{"id":"cb_question_bank:d443ef51-d0b2-4400-97aa-26776aeb1637","source_id":"cb_question_bank","source_item_id":"d443ef51-d0b2-4400-97aa-26776aeb1637","external_id":null,"ibn":"10864-DC","question_id":"3f8a701b","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.A.","skill":"Linear equations in one variable","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The equation 9 x plus 5, equals a, times, open parenthesis, x plus b, close parenthesis , where a and b are constants, has no solutions. Which of the following must be true? I. a, equals 9 II. b equals 5 III. b is not equal to five ninths","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">The equation <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bdf90888488e3348dcda0ef7caee5d65cd4d9d24dceb8e8b1d65955250f75bbf\" alt=\"9 x plus 5, equals a, times, open parenthesis, x plus b, close parenthesis\"></span></span>, where <span class=\"italic\">a</span> and <span class=\"italic\">b</span> are constants, has no solutions. Which of the following must be true?</p><p class=\"list_paragraph \">\n            <span class=\"math_expression \">I. <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/870dee33085d101927f4344d2907d50e004a4b6b543823d3e747127cdd9d7f1d\" alt=\"a, equals 9\"></span></span>\n          </p><p class=\"list_paragraph \">\n            <span class=\"math_expression \">II. <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/70bcc5029fa8429202856178a8f15b635fd9bd8c2d07442b86f0932b9655d401\" alt=\"b equals 5\"></span></span>\n          </p><p class=\"list_paragraph \">\n            <span class=\"math_expression \">III. <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/14e43c4c2b7bd7d320adc08d5a715b9d5773c0365400f3574f142a40b69996c2\" alt=\"b is not equal to five ninths\"></span></span>\n          </p><p></p>\n","rationale_html":"<p>Choice D is correct. For a linear equation in a form <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1064d43cc9f19a492a9dce5ab595c757b238d73e8baa6c823327927f9f1e8e55\" alt=\"a, x plus b, equals, c x plus d\"></span> to have no solutions, the <span class=\"italic\">x</span>-terms must have equal coefficients and the remaining terms must not be equal. Expanding the right-hand side of the given equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d89534bfcab6f281c7db16b20eabe39609517b0a658c1e73452e25e597aad185\" alt=\"9 x plus 5, equals, a, x plus a, b\"></span>. Inspecting the <span class=\"italic\">x</span>-terms, 9 must equal <span class=\"italic\">a, </span>so statement I must be true. Inspecting the remaining terms, 5 can’t equal <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f0acda42bd465f884b2048d44be56da94f6d20c983d76003733c1c5c6bfc98d7\" alt=\"9 b\"></span><span class=\"italic\">. </span>Dividing both of these quantities by 9 yields that <span class=\"italic\">b</span> can’t equal <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/af28828be78c7018132ef76885352666298c9fd5e429fbc7fd6cad80e0086dcf\" alt=\"five ninths\"></span>. Therefore, statement III must be true. Since <span class=\"italic\">b</span> can have any value other than <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/af28828be78c7018132ef76885352666298c9fd5e429fbc7fd6cad80e0086dcf\" alt=\"five ninths\"></span>, statement II may or may not be true.</p><p>Choice A is incorrect. For the given equation to have no solution, both <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/870dee33085d101927f4344d2907d50e004a4b6b543823d3e747127cdd9d7f1d\" alt=\"a, equals 9 \"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/14e43c4c2b7bd7d320adc08d5a715b9d5773c0365400f3574f142a40b69996c2\" alt=\"b is not equal to five ninths \"></span> must be true. Choice B is incorrect because it must also be true that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/14e43c4c2b7bd7d320adc08d5a715b9d5773c0365400f3574f142a40b69996c2\" alt=\"b is not equal to five ninths \"></span>. Choice C is incorrect because when <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/870dee33085d101927f4344d2907d50e004a4b6b543823d3e747127cdd9d7f1d\" alt=\"a, equals 9 \"></span>, there are many values of <span class=\"italic\">b </span>that lead to an equation having no solution. That is, <span class=\"italic\">b</span> might be 5, but <span class=\"italic\">b</span> isn’t required to be 5.</p><p></p>\n","correct_answer":["D"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.638000+00:00","source_updated_at":"2023-08-02T20:25:59.638000+00:00","options":[{"label":"A","content_html":"<p class=\"choice_paragraph \">None</p>\n","ord":0},{"label":"B","content_html":"<p class=\"choice_paragraph \">I only</p>\n","ord":1},{"label":"C","content_html":"<p class=\"choice_paragraph \">I and II only</p>\n","ord":2},{"label":"D","content_html":"<p class=\"choice_paragraph \">I and III only</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}}