{"id":"cb_question_bank:41005ea5-49ba-49df-a5aa-c5b501f529f9","source_id":"cb_question_bank","source_item_id":"41005ea5-49ba-49df-a5aa-c5b501f529f9","external_id":null,"ibn":"028796-DC","question_id":"6e02cd78","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":5,"answer_type":"spr","answer_source":"rationale","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In the xy -plane, what is the y -coordinate of the point of intersection of the graphs of y equals, open parenthesis, x minus 1, close parenthesis, squared and y equals, 2 x minus 3 ?","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">In the <span class=\"italic \">xy</span>-plane, what is the <span class=\"italic \">y</span>-coordinate of the point of intersection of the graphs of <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/cf8280cb4157fcf1507a5a6726702997faae44af8a43c0d97fb2094ea81bd65b\" alt=\"y equals, open parenthesis, x minus 1, close parenthesis, squared \"></span></span> and <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/07b5d2b00b4e5045e8a1323d829a593ef8ae8a8c1148882174c56cdae2c2f285\" alt=\"y equals, 2 x minus 3\"></span>&nbsp;</span>?</p>\n","rationale_html":"<p>The correct answer is 1. The point of intersection of the graphs of the given equations is the solution to the system of the two equations. Since <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/cf8280cb4157fcf1507a5a6726702997faae44af8a43c0d97fb2094ea81bd65b\" alt=\"y equals, open parenthesis, x minus 1, close parenthesis, squared \"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/07b5d2b00b4e5045e8a1323d829a593ef8ae8a8c1148882174c56cdae2c2f285\" alt=\"y equals, 2 x minus 3\"></span>, it follows that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/00c5d20ead07e5817b6cb24c9ae15296b282c8892e4fc4d649bf0db47497a03f\" alt=\"open parenthesis, x minus 1, close parenthesis, squared, equals, 2 x minus 3\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/3c75b7eb1fd8eb00101d7fff84080efe604f1824bb4bff060f49acbfd1510274\" alt=\"open parenthesis, x minus 1, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, equals, 2 x minus 3\"></span>. Applying the distributive property to the left-hand side of this equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/132ed85e1588451ec14836523dcd8df36c1a5f63188c397c9de7533cc52baf30\" alt=\"x squared, minus 2 x, plus 1, equals, 2 x minus 3\"></span>. Subtracting <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/3ad1ab966f135e8f8e49df526609c11ae015bcd74fa24b4bb565d5469f14ff35\" alt=\"2 x\"></span> from and adding 3 to both sides of this equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/065e7a24e9418bb5f2a39e5a9c4bb7e4e99ab63e9cdd8079a9687b7ff40e8fdb\" alt=\"x squared, minus 4 x, plus 4, equals 0\"></span>. Factoring the left-hand side of this equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/885097e78c0fa9b8ba24b03507d2e0affcb8044c032c7fe0b7b1a44a1b917173\" alt=\"open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x minus 2, close parenthesis, equals 0\"></span>. By the zero product property, if <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/885097e78c0fa9b8ba24b03507d2e0affcb8044c032c7fe0b7b1a44a1b917173\" alt=\"open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x minus 2, close parenthesis, equals 0\"></span>, it follows that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4ac57730081fd704588a223e1e63797c42339f131ed644c8b4cf701a2ad12a30\" alt=\"x minus 2, equals 0\"></span>. Adding 2 to both sides of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4ac57730081fd704588a223e1e63797c42339f131ed644c8b4cf701a2ad12a30\" alt=\"x minus 2, equals 0\"></span> yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/401169281090f4f1d9aafeeb928524e672d737cdce814b4e9256c5e06a42d325\" alt=\"x equals 2\"></span>. Substituting 2 for <span class=\"italic\">x</span> in either of the given equations yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/cc0d4aec836ba9b82085d477b7ea166b87d9abd1486c2dd63be7e952b395ed15\" alt=\"y equals 1\"></span>. For example, substituting 2 for <span class=\"italic\">x</span> in the second given equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/20a1f442fcb5b85f3219b3ae538f7acdaecaa88551a694fc70755979fd32fd5a\" alt=\"y equals, 2 times 2, minus 3\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/cc0d4aec836ba9b82085d477b7ea166b87d9abd1486c2dd63be7e952b395ed15\" alt=\"y equals 1\"></span>. Therefore, the point of intersection of the graphs of the given equations is <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/99480164454b02ecf792c33a133a14ca2e7231c181912988b6927bd32d27b345\" alt=\"the point with coordinates 2 comma 1\"></span>. The <span class=\"italic\">y</span>-coordinate of this point is 1.</p>\n","correct_answer":["1"],"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":[],"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}}