{"id":"cb_question_bank:4087b4a7-e8fd-4bc6-ae22-b3ecb9bf0737","source_id":"cb_question_bank","source_item_id":"4087b4a7-e8fd-4bc6-ae22-b3ecb9bf0737","external_id":"5c64450d-1ff5-486d-bc37-e47fa80d2f7d","ibn":null,"question_id":"6a87902f","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":"y equals 2 x plus 10 y equals 2 x minus 1 At how many points do the graphs of the given equations intersect in the xy -plane?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"y equals 2 x plus 10\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>2</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mn>10</mn>\n\t</mrow>\n</mrow>\n</math></p>\n<p style=\"text-align: center;\"><math alttext=\"y equals 2 x minus 1\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>2</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>-</mo>\n\t\t<mn>1</mn>\n\t</mrow>\n</mrow>\n</math></p>\n<p>At how many points do the graphs of the given equations intersect in the <em>xy</em>-plane?</p>","rationale_html":"<p style=\"text-align: left;\">Choice A is correct. A system of two linear equations in two variables, <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"y\"><mi>y</mi>\n</math>, has zero points of intersection if the lines represented by the equations in the <em>xy</em>-plane are distinct and parallel. The graphs of two lines in the <em>xy</em>-plane represented by equations 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>, are distinct if the <em>y</em>-coordinates of their <em>y</em>-intercepts, <math alttext=\"b\"><mi>b</mi>\n</math>, are different and are parallel if their slopes, <math alttext=\"m\"><mi>m</mi>\n</math>, are the same. For the two equations in the given system, <math alttext=\"y equals 2 x plus 10\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>2</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mn>10</mn>\n\t</mrow>\n</mrow>\n</math> and <math alttext=\"y equals 2 x minus 1\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>2</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>-</mo>\n\t\t<mn>1</mn>\n\t</mrow>\n</mrow>\n</math>, the values of <math alttext=\"b\"><mi>b</mi>\n</math> are <math alttext=\"10\"><mn>10</mn>\n</math> and <math alttext=\"negative 1\"><mo>-</mo><mn>1</mn>\n</math>, respectively, and the values of <math alttext=\"m\"><mi>m</mi>\n</math> are both <math alttext=\"2\"><mn>2</mn>\n</math>. Since the values of <math alttext=\"b\"><mi>b</mi>\n</math> are different, the graphs of these lines have different <em>y</em>-coordinates of the <em>y</em>-intercept and are distinct. Since the values of <math alttext=\"m\"><mi>m</mi>\n</math> are the same, the graphs of these lines have the same slope and are parallel. Therefore, the graphs of the given equations are lines that intersect at zero points in the <em>xy</em>-plane.</p>\n<p style=\"text-align: left;\">Choice B is incorrect. The graphs of a system of two linear equations have exactly one point of intersection if the lines represented by the equations have different slopes. Since the given equations represent lines with the same slope, there is not exactly one intersection point.</p>\n<p style=\"text-align: left;\">Choice C is incorrect. The graphs of a system of two linear equations can never have exactly two intersection points.</p>\n<p style=\"text-align: left;\">Choice D is incorrect. The graphs of a system of two linear equations have infinitely many intersection points when the lines represented by the equations have the same slope and the same <em>y</em>-coordinate of the <em>y</em>-intercept. Since the given equations represent lines with different <em>y</em>-coordinates of their <em>y</em>-intercepts, there are not infinitely many intersection points.</p>","correct_answer":["A"],"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>Zero</p>","ord":0},{"label":"B","content_html":"<p>Exactly one</p>","ord":1},{"label":"C","content_html":"<p>Exactly two</p>","ord":2},{"label":"D","content_html":"<p>Infinitely many</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}}