{"id":"cb_question_bank:a21ea22b-e8ec-445b-bf17-9418bb52b98e","source_id":"cb_question_bank","source_item_id":"a21ea22b-e8ec-445b-bf17-9418bb52b98e","external_id":"9d5bc04d-81bd-4fda-8c78-8c26fa0de2f0","ibn":null,"question_id":"153ee763","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.A.","skill":"Linear equations in one variable","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"minus 3 x plus 21 p x equals 84 In the given equation, p is a constant. The equation has no solution. What is the value of p ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"minus 3 x plus 21 p x equals 84\"><mo>-</mo><mrow><mn>3</mn></mrow><mi>x</mi><mo>+</mo><mrow><mn>21</mn></mrow><mi>p</mi><mi>x</mi><mo>=</mo><mrow><mn>84</mn></mrow></math></p>\n<p style=\"text-align: left;\">In the given equation, <math alttext=\"p\"><mi>p</mi>\n</math> is a constant. The equation has no solution. What is the value of <math alttext=\"p\"><mi>p</mi>\n</math>?</p>","rationale_html":"<p>Choice B is correct. A linear equation in one variable has no solution if and only if the equation is false; that is, when there is no value of <math alttext=\"x\"><mi>x</mi>\n</math> that produces a true statement. It's given that in the equation <math alttext=\"minus 3 x plus 21 p x equals 84\"><mo>-</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>21</mn><mi>p</mi><mi>x</mi><mo>=</mo><mn>84</mn></math>, <math alttext=\"p\"><mi>p</mi>\n</math> is a constant and the equation has no solution for <math alttext=\"x\"><mi>x</mi>\n</math>. Therefore, the value of the constant <math alttext=\"p\"><mi>p</mi>\n</math> is one that results in a false equation. Factoring out the common factor of <math alttext=\"minus 3 x\"><mrow>\n\t<mo>-</mo>\n\t<mn>3</mn>\n\t<mi>x</mi>\n</mrow>\n</math> on the left-hand side of the given equation yields <math alttext=\"minus 3 x left parenthesis 1 minus 7 p right parenthesis equals 84\"><mo>-</mo><mn>3</mn><mi>x</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mn>7</mn><mi>p</mi></mrow></mfenced><mo>=</mo><mn>84</mn></math>. Dividing both sides of this equation by <math alttext=\"negative 3\"><mo>-</mo><mn>3</mn>\n</math> yields <math alttext=\"x left parenthesis 1 minus 7 p right parenthesis equals negative 28\"><mi>x</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mn>7</mn><mi>p</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mn>28</mn></math>. Dividing both sides of this equation by&nbsp;<math alttext=\"left parenthesis 1 minus 7 p right parenthesis\"><mfenced><mrow><mn>1</mn><mo>-</mo><mn>7</mn><mi>p</mi></mrow></mfenced></math> yields <math alttext=\"x equals StartFraction negative 28 Over 1 minus 7 p EndFraction\"><mi>x</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>28</mn></mrow><mrow><mn>1</mn><mo>-</mo><mn>7</mn><mi>p</mi></mrow></mfrac></math>. This equation is false if and only if <math alttext=\"1 minus 7 p equals 0\"><mn>1</mn><mo>-</mo><mn>7</mn><mi>p</mi><mo>=</mo><mn>0</mn></math>. Adding <math alttext=\"7 p\"><mrow>\n\t<mn>7</mn>\n\t<mi>p</mi>\n</mrow>\n</math> to both sides of&nbsp;<math alttext=\"1 minus 7 p equals 0\"><mn>1</mn><mo>-</mo><mn>7</mn><mi>p</mi><mo>=</mo><mn>0</mn></math> yields <math alttext=\"1 equals 7 p\"><mn>1</mn><mo>=</mo><mn>7</mn><mi>p</mi></math>. Dividing both sides of this equation by <math alttext=\"7\"><mn>7</mn>\n</math> yields <math alttext=\"one seventh equals p\"><mfrac><mn>1</mn><mn>7</mn></mfrac><mo>=</mo><mi>p</mi></math>. It follows that the equation&nbsp;<math alttext=\"x equals StartFraction negative 28 Over 1 minus 7 p EndFraction\"><mi>x</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>28</mn></mrow><mrow><mn>1</mn><mo>-</mo><mn>7</mn><mi>p</mi></mrow></mfrac></math> is false if and only if <math alttext=\"p equals one seventh\"><mi>p</mi><mo>=</mo><mfrac><mn>1</mn><mn>7</mn></mfrac></math>. Therefore, the given equation has no solution if and only if the value of <math alttext=\"p\"><mi>p</mi>\n</math> is <math alttext=\"one seventh\"><mfrac><mn>1</mn><mn>7</mn></mfrac></math>.</p>\n<p>Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice C is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["B"],"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":[{"label":"A","content_html":"<p><math alttext=\"0\"><mn>0</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"one seventh\"><mfrac>\n\t<mn>1</mn>\n\t<mn>7</mn>\n</mfrac>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"four thirds\"><mfrac>\n\t<mn>4</mn>\n\t<mn>3</mn>\n</mfrac>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"4\"><mn>4</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}}