{"id":"cb_question_bank:277fda06-fa5b-483f-b234-fc7cb15e6e8e","source_id":"cb_question_bank","source_item_id":"277fda06-fa5b-483f-b234-fc7cb15e6e8e","external_id":"5014724c-eaa2-486b-8c20-ebe36ecc982b","ibn":null,"question_id":"84e5e36c","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":"E","score_band":3,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"y equals 76 y equals x squared minus 5 The graphs of the given equations in the xy -plane intersect at the point left parenthesis x comma y right parenthesis . What is a possible value of x ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"y equals 76\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mn>76</mn>\n</mrow>\n</math><br><math alttext=\"y equals x squared minus 5\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<msup>\n\t\t\t<mi>x</mi>\n\t\t\t<mn>2</mn>\n\t\t</msup>\n\t\t<mo>-</mo>\n\t\t<mn>5</mn>\n\t</mrow>\n</mrow>\n</math></p>\n<p>The graphs of the given equations in the&nbsp;<em>xy</em>-plane intersect at the point <math alttext=\"left parenthesis x comma y right parenthesis\"><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math>. What is a possible value of <math alttext=\"x\"><mi>x</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">Choice B is correct. Since the point <math alttext=\"left parenthesis x comma y right parenthesis\"><mfenced><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></math> is an intersection point of the graphs of the given equations in the <em>xy</em>-plane, the pair <math alttext=\"left parenthesis x comma y right parenthesis\"><mfenced><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></math> should satisfy both equations, and thus is a solution of the given system. According to the first equation, <math alttext=\"y equals 76\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mn>76</mn>\n</mrow>\n</math>. Substituting <math alttext=\"76\"><mn>76</mn>\n</math> in place of <math alttext=\"y\"><mi>y</mi>\n</math> in the second equation yields <math alttext=\"x squared minus 5 equals 76\"><mrow>\n\t<mrow>\n\t\t<msup>\n\t\t\t<mi>x</mi>\n\t\t\t<mn>2</mn>\n\t\t</msup>\n\t\t<mo>-</mo>\n\t\t<mn>5</mn>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>76</mn>\n</mrow>\n</math>. Adding <math alttext=\"5\"><mn>5</mn>\n</math> to both sides of this equation yields <math alttext=\"x squared equals 81\"><mrow>\n\t<msup>\n\t\t<mi>x</mi>\n\t\t<mn>2</mn>\n\t</msup>\n\t<mo>=</mo>\n\t<mn>81</mn>\n</mrow>\n</math>. Taking the square root of both sides of this equation yields two solutions: <math alttext=\"x equals 9\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>9</mn>\n</mrow>\n</math> and <math alttext=\"x equals negative 9\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>9</mn>\n</mrow>\n</math>. Of these two solutions, only <math alttext=\"negative 9\"><mo>-</mo><mn>9</mn>\n</math> is given as a choice.</p>\n<p style=\"text-align: left;\">Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice C is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice D is incorrect. This is the value of coordinate <math alttext=\"y\"><mi>y</mi>\n</math>, rather than <math alttext=\"x\"><mi>x</mi>\n</math>, of the intersection point <math alttext=\"left parenthesis x comma y right parenthesis\"><mfenced><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></math>.</p>","correct_answer":["B"],"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":"<p><math alttext=\"negative StartFraction 76 Over 5 EndFraction\"><mrow>\n\t<mo>-</mo>\n\t<mfrac>\n\t\t<mn>76</mn>\n\t\t<mn>5</mn>\n\t</mfrac>\n</mrow>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"negative 9\"><mo>-</mo><mn>9</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"5\"><mn>5</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"76\"><mn>76</mn>\n</math></p>","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}}