{"id":"cb_question_bank:5c336051-559f-4a09-9124-7c507e55bcf3","source_id":"cb_question_bank","source_item_id":"5c336051-559f-4a09-9124-7c507e55bcf3","external_id":"155657c1-cb62-489d-b3be-d87bbbf5aada","ibn":null,"question_id":"b9835972","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.C.","skill":"Linear equations in two variables","difficulty":"H","score_band":6,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In the xy -plane, line script l passes through the point left parenthesis 0 comma 0 right parenthesis and is parallel to the line represented by the equation y equals 8 x plus 2 . If line script l also passes through the point left parenthesis 3 comma d right parenthesis , what is the value of d ?","stimulus":"","stimulus_html":null,"stem_html":"<p>In the <em>xy</em>-plane, line <math alttext=\"script l\"><mi>l</mi></math> passes through the point <math alttext=\"left parenthesis 0 comma 0 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> and is parallel to the line represented by the equation <math alttext=\"y equals 8 x plus 2\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>8</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mn>2</mn>\n\t</mrow>\n</mrow>\n</math>. If line <math alttext=\"script l\"><mi>l</mi></math> also passes through the point <math alttext=\"left parenthesis 3 comma d right parenthesis\"><mfenced><mrow><mrow><mn>3</mn></mrow><mo>,</mo><mi>d</mi></mrow></mfenced></math>, what is the value of <math alttext=\"d\"><mi>d</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"24\"><mn>24</mn>\n</math>. A line in the <em>xy</em>-plane can be defined by the equation <math alttext=\"y equals m x plus b\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mi>m</mi>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mi>b</mi>\n\t</mrow>\n</mrow>\n</math>, where <math alttext=\"m\"><mi>m</mi>\n</math> is the slope of the line and <math alttext=\"b\"><mi>b</mi>\n</math> is the <em>y</em>-coordinate of the <em>y</em>-intercept of the line. It's given that line <math alttext=\"script l\"><mi>l</mi></math> passes through the point <math alttext=\"left parenthesis 0 comma 0 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math>. Therefore, the <em>y</em>-coordinate of the <em>y</em>-intercept of line <math alttext=\"script l\"><mi>l</mi></math> is <math alttext=\"0\"><mn>0</mn>\n</math>. It's given that line <math alttext=\"script l\"><mi>l</mi></math> is parallel to the line represented by the equation <math alttext=\"y equals 8 x plus 2\"><mi>y</mi><mo>=</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>2</mn></math>. Since parallel lines have the same slope, it follows that the slope of line <math alttext=\"script l\"><mi>l</mi></math> is <math alttext=\"8\"><mn>8</mn>\n</math>. Therefore, line <math alttext=\"script l\"><mi>l</mi></math> can be defined by an equation in the form <math alttext=\"y equals m x plus b\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mi>m</mi>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mi>b</mi>\n\t</mrow>\n</mrow>\n</math>, where <math alttext=\"m equals 8\"><mrow>\n\t<mi>m</mi>\n\t<mo>=</mo>\n\t<mn>8</mn>\n</mrow>\n</math> and <math alttext=\"b equals 0\"><mrow>\n\t<mi>b</mi>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math>. Substituting <math alttext=\"8\"><mn>8</mn>\n</math> for <math alttext=\"m\"><mi>m</mi>\n</math> and <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"b\"><mi>b</mi>\n</math> in <math alttext=\"y equals m x plus b\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mi>m</mi>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mi>b</mi>\n\t</mrow>\n</mrow>\n</math> yields the equation <math alttext=\"y equals 8 x plus 0\"><mi>y</mi><mo>=</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>0</mn></math>, or <math alttext=\"y equals 8 x\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mn>8</mn>\n\t\t<mi>x</mi>\n\t</mrow>\n</mrow>\n</math> . If line <math alttext=\"script l\"><mi>l</mi></math> passes through the point <math alttext=\"left parenthesis 3 comma d right parenthesis\"><mfenced><mrow><mn>3</mn><mo>,</mo><mi>d</mi></mrow></mfenced></math>, then when <math alttext=\"x equals 3\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>3</mn>\n</mrow>\n</math>, <math alttext=\"y equals d\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mi>d</mi>\n</mrow>\n</math> for the equation <math alttext=\"y equals 8 x\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mn>8</mn>\n\t\t<mi>x</mi>\n\t</mrow>\n</mrow>\n</math>. Substituting <math alttext=\"3\"><mn>3</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"d\"><mi>d</mi>\n</math> for <math alttext=\"y\"><mi>y</mi>\n</math> in the equation <math alttext=\"y equals 8 x\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mn>8</mn>\n\t\t<mi>x</mi>\n\t</mrow>\n</mrow>\n</math> yields <math alttext=\"d equals 8 left parenthesis 3 right parenthesis\"><mi>d</mi><mo>=</mo><mn>8</mn><mfenced><mn>3</mn></mfenced></math>, or <math alttext=\"d equals 24\"><mrow>\n\t<mi>d</mi>\n\t<mo>=</mo>\n\t<mn>24</mn>\n</mrow>\n</math>.</p>","correct_answer":["24"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.816000+00:00","source_updated_at":"2023-08-02T20:25:59.816000+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}}