{"id":"cb_question_bank:e883decd-767b-4a56-993b-e416ce05f2ed","source_id":"cb_question_bank","source_item_id":"e883decd-767b-4a56-993b-e416ce05f2ed","external_id":null,"ibn":"08143-DC","question_id":"2d2ab76b","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":"M","score_band":4,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"When the equations above are graphed in the xy -plane, what are the coordinates ( x , y ) of the points of intersection of the two graphs?","stimulus":"y equals, x squared, minus 1 and y equals 3","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/a1f06564cef0b3780d7476ccd375f78ad5ebc6333d16a6c1825dfd63d30d14d4\" alt=\"y equals, x squared, minus 1 and y equals 3\"></span></span>\n        </p>\n      </div>\n","stem_html":"<p class=\"stem_paragraph \">When the equations above are graphed in the <span class=\"italic\"><span class=\"formatted_text font_style:italic \">xy</span></span>-plane, what are the coordinates (<span class=\"formatted_text font_style:italic \"><span class=\"italic\">x</span>, <span class=\"italic\">y</span></span>) of the points of intersection of the two graphs?</p>\n","rationale_html":"<p>Choice A is correct. The two equations form a system of equations, and the solutions to the system correspond to the points of intersection of the graphs. The solutions to the system can be found by substitution. Since the second equation gives <span class=\"italic\">y</span> = 3, substituting 3 for <span class=\"italic\">y</span> in the first equation gives 3 = <span class=\"italic\">x</span><sup>2</sup> – 1. Adding 1 to both sides of the equation gives 4 = <span class=\"italic\">x</span><sup>2</sup>. Solving by taking the square root of both sides of the equation gives <span class=\"italic\">x</span> = ±2. Since <span class=\"italic\">y</span> = 3 for all values of <span class=\"italic\">x</span> for the second equation, the solutions are (2, 3) and (–2, 3). Therefore, the points of intersection of the two graphs are (2, 3) and (–2, 3).</p><p>Choices B, C, and D are incorrect and may be the result of calculation errors.</p><p></p>\n","correct_answer":["A"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.635000+00:00","source_updated_at":"2023-08-02T20:25:59.635000+00:00","options":[{"label":"A","content_html":"<span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/363e48748cb197dad493adc9e8061cc0a503ad60bd216c5862e3e4fefc7ed096\" alt=\"2 comma 3\"></span><p> and <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/078935707b2cbcfda587933a2af89f5fcdd712d683fc6d866b7e486ba1b07532\" alt=\"negative 2 comma 3\"></span></span>\n</p></span>\n","ord":0},{"label":"B","content_html":"<span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/51485fa5d3349b96da9cb9a0ba0f68ce64a97b24ef0f81b1aa741e2afe70eedd\" alt=\"2 comma 4\"></span><p> and <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2adbf77e3b2dc2d25e3860b7622183500739d691b99cda70e693d3be29117fec\" alt=\"negative 2 comma 4\"></span></span>\n</p></span>\n","ord":1},{"label":"C","content_html":"<span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ebdbc03908fc3a98281c131fe1c307bd68bec46ff9869400fa9699c0a64dbcbc\" alt=\"3 comma 8\"></span><p> and <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f14b09d23c2ad1bd61a24803328a5520333dcc2bbbb849a0e9d04c78223f919c\" alt=\"negative 3 comma 8\"></span></span>\n</p></span>\n","ord":2},{"label":"D","content_html":"<span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f3d73ff303d51093bd639de19e026ccd198b204035b537df83f6f99e28e8ca68\" alt=\"the square root of 2 comma 3\"></span><p> and <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/3822454f2a803c7ef444ceace66135d1a49a137321160ee7c39619d337c69411\" alt=\"the negative square root of 2 comma 3\"></span></span>\n</p></span>\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}}