{"id":"cb_question_bank:137e07a4-2592-4db4-a3d2-f299d59d4413","source_id":"cb_question_bank","source_item_id":"137e07a4-2592-4db4-a3d2-f299d59d4413","external_id":"6e2c4ed8-a7d5-4552-bdd7-42223af298ea","ibn":null,"question_id":"db422e7f","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":"Line p is defined by 4 y plus 8 x equals 6 . Line r is perpendicular to line p in the xy -plane. What is the slope of line r ?","stimulus":"","stimulus_html":null,"stem_html":"<p>Line <math alttext=\"p\"><mi>p</mi>\n</math> is defined by <math alttext=\"4 y plus 8 x equals 6\"><mrow><mn>4</mn></mrow><mi>y</mi><mo>+</mo><mrow><mn>8</mn></mrow><mi>x</mi><mo>=</mo><mrow><mn>6</mn></mrow></math>. Line <math alttext=\"r\"><mi>r</mi>\n</math> is perpendicular to line <math alttext=\"p\"><mi>p</mi>\n</math> in the <em>xy</em>-plane. What is the slope of line <math alttext=\"r\"><mi>r</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"one half\"><mfrac><mn>1</mn><mn>2</mn></mfrac></math>. For an equation in slope-intercept form <math alttext=\"y equals m x plus b\"><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi></math>, <math alttext=\"m\"><mi>m</mi>\n</math> represents the slope of the line in the <em>xy</em>-plane defined by this equation. It's given that line <math alttext=\"p\"><mi>p</mi>\n</math> is defined by <math alttext=\"4 y plus 8 x equals 6\"><mn>4</mn><mi>y</mi><mo>+</mo><mn>8</mn><mi>x</mi><mo>=</mo><mn>6</mn></math>. Subtracting <math alttext=\"8 x\"><mn>8</mn><mi>x</mi></math> from both sides of this equation yields <math alttext=\"4 y equals minus 8 x plus 6\"><mn>4</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>6</mn></math>. Dividing both sides of this equation by <math alttext=\"4\"><mn>4</mn>\n</math> yields <math alttext=\"y equals minus eight fourths x plus six fourths\"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>8</mn><mn>4</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>6</mn><mn>4</mn></mfrac></math>, or <math alttext=\"y equals minus 2 x plus three halves\"><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math>. Thus, the slope of line <math alttext=\"p\"><mi>p</mi>\n</math> is <math alttext=\"negative 2\"><mo>-</mo><mn>2</mn>\n</math>. If line <math alttext=\"r\"><mi>r</mi>\n</math> is perpendicular to line <math alttext=\"p\"><mi>p</mi>\n</math>, then the slope of line <math alttext=\"r\"><mi>r</mi>\n</math> is the negative reciprocal of the slope of line <math alttext=\"p\"><mi>p</mi>\n</math>. The negative reciprocal of <math alttext=\"negative 2\"><mo>-</mo><mn>2</mn>\n</math> is <math alttext=\"minus StartFraction 1 Over left parenthesis negative 2 right parenthesis EndFraction equals one half\"><mo>-</mo><mfrac><mn>1</mn><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>. Note that 1/2 and .5 are examples of ways to enter a correct answer.</p>","correct_answer":[".5","1/2"],"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}}