{"id":"cb_question_bank:fc24cdc5-c2e0-49f7-a1cb-7719f1519633","source_id":"cb_question_bank","source_item_id":"fc24cdc5-c2e0-49f7-a1cb-7719f1519633","external_id":null,"ibn":"027878-DC","question_id":"4b76c7f1","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.D.","skill":"Systems of two linear equations in two variables","difficulty":"E","score_band":3,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In the given system of equations, a is a constant. If the system has infinitely many solutions, what is the value of a ?","stimulus":"Equation 1: 2 x, plus 7 y, equals 9. Equation 2: 8 x, plus 28 y, equals a","stimulus_html":"<div class=\"stimulus_reference \">\n        <p class=\"standalone_statement style:1 \">\n          <span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/9c2ad97157f53305596c0417556496c492dec46d9617716c0a1778c0294edbeb\" alt=\"Equation 1: 2 x, plus 7 y, equals 9. Equation 2: 8 x, plus 28 y, equals a\"></span></span>\n        </p>\n      </div>\n","stem_html":"<p class=\"stem_paragraph \">In the given system of equations, <span class=\"italic\">a</span> is a constant. If the system has infinitely many solutions, what is the value of <span class=\"italic\">a</span> ?</p>\n","rationale_html":"<p>Choice C is correct. A system of two linear equations has infinitely many solutions if one equation is equivalent to the other. This means that when the two equations are written in the same form, each coefficient or constant in one equation is equal to the corresponding coefficient or constant in the other equation multiplied by the same number. The equations in the given system of equations are written in the same form, with<span class=\"italic\"> x</span> and<span class=\"italic\"> y</span> on the left-hand side of the equation and a constant on the right-hand side of the equation. The coefficients of <span class=\"italic\">x</span> and <span class=\"italic\">y</span> in the second equation are equal to the coefficients of <span class=\"italic\">x</span> and<span class=\"italic\"> y</span>, respectively, in the first equation multiplied by 4: <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4bdced638c363fbe5a0d259f5797e5c95fa19db50e738489bd57954385349d5c\" alt=\"8 equals, 2 times 4 \"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/171c373ab3cd0ee962381d1da9424d6b538f0841b1834b7df4acb7f92ae34db7\" alt=\"28 equals, 7 times 4\"></span>. Therefore, the constant in the second equation must be equal to 4 times the constant in the first equation: <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bd95dde13857ea5c57e47786082296f46b85c6382436b9776668c80ac2a879bd\" alt=\"a, equals, 9 times 4\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f617b2e3a2b32ed255c7237ecdd5c8ea28e24e4d311b6379830cdd20117f1b30\" alt=\"a, equals 36\"></span>.</p><p>Choices A, B, and D are incorrect. When <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/a6dcfb5d78d2a8aaea4e941281354ada11d9a4d74ead05891bc9376286558278\" alt=\"a, equals 4\"></span>, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/870dee33085d101927f4344d2907d50e004a4b6b543823d3e747127cdd9d7f1d\" alt=\"a, equals 9\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/63cb7dadbe084522d2a18cfda4789b85644f465a28c177d74422a2139510279a\" alt=\"a, equals 54\"></span>, the given system of equations has no solution.</p><p></p>\n","correct_answer":["C"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.622000+00:00","source_updated_at":"2023-08-02T20:25:59.622000+00:00","options":[{"label":"A","content_html":"<p class=\"choice_paragraph \">4</p>\n","ord":0},{"label":"B","content_html":"<p class=\"choice_paragraph \">9</p>\n","ord":1},{"label":"C","content_html":"<p class=\"choice_paragraph \">36</p>\n","ord":2},{"label":"D","content_html":"<p class=\"choice_paragraph \">54</p>\n","ord":3}],"attribution":{"source_id":"cb_question_bank","source_name":"College Board digital SAT question bank (community dump)","rights_holder":"College Board","canonical_url":"https://satsuitequestionbank.collegeboard.org/","retrieved_from":"https://raw.githubusercontent.com/mdn522/sat-question-bank/main/data/cb-digital-questions.json","attribution":"SAT practice questions © College Board, from the official SAT Suite Question Bank. Retrieved via the community dump at github.com/mdn522/sat-question-bank.","disclaimer":"SAT® and PSAT/NMSQT® are trademarks registered by the College Board, which is not affiliated with, does not sponsor, and does not endorse this project.","license":"College Board copyright; source repository carries no licence. Local use only.","redistributable":false,"item_count":2017}}