{"id":"cb_question_bank:cb007c32-1efa-4b00-8925-75c80e21fce8","source_id":"cb_question_bank","source_item_id":"cb007c32-1efa-4b00-8925-75c80e21fce8","external_id":null,"ibn":"09411-DC","question_id":"2c5c22d0","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.B.","skill":"Nonlinear equations in one variable and systems of equations in two variables","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"How many solutions are there to the system of equations above?","stimulus":"y equals, x squared, plus 3 x, minus 7, and, y minus 5 x, plus 8, equals 0","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/c1c5a3812f8125344521691333fd1043734f76f2915c73d8fc5963bae602b2d7\" alt=\"y equals, x squared, plus 3 x, minus 7, and, y minus 5 x, plus 8, equals 0\n\"></span></span>\n        </p>\n      </div>\n","stem_html":"<p class=\"stem_paragraph \">How many solutions are there to the system of equations above?</p>\n","rationale_html":"<p>Choice C is correct. The second equation of the system can be rewritten as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/7d05222a4c16684ed691a8dc5ed4a8ea37fa709704425933c4b35456676e753b\" alt=\"y equals, 5 x minus 8\"></span>. Substituting <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/aeb1fe1a06b5b322e412293a6d2e54b8bffff82571b3f90a152e315f871e32ae\" alt=\"5 x minus 8\"></span> for <span class=\"italic\">y</span> in the first equation gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/142a0d168b6c23b92fcd06242cc1f3e048d224a180ef2908ca72d45b4bbf8c04\" alt=\"5 x minus 8, equals, x squared, plus 3 x, minus 7\"></span>. This equation can be solved as shown below:</p><p>\n        <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/267a4831ed33308e46dbade160963f7b50c981b4f16f2effae4f192cf5ba9e75\" alt=\"x squared, plus 3 x, minus 7, minus 5 x, plus 8, equals 0\"></span></p><p>\n        <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c1262f19c0947c44f227eaa96524fd147bd49b61fa8bbec1798300187c34963d\" alt=\"x squared, minus 2 x, plus 1, equals 0\"></span></p><p>\n        <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2a168049d9e6212684e324e56584d8fd220f4710b7124829db86695970be4f58\" alt=\"open parenthesis, x minus 1, close parenthesis, squared, equals 0\"></span></p><p>\n        <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6e4319fe2c5755fc5aa3f69e1a43b989d061cbb17c450f24b51a6616f131de4b\" alt=\"x equals 1\"></span></p><p>Substituting 1 for <span class=\"italic\">x</span> in the equation <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/7d05222a4c16684ed691a8dc5ed4a8ea37fa709704425933c4b35456676e753b\" alt=\"y equals, 5 x minus 8\"></span> gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d8c887aafaf2e0e57173c2666d8f50a99ed9e2beff01e81c88790b2415dd4914\" alt=\"y equals negative 3\"></span>. Therefore, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/fd688fd2bbb64bfd4de599b38eb86e6fb96602aef7b5248717a4f8a8f1be3268\" alt=\"the ordered pair 1 comma negative 3\"></span> is the only solution to the system of equations.</p><p>Choice A is incorrect. In the <span class=\"italic\">xy</span>-plane, a parabola and a line can intersect at no more than two points. Since the graph of the first equation is a parabola and the graph of the second equation is a line, the system cannot have more than 2 solutions. Choice B is incorrect. There is a single ordered pair <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/338db7e3a31d3afebe84d6ca556e60cbe33b3baf49002acc912917c4ff25d31e\" alt=\"x comma y\"></span> that satisfies both equations of the system. Choice D is incorrect because the ordered pair <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/fd688fd2bbb64bfd4de599b38eb86e6fb96602aef7b5248717a4f8a8f1be3268\" alt=\"1 comma negative 3\"></span> satisfies both equations of the system.</p><p></p>\n","correct_answer":["C"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.637000+00:00","source_updated_at":"2023-08-02T20:25:59.637000+00:00","options":[{"label":"A","content_html":"<p>There are exactly 4 solutions.</p>\n","ord":0},{"label":"B","content_html":"<p>There are exactly 2 solutions.</p>\n","ord":1},{"label":"C","content_html":"<p>There is exactly 1 solution.</p>\n","ord":2},{"label":"D","content_html":"<p>There are no solutions.</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}}