{"id":"cb_question_bank:2615b496-fb9e-49ce-a810-2ebf29429a8e","source_id":"cb_question_bank","source_item_id":"2615b496-fb9e-49ce-a810-2ebf29429a8e","external_id":"07807ef4-fe53-474a-92e4-59c6ea6c579e","ibn":null,"question_id":"0dd6227f","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.D.","skill":"Systems of two 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":"At how many points do the graphs of the equations y equals x plus 20 and y equals 8 x intersect in the xy -plane?","stimulus":"","stimulus_html":null,"stem_html":"<p>At how many points do the graphs of the equations <math alttext=\"y equals x plus 20\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mi>x</mi>\n\t\t<mo>+</mo>\n\t\t<mn>20</mn>\n\t</mrow>\n</mrow>\n</math> and <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> intersect in the <em>xy</em>-plane?</p>","rationale_html":"<p style=\"text-align: left;\">Choice B is correct. Each given equation is written 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> is the slope and&nbsp;<math alttext=\"left parenthesis 0 comma b right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mi>b</mi></mrow></mfenced></math> is the <em>y</em>-intercept of the graph of the equation in the <em>xy</em>-plane. The graphs of two lines that have different slopes will intersect at exactly one point. The graph of the first equation is a line with slope <math alttext=\"1\"><mn>1</mn>\n</math>. The graph of the second equation is a line with slope <math alttext=\"8\"><mn>8</mn>\n</math>. Since the graphs are lines with different slopes, they will intersect at exactly one point.</p>\n<p style=\"text-align: left;\">Choice A is incorrect because two graphs of linear equations have <math alttext=\"0\"><mn>0</mn>\n</math> intersection points only if they are parallel and therefore have the same slope.</p>\n<p style=\"text-align: left;\">Choice C is incorrect because two graphs of linear equations in the <em>xy</em>-plane can have only <math alttext=\"0\"><mn>0</mn>\n</math>, <math alttext=\"1\"><mn>1</mn>\n</math>, or infinitely many points of intersection.</p>\n<p style=\"text-align: left;\">Choice D is incorrect because two graphs of linear equations in the <em>xy</em>-plane can have only <math alttext=\"0\"><mn>0</mn>\n</math>, <math alttext=\"1\"><mn>1</mn>\n</math>, or infinitely many points of intersection.</p>","correct_answer":["B"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.818000+00:00","source_updated_at":"2023-08-02T20:25:59.818000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"0\"><mn>0</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"1\"><mn>1</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"2\"><mn>2</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"8\"><mn>8</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}}