{"id":"cb_question_bank:71cdf101-a40a-49b2-b6ae-70a8c433d340","source_id":"cb_question_bank","source_item_id":"71cdf101-a40a-49b2-b6ae-70a8c433d340","external_id":"c17521a8-2834-4b36-9f1a-dca9bf6e906d","ibn":null,"question_id":"7d5d1b32","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.A.","skill":"Linear equations in one variable","difficulty":"H","score_band":7,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"2 left parenthesis k x minus n right parenthesis equals negative StartFraction 28 Over 15 EndFraction x minus StartFraction 36 Over 19 EndFraction In the given equation, k and n are constants and n greater than 1 . The equation has no solution. What is the value of k ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"2 left parenthesis k x minus n right parenthesis equals negative StartFraction 28 Over 15 EndFraction x minus StartFraction 36 Over 19 EndFraction\"><mrow><mn>2</mn></mrow><mfenced><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mi>n</mi></mrow></mfenced><mo>=</mo><mrow>\n\t<mo>-</mo>\n\t<mfrac>\n\t\t<mn>28</mn>\n\t\t<mn>15</mn>\n\t</mfrac>\n</mrow>\n<mrow>\n\t<mi>x</mi>\n\t<mo>-</mo>\n\t<mfrac>\n\t\t<mn>36</mn>\n\t\t<mn>19</mn>\n\t</mfrac>\n</mrow>\n</math></p>\n<p style=\"text-align: left;\">In the given equation, <math alttext=\"k\"><mi>k</mi>\n</math> and <math alttext=\"n\"><mi>n</mi>\n</math> are constants and&nbsp;<math alttext=\"n greater than 1\"><mi>n</mi><mo>&gt;</mo><mn>1</mn></math>. The equation has no solution. What is the value of <math alttext=\"k\"><mi>k</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"negative StartFraction 14 Over 15 EndFraction\"><mo>-</mo><mfrac><mn>14</mn><mn>15</mn></mfrac></math>. A linear equation in the form <math alttext=\"a x plus b equals c x plus d\"><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo>=</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi></math> has no solution only when the coefficients of <math alttext=\"x\"><mi>x</mi>\n</math> on each side of the equation are equal and the constant terms are not equal. Dividing both sides of the given equation by <math alttext=\"2\"><mn>2</mn>\n</math> yields <math alttext=\"k x minus n equals minus StartFraction 28 Over 30 EndFraction x minus StartFraction 36 Over 38 EndFraction\"><mi>k</mi><mi>x</mi><mo>-</mo><mi>n</mi><mo>=</mo><mo>-</mo><mfrac><mn>28</mn><mn>30</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>36</mn><mn>38</mn></mfrac></math>, or <math alttext=\"k x minus n equals minus StartFraction 14 Over 15 EndFraction x minus StartFraction 18 Over 19 EndFraction\"><mi>k</mi><mi>x</mi><mo>-</mo><mi>n</mi><mo>=</mo><mo>-</mo><mfrac><mn>14</mn><mn>15</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>18</mn><mn>19</mn></mfrac></math>. Since it’s given that the equation has no solution, the coefficient of <math alttext=\"x\"><mi>x</mi>\n</math> on both sides of this equation must be equal, and the constant terms on both sides of this equation must not be equal. Since&nbsp;<math alttext=\"StartFraction 18 Over 19 EndFraction less than 1\"><mfrac><mn>18</mn><mn>19</mn></mfrac><mo>&lt;</mo><mn>1</mn></math>, and it's given that <math alttext=\"n greater than 1\"><mi>n</mi><mo>&gt;</mo><mn>1</mn></math>, the second condition is true. Thus, <math alttext=\"k\"><mi>k</mi>\n</math> must be equal to <math alttext=\"negative StartFraction 14 Over 15 EndFraction\"><mo>-</mo><mfrac><mn>14</mn><mn>15</mn></mfrac></math>. Note that -14/15, -.9333, and -0.933 are examples of ways to enter a correct answer.</p>","correct_answer":["-.9333","-14/15"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.809000+00:00","source_updated_at":"2023-08-02T20:25:59.809000+00:00","options":[],"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}}