{"id":"cb_question_bank:b869369c-1afb-495b-8525-724acfb5051a","source_id":"cb_question_bank","source_item_id":"b869369c-1afb-495b-8525-724acfb5051a","external_id":"33e5c79f-6eb4-44bc-917c-7090df8d4158","ibn":null,"question_id":"e77a76ce","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":5,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"Which of the following systems of linear equations has no solution?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">Which of the following systems of linear equations has no solution?</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 no solution if the graphs of the lines represented by the equations in the <em>xy</em>-plane are distinct and parallel. The graphs of two lines in the&nbsp;<em>xy</em>-plane represented by equations 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>, where <math alttext=\"m\"><mi>m</mi>\n</math> and <math alttext=\"b\"><mi>b</mi>\n</math> are constants, are parallel if their slopes, <math alttext=\"m\"><mi>m</mi>\n</math>, are the same and are distinct if their <em>y</em>-coordinates of the <em>y</em>-intercepts, <math alttext=\"b\"><mi>b</mi>\n</math>, are different. In the equations&nbsp;<math alttext=\"y equals 6 x plus 3\"><mi>y</mi><mo>=</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>3</mn></math> and&nbsp;<math alttext=\"y equals 6 x plus 9\"><mi>y</mi><mo>=</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>9</mn></math>, the values of <math alttext=\"m\"><mi>m</mi>\n</math> are each <math alttext=\"6\"><mn>6</mn>\n</math>, and the values of <math alttext=\"b\"><mi>b</mi>\n</math> are <math alttext=\"3\"><mn>3</mn>\n</math> and <math alttext=\"9\"><mn>9</mn>\n</math>, respectively. Since the slopes of these lines are the same and the<em> y</em>-coordinates of the <em>y</em>-intercepts are different, it follows that the system of linear equations in choice A has no solution.</p>\n<p style=\"text-align: left;\">Choice B is incorrect. The two lines represented by these equations are a horizontal line and a line with a slope of <math alttext=\"10\"><mn>10</mn>\n</math> that have the same <em>y</em>-coordinate of the <em>y</em>-intercept. Therefore, this system has a solution, <math alttext=\"left parenthesis 0 comma 10 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>10</mn></mrow></mfenced></math>, rather than no solution.</p>\n<p style=\"text-align: left;\">Choice C is incorrect. The two lines represented by these equations have different slopes and the same <em>y</em>-coordinate of the&nbsp;<em>y</em>-intercept. Therefore, this system has a solution, <math alttext=\"left parenthesis 0 comma 14 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>14</mn></mrow></mfenced></math>, rather than no solution.</p>\n<p style=\"text-align: left;\">Choice D is incorrect. The two lines represented by these equations are a vertical line and a horizontal line. Therefore, this system has a solution, <math alttext=\"left parenthesis 3 comma 10 right parenthesis\"><mfenced><mrow><mn>3</mn><mo>,</mo><mn>10</mn></mrow></mfenced></math>, rather than no solution.</p>","correct_answer":["A"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.817000+00:00","source_updated_at":"2023-08-02T20:25:59.817000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"y equals 6 x plus 3\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>6</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mn>3</mn>\n\t</mrow>\n</mrow>\n</math></p>\n<p><math alttext=\"y equals 6 x plus 9\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>6</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></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"y equals 10\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mn>10</mn>\n</mrow>\n</math></p>\n<p><math alttext=\"y equals 10 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>10</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>","ord":1},{"label":"C","content_html":"<p><math alttext=\"y equals 14 x plus 14\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>14</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mn>14</mn>\n\t</mrow>\n</mrow>\n</math></p>\n<p><math alttext=\"y equals 10 x plus 14\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>10</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mn>14</mn>\n\t</mrow>\n</mrow>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"x equals 3\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>3</mn>\n</mrow>\n</math></p>\n<p><math alttext=\"y equals 10\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mn>10</mn>\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}}