{"id":"cb_question_bank:4f8f7f96-f248-44e4-aa38-5b9173894767","source_id":"cb_question_bank","source_item_id":"4f8f7f96-f248-44e4-aa38-5b9173894767","external_id":"b026d255-ee45-4659-b390-1b694d47296b","ibn":null,"question_id":"17d0e87d","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":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"StartFraction 14 x Over 7 y EndFraction equals 2 StartRoot w plus 19 EndRoot The given equation relates the distinct positive real numbers w , x , and y . Which equation correctly expresses w in terms of x and y ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"StartFraction 14 x Over 7 y EndFraction equals 2 StartRoot w plus 19 EndRoot\"><mfrac><mrow><mrow><mn>14</mn></mrow><mi>x</mi></mrow><mrow><mrow><mn>7</mn></mrow><mi>y</mi></mrow></mfrac><mo>=</mo><mn>2</mn><msqrt><mi>w</mi><mo>+</mo><mrow><mn>19</mn></mrow></msqrt></math></p>\n<p style=\"text-align: left;\">The given equation relates the distinct positive real numbers <math alttext=\"w\"><mi>w</mi>\n</math>, <math alttext=\"x\"><mi>x</mi>\n</math>, and <math alttext=\"y\"><mi>y</mi>\n</math>. Which equation correctly expresses <math alttext=\"w\"><mi>w</mi>\n</math> in terms of <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"y\"><mi>y</mi>\n</math>?</p>","rationale_html":"<p>Choice C is correct. Dividing each side of the given equation by <math alttext=\"2\"><mn>2</mn>\n</math> yields <math alttext=\"StartFraction 14 x Over 14 y EndFraction equals StartFraction 2 StartRoot w plus 19 EndRoot Over 2 EndFraction\"><mfrac><mrow><mn>14</mn><mi>x</mi></mrow><mrow><mn>14</mn><mi>y</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><msqrt><mi>w</mi><mo>+</mo><mn>19</mn></msqrt></mrow><mn>2</mn></mfrac></math>, or&nbsp;<math alttext=\"StartFraction x Over y EndFraction equals StartRoot w plus 19 EndRoot\"><mfrac><mi>x</mi><mi>y</mi></mfrac><mo>=</mo><msqrt><mi>w</mi><mo>+</mo><mn>19</mn></msqrt></math>. Because it's given that each of the variables is positive, squaring each side of this equation yields the equivalent equation&nbsp;<math alttext=\"left parenthesis StartFraction x Over y EndFraction right parenthesis squared equals w plus 19\"><msup><mfenced><mfrac><mi>x</mi><mi>y</mi></mfrac></mfenced><mn>2</mn></msup><mo>=</mo><mi>w</mi><mo>+</mo><mn>19</mn></math>. Subtracting <math alttext=\"19\"><mn>19</mn>\n</math> from each side of this equation yields <math alttext=\"left parenthesis StartFraction x Over y EndFraction right parenthesis squared minus 19 equals w\"><msup><mfenced><mfrac><mi>x</mi><mi>y</mi></mfrac></mfenced><mn>2</mn></msup><mo>-</mo><mn>19</mn><mo>=</mo><mi>w</mi></math>, or&nbsp;<math alttext=\"w equals left parenthesis StartFraction x Over y EndFraction right parenthesis squared minus 19\"><mi>w</mi><mo>=</mo><msup><mfenced><mfrac><mi>x</mi><mi>y</mi></mfrac></mfenced><mn>2</mn></msup><mo>-</mo><mn>19</mn></math>.</p>\n<p>Choice A is incorrect. This equation isn't equivalent to the given equation.&nbsp;</p>\n<p>Choice B is incorrect. This equation isn't equivalent to the given equation.&nbsp;</p>\n<p>Choice D is incorrect. This equation isn't equivalent to the given equation.&nbsp;</p>","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":"<p><math alttext=\"w equals StartRoot StartFraction x Over y EndFraction EndRoot minus 19\"><mi>w</mi><mo>=</mo><msqrt><mfrac><mi>x</mi><mi>y</mi></mfrac></msqrt><mo>-</mo><mrow><mn>19</mn></mrow></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"w equals StartRoot StartFraction 28 x Over 14 y EndFraction EndRoot minus 19\"><mi>w</mi><mo>=</mo><msqrt><mfrac><mrow><mrow><mn>28</mn></mrow><mi>x</mi></mrow><mrow><mrow><mn>14</mn></mrow><mi>y</mi></mrow></mfrac></msqrt><mo>-</mo><mrow><mn>19</mn></mrow></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"w equals left parenthesis StartFraction x Over y EndFraction right parenthesis squared minus 19\"><mi>w</mi><mo>=</mo><msup><mfenced><mfrac><mi>x</mi><mi>y</mi></mfrac></mfenced><mn>2</mn></msup><mo>-</mo><mrow><mn>19</mn></mrow></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"w equals left parenthesis StartFraction 28 x Over 14 y EndFraction right parenthesis squared minus 19\"><mi>w</mi><mo>=</mo><msup><mfenced><mfrac><mrow><mrow><mn>28</mn></mrow><mi>x</mi></mrow><mrow><mrow><mn>14</mn></mrow><mi>y</mi></mrow></mfrac></mfenced><mn>2</mn></msup><mo>-</mo><mrow><mn>19</mn></mrow></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}}