{"id":"cb_question_bank:6c655606-6c41-45e5-91e1-b481ba74a2f7","source_id":"cb_question_bank","source_item_id":"6c655606-6c41-45e5-91e1-b481ba74a2f7","external_id":null,"ibn":"08050-DC","question_id":"70e29454","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.A.","skill":"Linear equations in one variable","difficulty":"M","score_band":5,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In the equation above, a and b are constants. If the equation has infinitely many solutions, what are the values of a and b ?","stimulus":"","stimulus_html":"<div class=\"stimulus_reference \">\n\t<p class=\"standalone_statement style:1 \"><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/606949c48839874ddc055177ee771098758354e0c484b570ac9df3b28ad00b7d\" alt=\"\"></span></span></p>\n</div>\n","stem_html":"<p class=\"stem_paragraph \">In the equation above, <span class=\"italic\">a</span> and <span class=\"italic\">b</span> are constants. If the equation has infinitely many solutions, what are the values of <span class=\"italic\">a</span> and <span class=\"italic\">b</span> ?</p>\n","rationale_html":"<p>Choice B is correct. Distributing the <span class=\"italic\">a</span> on the left-hand side of the equation gives 3<span class=\"italic\">a</span>&nbsp;– <span class=\"italic\">b</span>&nbsp;–&nbsp;<span class=\"italic\">ax</span> = –1 – 2<span class=\"italic\">x</span>. Rearranging the terms in each side of the equation yields –<span class=\"italic\">ax</span> + 3<span class=\"italic\">a</span> –<span class=\"italic\"> b</span> = –2<span class=\"italic\">x</span> –1. Since the equation has infinitely many solutions, it follows that the coefficients of <span class=\"italic\">x</span> and the free terms on both sides must be equal. That is,&nbsp; –<span class=\"italic\">a</span>&nbsp;= –2, or <span class=\"italic\">a</span> = 2, and 3<span class=\"italic\">a</span> – <span class=\"italic\">b</span> = –1. Substituting 2 for <span class=\"italic\">a</span> in the equation 3<span class=\"italic\">a</span>&nbsp;– <span class=\"italic\">b</span> =&nbsp;–1 gives 3(2) <span class=\"italic\">– b</span> = –1, so&nbsp; <span class=\"italic\">b</span> = 7.</p><p>Choice A is incorrect and may be the result of a conceptual error when finding the value of <span class=\"italic\">b</span>. Choices C and D are incorrect and may result from making a sign error when simplifying.</p><p></p>\n","correct_answer":["B"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.634000+00:00","source_updated_at":"2023-08-02T20:25:59.634000+00:00","options":[{"label":"A","content_html":"<p><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/3925e44374c7731a4f6034b2cb3cd0be1061adba8c3a20d5bfaf6c60c879d9da\" alt=\"a equals 2\"></span></span> and <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e7591c90d04d93ac61bcfeb9f58ebd5ee6cdc220f79042ac9f5902b21b97363f\" alt=\"b equals 1\"></span></span></p>\n","ord":0},{"label":"B","content_html":"<p><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/3925e44374c7731a4f6034b2cb3cd0be1061adba8c3a20d5bfaf6c60c879d9da\" alt=\"a equals 2\"></span></span> and <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f7fb1cd7296414a04b0099a68032f7f0631082149258d462aefd6958938ea14b\" alt=\"b equals 7\"></span></span></p>\n","ord":1},{"label":"C","content_html":"<p><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e394caa237e4af4b3ff3592b6fdc58418bcfc5aa0a617b00520f7cdb89c49350\" alt=\"a equals negative 2\"></span></span> and <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/86ed55617b0324e1e688dfa3af8314501ab2cd7a1660495779c6409d8aea00fb\" alt=\"b equals 5\"></span></span></p>\n","ord":2},{"label":"D","content_html":"<p><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e394caa237e4af4b3ff3592b6fdc58418bcfc5aa0a617b00520f7cdb89c49350\" alt=\"a equals negative 2\"></span></span> and <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/acfede8f12a1a37fe3db182b3e366cf5c1918baebb6e7230d0f9727373d41c36\" alt=\"b equals negative 5\"></span></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}}