{"id":"cb_question_bank:b2cfac8c-e812-499a-a440-ed0f458ba864","source_id":"cb_question_bank","source_item_id":"b2cfac8c-e812-499a-a440-ed0f458ba864","external_id":null,"ibn":"06323-DC","question_id":"1697ffcf","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":"spr","answer_source":"rationale","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In the xy -plane, the graph of y equals 3 x squared minus 14 x intersects the graph of y equals x at the points with coordinates zero comma zero and a, comma a . What is the value of a ?","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">In the<span class=\"italic\"><span class=\"italic \"> xy</span></span>-plane, the graph of <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f622453d9165b80ae96ab150c9554cf6145280e2f4cffb36c4592139830013cb\" alt=\"y equals 3 x squared minus 14 x\"></span></span> intersects the graph of <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1bdcfc738c339d45c2a78f9584c9c000a6555d7c13f624184822702800a07645\" alt=\"y equals x\"></span></span> at the points <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/970784ff7a9c2b48a6f39d0559ad70393a7133391e6e97a52ec6cdaebf220a0c\" alt=\"with coordinates zero comma zero\"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/22de8e8d593ddebeb41020f3e3fbc35c6867ccd3fc0ee78acecf5fbf5df31ea5\" alt=\"a, comma a\"></span><span class=\"math_expression \"></span>. What is the value of <span class=\"italic\">a</span> ?</p>\n","rationale_html":"<p>The correct answer is 5. The intersection points of the graphs of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f622453d9165b80ae96ab150c9554cf6145280e2f4cffb36c4592139830013cb\" alt=\"y equals, 3 x squared, minus 14 x\"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1bdcfc738c339d45c2a78f9584c9c000a6555d7c13f624184822702800a07645\" alt=\"y equals x\"></span> can be found by solving the system consisting of these two equations. To solve the system, substitute <span class=\"italic\">x</span> for <span class=\"italic\">y</span> in the first equation. This gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/11eb73cf1a1ca086ac3cb65de8680fc2d4a97369f64880bbc5e221d8ec066c74\" alt=\"x equals, 3 x squared, minus 14 x\"></span>. Subtracting <span class=\"italic\">x</span> from both sides of the equation gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/8037c80fa3fbf1edb3de0bf643e600c3eb64427765fb5192b149468f6fcf5bb1\" alt=\"0 equals, 3 x squared, minus 15 x\"></span>. Factoring <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/69a2a80c807e391764b4352ceabcc097fb477aedbcbc956d1cdf0ec68f19f8e4\" alt=\"3 x\"></span> out of each term on the left-hand side of the equation gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/64f0187dfa8fc105987779569c7baa441d8da0ab33fd470a45b890b0b58ca7ef\" alt=\"0 equals, 3 x times, open parenthesis, x minus 5, close parenthesis\"></span>. Therefore, the possible values for <span class=\"italic\">x</span> are 0 and 5. Since <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1bdcfc738c339d45c2a78f9584c9c000a6555d7c13f624184822702800a07645\" alt=\"y equals x\"></span>, the two intersection points are <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5d9546145044d67f4c5d0648e8663a351f932cd21526a7309901dbb9e25d27fb\" alt=\"the points with coordinates 0 comma 0\"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/3d0b4d4523057d936547580354e45d7aa801b7155efdd47f28da0d3d587ef5d3\" alt=\"5 comma 5\"></span>. Therefore, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/226beed4da982a407b56e0dbe36d299218cf28710731f2bfb68637f4175e84b5\" alt=\"a, equals 5\"></span>.</p>\n","correct_answer":["5"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.630000+00:00","source_updated_at":"2023-08-02T20:25:59.630000+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}}