{"id":"cb_question_bank:13c27936-b67a-4f69-84ee-16cdc3fa934f","source_id":"cb_question_bank","source_item_id":"13c27936-b67a-4f69-84ee-16cdc3fa934f","external_id":"ab826e08-7891-4957-94db-0152ef740dff","ibn":null,"question_id":"01682aa5","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.C.","skill":"Linear equations in two variables","difficulty":"M","score_band":5,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"Line p is defined by 2 y plus 18 x equals 9 . 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 style=\"text-align: left;\">Line <math alttext=\"p\"><mi>p</mi>\n</math> is defined by <math alttext=\"2 y plus 18 x equals 9\"><mn>2</mn><mi>y</mi><mo>+</mo><mrow><mn>18</mn></mrow><mi>x</mi><mo>=</mo><mrow><mn>9</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;\">Choice C is correct. It’s given that 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. This means that 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>. If the equation for line <math alttext=\"p\"><mi>p</mi>\n</math> is rewritten in slope-intercept 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\"><mi>m</mi>\n</math> and <math alttext=\"b\"><mi>b</mi>\n</math> are constants, then <math alttext=\"m\"><mi>m</mi>\n</math> is the slope of the line and <math alttext=\"left parenthesis 0 comma b right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mi>b</mi></mrow></mfenced></math> is its <em>y</em>-intercept. Subtracting <math alttext=\"18 x\"><mrow>\n\t<mn>18</mn>\n\t<mi>x</mi>\n</mrow>\n</math> from both sides of the equation <math alttext=\"2 y plus 18 x equals 9\"><mn>2</mn><mi>y</mi><mo>+</mo><mn>18</mn><mi>x</mi><mo>=</mo><mn>9</mn></math>&nbsp;yields <math alttext=\"2 y equals minus 18 x plus 9\"><mrow>\n\t<mrow>\n\t\t<mn>2</mn>\n\t\t<mi>y</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mo>-</mo>\n\t\t\t<mn>18</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mn>9</mn>\n\t</mrow>\n</mrow>\n</math>. Dividing both sides of this equation by <math alttext=\"2\"><mn>2</mn>\n</math> yields <math alttext=\"y equals minus 9 x plus nine halves\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mo>-</mo>\n\t\t\t<mn>9</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mfrac>\n\t\t\t<mn>9</mn>\n\t\t\t<mn>2</mn>\n\t\t</mfrac>\n\t</mrow>\n</mrow>\n</math>. It follows that the slope of line <math alttext=\"p\"><mi>p</mi>\n</math> is <math alttext=\"negative 9\"><mo>-</mo><mn>9</mn>\n</math>. The negative reciprocal of a number is <math alttext=\"negative 1\"><mo>-</mo><mn>1</mn>\n</math> divided by the number. Therefore, the negative reciprocal of <math alttext=\"negative 9\"><mo>-</mo><mn>9</mn>\n</math> is&nbsp;<math alttext=\"StartFraction negative 1 Over negative 9 EndFraction\"><mfrac><mrow><mo>-</mo><mn>1</mn></mrow><mrow><mo>-</mo><mn>9</mn></mrow></mfrac></math>, or <math alttext=\"one ninth\"><mfrac>\n\t<mn>1</mn>\n\t<mn>9</mn>\n</mfrac>\n</math>. Thus, the slope of line <math alttext=\"r\"><mi>r</mi>\n</math> is <math alttext=\"one ninth\"><mfrac>\n\t<mn>1</mn>\n\t<mn>9</mn>\n</mfrac>\n</math>.</p>\n<p style=\"text-align: left;\">Choice A is incorrect. This is the slope of line <math alttext=\"p\"><mi>p</mi>\n</math>, not line <math alttext=\"r\"><mi>r</mi>\n</math>.</p>\n<p style=\"text-align: left;\">Choice B is incorrect. This is the reciprocal, not the negative reciprocal, of the slope of line <math alttext=\"p\"><mi>p</mi>\n</math>.</p>\n<p style=\"text-align: left;\">Choice D is incorrect. This is the negative, not the negative reciprocal, of the slope of line <math alttext=\"p\"><mi>p</mi>\n</math>.</p>","correct_answer":["C"],"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":[{"label":"A","content_html":"<p><math alttext=\"negative 9\"><mo>-</mo><mn>9</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"negative one ninth\"><mrow>\n\t<mo>-</mo>\n\t<mfrac>\n\t\t<mn>1</mn>\n\t\t<mn>9</mn>\n\t</mfrac>\n</mrow>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"one ninth\"><mfrac>\n\t<mn>1</mn>\n\t<mn>9</mn>\n</mfrac>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"9\"><mn>9</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}}