{"id":"cb_question_bank:11071bc4-2682-42c3-a2df-894e727fcfdc","source_id":"cb_question_bank","source_item_id":"11071bc4-2682-42c3-a2df-894e727fcfdc","external_id":null,"ibn":"05327-DC","question_id":"4661e2a9","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":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"Which ordered pair is a solution to the system of equations above?","stimulus":"Equation 1: x minus y, equals 1. Equation 2: x plus y equals, x squared minus 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/c6b24713280df2c221661eaf2db262f984d0d4f86e64bac05fcb758bf1c39902\" alt=\"Equation 1: x minus y, equals 1. Equation 2: x plus y equals, x squared minus 3\"></span></span>\n        </p>\n      </div>\n","stem_html":"<p class=\"stem_paragraph \">Which ordered pair is a solution to the system of equations above?</p>\n","rationale_html":"<p>Choice A is correct. The solution to the given system of equations can be found by solving the first equation for <span class=\"italic\">x</span>, which gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0904b9c72f38d79002e62616cf4b6ced9eba984a7f1eeb3b570c7e517d17d33c\" alt=\"x equals, y plus 1\"></span>, and substituting that value of <span class=\"italic\">x</span> into the second equation which gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/a9c565feaa98ee32ac4cd8138b749e2afffabc53ed532b17d1816a96d2acbfe3\" alt=\"y plus 1, plus y, equals, open parenthesis, y plus 1, close parenthesis, squared, minus 3\"></span>. Rewriting this equation by adding like terms and expanding <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/20e12c0b3dac1c64f40c84f274e7ff7809fdf2045e4b4f4ef618d04f1593f9ef\" alt=\"open parenthesis, y plus 1, close parenthesis, squared\"></span> gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/76a38e6d6a7821c2406910dc4568db1a98ee0a23524110cb9b3c30fdb78ec8d8\" alt=\"2 y plus 1, equals, y squared plus 2 y, minus 2\"></span>. Subtracting <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/859a2a2854fb0d081dfe8aa77ec2c8669e51439a083217044da2e0cb9e755e8c\" alt=\"2 y\"></span> from both sides of this equation gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/654bfdfc77ce23b009e37a3f764a00c89b5abae705d6e8ee31827bd3f80c1248\" alt=\"1 equals, y squared minus 2\"></span>. Adding to 2 to both sides of this equation gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/858003dc972b677814a173ff618dc21e32be2b8c542af0f51879ce47030e0a9d\" alt=\"3 equals y squared\"></span>. Therefore, it follows that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ea76f03b6f03ad5e5a57a36d41d6f17598d461f810dc610d89952beae5cdef6e\" alt=\"y equals, plus or minus the square root of 3\"></span>. Substituting <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/548f291932b05f462911ef8ecc28c868d5c8cd912ea251c39ba1e00283050296\" alt=\"the square root of 3\"></span> for <span class=\"italic\">y</span> in the first equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c68696e0183e918fccd94962daa99cf317dfae745dc1fba36ed92e45b64f2a3a\" alt=\"x minus the square root of 3, equals 1\"></span>. Adding <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/548f291932b05f462911ef8ecc28c868d5c8cd912ea251c39ba1e00283050296\" alt=\"the square root of 3 \"></span> to both sides of this equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/cc693c35972ddb88113cdd3a90f5a5ed81d46abe6adc518b95ffc4484540902a\" alt=\"x equals, 1 plus the square root of 3\"></span>. Therefore, the ordered pair <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4bdbf8ce6dc5872ba07b05be9ce409ca0e32e082fa2496b268a6191f965e6395\" alt=\"1 plus the square root of 3, end root, comma the square root of 3\"></span> is a solution to the given system of equations.</p><p>Choice B is incorrect. Substituting <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/548f291932b05f462911ef8ecc28c868d5c8cd912ea251c39ba1e00283050296\" alt=\"the square root of 3 \"></span> for <span class=\"italic\">x</span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2c5292d7b08f36418c22bdaaa6e046261e53aa0a141723538b28771d9e1d8e3d\" alt=\"the negative of the square root of 3 \"></span> for <span class=\"italic\">y</span> in the first equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/21a06264072d18cfd13f90bbdaaca21384aae58302936d17680bb513813f0fa7\" alt=\"the square root of 3, end root, minus, the negative of the square root of 3, equals 1\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d17c2ab5f047c69e00cf72ebb96c1230e07fac47aa0cca2754ef42f3e06bd585\" alt=\"2 times the square root of 3, equals 1\"></span>, which isn’t a true statement. Choice C is incorrect. Substituting <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f39d35ae58a33a473d1de4f6dc2e8b288550b28c10744647ffe1017ade0c86fe\" alt=\"1 plus the square root of 5 \"></span> for <span class=\"italic\">x</span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/aaaaab78195c53e195e4d77973b5a6313ddaef72fc8548189167ac02255d29ed\" alt=\"the square root of 5 \"></span> for <span class=\"italic\">y</span> in the second equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b249df225f0f4368e906702bc807c39a046482ce858883fb9850ac09ad6bd076\" alt=\"open parenthesis, 1 plus the square root of 5, close parenthesis, plus the square root of 5, equals, open parenthesis, 1 plus the square root of 5, close parenthesis, squared, minus 3\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e937ed36a77de0211df57148cad7e6fcddca032a7d1d13192531f811c1eee26e\" alt=\"1 plus, 2 times the square root of 5, equals, 2 times the square root of 5, plus 3\"></span>, which isn’t a true statement. Choice D is incorrect. Substituting <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/aaaaab78195c53e195e4d77973b5a6313ddaef72fc8548189167ac02255d29ed\" alt=\"the square root of 5\"></span> for <span class=\"italic\">x</span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/61b864bbe137f091d81c16c163e82412fb9af1469a313ecfe8bbd23aa3637b3f\" alt=\"open parenthesis, negative 1 plus the square root of 5, close parenthesis \"></span> for <span class=\"italic\">y</span> in the second equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/7468f420d38b7236178d98dbb4f4b353eb26936c0e6249dfa2b389095badcaea\" alt=\"the square root of 5 plus, open parenthesis, negative 1 plus the square root of 5, close parenthesis, equals, the square root of 5, squared, minus 3\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6572b3802e5769a81de43c09a13b870d1a15c6c6f277a31e5fcf27ed2be032ff\" alt=\"2 times the square root of 5, minus 1, equals 2\"></span>, which isn’t a true statement.</p><p></p>\n","correct_answer":["A"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.629000+00:00","source_updated_at":"2023-08-02T20:25:59.629000+00:00","options":[{"label":"A","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/cb1a979191aedcff0931260c8c079c19398152e80775ee0c96764f14ddfb4716\" alt=\"1 plus the square root of 3, comma, the square root of 3\"></span></p>\n","ord":0},{"label":"B","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5a4bbe060366c997785bf4dcea7b7f35802467c23410e4f070c3c6aa45bff825\" alt=\"the square root of 3, comma, the negative of the square root of 3\"></span></p>\n","ord":1},{"label":"C","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1d9c9db354c5eefeaa750de3c6cb2f954012fa822c3bb008e22dc1199232328d\" alt=\"1 plus the square root of 5, comma, the square root of 5\"></span></p>\n","ord":2},{"label":"D","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/dfd92701dcb277274259719d13e19924f59b7f3bfac6fcd358ae539934b600e2\" alt=\"the square root of 5, comma, negative 1 plus the square root of 5\"></span></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}}