{"id":"cb_question_bank:a9592ed8-9c71-4456-84b0-f1ab957d883f","source_id":"cb_question_bank","source_item_id":"a9592ed8-9c71-4456-84b0-f1ab957d883f","external_id":"02dc7dfd-b457-45d1-8655-2bcfcc4dfc17","ibn":null,"question_id":"f81a0503","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.C.","skill":"Linear equations in two variables","difficulty":"M","score_band":4,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In the xy -plane, line k passes through the points left parenthesis 0 comma negative 5 right parenthesis and left parenthesis 1 comma negative 1 right parenthesis . Which equation defines line k ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">In the&nbsp;<em>xy</em>-plane, line <math alttext=\"k\"><mi>k</mi>\n</math> passes through the points&nbsp;<math alttext=\"left parenthesis 0 comma negative 5 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mn>5</mn></mrow></mfenced></math> and&nbsp;<math alttext=\"left parenthesis 1 comma negative 1 right parenthesis\"><mfenced><mrow><mn>1</mn><mo>,</mo><mo>-</mo><mn>1</mn></mrow></mfenced></math>. Which equation defines line <math alttext=\"k\"><mi>k</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">Choice D is correct. An equation defining a line in the <em>xy</em>-plane can be written 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\"><mi>m</mi>\n</math> represents the slope and <math alttext=\"left parenthesis 0 comma b right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mi>b</mi></mrow></mfenced></math> represents the <em>y</em>-intercept of the line. It’s given that line <math alttext=\"k\"><mi>k</mi>\n</math> passes through the point <math alttext=\"left parenthesis 0 comma negative 5 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mn>5</mn></mrow></mfenced></math>; therefore, <math alttext=\"b equals negative 5\"><mrow>\n\t<mi>b</mi>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>5</mn>\n</mrow>\n</math>. The slope, <math alttext=\"m\"><mi>m</mi>\n</math>, of a line can be found using any two points on the line,&nbsp;<math alttext=\"left parenthesis x 1 comma y 1 right parenthesis\"><mfenced><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>y</mi><mn>1</mn></msub></mrow></mfenced></math> and <math alttext=\"left parenthesis x 2 comma y 2 right parenthesis\"><mfenced><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><msub><mi>y</mi><mn>2</mn></msub></mrow></mfenced></math>, and the slope formula <math alttext=\"m equals StartFraction y 2 minus y 1 Over x 2 minus x 1 EndFraction\"><mi>m</mi><mo>=</mo><mfrac><mrow><msub><mi>y</mi><mn>2</mn></msub><mo>-</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>-</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></math>. Substituting the points&nbsp;<math alttext=\"left parenthesis 0 comma negative 5 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mn>5</mn></mrow></mfenced></math> and <math alttext=\"left parenthesis 1 comma negative 1 right parenthesis\"><mfenced><mrow><mn>1</mn><mo>,</mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> for <math alttext=\"left parenthesis x 1 comma y 1 right parenthesis\"><mfenced><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>y</mi><mn>1</mn></msub></mrow></mfenced></math> and <math alttext=\"left parenthesis x 2 comma y 2 right parenthesis\"><mfenced><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><msub><mi>y</mi><mn>2</mn></msub></mrow></mfenced></math>, respectively, in the slope formula yields <math alttext=\"m equals StartFraction left parenthesis negative 1 minus left parenthesis negative 5 right parenthesis right parenthesis Over left parenthesis 1 minus 0 right parenthesis EndFraction\"><mi>m</mi><mo>=</mo><mfrac><mfenced><mrow><mo>-</mo><mn>1</mn><mo>-</mo><mfenced><mrow><mo>-</mo><mn>5</mn></mrow></mfenced></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mn>0</mn></mrow></mfenced></mfrac></math>, or <math alttext=\"m equals 4\"><mrow>\n\t<mi>m</mi>\n\t<mo>=</mo>\n\t<mn>4</mn>\n</mrow>\n</math>. Substituting <math alttext=\"4\"><mn>4</mn>\n</math> for <math alttext=\"m\"><mi>m</mi>\n</math> and <math alttext=\"negative 5\"><mo>-</mo><mn>5</mn>\n</math> for <math alttext=\"b\"><mi>b</mi>\n</math> in 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> yields <math alttext=\"y equals 4 x minus 5\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>4</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>-</mo>\n\t\t<mn>5</mn>\n\t</mrow>\n</mrow>\n</math>.&nbsp;</p>\n<p style=\"text-align: left;\">Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice C is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["D"],"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=\"y equals negative x plus one fourth\"><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<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mfrac>\n\t\t\t<mn>1</mn>\n\t\t\t<mn>4</mn>\n\t\t</mfrac>\n\t</mrow>\n</mrow>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"y equals one fourth x minus 5\"><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi><mo>-</mo><mn>5</mn></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"y equals negative x plus 4\"><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<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mn>4</mn>\n\t</mrow>\n</mrow>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"y equals 4 x minus 5\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>4</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>-</mo>\n\t\t<mn>5</mn>\n\t</mrow>\n</mrow>\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}}