{"id":"cb_question_bank:e31ea41f-be85-4ffe-9d6c-2ebf13f0d90e","source_id":"cb_question_bank","source_item_id":"e31ea41f-be85-4ffe-9d6c-2ebf13f0d90e","external_id":"673c8b0c-840e-41ac-bc4e-82b2f8b2a747","ibn":null,"question_id":"5e08a055","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.D.","skill":"Systems of two linear equations in two variables","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"y equals 6 x plus 18 One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"y equals 6 x plus 18\"><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>18</mn>\n\t</mrow>\n</mrow>\n</math></p>\n<p style=\"text-align: left;\">One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?</p>","rationale_html":"<p>Choice B 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 lines represented by the equations in the <em>xy</em>-plane are parallel and distinct. Lines represented by equations in standard form, <math alttext=\"upper A x plus upper B y equals upper C\"><mi>A</mi><mi>x</mi><mo>+</mo><mi>B</mi><mi>y</mi><mo>=</mo><mi>C</mi></math> and <math alttext=\"upper D x plus upper E y equals upper F\"><mi>D</mi><mi>x</mi><mo>+</mo><mi>E</mi><mi>y</mi><mo>=</mo><mi>F</mi></math>, are parallel if the coefficients for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"y\"><mi>y</mi>\n</math> in one equation are proportional to the corresponding coefficients in the other equation, meaning <math alttext=\"StartFraction upper D Over upper A EndFraction equals StartFraction upper E Over upper B EndFraction\"><mfrac><mi>D</mi><mi>A</mi></mfrac><mo>=</mo><mfrac><mi>E</mi><mi>B</mi></mfrac></math>; and the lines are distinct if the constants are not proportional, meaning <math alttext=\"StartFraction upper F Over upper C EndFraction\"><mfrac><mi>F</mi><mi>C</mi></mfrac></math> is not equal to <math alttext=\"StartFraction upper D Over upper A EndFraction\"><mfrac><mi>D</mi><mi>A</mi></mfrac></math> or <math alttext=\"StartFraction upper E Over upper B EndFraction\"><mfrac><mi>E</mi><mi>B</mi></mfrac></math>. The given equation, <math alttext=\"y equals 6 x plus 18\"><mi>y</mi><mo>=</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>18</mn></math>, can be written in standard form by subtracting <math alttext=\"6 x\"><mn>6</mn><mi>x</mi></math> from both sides of the equation to yield <math alttext=\"minus 6 x plus y equals 18\"><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>18</mn></math>. Therefore, the given equation can be written in the form <math alttext=\"upper A x plus upper B y equals upper C\"><mi>A</mi><mi>x</mi><mo>+</mo><mi>B</mi><mi>y</mi><mo>=</mo><mi>C</mi></math>, where <math alttext=\"upper A equals negative 6\"><mi>A</mi><mo>=</mo><mo>-</mo><mn>6</mn></math>, <math alttext=\"upper B equals 1\"><mi>B</mi><mo>=</mo><mn>1</mn></math>, and <math alttext=\"upper C equals 18\"><mi>C</mi><mo>=</mo><mn>18</mn></math>. The equation in choice B, <math alttext=\"minus 6 x plus y equals 22\"><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>22</mn></math>, is written in the form <math alttext=\"upper D x plus upper E y equals upper F\"><mi>D</mi><mi>x</mi><mo>+</mo><mi>E</mi><mi>y</mi><mo>=</mo><mi>F</mi></math>, where <math alttext=\"upper D equals negative 6\"><mi>D</mi><mo>=</mo><mo>-</mo><mn>6</mn></math>, <math alttext=\"upper E equals 1\"><mi>E</mi><mo>=</mo><mn>1</mn></math>, and <math alttext=\"upper F equals 22\"><mi>F</mi><mo>=</mo><mn>22</mn></math>. Therefore,&nbsp; <math alttext=\"StartFraction upper D Over upper A EndFraction equals StartFraction negative 6 Over negative 6 EndFraction\"><mfrac><mi>D</mi><mi>A</mi></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mn>6</mn></mrow><mrow><mo>-</mo><mn>6</mn></mrow></mfrac></math>, which can be rewritten as <math alttext=\"StartFraction upper D Over upper A EndFraction equals 1\"><mfrac><mi>D</mi><mi>A</mi></mfrac><mo>=</mo><mn>1</mn></math>; <math alttext=\"StartFraction upper E Over upper B EndFraction equals one oneth\"><mfrac><mi>E</mi><mi>B</mi></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>1</mn></mfrac></math>, which can be rewritten as <math alttext=\"StartFraction upper E Over upper B EndFraction equals 1\"><mfrac><mi>E</mi><mi>B</mi></mfrac><mo>=</mo><mn>1</mn></math>; and <math alttext=\"StartFraction upper F Over upper C EndFraction equals StartFraction 22 Over 18 EndFraction\"><mfrac><mi>F</mi><mi>C</mi></mfrac><mo>=</mo><mfrac><mn>22</mn><mn>18</mn></mfrac></math>, which can be rewritten as <math alttext=\"StartFraction upper F Over upper C EndFraction equals StartFraction 11 Over 9 EndFraction\"><mfrac><mi>F</mi><mi>C</mi></mfrac><mo>=</mo><mfrac><mn>11</mn><mn>9</mn></mfrac></math>. Since <math alttext=\"StartFraction upper D Over upper A EndFraction equals 1\"><mfrac><mi>D</mi><mi>A</mi></mfrac><mo>=</mo><mn>1</mn></math>,&nbsp;<math alttext=\"StartFraction upper E Over upper B EndFraction equals 1\"><mfrac><mi>E</mi><mi>B</mi></mfrac><mo>=</mo><mn>1</mn></math>, and <math alttext=\"StartFraction upper F Over upper C EndFraction\"><mfrac><mi>F</mi><mi>C</mi></mfrac></math> is not equal to <math alttext=\"1\"><mn>1</mn>\n</math>, it follows that the given equation and the equation <math alttext=\"minus 6 x plus y equals 22\"><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>22</mn></math> are parallel and distinct. Therefore, a system of two linear equations consisting of the given equation and the equation <math alttext=\"minus 6 x plus y equals 22\"><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>22</mn></math> has no solution. Thus, the&nbsp;equation in choice B could be the second equation in the system.</p>\n<p>Choice A is incorrect. The equation&nbsp;<math alttext=\"minus 6 x plus y equals 18\"><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>18</mn></math> and the given equation represent the same line in the <em>xy</em>-plane. Therefore, a system of these linear equations would have infinitely many solutions, rather than no solution.&nbsp;</p>\n<p>Choice C is incorrect. The equation <math alttext=\"minus 12 x plus y equals 36\"><mo>-</mo><mn>12</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>36</mn></math> and the given equation represent lines in the <em>xy</em>-plane that are distinct and not parallel. Therefore, a system of these linear equations would have exactly one solution, rather than no solution.</p>\n<p>Choice D is incorrect. The equation <math alttext=\"minus 12 x plus y equals 18\"><mo>-</mo><mn>12</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>18</mn></math> and the given equation represent lines in the <em>xy</em>-plane that are distinct and not parallel. Therefore, a system of these linear equations would have&nbsp;exactly one solution, rather than no solution.</p>","correct_answer":["B"],"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=\"minus 6 x plus y equals 18\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mo>-</mo>\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<mi>y</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>18</mn>\n</mrow>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"minus 6 x plus y equals 22\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mo>-</mo>\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<mi>y</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>22</mn>\n</mrow>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"minus 12 x plus y equals 36\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mo>-</mo>\n\t\t\t<mn>12</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mi>y</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>36</mn>\n</mrow>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"minus 12 x plus y equals 18\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mo>-</mo>\n\t\t\t<mn>12</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mi>y</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>18</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}}