{"id":"cb_question_bank:69e74fe9-0dd8-458d-9bd0-f9ffc0d44b0b","source_id":"cb_question_bank","source_item_id":"69e74fe9-0dd8-458d-9bd0-f9ffc0d44b0b","external_id":null,"ibn":"09289-DC","question_id":"e9349667","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":"If the ordered pair x subscript 1 comma y subscript 1 and the ordered pair x subscript 2 comma y subscript 2 are the two solutions to the system of equations above, what is the value of y subscript 1, plus, y subscript 2 ?","stimulus":"y equals, x squared plus 2 x plus 1, and, x plus y plus 1, equals 0","stimulus_html":"
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If \"the and \"the are the two solutions to the system of equations above, what is the value of \"y ?

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Choice D is correct. The system of equations can be solved using the substitution method. Solving the second equation for y gives y = –x – 1. Substituting the expression –x – 1 for y into the first equation gives –x – 1 = x2 + 2x + 1. Adding x + 1 to both sides of the equation yields x2 + 3x + 2 = 0. The left-hand side of the equation can be factored by finding two numbers whose sum is 3 and whose product is 2, which gives (x + 2)(x + 1) = 0. Setting each factor equal to 0 yields x + 2 = 0 and x + 1 = 0, and solving for x yields x = –2 or x = –1. These values of x can be substituted for x in the equation y = –x – 1 to find the corresponding y-values: y = –(–2) – 1 = 2 – 1 = 1 and y = –(–1) – 1 = 1 – 1 = 0. It follows that (–2, 1) and (–1, 0) are the solutions to the given system of equations. Therefore, (x1, y1) = (–2, 1), (x2, y2) = (–1, 0), and y1 + y2 = 1 + 0 = 1.

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Choice A is incorrect. The solutions to the system of equations are (x1, y1) = (–2, 1) and (x2, y2) = (–1, 0). Therefore, –3 is the sum of the x-coordinates of the solutions, not the sum of the y-coordinates of the solutions. Choices B and C are incorrect and may be the result of computation or substitution errors.

\n","correct_answer":["D"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.636000+00:00","source_updated_at":"2023-08-02T20:25:59.636000+00:00","options":[{"label":"A","content_html":"\n \"negative\n","ord":0},{"label":"B","content_html":"\n \"negative\n","ord":1},{"label":"C","content_html":"\n \"negative\n","ord":2},{"label":"D","content_html":"\n \"1\"\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}}