{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0001","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8a4122adbb253c72d14992730032fa38e9ce7d7cc1b5168c30a4748f16a4d14","family_id":"qf-density14-equation-fraction-side-check","seed":202610143001,"semantic_template":"qf-density14-equation-fraction-side-check/[\"0/1\",\"0/1\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((0)x+(0))/(1), R(x)=(0)x+(0), q≠0, x=0/1. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, 0/1, 0/1, Yes)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (0/1, 0/1, 0/1, Yes)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0002","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f4b95f494f5a86a8c3c6498a9dcae06e109ba93c63d5f0c5a5958fbd98176f4","family_id":"qf-density14-equation-fraction-side-check","seed":202610143002,"semantic_template":"qf-density14-equation-fraction-side-check/[\"-2/3\",\"1/2\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((2)x+(1))/(-3), R(x)=(1)x+(0), q≠0, x=1/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-2/3, 1/2, -7/6, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-2/3, 1/2, -7/6, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0003","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df470f00e3b951beaa223074567393c66ad4a15813692d50ef34dd535bbf3177","family_id":"qf-density14-equation-fraction-side-check","seed":202610143003,"semantic_template":"qf-density14-equation-fraction-side-check/[\"-4/1\",\"-1/2\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((-10)x+(3))/(3), R(x)=(1)x+(-2), q≠0, x=3/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-4/1, -1/2, -7/2, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-4/1, -1/2, -7/2, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0004","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69d30d7b8db6abac0913a9c4488f9e3ad695ab2b2a6d99f856b40e719e11dc88","family_id":"qf-density14-equation-fraction-side-check","seed":202610143004,"semantic_template":"qf-density14-equation-fraction-side-check/[\"7/1\",\"-5/1\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((-9)x+(4))/(-2), R(x)=(-1)x+(-3), q≠0, x=4/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(7/1, -5/1, 12/1, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (7/1, -5/1, 12/1, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0005","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9f8988cf5d5ca886d259b9f5b042a5822d7bc2c298fd705f8fbabbdf9506341","family_id":"qf-density14-equation-fraction-side-check","seed":202610143005,"semantic_template":"qf-density14-equation-fraction-side-check/[\"-5/1\",\"-4/1\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((-8)x+(5))/(3), R(x)=(0)x+(-4), q≠0, x=5/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-5/1, -4/1, -1/1, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-5/1, -4/1, -1/1, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0006","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68ce4bf651f9f4e542f75b92877c5b5814aaa85ca90b5cf3ad9f1e56fc32a487","family_id":"qf-density14-equation-fraction-side-check","seed":202610143006,"semantic_template":"qf-density14-equation-fraction-side-check/[\"15/2\",\"-2/1\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((-7)x+(6))/(-2), R(x)=(1)x+(-5), q≠0, x=6/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(15/2, -2/1, 19/2, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (15/2, -2/1, 19/2, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0007","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d6f79e3d2bcc6f455111463ea533457a88ca3653a40e2973cabaa09dd37d713","family_id":"qf-density14-equation-fraction-side-check","seed":202610143007,"semantic_template":"qf-density14-equation-fraction-side-check/[\"-14/3\",\"-19/2\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((-6)x+(7))/(3), R(x)=(-1)x+(-6), q≠0, x=7/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-14/3, -19/2, 29/6, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-14/3, -19/2, 29/6, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0008","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"469f7969845022b076490630114f5048f3bd25d0bb939dec324d37d953260d15","family_id":"qf-density14-equation-fraction-side-check","seed":202610143008,"semantic_template":"qf-density14-equation-fraction-side-check/[\"6/1\",\"-7/1\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((-5)x+(8))/(-2), R(x)=(0)x+(-7), q≠0, x=8/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(6/1, -7/1, 13/1, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (6/1, -7/1, 13/1, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0009","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a507b12325c40f59b1210305f6406047f913ee82e7853af630b0726ccfc38dd0","family_id":"qf-density14-equation-fraction-side-check","seed":202610143009,"semantic_template":"qf-density14-equation-fraction-side-check/[\"-3/1\",\"-7/2\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((-4)x+(9))/(3), R(x)=(1)x+(-8), q≠0, x=9/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-3/1, -7/2, 1/2, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-3/1, -7/2, 1/2, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0010","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8744d113f43291b103ed77cb1c7dc23219ba553ee68aeaf0e842641dde26ef4c","family_id":"qf-density14-equation-fraction-side-check","seed":202610143010,"semantic_template":"qf-density14-equation-fraction-side-check/[\"5/2\",\"-14/1\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((-3)x+(10))/(-2), R(x)=(-1)x+(-9), q≠0, x=10/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(5/2, -14/1, 33/2, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (5/2, -14/1, 33/2, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0011","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08186bbad33df4f48d85e9297e1302edbd0fc3bc432a1c7040a32bfc4e1fe871","family_id":"qf-density14-equation-fraction-side-check","seed":202610143011,"semantic_template":"qf-density14-equation-fraction-side-check/[\"0/1\",\"-10/1\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((-2)x+(11))/(3), R(x)=(0)x+(-10), q≠0, x=11/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, -10/1, 10/1, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (0/1, -10/1, 10/1, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0012","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4c31f18124649e6718a483d9693e124c4f34b00bc7cf1847a3155062e1f3f0f","family_id":"qf-density14-equation-fraction-side-check","seed":202610143012,"semantic_template":"qf-density14-equation-fraction-side-check/[\"-3/1\",\"-5/1\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((-1)x+(12))/(-2), R(x)=(1)x+(-11), q≠0, x=12/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-3/1, -5/1, 2/1, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-3/1, -5/1, 2/1, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0013","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53f41244468141df646188983600e72c25fd75f89c13c2f03eb7d64d3712e8b3","family_id":"qf-density14-equation-fraction-side-check","seed":202610143013,"semantic_template":"qf-density14-equation-fraction-side-check/[\"13/3\",\"-37/2\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((0)x+(13))/(3), R(x)=(-1)x+(-12), q≠0, x=13/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(13/3, -37/2, 137/6, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (13/3, -37/2, 137/6, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0014","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5eab8da84ea81f69f9d0ad02f095eecfa03f4bd826f6c767435e3dd868fa02de","family_id":"qf-density14-equation-fraction-side-check","seed":202610143014,"semantic_template":"qf-density14-equation-fraction-side-check/[\"-21/2\",\"-13/1\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((1)x+(14))/(-2), R(x)=(0)x+(-13), q≠0, x=14/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-21/2, -13/1, 5/2, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-21/2, -13/1, 5/2, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0015","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c55bcf4dae1915cd7c07d4d495d80cf3d34de66d7e6f991f53ba5d714dd3985c","family_id":"qf-density14-equation-fraction-side-check","seed":202610143015,"semantic_template":"qf-density14-equation-fraction-side-check/[\"10/1\",\"-13/2\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((2)x+(15))/(3), R(x)=(1)x+(-14), q≠0, x=15/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(10/1, -13/2, 33/2, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (10/1, -13/2, 33/2, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0016","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de503b6ee4260dd00804450ed1408d22f1a4ea50e21906884db7206d16bbc3c9","family_id":"qf-density14-equation-fraction-side-check","seed":202610143016,"semantic_template":"qf-density14-equation-fraction-side-check/[\"-20/1\",\"-23/1\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((3)x+(16))/(-2), R(x)=(-1)x+(-15), q≠0, x=16/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-20/1, -23/1, 3/1, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-20/1, -23/1, 3/1, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0017","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"687e1be6a7e50401df955159055b2a36c42fad06491015d6f44e422049fe22a2","family_id":"qf-density14-equation-fraction-side-check","seed":202610143017,"semantic_template":"qf-density14-equation-fraction-side-check/[\"17/1\",\"-16/1\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((4)x+(17))/(3), R(x)=(0)x+(-16), q≠0, x=17/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(17/1, -16/1, 33/1, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (17/1, -16/1, 33/1, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0018","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d1c280bbe6c0e59ed453434a119a4afaf07ac100cf5bf45dc985c63097bf3ce","family_id":"qf-density14-equation-fraction-side-check","seed":202610143018,"semantic_template":"qf-density14-equation-fraction-side-check/[\"-63/2\",\"-8/1\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((5)x+(18))/(-2), R(x)=(1)x+(-17), q≠0, x=18/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-63/2, -8/1, -47/2, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-63/2, -8/1, -47/2, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0019","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2531e92b02cea4bb648dd8a2e8ee4da7fe5fdf42e37bfccfb9a5677571b7b50","family_id":"qf-density14-equation-fraction-side-check","seed":202610143019,"semantic_template":"qf-density14-equation-fraction-side-check/[\"76/3\",\"-55/2\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((6)x+(19))/(3), R(x)=(-1)x+(-18), q≠0, x=19/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(76/3, -55/2, 317/6, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (76/3, -55/2, 317/6, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0020","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e764026192267ea81ea818b67a1cb2abb4b5507b19a6064ab39f188b0577731c","family_id":"qf-density14-equation-fraction-side-check","seed":202610143020,"semantic_template":"qf-density14-equation-fraction-side-check/[\"-45/1\",\"-19/1\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((7)x+(20))/(-2), R(x)=(0)x+(-19), q≠0, x=20/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-45/1, -19/1, -26/1, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-45/1, -19/1, -26/1, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0021","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3461d1e9e7e3f1ac17a03de6a8223290ee1be784b00fd437916ba11ac567abb9","family_id":"qf-density14-equation-fraction-side-check","seed":202610143021,"semantic_template":"qf-density14-equation-fraction-side-check/[\"35/1\",\"-19/2\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((8)x+(21))/(3), R(x)=(1)x+(-20), q≠0, x=21/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(35/1, -19/2, 89/2, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (35/1, -19/2, 89/2, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0022","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ecf3989bcd32f50dd7f4aeca18a988ad19d7cc703745859ee544f4b3f9e35a5","family_id":"qf-density14-equation-fraction-side-check","seed":202610143022,"semantic_template":"qf-density14-equation-fraction-side-check/[\"-121/2\",\"-32/1\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((9)x+(22))/(-2), R(x)=(-1)x+(-21), q≠0, x=22/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-121/2, -32/1, -57/2, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-121/2, -32/1, -57/2, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0023","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f74efda00e8e586d9c4f781cc04017321df6729ec0f7145583175cd337723396","family_id":"qf-density14-equation-fraction-side-check","seed":202610143023,"semantic_template":"qf-density14-equation-fraction-side-check/[\"46/1\",\"-22/1\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((10)x+(23))/(3), R(x)=(0)x+(-22), q≠0, x=23/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(46/1, -22/1, 68/1, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (46/1, -22/1, 68/1, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0024","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36230f43262be095a72c7fe5a7a5c071be00f00054bd64b8eb35c98b15a571b5","family_id":"qf-density14-equation-fraction-side-check","seed":202610143024,"semantic_template":"qf-density14-equation-fraction-side-check/[\"-78/1\",\"-11/1\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((11)x+(24))/(-2), R(x)=(1)x+(-23), q≠0, x=24/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-78/1, -11/1, -67/1, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-78/1, -11/1, -67/1, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0025","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75cf4535b4f4071f27ebae58e69802ff2c672b4cf913f393002d3efaca2a4375","family_id":"qf-density14-equation-fraction-side-check","seed":202610143025,"semantic_template":"qf-density14-equation-fraction-side-check/[\"175/3\",\"-73/2\"]"},"fields":{"title":"Original Rational Equation Sides and Defect","stem":"Use the original equation’s common rational domain. L(x)=((12)x+(25))/(3), R(x)=(-1)x+(-24), q≠0, x=25/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(175/3, -73/2, 569/6, No)","explanation":"Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (175/3, -73/2, 569/6, No)."},"skills":["Original Rational Equation Sides and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0026","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3fa846b9dd10f03afe20c5481fb492d7da68eb47f3fdc95b08139e0307fafbe","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143026,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"全Q\",\"全Q\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (0)x+(0)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(all Q, all Q, Yes)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (all Q, all Q, Yes)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0027","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c02ce56395463fd4f5a24577f54dd3252225d77d49c6aa941bbb30e9f96cbfa","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143027,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"無解\",\"全Q\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (0)x+(1)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(no-solution, all Q, No)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (no-solution, all Q, No)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0028","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6b0474384d8cc819456ab0a8a21a43f009b2d8b704ad950220069aaf24a85f0","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143028,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"無解\",\"無解\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (0)x+(1)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(no-solution, no-solution, Yes)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (no-solution, no-solution, Yes)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0029","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"940af48d11d571d95bacaa8c41a52d8c17e6bc8f40fa64f3373cedc54d0fe3de","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143029,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 0/1\",\"全Q\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(0)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 0/1, all Q, No)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 0/1, all Q, No)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0030","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad42a04bef97c1a4c82ae8a0281a63019a821ef8ef1d19d4a53da97ecd6ffd13","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143030,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 1/2\",\"單解 1/2\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-1)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 1/2, single-solution 1/2, Yes)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 1/2, single-solution 1/2, Yes)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0031","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33154dd601ab13968a61f65b7f7b7a48f6644835551b9f210be9399699d0f528","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143031,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 1/1\",\"全Q\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-2)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 1/1, all Q, No)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 1/1, all Q, No)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0032","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a16b7abe12f9702ef6aae89a9dc363e2adab568d7ca6c27d267e03f87e1fb5ed","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143032,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 3/2\",\"單解 3/2\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-3)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 3/2, single-solution 3/2, Yes)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 3/2, single-solution 3/2, Yes)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0033","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fc69a05152b8e674c68c9362fbf335392f229ed1a9898526d66e93f75fc3d28","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143033,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 2/1\",\"全Q\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-4)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 2/1, all Q, No)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 2/1, all Q, No)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0034","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a192fb64c8c26f2fd53b6e6e90e7c0ce3cda3b9f5f6ce6b05e6b74f749fc4678","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143034,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 5/2\",\"單解 5/2\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-5)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 5/2, single-solution 5/2, Yes)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 5/2, single-solution 5/2, Yes)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0035","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1eef61b4196ffd41ea7d6ecd3e9e84c5540b5c26a61ad45b7b5ddb75c9e88ff","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143035,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 3/1\",\"全Q\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-6)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 3/1, all Q, No)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 3/1, all Q, No)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0036","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"659c305dbe2c49dacaf43b21cbe2aa763093f5450c5da0ce885ba1c127742058","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143036,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 7/2\",\"單解 7/2\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-7)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 7/2, single-solution 7/2, Yes)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 7/2, single-solution 7/2, Yes)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0037","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9391eb43453e638d16e392c6d585f266fef5a83079d8951033c1a4b4e68ddff3","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143037,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 4/1\",\"全Q\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-8)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 4/1, all Q, No)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 4/1, all Q, No)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0038","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0aaaac89b6cac01d250db95be0add39b323b10453718bead84d572517a8514ee","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143038,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 9/2\",\"單解 9/2\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-9)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 9/2, single-solution 9/2, Yes)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 9/2, single-solution 9/2, Yes)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0039","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4742c24fda16c7c2370cd1cb6ad2b055106bf0e5f96540c3661c2682d36bdb2","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143039,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 5/1\",\"全Q\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-10)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 5/1, all Q, No)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 5/1, all Q, No)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0040","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ebd442bf96f32068ea54c4d93d5915c7d3151b1a684e6cb0d303df317f1a8af","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143040,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 11/2\",\"單解 11/2\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-11)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 11/2, single-solution 11/2, Yes)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 11/2, single-solution 11/2, Yes)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0041","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73562cc2fce4bbc8d9fa0a590101a9a96ef8838dfc98dc1dc0aa61a36107c049","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143041,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 6/1\",\"全Q\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-12)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 6/1, all Q, No)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 6/1, all Q, No)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0042","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee5e83bb8f0f7351c0bce51f128ee31d75c08a6698ff43f63a65126583f9b6f6","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143042,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 13/2\",\"單解 13/2\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-13)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 13/2, single-solution 13/2, Yes)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 13/2, single-solution 13/2, Yes)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0043","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45fa31dd17bf640ccbd73caf4892c54279ef61934d8feb349ec1232a17070728","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143043,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 7/1\",\"全Q\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-14)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 7/1, all Q, No)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 7/1, all Q, No)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0044","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c62e71ff1e081b549953b0601ea0fd550749963a36c7e206e966bc612ab0d29","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143044,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 15/2\",\"單解 15/2\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-15)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 15/2, single-solution 15/2, Yes)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 15/2, single-solution 15/2, Yes)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0045","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4617a5065257c889f3bbbf367e97e3cb8a42ed478083f75ff7e21484eb296417","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143045,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 8/1\",\"全Q\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-16)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 8/1, all Q, No)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 8/1, all Q, No)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0046","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d46de5fcc0587ba995c2ce0f0146033e3941e2d73c9b8359be26c44b7b0654a","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143046,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 17/2\",\"單解 17/2\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-17)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 17/2, single-solution 17/2, Yes)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 17/2, single-solution 17/2, Yes)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0047","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"889804ec40a210bd49b3ea3c1416d1d686f278d531b888337aea3e820966e7e4","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143047,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 9/1\",\"全Q\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-18)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 9/1, all Q, No)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 9/1, all Q, No)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0048","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3112ba0df01dd48954673c14c4650ca0d1f5794f31488cf78a630cd9adfae2b5","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143048,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 19/2\",\"單解 19/2\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-19)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 19/2, single-solution 19/2, Yes)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 19/2, single-solution 19/2, Yes)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0049","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16a7b1838e2eac068185ee23ae50b7bb830e6e3fcd5f392818ba35dca388bb89","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143049,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 10/1\",\"全Q\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-20)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 10/1, all Q, No)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 10/1, all Q, No)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0050","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"105b5f2901b3afe11c28bd7e70a60a5903e44dc5730fc74933b5e383373134d1","family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143050,"semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 21/2\",\"單解 21/2\"]"},"fields":{"title":"Solution Classes Before and After Scaling","stem":"Over Q, the original equation is (2)x+(-21)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.","answer":{"kind":"expression","value":"(single-solution 21/2, single-solution 21/2, Yes)","explanation":"Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 21/2, single-solution 21/2, Yes)."},"skills":["Solution Classes Before and After Scaling"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0051","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"019ac5fa4ffa123f113a1ad5b79fa86315fa3008991fa6af91e94f2d2d6fe940","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143051,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"0/1\"],[true,\"0/1\",\"0/1\"],[true,\"0/1\",\"0/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((0)x+(0))/(x−(0))=0. Exclude x=h=0; ordered queries=[0/1, 1/1, -1/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 0/1, 0/1], [0/1, 0/1, 0/1], [No, Yes, Yes])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 0/1, 0/1], [0/1, 0/1, 0/1], [No, Yes, Yes])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0052","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61f32367a5847c796483312b21b387b2c7486ff78dcf82a3e177b4e1a2cce84f","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143052,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"0/1\"],[true,\"2/1\",\"2/1\"],[true,\"2/1\",\"-2/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((2)x+(-2))/(x−(1))=0. Exclude x=h=1; ordered queries=[1/1, 2/1, 0/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 2/1, 2/1], [0/1, 2/1, -2/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 2/1, 2/1], [0/1, 2/1, -2/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0053","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86d53d4fa5a2626abd30c52f6b9d279ed362f29c0ae75cb6cfb6ab883b74b458","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143053,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"3/1\"],[true,\"-5/1\",\"-5/2\"],[true,\"-19/2\",\"-19/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((-10)x+(3))/(x−(0))=1. Exclude x=h=0; ordered queries=[0/1, 1/2, 2/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, -5/1, -19/2], [3/1, -5/2, -19/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, -5/1, -19/2], [3/1, -5/2, -19/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0054","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26c8f9cfe03154d8009f441d0b9aa6d0c2fe24be58c1e6580a70d5068362f94a","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143054,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"-5/1\"],[true,\"-18/1\",\"-9/1\"],[true,\"-21/2\",\"-21/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((-9)x+(4))/(x−(1))=-1. Exclude x=h=1; ordered queries=[1/1, 3/2, 3/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, -18/1, -21/2], [-5/1, -9/1, -21/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, -18/1, -21/2], [-5/1, -9/1, -21/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0055","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e9921aeb32bfa5b34a0dac26198f1701d01792db3daff6630c90daa345a1153","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143055,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"-11/1\"],[true,\"-30/1\",\"-15/1\"],[true,\"-27/2\",\"-27/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((-8)x+(5))/(x−(2))=0. Exclude x=h=2; ordered queries=[2/1, 5/2, 4/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, -30/1, -27/2], [-11/1, -15/1, -27/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, -30/1, -27/2], [-11/1, -15/1, -27/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0056","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69c9b40bf2b4af4b14e32d33e254ceae035099fcc987773379fea5b3a316a3b4","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143056,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"20/1\"],[true,\"32/1\",\"16/1\"],[true,\"2/1\",\"4/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((-7)x+(6))/(x−(-2))=1. Exclude x=h=-2; ordered queries=[-2/1, -3/2, 0/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 32/1, 2/1], [20/1, 16/1, 4/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 32/1, 2/1], [20/1, 16/1, 4/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0057","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6a379db3b5179d9463f0124c09f3bfb4f8b7dfd90fca8efce2355fc45550720","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143057,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"13/1\"],[true,\"21/1\",\"21/2\"],[true,\"3/2\",\"3/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((-6)x+(7))/(x−(-1))=-1. Exclude x=h=-1; ordered queries=[-1/1, -1/2, 1/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 21/1, 3/2], [13/1, 21/2, 3/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 21/1, 3/2], [13/1, 21/2, 3/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0058","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bf25d2e6125c7a3cf4fe26b685417ec02d53d6a8548b3f83f70a65c9fa274ba","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143058,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"8/1\"],[true,\"11/1\",\"11/2\"],[true,\"-1/1\",\"-2/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((-5)x+(8))/(x−(0))=0. Exclude x=h=0; ordered queries=[0/1, 1/2, 2/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 11/1, -1/1], [8/1, 11/2, -2/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 11/1, -1/1], [8/1, 11/2, -2/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0059","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fc0c20a625874d4a8514b48e338ec8c1a4196495b702710c164c8ff50437539","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143059,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"5/1\"],[true,\"5/1\",\"5/2\"],[true,\"-5/2\",\"-5/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((-4)x+(9))/(x−(1))=1. Exclude x=h=1; ordered queries=[1/1, 3/2, 3/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 5/1, -5/2], [5/1, 5/2, -5/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 5/1, -5/2], [5/1, 5/2, -5/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0060","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b4b05f0c122db8b271c56760b5b1a9303e2f230ebe33c9004727faa082a1214","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143060,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"4/1\"],[true,\"6/1\",\"3/1\"],[true,\"0/1\",\"0/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((-3)x+(10))/(x−(2))=-1. Exclude x=h=2; ordered queries=[2/1, 5/2, 4/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 6/1, 0/1], [4/1, 3/1, 0/1], [No, No, Yes])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 6/1, 0/1], [4/1, 3/1, 0/1], [No, No, Yes])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0061","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed7f0f181532d42cdb14cbe27c6e536a437c0dd1cb78e09ab6e5dd26368b279b","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143061,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"15/1\"],[true,\"28/1\",\"14/1\"],[true,\"11/2\",\"11/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((-2)x+(11))/(x−(-2))=0. Exclude x=h=-2; ordered queries=[-2/1, -3/2, 0/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 28/1, 11/2], [15/1, 14/1, 11/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 28/1, 11/2], [15/1, 14/1, 11/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0062","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a458510c98c8d65dd57a506ac5af93f4d8ff3850843fe8b829b68cb57a1b5f3e","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143062,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"13/1\"],[true,\"24/1\",\"12/1\"],[true,\"9/2\",\"9/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((-1)x+(12))/(x−(-1))=1. Exclude x=h=-1; ordered queries=[-1/1, -1/2, 1/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 24/1, 9/2], [13/1, 12/1, 9/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 24/1, 9/2], [13/1, 12/1, 9/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0063","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75c30ef45a13dd1915a78a89bc5c39b273e21f2d12028b401f14b8e7a046d0a8","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143063,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"13/1\"],[true,\"27/1\",\"27/2\"],[true,\"15/2\",\"15/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((0)x+(13))/(x−(0))=-1. Exclude x=h=0; ordered queries=[0/1, 1/2, 2/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 27/1, 15/2], [13/1, 27/2, 15/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 27/1, 15/2], [13/1, 27/2, 15/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0064","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"191e7db69898cb9265f4186f83f368c0a5bf6fd9ca5b2a74cf5bdec7209c1973","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143064,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"15/1\"],[true,\"31/1\",\"31/2\"],[true,\"17/2\",\"17/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((1)x+(14))/(x−(1))=0. Exclude x=h=1; ordered queries=[1/1, 3/2, 3/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 31/1, 17/2], [15/1, 31/2, 17/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 31/1, 17/2], [15/1, 31/2, 17/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0065","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6c20c6330db7933dd465cff8f0cc16983e07a55a87e8ee47815cf7597c38063","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143065,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"19/1\"],[true,\"39/1\",\"39/2\"],[true,\"21/2\",\"21/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((2)x+(15))/(x−(2))=1. Exclude x=h=2; ordered queries=[2/1, 5/2, 4/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 39/1, 21/2], [19/1, 39/2, 21/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 39/1, 21/2], [19/1, 39/2, 21/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0066","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"091c25109123dbe9080556eaeac577e7aeca13f71a61d71948625bd230d07f36","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143066,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"10/1\"],[true,\"24/1\",\"12/1\"],[true,\"9/1\",\"18/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((3)x+(16))/(x−(-2))=-1. Exclude x=h=-2; ordered queries=[-2/1, -3/2, 0/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 24/1, 9/1], [10/1, 12/1, 18/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 24/1, 9/1], [10/1, 12/1, 18/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0067","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8970383aad228960540ec50dfd406e6975910599885c8371f1b0bd073615cd06","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143067,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"13/1\"],[true,\"30/1\",\"15/1\"],[true,\"21/2\",\"21/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((4)x+(17))/(x−(-1))=0. Exclude x=h=-1; ordered queries=[-1/1, -1/2, 1/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 30/1, 21/2], [13/1, 15/1, 21/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 30/1, 21/2], [13/1, 15/1, 21/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0068","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25bf0f5994aa5dab9ef6bd74c07cb8fe4c3c903c4fa838c57749c918d772ae3e","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143068,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"18/1\"],[true,\"40/1\",\"20/1\"],[true,\"13/1\",\"26/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((5)x+(18))/(x−(0))=1. Exclude x=h=0; ordered queries=[0/1, 1/2, 2/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 40/1, 13/1], [18/1, 20/1, 26/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 40/1, 13/1], [18/1, 20/1, 26/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0069","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6236f5bfa81d76595d79bfdfe75a77251c9c95d5eb31aa17bc31252313eb11be","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143069,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"25/1\"],[true,\"57/1\",\"57/2\"],[true,\"39/2\",\"39/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((6)x+(19))/(x−(1))=-1. Exclude x=h=1; ordered queries=[1/1, 3/2, 3/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 57/1, 39/2], [25/1, 57/2, 39/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 57/1, 39/2], [25/1, 57/2, 39/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0070","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"459a12de6a3c0dc9b474a67a8ef9ac89c11c2de2f28db3b7b89157392ddeff71","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143070,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"34/1\"],[true,\"75/1\",\"75/2\"],[true,\"24/1\",\"48/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((7)x+(20))/(x−(2))=0. Exclude x=h=2; ordered queries=[2/1, 5/2, 4/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 75/1, 24/1], [34/1, 75/2, 48/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 75/1, 24/1], [34/1, 75/2, 48/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0071","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dff17c2559b3e4c5223e4cf24a1716632336ee62328893fa86193aafab3628e5","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143071,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"5/1\"],[true,\"17/1\",\"17/2\"],[true,\"19/2\",\"19/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((8)x+(21))/(x−(-2))=1. Exclude x=h=-2; ordered queries=[-2/1, -3/2, 0/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 17/1, 19/2], [5/1, 17/2, 19/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 17/1, 19/2], [5/1, 17/2, 19/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0072","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89a10a97efa179af12b0fe961f93f3754c6721b0a315bc06ce55bf1a216da63a","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143072,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"13/1\"],[true,\"36/1\",\"18/1\"],[true,\"33/2\",\"33/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((9)x+(22))/(x−(-1))=-1. Exclude x=h=-1; ordered queries=[-1/1, -1/2, 1/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 36/1, 33/2], [13/1, 18/1, 33/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 36/1, 33/2], [13/1, 18/1, 33/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0073","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5bb3f2d48e0c30b840b96db7dcf9b467c4f0472cf185c124c4c04f5accb8007","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143073,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"23/1\"],[true,\"56/1\",\"28/1\"],[true,\"43/2\",\"43/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((10)x+(23))/(x−(0))=0. Exclude x=h=0; ordered queries=[0/1, 1/2, 2/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 56/1, 43/2], [23/1, 28/1, 43/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 56/1, 43/2], [23/1, 28/1, 43/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0074","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e07998242a6908c062b52c25a48cce233a2adbe93a3d50caa29c445c4769853f","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143074,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"35/1\"],[true,\"80/1\",\"40/1\"],[true,\"55/2\",\"55/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((11)x+(24))/(x−(1))=1. Exclude x=h=1; ordered queries=[1/1, 3/2, 3/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 80/1, 55/2], [35/1, 40/1, 55/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 80/1, 55/2], [35/1, 40/1, 55/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0075","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6013d6de9320d21b1ac73587284aa4daaf622af3857a90603a657a1008248e31","family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143075,"semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"49/1\"],[true,\"111/1\",\"111/2\"],[true,\"75/2\",\"75/1\"]]"},"fields":{"title":"Original Denominator Guard versus Cross-Multiplication","stem":"Over Q, the original equation is ((12)x+(25))/(x−(2))=-1. Exclude x=h=2; ordered queries=[2/1, 5/2, 4/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [undefined, 111/1, 75/2], [49/1, 111/2, 75/1], [No, No, No])","explanation":"Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 111/1, 75/2], [49/1, 111/2, 75/1], [No, No, No])."},"skills":["Original Denominator Guard versus Cross-Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0076","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18b9b9401afa960f44570b58a2ca9796e3fc5bf193d63edb255e8d418b25a84f","family_id":"qf-density14-equation-common-shift-check","seed":202610143076,"semantic_template":"qf-density14-equation-common-shift-check/[\"1/1\",\"1/1\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(0)x+(0), R=(0)x+(0), x=0/1, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, 1/1, 1/1, 0/1, Yes)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (0/1, 1/1, 1/1, 0/1, Yes)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0077","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"182b0ad3fa593639091f1f213c8f86d5d29c9501b2f86594e4fd7d08cb0f8595","family_id":"qf-density14-equation-common-shift-check","seed":202610143077,"semantic_template":"qf-density14-equation-common-shift-check/[\"-1/2\",\"3/2\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-1), R=(-1)x+(2), x=1/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-2/1, -1/2, 3/2, -2/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-2/1, -1/2, 3/2, -2/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0078","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd53d7d0b19581406f755740d819d11d9f8ecf865579730287d6bca15d98ca52","family_id":"qf-density14-equation-common-shift-check","seed":202610143078,"semantic_template":"qf-density14-equation-common-shift-check/[\"0/1\",\"3/1\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-2), R=(-1)x+(3), x=2/2, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-3/1, 0/1, 3/1, -3/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-3/1, 0/1, 3/1, -3/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0079","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb97191c0abcfc96a0faaeaa470c1ec413c9ae83fe827dc0c69e50ce8473c507","family_id":"qf-density14-equation-common-shift-check","seed":202610143079,"semantic_template":"qf-density14-equation-common-shift-check/[\"-5/2\",\"3/2\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-3), R=(-1)x+(4), x=3/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-4/1, -5/2, 3/2, -4/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-4/1, -5/2, 3/2, -4/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0080","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c755a9bd99d1528ef4762518be1d95bd5067d09141d1f9c34310cb18981921c","family_id":"qf-density14-equation-common-shift-check","seed":202610143080,"semantic_template":"qf-density14-equation-common-shift-check/[\"-2/1\",\"3/1\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-4), R=(-1)x+(5), x=4/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-5/1, -2/1, 3/1, -5/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-5/1, -2/1, 3/1, -5/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0081","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f339ac122d1448114637bbd1f536ebf18ab7450025a68b3dd5f9011a1c22d7e","family_id":"qf-density14-equation-common-shift-check","seed":202610143081,"semantic_template":"qf-density14-equation-common-shift-check/[\"-3/2\",\"9/2\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-5), R=(-1)x+(6), x=5/2, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-6/1, -3/2, 9/2, -6/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-6/1, -3/2, 9/2, -6/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0082","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a9bd582de143f63ac76678e92ee8b217af892e97e6db09b55df4adb04ebe8ab","family_id":"qf-density14-equation-common-shift-check","seed":202610143082,"semantic_template":"qf-density14-equation-common-shift-check/[\"-4/1\",\"3/1\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-6), R=(-1)x+(7), x=6/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-7/1, -4/1, 3/1, -7/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-7/1, -4/1, 3/1, -7/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0083","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3917a30cbdc633aa5c4cc9e6013e6646a06c43cb17d55bfbef8faa808396899f","family_id":"qf-density14-equation-common-shift-check","seed":202610143083,"semantic_template":"qf-density14-equation-common-shift-check/[\"-7/2\",\"9/2\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-7), R=(-1)x+(8), x=7/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-8/1, -7/2, 9/2, -8/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-8/1, -7/2, 9/2, -8/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0084","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6372dfc2214503eccec391c9f32b7204850f2dfedec620da2ed72fedf6c07466","family_id":"qf-density14-equation-common-shift-check","seed":202610143084,"semantic_template":"qf-density14-equation-common-shift-check/[\"-3/1\",\"6/1\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-8), R=(-1)x+(9), x=8/2, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-9/1, -3/1, 6/1, -9/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-9/1, -3/1, 6/1, -9/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0085","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc89e6c17c32252118d49a8944708c688eb7a9d2d6e9e8ce9461a0ea38dbf299","family_id":"qf-density14-equation-common-shift-check","seed":202610143085,"semantic_template":"qf-density14-equation-common-shift-check/[\"-11/2\",\"9/2\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-9), R=(-1)x+(10), x=9/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-10/1, -11/2, 9/2, -10/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-10/1, -11/2, 9/2, -10/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0086","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6dd040053c6c799cf4cf8706754ca1b97be432671c1b63b3776447739abecd6","family_id":"qf-density14-equation-common-shift-check","seed":202610143086,"semantic_template":"qf-density14-equation-common-shift-check/[\"-5/1\",\"6/1\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-10), R=(-1)x+(11), x=10/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-11/1, -5/1, 6/1, -11/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-11/1, -5/1, 6/1, -11/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0087","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd0e8c7065ecc45f07a446e00879f1f1d586ee8abd7504c1eaed43acc36e92fe","family_id":"qf-density14-equation-common-shift-check","seed":202610143087,"semantic_template":"qf-density14-equation-common-shift-check/[\"-9/2\",\"15/2\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-11), R=(-1)x+(12), x=11/2, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-12/1, -9/2, 15/2, -12/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-12/1, -9/2, 15/2, -12/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0088","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc933a29384896cad6aec0c5d9ae1b39a818cb83460af9c2bfbe22f3a67feceb","family_id":"qf-density14-equation-common-shift-check","seed":202610143088,"semantic_template":"qf-density14-equation-common-shift-check/[\"-7/1\",\"6/1\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-12), R=(-1)x+(13), x=12/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-13/1, -7/1, 6/1, -13/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-13/1, -7/1, 6/1, -13/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0089","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db112f8e9180b39ee51350b965e0987de0a2ae565ca5c26ed35759d9192dee3b","family_id":"qf-density14-equation-common-shift-check","seed":202610143089,"semantic_template":"qf-density14-equation-common-shift-check/[\"-13/2\",\"15/2\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-13), R=(-1)x+(14), x=13/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-14/1, -13/2, 15/2, -14/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-14/1, -13/2, 15/2, -14/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0090","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a12170aff3702edefa28fcb945989b9dbe871fd87f424c0d07c92a7cecaee5b3","family_id":"qf-density14-equation-common-shift-check","seed":202610143090,"semantic_template":"qf-density14-equation-common-shift-check/[\"-6/1\",\"9/1\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-14), R=(-1)x+(15), x=14/2, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-15/1, -6/1, 9/1, -15/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-15/1, -6/1, 9/1, -15/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0091","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85c208db14224fd925bdaeb36326588446498038349415b8f985a9a5cf8ad727","family_id":"qf-density14-equation-common-shift-check","seed":202610143091,"semantic_template":"qf-density14-equation-common-shift-check/[\"-17/2\",\"15/2\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-15), R=(-1)x+(16), x=15/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-16/1, -17/2, 15/2, -16/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-16/1, -17/2, 15/2, -16/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0092","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79d723742b75c9389b02301fb8d7e7776d99c3ccf2daef5438f46080658b0da3","family_id":"qf-density14-equation-common-shift-check","seed":202610143092,"semantic_template":"qf-density14-equation-common-shift-check/[\"-8/1\",\"9/1\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-16), R=(-1)x+(17), x=16/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-17/1, -8/1, 9/1, -17/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-17/1, -8/1, 9/1, -17/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0093","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3f2f0283f20f97a2c231d0db0c48562661fd80cce720be79f2ee9659e98e8d5","family_id":"qf-density14-equation-common-shift-check","seed":202610143093,"semantic_template":"qf-density14-equation-common-shift-check/[\"-15/2\",\"21/2\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-17), R=(-1)x+(18), x=17/2, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-18/1, -15/2, 21/2, -18/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-18/1, -15/2, 21/2, -18/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0094","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5a0124f0fb0885e54a9908d05b4545776a07178b518ee5e2afce320b7f993bc","family_id":"qf-density14-equation-common-shift-check","seed":202610143094,"semantic_template":"qf-density14-equation-common-shift-check/[\"-10/1\",\"9/1\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-18), R=(-1)x+(19), x=18/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-19/1, -10/1, 9/1, -19/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-19/1, -10/1, 9/1, -19/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0095","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff533b63876ebfd864b26a6159a9d24f6fd9c86f4a3996c1d4dccefb6d01aef7","family_id":"qf-density14-equation-common-shift-check","seed":202610143095,"semantic_template":"qf-density14-equation-common-shift-check/[\"-19/2\",\"21/2\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-19), R=(-1)x+(20), x=19/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-20/1, -19/2, 21/2, -20/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-20/1, -19/2, 21/2, -20/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0096","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9163c692d2322e01f87f203a3a555af5a41eb2ca4bfcf409190694e7bb3bf64f","family_id":"qf-density14-equation-common-shift-check","seed":202610143096,"semantic_template":"qf-density14-equation-common-shift-check/[\"-9/1\",\"12/1\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-20), R=(-1)x+(21), x=20/2, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-21/1, -9/1, 12/1, -21/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-21/1, -9/1, 12/1, -21/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0097","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7b46f3734b515772536f368a6af59b86b3c4e22e578152f512a25c9095e07df","family_id":"qf-density14-equation-common-shift-check","seed":202610143097,"semantic_template":"qf-density14-equation-common-shift-check/[\"-23/2\",\"21/2\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-21), R=(-1)x+(22), x=21/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-22/1, -23/2, 21/2, -22/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-22/1, -23/2, 21/2, -22/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0098","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c6a041df5f2604aac6773f0edcced19fd7d89ba44667465ddcbb31c41861cff","family_id":"qf-density14-equation-common-shift-check","seed":202610143098,"semantic_template":"qf-density14-equation-common-shift-check/[\"-11/1\",\"12/1\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-22), R=(-1)x+(23), x=22/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-23/1, -11/1, 12/1, -23/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-23/1, -11/1, 12/1, -23/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0099","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91e77e8ee14692333ec4888ad20ff13a84f38810ac531a84f8e9bb1d123647d5","family_id":"qf-density14-equation-common-shift-check","seed":202610143099,"semantic_template":"qf-density14-equation-common-shift-check/[\"-21/2\",\"27/2\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-23), R=(-1)x+(24), x=23/2, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-24/1, -21/2, 27/2, -24/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-24/1, -21/2, 27/2, -24/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r03-equation-0100","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18c1e0cccfeaaff72fb30e7d668822f2ba5bbd2e662238159747f4d7aa2098b9","family_id":"qf-density14-equation-common-shift-check","seed":202610143100,"semantic_template":"qf-density14-equation-common-shift-check/[\"-13/1\",\"12/1\"]"},"fields":{"title":"Both Equation Sides after a Common Shift","stem":"Over Q, L=(1)x+(-24), R=(-1)x+(25), x=24/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-25/1, -13/1, 12/1, -25/1, No)","explanation":"Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-25/1, -13/1, 12/1, -25/1, No)."},"skills":["Both Equation Sides after a Common Shift"],"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."}}