{"id":"cb_question_bank:a5e69f0d-e3fd-4307-8261-7f31b6f986ed","source_id":"cb_question_bank","source_item_id":"a5e69f0d-e3fd-4307-8261-7f31b6f986ed","external_id":"91e78971-fbd4-44f8-ad04-97fb96cf9931","ibn":null,"question_id":"77c0cced","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":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"y equals 2 x squared minus 21 x plus 64 y equals 3 x plus a In the given system of equations, a is a constant. The graphs of the equations in the given system intersect at exactly one point, left parenthesis x comma y right parenthesis , in the xy -plane. What is the value of x ?","stimulus":"","stimulus_html":null,"stem_html":"

\n\ty\n\t=\n\t\n\t\t\n\t\t\t2\n\t\t\t\n\t\t\t\tx\n\t\t\t\t2\n\t\t\t\n\t\t\n\t\t-\n\t\t\n\t\t\t21\n\t\t\tx\n\t\t\n\t\t+\n\t\t64\n\t\n\n

\n

\n\ty\n\t=\n\t\n\t\t\n\t\t\t3\n\t\t\tx\n\t\t\n\t\t+\n\t\ta\n\t\n\n

\n

In the given system of equations, a\n is a constant. The graphs of the equations in the given system intersect at exactly one point, (x,y), in the xy-plane. What is the value of x\n?

","rationale_html":"

Choice C is correct. It's given that the graphs of the equations in the given system intersect at exactly one point, x,y, in the xy-plane. Therefore, x,y is the only solution to the given system of equations. The given system of equations can be solved by subtracting the second equation, y=3x+a, from the first equation, y=2x2-21x+64. This yields y-y=2x2-21x+64-3x+a, or 0=2x2-24x+64-a. Since the given system has only one solution, this equation has only one solution. A quadratic equation in the form rx2+sx+t=0, where r\n, s\n, and t\n are constants, has one solution if and only if the discriminant, s2-4rt, is equal to zero. Substituting 2\n for r\n, -24 for s\n, and \n\t\n\t\t-\n\t\ta\n\t\n\t+\n\t64\n\n for t\n in the expression s2-4rt yields -242-4264-a. Setting this expression equal to zero yields -242-4264-a=0, or \n\t\n\t\t\n\t\t\t8\n\t\t\ta\n\t\t\n\t\t+\n\t\t64\n\t\n\t=\n\t0\n\n. Subtracting 64\n from both sides of this equation yields 8a=-64. Dividing both sides of this equation by 8\n yields a=-8. Substituting -8 for a\n in the equation 0=2x2-24x+64-a yields 0=2x2-24x+64+8, or 0=2x2-24x+72. Factoring 2\n from the right-hand side of this equation yields 0=2x2-12x+36. Dividing both sides of this equation by 2\n yields 0=x2-12x+36, which is equivalent to 0=x-6x-6, or 0=x-62. Taking the square root of both sides of this equation yields \n\t0\n\t=\n\t\n\t\tx\n\t\t-\n\t\t6\n\t\n\n. Adding 6\n to both sides of this equation yields \n\tx\n\t=\n\t6\n\n. 

\n

Choice A is incorrect. This is the value of a\n, not x\n.

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Choice B 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":["C"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.824000+00:00","source_updated_at":"2023-08-02T20:25:59.824000+00:00","options":[{"label":"A","content_html":"

-8\n

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

-6\n

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

6\n

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

8\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}}