06b4c3d8a895e4e0946331209c12433e366f7dc25eb7f07befa5a9dd50ac7","family_id":"qf-linear-system-determinant","seed":202610046007,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-6\",\"1\"],[\"1\",\"-1\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (-6)x+(1)y=-1 and (1)x+(-1)y=3 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"5","explanation":"The coefficient determinant is D=5, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0008","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"791d3b45c25b49c56d733824bda92cfadfe218435978a544b68a7d739f1c7023","family_id":"qf-linear-system-determinant","seed":202610046008,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-5\",\"1\"],[\"1\",\"0\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (-5)x+(1)y=0 and (1)x+(0)y=-3 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"-1","explanation":"The coefficient determinant is D=-1, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0009","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be317b056ad98ec7dd6ef179fdc9330480b5aeb1ab197af418985a4c20bc6594","family_id":"qf-linear-system-determinant","seed":202610046009,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-4\",\"1\"],[\"1\",\"1\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (-4)x+(1)y=1 and (1)x+(1)y=-2 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"-5","explanation":"The coefficient determinant is D=-5, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0010","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2938a6ce0cdce3987c0f4882229ce86274de8d4ca497658462c46bcb2132a30","family_id":"qf-linear-system-determinant","seed":202610046010,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-3\",\"1\"],[\"1\",\"2\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (-3)x+(1)y=2 and (1)x+(2)y=-1 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"-7","explanation":"The coefficient determinant is D=-7, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0011","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c94d83f8d814655a6c7df7eb9857738eb3ceebf3d1fe41fc5e15620607a5b9d4","family_id":"qf-linear-system-determinant","seed":202610046011,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-2\",\"1\"],[\"1\",\"-2\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (-2)x+(1)y=-2 and (1)x+(-2)y=0 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"3","explanation":"The coefficient determinant is D=3, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0012","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bacd0856a7e9f771be8cbe850007919a449153e15ed01cc9672e4d12886d1498","family_id":"qf-linear-system-determinant","seed":202610046012,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"-1\",\"1\"],[\"1\",\"-1\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (-1)x+(1)y=-1 and (1)x+(-1)y=1 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"0","explanation":"The coefficient determinant is D=0, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0013","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8280ec9d0d48477385035cd9639cef24e64edb1d439b2562a74b1e791a54ea95","family_id":"qf-linear-system-determinant","seed":202610046013,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"0\",\"1\"],[\"1\",\"0\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (0)x+(1)y=0 and (1)x+(0)y=2 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"-1","explanation":"The coefficient determinant is D=-1, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0014","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d75bfc03ee5f24f1b579a6646ddd4756cfb2977dca8fc27c23caef25c394cf8","family_id":"qf-linear-system-determinant","seed":202610046014,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"1\",\"1\"],[\"1\",\"1\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (1)x+(1)y=1 and (1)x+(1)y=3 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"0","explanation":"The coefficient determinant is D=0, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0015","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f190f6f701d05a6955ca50e9a10b93e60a19c49c82d63ce60397abe4a70ba85","family_id":"qf-linear-system-determinant","seed":202610046015,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"2\",\"1\"],[\"1\",\"2\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (2)x+(1)y=2 and (1)x+(2)y=-3 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"3","explanation":"The coefficient determinant is D=3, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0016","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fab0e4fa404e9f7d59fa1af9ab32e0626b14204880eb981cd50351879b1149a","family_id":"qf-linear-system-determinant","seed":202610046016,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"3\",\"1\"],[\"1\",\"-2\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (3)x+(1)y=-2 and (1)x+(-2)y=-2 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"-7","explanation":"The coefficient determinant is D=-7, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0017","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f91a9e01e1137b88bdaab86658636497088e0578cedf442282fc16edd67549b","family_id":"qf-linear-system-determinant","seed":202610046017,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"4\",\"1\"],[\"1\",\"-1\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (4)x+(1)y=-1 and (1)x+(-1)y=-1 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"-5","explanation":"The coefficient determinant is D=-5, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0018","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e987df71e9ba4e5895ed2f5dec8d144d6707b8155cc262d3673c4d0c4a6d67d6","family_id":"qf-linear-system-determinant","seed":202610046018,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"5\",\"1\"],[\"1\",\"0\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (5)x+(1)y=0 and (1)x+(0)y=0 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"-1","explanation":"The coefficient determinant is D=-1, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0019","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c31797f4b40da978b22012737e0afca7f4bd417f1334e9fcfd315e68905187e8","family_id":"qf-linear-system-determinant","seed":202610046019,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"6\",\"1\"],[\"1\",\"1\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (6)x+(1)y=1 and (1)x+(1)y=1 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"5","explanation":"The coefficient determinant is D=5, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0020","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acb1f39251f082ac34aad7a660ea17974adf20a790188f6d09b74d5e444f5b7c","family_id":"qf-linear-system-determinant","seed":202610046020,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"7\",\"1\"],[\"1\",\"2\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (7)x+(1)y=2 and (1)x+(2)y=2 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"13","explanation":"The coefficient determinant is D=13, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0021","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8deb09c6920fd84b942300a9d4b001742868d1f24234cfd52f4592fd963e7d8","family_id":"qf-linear-system-determinant","seed":202610046021,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"8\",\"1\"],[\"1\",\"-2\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (8)x+(1)y=-2 and (1)x+(-2)y=3 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"-17","explanation":"The coefficient determinant is D=-17, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0022","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e62a7916510b0b4b0135a8836cccb2dda65a92a9fa29ae7802ddbf1d1935b6f6","family_id":"qf-linear-system-determinant","seed":202610046022,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"9\",\"1\"],[\"1\",\"-1\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (9)x+(1)y=-1 and (1)x+(-1)y=-3 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"-10","explanation":"The coefficient determinant is D=-10, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0023","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ed2bbbd6524e383d78298a6ec7c89c95a925665f238d19a36e0eb962b243f17","family_id":"qf-linear-system-determinant","seed":202610046023,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"10\",\"1\"],[\"1\",\"0\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (10)x+(1)y=0 and (1)x+(0)y=-2 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"-1","explanation":"The coefficient determinant is D=-1, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0024","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b26db05abf756ab419a3b04b5d548a9fa420a368a58986581019c8f781ce49c","family_id":"qf-linear-system-determinant","seed":202610046024,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"11\",\"1\"],[\"1\",\"1\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (11)x+(1)y=1 and (1)x+(1)y=-1 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"10","explanation":"The coefficient determinant is D=10, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0025","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0430a574c1deaa9b5bf0f068a681ccc62166e984165a87da91d538c0375ea16","family_id":"qf-linear-system-determinant","seed":202610046025,"semantic_template":"qf-linear-system-determinant/{\"domain\":\"reals\",\"operation\":\"determinant\",\"rows\":[[\"12\",\"1\"],[\"1\",\"2\"]]}"},"fields":{"title":"Determinant of a Two-Variable Linear System","stem":"Over the reals, the constraints (12)x+(1)y=2 and (1)x+(2)y=0 must both hold. Find the coefficient determinant D=ae−bd.","answer":{"kind":"numeric","value":"23","explanation":"The coefficient determinant is D=23, the main-diagonal product minus the other diagonal product. Right-side constants are not included."},"skills":["Determinant of a Two-Variable Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0026","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3523c460716b9b2fb32088e37251f68eff1b2cc52cc60ac8831cc9f8af2af4e","family_id":"qf-linear-system-solution-class","seed":202610046026,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"0\",\"0\",\"0\"],[\"0\",\"0\",\"0\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (0)x+(0)y=0 and (0)x+(0)y=0 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"infinitely many solutions","explanation":"D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0027","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48b2230983a78c325a8aefd4e05159c672c83e5a02d92d49594833fb02d68f2e","family_id":"qf-linear-system-solution-class","seed":202610046027,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"0\",\"0\",\"0\"],[\"0\",\"0\",\"1\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (0)x+(0)y=1 and (0)x+(0)y=0 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"no solution","explanation":"D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0028","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9543f9cb343742fd92180c871dcd01febd730e7fd4322fa3c9340173e3ef7665","family_id":"qf-linear-system-solution-class","seed":202610046028,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"0\",\"0\",\"0\"],[\"1\",\"0\",\"2\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (1)x+(0)y=2 and (0)x+(0)y=0 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"infinitely many solutions","explanation":"D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0029","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04ee9249d96e5c447540715f396c6e4570218cb604afbfe728d3a3ff05d7fa1a","family_id":"qf-linear-system-solution-class","seed":202610046029,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"2\",\"3\"],[\"1\",\"2\",\"3\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (1)x+(2)y=3 and (2)x+(4)y=6 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"infinitely many solutions","explanation":"D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0030","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f9399b4d93290070b9ae00e66ee1f79d6e00695171b0e24be0e0688e749c9de","family_id":"qf-linear-system-solution-class","seed":202610046030,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"2\",\"3\"],[\"2\",\"4\",\"7\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (1)x+(2)y=3 and (2)x+(4)y=7 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"no solution","explanation":"D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0031","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"706caf6e01e4bcbadea9609951a45815c9c04539c867fc691d5284cdd9d02803","family_id":"qf-linear-system-solution-class","seed":202610046031,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"0\",\"1\",\"2\"],[\"0\",\"2\",\"5\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (0)x+(1)y=2 and (0)x+(2)y=5 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"no solution","explanation":"D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0032","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81b07d926d9f7e83b2ade618326c15c2566a0d75a40b3e467be3aa2df4cad7b0","family_id":"qf-linear-system-solution-class","seed":202610046032,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"0\",\"1\",\"0\"],[\"1\",\"0\",\"0\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (1)x+(0)y=0 and (0)x+(-1)y=0 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"unique solution","explanation":"D=-1 and the classification is unique solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0033","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92360836947486ddf31caa02c7f94b6ede35103eb491c1c82644203459e9e544","family_id":"qf-linear-system-solution-class","seed":202610046033,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"0\",\"0\",\"1\"],[\"1\",\"2\",\"0\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (1)x+(2)y=0 and (0)x+(0)y=4 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"no solution","explanation":"D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0034","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed0ffe134fb76b51284c4fb372e3be6cd35bd476f56b8f377a1a0dce49a8b212","family_id":"qf-linear-system-solution-class","seed":202610046034,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"-1\",\"-2\"],[\"2\",\"-1\",\"-1\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (-2)x+(1)y=1 and (1)x+(-1)y=-2 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"unique solution","explanation":"D=1 and the classification is unique solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0035","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5f7ae5f5b01ddced995fc16876fc41477275998ba645a9a8e2223ff58e671ea","family_id":"qf-linear-system-solution-class","seed":202610046035,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"3\",\"5\"],[\"1\",\"3\",\"5\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (1)x+(3)y=5 and (2)x+(6)y=10 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"infinitely many solutions","explanation":"D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0036","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97d4b9897d88b2a359f5ed09b97350a1a695b3e55e3a184e57d9995909cb7862","family_id":"qf-linear-system-solution-class","seed":202610046036,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"4\",\"6\"],[\"2\",\"8\",\"13\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (1)x+(4)y=6 and (2)x+(8)y=13 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"no solution","explanation":"D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0037","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23587377d6c9a4a6d42996b40a8dff9355a5bdc0453b8d8bd7fc64eba9c9fbf5","family_id":"qf-linear-system-solution-class","seed":202610046037,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"-1\",\"1\"],[\"1\",\"1\",\"-1\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (1)x+(1)y=-1 and (1)x+(-1)y=1 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"unique solution","explanation":"D=-2 and the classification is unique solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0038","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cb72a25f1c01e2e8139c22031ddb9ced74a842db5fd836b936eab48584931c6","family_id":"qf-linear-system-solution-class","seed":202610046038,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"6\",\"8\"],[\"1\",\"6\",\"8\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (1)x+(6)y=8 and (2)x+(12)y=16 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"infinitely many solutions","explanation":"D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0039","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af66c465a63daa6efe0e81994feeaf8abc7701e19eaedd9158f4456e7e823c11","family_id":"qf-linear-system-solution-class","seed":202610046039,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"7\",\"9\"],[\"2\",\"14\",\"19\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (1)x+(7)y=9 and (2)x+(14)y=19 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"no solution","explanation":"D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0040","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"975ae30c7e353a26e4a50782f909f6cd7c3644c3587af02d68f1297c81b22553","family_id":"qf-linear-system-solution-class","seed":202610046040,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"-1\",\"-3\"],[\"4\",\"1\",\"2\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (4)x+(1)y=2 and (1)x+(-1)y=-3 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"unique solution","explanation":"D=-5 and the classification is unique solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0041","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ca1f42380dd3a99990e95ccf8741b93e3f861b9a5be831ad6ab76201bc97006","family_id":"qf-linear-system-solution-class","seed":202610046041,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"9\",\"11\"],[\"1\",\"9\",\"11\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (1)x+(9)y=11 and (2)x+(18)y=22 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"infinitely many solutions","explanation":"D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0042","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29f283283592e334d77abf02ac122ddf9ab71e651d00c5380929e5a155552532","family_id":"qf-linear-system-solution-class","seed":202610046042,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"10\",\"12\"],[\"2\",\"20\",\"25\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (1)x+(10)y=12 and (2)x+(20)y=25 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"no solution","explanation":"D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0043","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b1e7a876e69c4fe2697a3da29f8bf50c58a475f86a681b05334c9b3b214b766","family_id":"qf-linear-system-solution-class","seed":202610046043,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"-1\",\"0\"],[\"7\",\"1\",\"0\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (7)x+(1)y=0 and (1)x+(-1)y=0 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"unique solution","explanation":"D=-8 and the classification is unique solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0044","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afff09bc8dc29f4d289ba5dd47df92f02e00b4a4ef927375658a0c21e0d99d57","family_id":"qf-linear-system-solution-class","seed":202610046044,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"12\",\"14\"],[\"1\",\"12\",\"14\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (1)x+(12)y=14 and (2)x+(24)y=28 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"infinitely many solutions","explanation":"D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0045","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bdb26c5bf54a4acdec450db3a9ec2ad0eebf2d70ff08dc33abc73adff8d6aa5","family_id":"qf-linear-system-solution-class","seed":202610046045,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"13\",\"15\"],[\"2\",\"26\",\"31\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (1)x+(13)y=15 and (2)x+(26)y=31 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"no solution","explanation":"D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0046","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"602f35e6b226831007eabc3bd6f21b8ffffa14a54987c5988ee9065fd8b295dc","family_id":"qf-linear-system-solution-class","seed":202610046046,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"-1\",\"3\"],[\"10\",\"1\",\"-2\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (10)x+(1)y=-2 and (1)x+(-1)y=3 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"unique solution","explanation":"D=-11 and the classification is unique solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0047","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab696825e805c43e80983985352deec482a64a96c467291a027c91dba457e5a8","family_id":"qf-linear-system-solution-class","seed":202610046047,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"15\",\"17\"],[\"1\",\"15\",\"17\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (1)x+(15)y=17 and (2)x+(30)y=34 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"infinitely many solutions","explanation":"D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0048","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29736eb77f04c16986635b7cbc97be078161015bfa005241da2721298dcafdee","family_id":"qf-linear-system-solution-class","seed":202610046048,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"16\",\"18\"],[\"2\",\"32\",\"37\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (1)x+(16)y=18 and (2)x+(32)y=37 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"no solution","explanation":"D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0049","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17456ee1e0189fa262dad3cbdc75ef58466cb579ae6eb4fb446a671f14212aa0","family_id":"qf-linear-system-solution-class","seed":202610046049,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"-1\",\"-1\"],[\"13\",\"1\",\"1\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (13)x+(1)y=1 and (1)x+(-1)y=-1 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"unique solution","explanation":"D=-14 and the classification is unique solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0050","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c21d4b560133c73d3cee7b4bef3c7002f38542c3c31c3a934341afd588c270d3","family_id":"qf-linear-system-solution-class","seed":202610046050,"semantic_template":"qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"18\",\"20\"],[\"1\",\"18\",\"20\"]]}"},"fields":{"title":"Solution Classification of a Linear System","stem":"Over the reals, the constraints (1)x+(18)y=20 and (2)x+(36)y=40 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.","answer":{"kind":"short_text","value":"infinitely many solutions","explanation":"D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0."},"skills":["Solution Classification of a Linear System"],"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-20261003-advance-20261004-r06-system-linear-equations-0051","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9aa6f2a3989017bb3f74eb135fc3012f86215ea24732102c6a742b1942a6cb5","family_id":"qf-linear-system-solution-verification","seed":202610046051,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"0\",\"0\",\"0\"],[\"0\",\"0\",\"0\"]],\"x\":[0,1],\"y\":[0,1]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (0)x+(0)y=0 and (0)x+(0)y=0 must both hold. Does the reduced positive-denominator candidate x=0/1, y=0/1 solve the entire system?","answer":{"kind":"short_text","value":"Yes","explanation":"After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0052","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31d276e4e3b9c884d96db5eab3916b6f0ce5d9ad14775a7269bd1d1007649076","family_id":"qf-linear-system-solution-verification","seed":202610046052,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"0\",\"0\",\"0\"],[\"0\",\"0\",\"1\"]],\"x\":[0,1],\"y\":[0,1]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (0)x+(0)y=1 and (0)x+(0)y=0 must both hold. Does the reduced positive-denominator candidate x=0/1, y=0/1 solve the entire system?","answer":{"kind":"short_text","value":"No","explanation":"After clearing the nonzero denominators, the two residuals are (-1, 0). Both must be zero, giving No; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0053","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a090a6a6a167c023132d4c398d63589ed93666d9b1b30e46e1d97731fd2141dc","family_id":"qf-linear-system-solution-verification","seed":202610046053,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"1\",\"2\",\"0\"],[\"1\",\"2\",\"0\"]],\"x\":[2,1],\"y\":[-1,1]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (1)x+(2)y=0 and (2)x+(4)y=0 must both hold. Does the reduced positive-denominator candidate x=2/1, y=-1/1 solve the entire system?","answer":{"kind":"short_text","value":"Yes","explanation":"After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0054","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"244555f06b7cd39f22f614a939c6c7d58adb74eeabb7bc75927f1d87241c68aa","family_id":"qf-linear-system-solution-verification","seed":202610046054,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"0\",\"1\",\"1\"],[\"1\",\"0\",\"0\"]],\"x\":[0,1],\"y\":[0,1]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (1)x+(0)y=0 and (0)x+(1)y=1 must both hold. Does the reduced positive-denominator candidate x=0/1, y=0/1 solve the entire system?","answer":{"kind":"short_text","value":"No","explanation":"After clearing the nonzero denominators, the two residuals are (0, -1). Both must be zero, giving No; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0055","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e03448f0a5ef1db1e8d1c4956172f4e1c5692f8b9fa1296da21ca37d47e798f","family_id":"qf-linear-system-solution-verification","seed":202610046055,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"-7\"],[\"4\",\"-3\",\"-17\"]],\"x\":[-4,1],\"y\":[1,3]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=-7 and (4)x+(-3)y=-17 must both hold. Does the reduced positive-denominator candidate x=-4/1, y=1/3 solve the entire system?","answer":{"kind":"short_text","value":"Yes","explanation":"After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0056","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d1dc21b6a1674ff0da748c494ef0431278ff7a586b7e1de77a998ef12a43fa6","family_id":"qf-linear-system-solution-verification","seed":202610046056,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"-5\"],[\"4\",\"-3\",\"-16\"]],\"x\":[-3,1],\"y\":[2,3]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=-5 and (4)x+(-3)y=-16 must both hold. Does the reduced positive-denominator candidate x=-3/1, y=2/3 solve the entire system?","answer":{"kind":"short_text","value":"No","explanation":"After clearing the nonzero denominators, the two residuals are (3, 6). Both must be zero, giving No; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0057","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22269db5c2cdb25395318a3b52ff6490d2b832df61fbae4fbe966e7fb56f9b10","family_id":"qf-linear-system-solution-verification","seed":202610046057,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"-3\"],[\"4\",\"-3\",\"-15\"]],\"x\":[-3,1],\"y\":[1,1]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=-3 and (4)x+(-3)y=-15 must both hold. Does the reduced positive-denominator candidate x=-3/1, y=1/1 solve the entire system?","answer":{"kind":"short_text","value":"Yes","explanation":"After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0058","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e5c114237220f6a1bd5711f1bcb6e2231b53320b5690d7e7a2c0a7630bc788e","family_id":"qf-linear-system-solution-verification","seed":202610046058,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"-8\"],[\"4\",\"-3\",\"-7\"]],\"x\":[-2,1],\"y\":[-1,1]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=-8 and (4)x+(-3)y=-7 must both hold. Does the reduced positive-denominator candidate x=-2/1, y=-1/1 solve the entire system?","answer":{"kind":"short_text","value":"No","explanation":"After clearing the nonzero denominators, the two residuals are (1, 2). Both must be zero, giving No; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0059","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cae42b5b1a1e19e566a5c60092d6037282de450f8f86248c7272625cc4e021c1","family_id":"qf-linear-system-solution-verification","seed":202610046059,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"-6\"],[\"4\",\"-3\",\"-6\"]],\"x\":[-2,1],\"y\":[-2,3]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=-6 and (4)x+(-3)y=-6 must both hold. Does the reduced positive-denominator candidate x=-2/1, y=-2/3 solve the entire system?","answer":{"kind":"short_text","value":"Yes","explanation":"After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0060","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41ef90f4ab66a15811c7a240873baa0ee8ea896fecf0e1e59f5fcaafe402b3b9","family_id":"qf-linear-system-solution-verification","seed":202610046060,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"-4\"],[\"4\",\"-3\",\"-5\"]],\"x\":[-1,1],\"y\":[-1,3]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=-4 and (4)x+(-3)y=-5 must both hold. Does the reduced positive-denominator candidate x=-1/1, y=-1/3 solve the entire system?","answer":{"kind":"short_text","value":"No","explanation":"After clearing the nonzero denominators, the two residuals are (3, 6). Both must be zero, giving No; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0061","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b356c65dc70aaddc40150434986d1b78479885c0bc2b8e97e35309baf52fac98","family_id":"qf-linear-system-solution-verification","seed":202610046061,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"-2\"],[\"4\",\"-3\",\"-4\"]],\"x\":[-1,1],\"y\":[0,1]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=-2 and (4)x+(-3)y=-4 must both hold. Does the reduced positive-denominator candidate x=-1/1, y=0/1 solve the entire system?","answer":{"kind":"short_text","value":"Yes","explanation":"After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0062","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc43be7d43630f896c12d64856d739136b063d831ecb3be0973479c84a00b7d2","family_id":"qf-linear-system-solution-verification","seed":202610046062,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"0\"],[\"4\",\"-3\",\"-3\"]],\"x\":[0,1],\"y\":[1,3]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=0 and (4)x+(-3)y=-3 must both hold. Does the reduced positive-denominator candidate x=0/1, y=1/3 solve the entire system?","answer":{"kind":"short_text","value":"No","explanation":"After clearing the nonzero denominators, the two residuals are (3, 6). Both must be zero, giving No; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0063","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d03315e5f8adfbc559b41d4021737e3a4526ec93f3981abbeb5893441ea8b33","family_id":"qf-linear-system-solution-verification","seed":202610046063,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"2\"],[\"4\",\"-3\",\"-2\"]],\"x\":[0,1],\"y\":[2,3]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=2 and (4)x+(-3)y=-2 must both hold. Does the reduced positive-denominator candidate x=0/1, y=2/3 solve the entire system?","answer":{"kind":"short_text","value":"Yes","explanation":"After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0064","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1407ff772eabdc2eeff1f8107c23ae05437aa21ca189fa9f46f73b3b0b0ac152","family_id":"qf-linear-system-solution-verification","seed":202610046064,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"4\"],[\"4\",\"-3\",\"-1\"]],\"x\":[1,1],\"y\":[1,1]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=4 and (4)x+(-3)y=-1 must both hold. Does the reduced positive-denominator candidate x=1/1, y=1/1 solve the entire system?","answer":{"kind":"short_text","value":"No","explanation":"After clearing the nonzero denominators, the two residuals are (1, 2). Both must be zero, giving No; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0065","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cc31d904f12fe028fcf524db851374bc811d53ea30abae4e7aaaf9dace606af","family_id":"qf-linear-system-solution-verification","seed":202610046065,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"-1\"],[\"4\",\"-3\",\"7\"]],\"x\":[1,1],\"y\":[-1,1]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=-1 and (4)x+(-3)y=7 must both hold. Does the reduced positive-denominator candidate x=1/1, y=-1/1 solve the entire system?","answer":{"kind":"short_text","value":"Yes","explanation":"After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0066","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"306ec5a8c046b91c96926611bc98a32fc4805f0b03192caf1a1de95644ef2d43","family_id":"qf-linear-system-solution-verification","seed":202610046066,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"1\"],[\"4\",\"-3\",\"8\"]],\"x\":[2,1],\"y\":[-2,3]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=1 and (4)x+(-3)y=8 must both hold. Does the reduced positive-denominator candidate x=2/1, y=-2/3 solve the entire system?","answer":{"kind":"short_text","value":"No","explanation":"After clearing the nonzero denominators, the two residuals are (3, 6). Both must be zero, giving No; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0067","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"058c2144a058c0a08d7e96cc46a84310f59861059cf28503c0e43a01be575f19","family_id":"qf-linear-system-solution-verification","seed":202610046067,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"3\"],[\"4\",\"-3\",\"9\"]],\"x\":[2,1],\"y\":[-1,3]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=3 and (4)x+(-3)y=9 must both hold. Does the reduced positive-denominator candidate x=2/1, y=-1/3 solve the entire system?","answer":{"kind":"short_text","value":"Yes","explanation":"After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0068","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19d80b3d7131b778d267f326aa877d14edd2da3b5832a8a9747e32436d611665","family_id":"qf-linear-system-solution-verification","seed":202610046068,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"5\"],[\"4\",\"-3\",\"10\"]],\"x\":[3,1],\"y\":[0,1]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=5 and (4)x+(-3)y=10 must both hold. Does the reduced positive-denominator candidate x=3/1, y=0/1 solve the entire system?","answer":{"kind":"short_text","value":"No","explanation":"After clearing the nonzero denominators, the two residuals are (1, 2). Both must be zero, giving No; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0069","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fcfc78adcea61ee629dbfbc794c72e784d67684d46de6b834da73d928960439","family_id":"qf-linear-system-solution-verification","seed":202610046069,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"7\"],[\"4\",\"-3\",\"11\"]],\"x\":[3,1],\"y\":[1,3]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=7 and (4)x+(-3)y=11 must both hold. Does the reduced positive-denominator candidate x=3/1, y=1/3 solve the entire system?","answer":{"kind":"short_text","value":"Yes","explanation":"After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0070","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f818bb13493a74dbb9fe4fc03ecba03e112d357eb57ebe26c14691381261ddd","family_id":"qf-linear-system-solution-verification","seed":202610046070,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"9\"],[\"4\",\"-3\",\"12\"]],\"x\":[4,1],\"y\":[2,3]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=9 and (4)x+(-3)y=12 must both hold. Does the reduced positive-denominator candidate x=4/1, y=2/3 solve the entire system?","answer":{"kind":"short_text","value":"No","explanation":"After clearing the nonzero denominators, the two residuals are (3, 6). Both must be zero, giving No; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0071","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdf447760fae3bc75e4522ec341a21defddf604418469e1e46e61812378c559c","family_id":"qf-linear-system-solution-verification","seed":202610046071,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"11\"],[\"4\",\"-3\",\"13\"]],\"x\":[4,1],\"y\":[1,1]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=11 and (4)x+(-3)y=13 must both hold. Does the reduced positive-denominator candidate x=4/1, y=1/1 solve the entire system?","answer":{"kind":"short_text","value":"Yes","explanation":"After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0072","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4d50a9a93a8d022c34e1575aaf17a7f6f9bf12007ed1fdacfe8c7239a90e152","family_id":"qf-linear-system-solution-verification","seed":202610046072,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"6\"],[\"4\",\"-3\",\"21\"]],\"x\":[5,1],\"y\":[-1,1]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=6 and (4)x+(-3)y=21 must both hold. Does the reduced positive-denominator candidate x=5/1, y=-1/1 solve the entire system?","answer":{"kind":"short_text","value":"No","explanation":"After clearing the nonzero denominators, the two residuals are (1, 2). Both must be zero, giving No; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0073","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70ab61ba4ee6fa9961bffcdd19afab696ac45d2dfb319e44c0e9da809e0fa861","family_id":"qf-linear-system-solution-verification","seed":202610046073,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"8\"],[\"4\",\"-3\",\"22\"]],\"x\":[5,1],\"y\":[-2,3]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=8 and (4)x+(-3)y=22 must both hold. Does the reduced positive-denominator candidate x=5/1, y=-2/3 solve the entire system?","answer":{"kind":"short_text","value":"Yes","explanation":"After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0074","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21deef37be972d5b82f983114b5b0fa6b60df7a917116544218b0a870560d104","family_id":"qf-linear-system-solution-verification","seed":202610046074,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"10\"],[\"4\",\"-3\",\"23\"]],\"x\":[6,1],\"y\":[-1,3]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=10 and (4)x+(-3)y=23 must both hold. Does the reduced positive-denominator candidate x=6/1, y=-1/3 solve the entire system?","answer":{"kind":"short_text","value":"No","explanation":"After clearing the nonzero denominators, the two residuals are (3, 6). Both must be zero, giving No; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0075","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbc8d6830d90e423b6dbbfc4fb0c6a39d2de9acddd42ff35db9f6c62f1da5f68","family_id":"qf-linear-system-solution-verification","seed":202610046075,"semantic_template":"qf-linear-system-solution-verification/{\"domain\":\"reals\",\"operation\":\"verify_solution\",\"rows\":[[\"2\",\"3\",\"12\"],[\"4\",\"-3\",\"24\"]],\"x\":[6,1],\"y\":[0,1]}"},"fields":{"title":"Checking a Rational Candidate in Both Rows","stem":"Over the reals, the constraints (2)x+(3)y=12 and (4)x+(-3)y=24 must both hold. Does the reduced positive-denominator candidate x=6/1, y=0/1 solve the entire system?","answer":{"kind":"short_text","value":"Yes","explanation":"After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient."},"skills":["Checking a Rational Candidate in Both Rows"],"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-20261003-advance-20261004-r06-system-linear-equations-0076","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5da9742d205e446e5b18d62a30cb408e6889d28bd32fea60deee1582601724a3","family_id":"qf-linear-system-unique-solve","seed":202610046076,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"0\",\"1\",\"0\"],[\"1\",\"0\",\"0\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(0)y=0 and (0)x+(1)y=0 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(0/1, 0/1)","explanation":"D=1≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (0/1, 0/1). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0077","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99c4d57b8ac97e311907651633583792e0e7309e8a2aa51381d857c582767a60","family_id":"qf-linear-system-unique-solve","seed":202610046077,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-1\",\"0\"],[\"1\",\"1\",\"1\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=1 and (1)x+(-1)y=0 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(1/2, 1/2)","explanation":"D=-2≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (1/2, 1/2). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0078","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95cd2db97b1ac6ac39dddb05d0aba1fe81e47d276ae7ab51d807a3ef8f48ba0b","family_id":"qf-linear-system-unique-solve","seed":202610046078,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"2\",\"-4\"],[\"2\",\"-1\",\"-3\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (-2)x+(1)y=3 and (1)x+(2)y=-4 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(-2/1, -1/1)","explanation":"D=-5≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-2/1, -1/1). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0079","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"068a490dc091a8e538b2b5a71c8b651131ff0b22802290909d97a12bdcd8912e","family_id":"qf-linear-system-unique-solve","seed":202610046079,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"1\"],[\"1\",\"1\",\"-9\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=-9 and (1)x+(-2)y=1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(-17/3, -10/3)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-17/3, -10/3). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0080","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18a0ed16a435dae3a4e8cd3d5fe809483ad17a4d117589f09df6cc06ace257ae","family_id":"qf-linear-system-unique-solve","seed":202610046080,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"2\"],[\"1\",\"1\",\"-8\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=-8 and (1)x+(-2)y=2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(-14/3, -10/3)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-14/3, -10/3). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0081","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbc93ccf3725edec22f7d09f13e17aa365af73dcd47a5ef5ef19eceb69e5a70b","family_id":"qf-linear-system-unique-solve","seed":202610046081,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"-2\"],[\"1\",\"1\",\"-7\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=-7 and (1)x+(-2)y=-2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(-16/3, -5/3)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-16/3, -5/3). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0082","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16be1b85c2536ba775e3bf6dfbc344e11aab80c82eff3dbfb6a5cc97e6cd3ffb","family_id":"qf-linear-system-unique-solve","seed":202610046082,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"-1\"],[\"1\",\"1\",\"-6\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=-6 and (1)x+(-2)y=-1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(-13/3, -5/3)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-13/3, -5/3). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0083","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1549b2de39267551c042746f99b027e44f3da8bcf67bd6280e6d3fbfeb94dfbe","family_id":"qf-linear-system-unique-solve","seed":202610046083,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"0\"],[\"1\",\"1\",\"-5\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=-5 and (1)x+(-2)y=0 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(-10/3, -5/3)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-10/3, -5/3). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0084","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48983c843720509a8bd5e1640c090ae1cd812b62c8c73d746c1eeb2a51a39ab2","family_id":"qf-linear-system-unique-solve","seed":202610046084,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"1\"],[\"1\",\"1\",\"-4\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=-4 and (1)x+(-2)y=1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(-7/3, -5/3)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-7/3, -5/3). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0085","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3404b18894eba0da8ac3789666cc260eacfd8e48b8c2aadcc9eef1bb02ba464b","family_id":"qf-linear-system-unique-solve","seed":202610046085,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"2\"],[\"1\",\"1\",\"-3\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=-3 and (1)x+(-2)y=2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(-4/3, -5/3)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-4/3, -5/3). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0086","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f9cadccb2fb05ee48c854d877fc5076003835f255177e1f8d0b7c325be8a72d","family_id":"qf-linear-system-unique-solve","seed":202610046086,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"-2\"],[\"1\",\"1\",\"-2\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=-2 and (1)x+(-2)y=-2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(-2/1, 0/1)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-2/1, 0/1). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0087","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4507afc749b42f00865b8516839698b04a29e9a2f328b3fd53a920b5e37dda0","family_id":"qf-linear-system-unique-solve","seed":202610046087,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"-1\"],[\"1\",\"1\",\"-1\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=-1 and (1)x+(-2)y=-1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(-1/1, 0/1)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-1/1, 0/1). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0088","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f1c3cc2bdbb10fd928c0be5cf7c48821bd67666058e977e7d048706e664d1f8","family_id":"qf-linear-system-unique-solve","seed":202610046088,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"0\"],[\"1\",\"1\",\"0\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=0 and (1)x+(-2)y=0 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(0/1, 0/1)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (0/1, 0/1). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0089","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f8519f8fc2ad86ba2b1b28120c453aef1e1ca2048155ef7f609c42045aaa0df","family_id":"qf-linear-system-unique-solve","seed":202610046089,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"1\"],[\"1\",\"1\",\"1\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=1 and (1)x+(-2)y=1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(1/1, 0/1)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (1/1, 0/1). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0090","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b76fe5249d78b279c4a2fed4e4816ae21dac267202f800dc1476408eae6e2b4","family_id":"qf-linear-system-unique-solve","seed":202610046090,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"2\"],[\"1\",\"1\",\"2\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=2 and (1)x+(-2)y=2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(2/1, 0/1)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (2/1, 0/1). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0091","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a4d0fa42049a13ba44197ae7d0d842ff95fe25e5d039653fac420fee73a8211","family_id":"qf-linear-system-unique-solve","seed":202610046091,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"-2\"],[\"1\",\"1\",\"3\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=3 and (1)x+(-2)y=-2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(4/3, 5/3)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (4/3, 5/3). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0092","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55e9e4dfb0592134ecde3c53a5367ceb9c12f16d63a9177770b8b1fe6fa48539","family_id":"qf-linear-system-unique-solve","seed":202610046092,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"-1\"],[\"1\",\"1\",\"4\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=4 and (1)x+(-2)y=-1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(7/3, 5/3)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (7/3, 5/3). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0093","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6693115cad98030c16007b0141e84ceb2893fa9d4c2ef348b301c6f924f9d87f","family_id":"qf-linear-system-unique-solve","seed":202610046093,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"0\"],[\"1\",\"1\",\"5\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=5 and (1)x+(-2)y=0 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(10/3, 5/3)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (10/3, 5/3). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0094","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27cdb11e745e09d6642bdc0c0e1a8b13f8996f79f3c7cf2a0e4f29e89dc58809","family_id":"qf-linear-system-unique-solve","seed":202610046094,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"1\"],[\"1\",\"1\",\"6\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=6 and (1)x+(-2)y=1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(13/3, 5/3)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (13/3, 5/3). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0095","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5dbc9bfa74102771893574b2841c53aabdcc8e53083374cd8c9f7a1c5c53ec7","family_id":"qf-linear-system-unique-solve","seed":202610046095,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"2\"],[\"1\",\"1\",\"7\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=7 and (1)x+(-2)y=2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(16/3, 5/3)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (16/3, 5/3). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0096","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a708b90a91e5c060127fe1f0c219a53e3b9170bd8360010497d754a208f4cec2","family_id":"qf-linear-system-unique-solve","seed":202610046096,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"-2\"],[\"1\",\"1\",\"8\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=8 and (1)x+(-2)y=-2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(14/3, 10/3)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (14/3, 10/3). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0097","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"115a0c9ddbae3d53c36733c88da12fd6f675795adda7b822f22d263a00be34f8","family_id":"qf-linear-system-unique-solve","seed":202610046097,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"-1\"],[\"1\",\"1\",\"9\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=9 and (1)x+(-2)y=-1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(17/3, 10/3)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (17/3, 10/3). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0098","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2eeb086c24316f83e70842c55c8e62875875207bc0cc394fc5ea8290f67745b5","family_id":"qf-linear-system-unique-solve","seed":202610046098,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"0\"],[\"1\",\"1\",\"10\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=10 and (1)x+(-2)y=0 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(20/3, 10/3)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (20/3, 10/3). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0099","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8e1ed5d3bbcbbf819e63a544907ffc929f410bc629d9714ed2e2db3c2353d62","family_id":"qf-linear-system-unique-solve","seed":202610046099,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"1\"],[\"1\",\"1\",\"11\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=11 and (1)x+(-2)y=1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(23/3, 10/3)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (23/3, 10/3). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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-20261003-advance-20261004-r06-system-linear-equations-0100","batch_id":"batch-20261003-advance-20261004-r06-system-linear-equations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3e4a8041f2961e303cb433da2a56aa24c16a25796566f70ee6f98e56ca77920","family_id":"qf-linear-system-unique-solve","seed":202610046100,"semantic_template":"qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"2\"],[\"1\",\"1\",\"12\"]]}"},"fields":{"title":"Exact Unique Solution with a Nonzero Determinant","stem":"Over the reals, the constraints (1)x+(1)y=12 and (1)x+(-2)y=2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"(26/3, 10/3)","explanation":"D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (26/3, 10/3). The D=0 cases are outside this unique-solution exercise."},"skills":["Exact Unique Solution with a Nonzero Determinant"],"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."}}