{"id":"cb_question_bank:7756c259-e1f6-4608-ae5a-1c2828d6a560","source_id":"cb_question_bank","source_item_id":"7756c259-e1f6-4608-ae5a-1c2828d6a560","external_id":"f3a8f6e5-3640-4af9-a194-e13afbea670e","ibn":null,"question_id":"50f4cb9c","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.B.","skill":"Linear functions","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"x f left parenthesis x right parenthesis 1 negative 64 2 0 3 64 For the linear function f , the table shows three values of x and their corresponding values of f left parenthesis x right parenthesis . Function f is defined by f left parenthesis x right parenthesis equals a x plus b , where a and b are constants. What is the value of a minus b ?","stimulus":"","stimulus_html":null,"stem_html":"<figure class=\"table\"><figure class=\"table\"><figure class=\"table\"><table border=\"1\">\n<thead>\n<tr>\n<th style=\"width: 47.3489%; text-align: center;\" scope=\"col\"><math alttext=\"x\"><mi>x</mi>\n</math></th>\n<th style=\"width: 47.3489%; text-align: center;\" scope=\"col\"><math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math></th>\n</tr>\n</thead>\n<tbody>\n<tr>\n<td style=\"width: 47.3489%; text-align: center;\"><math alttext=\"1\"><mn>1</mn>\n</math></td>\n<td style=\"width: 47.3489%; text-align: center;\"><math alttext=\"negative 64\"><mo>-</mo><mn>64</mn>\n</math></td>\n</tr>\n<tr>\n<td style=\"width: 47.3489%; text-align: center;\"><math alttext=\"2\"><mn>2</mn>\n</math></td>\n<td style=\"width: 47.3489%; text-align: center;\"><math alttext=\"0\"><mn>0</mn>\n</math></td>\n</tr>\n<tr>\n<td style=\"width: 47.3489%; text-align: center;\"><math alttext=\"3\"><mn>3</mn>\n</math></td>\n<td style=\"width: 47.3489%; text-align: center;\"><math alttext=\"64\"><mn>64</mn>\n</math></td>\n</tr>\n</tbody>\n</table></figure></figure></figure>\n<p style=\"text-align: left;\">For the linear function <math alttext=\"f\"><mi>f</mi>\n</math>, the table shows three values of <math alttext=\"x\"><mi>x</mi>\n</math> and their corresponding values of <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math>. Function <math alttext=\"f\"><mi>f</mi>\n</math> is defined by <math alttext=\"f left parenthesis x right parenthesis equals a x plus b\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math>, where <math alttext=\"a\"><mi>a</mi>\n</math> and <math alttext=\"b\"><mi>b</mi>\n</math> are constants. What is the value of <math alttext=\"a minus b\"><mrow>\n\t<mi>a</mi>\n\t<mo>-</mo>\n\t<mi>b</mi>\n</mrow>\n</math>?</p>","rationale_html":"<p>Choice D is correct. The table gives that <math alttext=\"f left parenthesis x right parenthesis equals 0\"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></math> when <math alttext=\"x equals 2\"><mi>x</mi><mo>=</mo><mn>2</mn></math>. Substituting 0 for <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> and <math alttext=\"2\"><mn>2</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> into the equation <math alttext=\"f left parenthesis x right parenthesis equals a x plus b\"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo>&nbsp;</mo></math>yields <math alttext=\"0 equals 2 a plus b\"><mn>0</mn><mo>=</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>b</mi></math>. Subtracting <math alttext=\"2 a\"><mrow>\n\t<mn>2</mn>\n\t<mi>a</mi>\n</mrow>\n</math> from both sides of this equation yields <math alttext=\"b equals minus 2 a\"><mi>b</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>a</mi></math>. The table gives that <math alttext=\"f left parenthesis x right parenthesis equals negative 64\"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>-</mo><mn>64</mn></math> when <math alttext=\"x equals 1\"><mi>x</mi><mo>=</mo><mn>1</mn></math>. Substituting <math alttext=\"minus 2 a\"><mo>-</mo><mn>2</mn><mi>a</mi></math>&nbsp;for <math alttext=\"b\"><mi>b</mi>\n</math>, <math alttext=\"negative 64\"><mo>-</mo><mn>64</mn></math> for <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math>, and <math alttext=\"1\"><mn>1</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math>&nbsp;into the equation <math alttext=\"f left parenthesis x right parenthesis equals a x plus b\"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math> yields&nbsp;<math alttext=\"negative 64 equals a left parenthesis 1 right parenthesis plus left parenthesis minus 2 a right parenthesis\"><mo>-</mo><mn>64</mn><mo>=</mo><mi>a</mi><mfenced><mn>1</mn></mfenced><mo>+</mo><mfenced><mrow><mo>-</mo><mn>2</mn><mi>a</mi></mrow></mfenced></math>. Combining like terms yields&nbsp;<math alttext=\"negative 64 equals negative a\"><mo>-</mo><mn>64</mn><mo>=</mo><mo>-</mo><mi>a</mi></math>, or <math alttext=\"a equals 64\"><mi>a</mi><mo>=</mo><mn>64</mn></math>. Since <math alttext=\"b equals minus 2 a\"><mi>b</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>a</mi></math>, substituting <math alttext=\"64\"><mn>64</mn>\n</math> for <math alttext=\"a\"><mi>a</mi>\n</math> into this equation gives <math alttext=\"b equals left parenthesis negative 2 right parenthesis left parenthesis 64 right parenthesis\"><mi>b</mi><mo>=</mo><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mn>64</mn></mfenced></math>, which yields&nbsp;<math alttext=\"b equals negative 128\"><mi>b</mi><mo>=</mo><mo>-</mo><mn>128</mn></math>. Thus, the value of&nbsp;<math alttext=\"a minus b\"><mi>a</mi><mo>-</mo><mi>b</mi></math> can be written as&nbsp;<math alttext=\"64 minus left parenthesis negative 128 right parenthesis\"><mn>64</mn><mo>-</mo><mo>(</mo><mo>-</mo><mn>128</mn><mo>)</mo></math>, which is <math alttext=\"192\"><mn>192</mn>\n</math>.</p>\n<p>Choice A is incorrect. This is the value of&nbsp;<math alttext=\"a plus b\"><mi>a</mi><mo>+</mo><mi>b</mi></math>, not&nbsp;<math alttext=\"a minus b\"><mi>a</mi><mo>-</mo><mi>b</mi></math>.</p>\n<p>Choice B is incorrect. This is the value of&nbsp;<math alttext=\"a minus 2\"><mi>a</mi><mo>-</mo><mn>2</mn></math>, not&nbsp;<math alttext=\"a minus b\"><mi>a</mi><mo>-</mo><mi>b</mi></math>.</p>\n<p>Choice C is incorrect. This is the value of&nbsp;<math alttext=\"2 a\"><mn>2</mn><mi>a</mi></math>, not <math alttext=\"a minus b\"><mi>a</mi><mo>-</mo><mi>b</mi></math>.</p>","correct_answer":["D"],"has_media":false,"has_mathml":true,"has_table":true,"source_created_at":"2023-08-02T20:25:59.811000+00:00","source_updated_at":"2023-08-02T20:25:59.811000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"negative 64\"><mo>-</mo><mn>64</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"62\"><mn>62</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"128\"><mn>128</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"192\"><mn>192</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}}