{"id":"cb_question_bank:366cd449-d396-4472-8c3c-ca67129a4b11","source_id":"cb_question_bank","source_item_id":"366cd449-d396-4472-8c3c-ca67129a4b11","external_id":"a24605a1-dcc2-4d33-8f1f-a5e3f3904603","ibn":null,"question_id":"15daa8d6","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.A.","skill":"Linear equations in one variable","difficulty":"M","score_band":4,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"2 x plus 16 equals a left parenthesis x plus 8 right parenthesis In the given equation, a is a constant. If the equation has infinitely many solutions, what is the value of a ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"2 x plus 16 equals a left parenthesis x plus 8 right parenthesis\"><mrow><mn>2</mn></mrow><mi>x</mi><mo>+</mo><mrow><mn>16</mn></mrow><mo>=</mo><mi>a</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mrow><mn>8</mn></mrow></mrow></mfenced></math></p>\n<p>In the given equation, <math alttext=\"a\"><mi>a</mi>\n</math> is a constant. If the equation has infinitely many solutions, what is the value of <math alttext=\"a\"><mi>a</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"2\"><mn>2</mn>\n</math>. An equation with one variable, <math alttext=\"x\"><mi>x</mi>\n</math>, has infinitely many solutions only when both sides of the equation are equal for any defined value of <math alttext=\"x\"><mi>x</mi>\n</math>. It's given that <math alttext=\"2 x plus 16 equals a left parenthesis x plus 8 right parenthesis\"><mn>2</mn><mi>x</mi><mo>+</mo><mn>16</mn><mo>=</mo><mi>a</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfenced></math>, where <math alttext=\"a\"><mi>a</mi>\n</math> is a constant. This equation can be rewritten as <math alttext=\"2 left parenthesis x plus 8 right parenthesis equals a left parenthesis x plus 8 right parenthesis\"><mn>2</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfenced><mo>=</mo><mi>a</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfenced></math>. If this equation has infinitely many solutions, then both sides of this equation are equal for any defined value of <math alttext=\"x\"><mi>x</mi>\n</math>. Both sides of this equation are equal for any defined value of <math alttext=\"x\"><mi>x</mi>\n</math> when <math alttext=\"2 equals a\"><mn>2</mn><mo>=</mo><mi>a</mi></math>. Therefore, if the equation has infinitely many solutions, the value of <math alttext=\"a\"><mi>a</mi>\n</math> is <math alttext=\"2\"><mn>2</mn>\n</math>.</p>\n<p style=\"text-align: left;\">Alternate approach: If the given equation,&nbsp;<math alttext=\"2 x plus 16 equals a left parenthesis x plus 8 right parenthesis\"><mn>2</mn><mi>x</mi><mo>+</mo><mn>16</mn><mo>=</mo><mi>a</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfenced></math>, has infinitely many solutions, then both sides of this equation are equal for any value of <math alttext=\"x\"><mi>x</mi>\n</math>. If <math alttext=\"x equals 0\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math>, then substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> in <math alttext=\"2 x plus 16 equals a left parenthesis x plus 8 right parenthesis\"><mn>2</mn><mi>x</mi><mo>+</mo><mn>16</mn><mo>=</mo><mi>a</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfenced></math> yields <math alttext=\"2 left parenthesis 0 right parenthesis plus 16 equals a left parenthesis 0 plus 8 right parenthesis\"><mn>2</mn><mfenced><mn>0</mn></mfenced><mo>+</mo><mn>16</mn><mo>=</mo><mi>a</mi><mfenced><mrow><mn>0</mn><mo>+</mo><mn>8</mn></mrow></mfenced></math>, or <math alttext=\"16 equals 8 a\"><mn>16</mn><mo>=</mo><mn>8</mn><mi>a</mi></math>. Dividing both sides of this equation by <math alttext=\"8\"><mn>8</mn>\n</math> yields <math alttext=\"2 equals a\"><mn>2</mn><mo>=</mo><mi>a</mi></math>.</p>","correct_answer":["2"],"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}}