{"id":"cb_question_bank:afcaaf12-05c7-4160-ab50-3ea549347478","source_id":"cb_question_bank","source_item_id":"afcaaf12-05c7-4160-ab50-3ea549347478","external_id":"c4e7a98c-fecd-45d1-ad5b-69c978af6bf7","ibn":null,"question_id":"f03465dc","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.D.","skill":"Systems of two linear equations in two variables","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"8 x plus 7 y equals 9 24 x plus 21 y equals 27 For each real number r , which of the following points lies on the graph of each equation in the xy -plane for the given system?","stimulus":"","stimulus_html":null,"stem_html":"

\n\t\n\t\t\n\t\t\t8\n\t\t\tx\n\t\t\n\t\t+\n\t\t\n\t\t\t7\n\t\t\ty\n\t\t\n\t\n\t=\n\t9\n\n
\n\t\n\t\t\n\t\t\t24\n\t\t\tx\n\t\t\n\t\t+\n\t\t\n\t\t\t21\n\t\t\ty\n\t\t\n\t\n\t=\n\t27\n\n

\n

For each real number r\n, which of the following points lies on the graph of each equation in the xy-plane for the given system?

","rationale_html":"

Choice A is correct. Dividing both sides of the second equation in the given system by 3\n yields \n\t\n\t\t\n\t\t\t8\n\t\t\tx\n\t\t\n\t\t+\n\t\t\n\t\t\t7\n\t\t\ty\n\t\t\n\t\n\t=\n\t9\n\n, which is the first equation in the given system. Therefore, the first and second equations represent the same line in the xy-plane. If the x- and y-coordinates of a point satisfy an equation, the point lies on the graph of the equation in the xy-plane. Choice A is a point with x-coordinate r\n and y-coordinate \n\t\n\t\t-\n\t\t\n\t\t\t\n\t\t\t\t8\n\t\t\t\tr\n\t\t\t\n\t\t\t7\n\t\t\n\t\n\t+\n\t\n\t\t9\n\t\t7\n\t\n\n. Substituting r\n for x\n and \n\t\n\t\t-\n\t\t\n\t\t\t\n\t\t\t\t8\n\t\t\t\tr\n\t\t\t\n\t\t\t7\n\t\t\n\t\n\t+\n\t\n\t\t9\n\t\t7\n\t\n\n for y\n in the equation \n\t\n\t\t\n\t\t\t8\n\t\t\tx\n\t\t\n\t\t+\n\t\t\n\t\t\t7\n\t\t\ty\n\t\t\n\t\n\t=\n\t9\n\n yields 8r+7-87r+97=9. Applying the distributive property to the left-hand side of this equation yields 8r-8r+9=9. Combining like terms on the left-hand side of this equation yields 9=9, so the coordinates of the point r,-87r+97 satisfy both equations in the given system. Therefore, for each real number r\n, the point r,-87r+97 lies on the graph of each equation in the xy-plane for the given system.

\n

Choice B is incorrect and may result from conceptual or calculation errors.

\n

Choice C is incorrect and may result from conceptual or calculation errors.

\n

Choice D is incorrect and may result from conceptual or calculation errors.

","correct_answer":["A"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.818000+00:00","source_updated_at":"2023-08-02T20:25:59.818000+00:00","options":[{"label":"A","content_html":"

r,-8r7+97

","ord":0},{"label":"B","content_html":"

-8r7+97,r

","ord":1},{"label":"C","content_html":"

-8r7+9,8r7+27

","ord":2},{"label":"D","content_html":"

r3+9,-r3+27

","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}}