{"id":"cb_question_bank:76c9622d-9ed9-4bb4-858b-8a00a13f0326","source_id":"cb_question_bank","source_item_id":"76c9622d-9ed9-4bb4-858b-8a00a13f0326","external_id":"64207213-df09-436f-aeeb-a588aafa2b18","ibn":null,"question_id":"45a534d0","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":7,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"48 x minus 72 y equals 30 y plus 24 r y equals one sixth minus 16 x In the given system of equations, r is a constant. If the system has no solution, what is the value of r ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"48 x minus 72 y equals 30 y plus 24\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>48</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>-</mo>\n\t\t<mrow>\n\t\t\t<mn>72</mn>\n\t\t\t<mi>y</mi>\n\t\t</mrow>\n\t</mrow>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>30</mn>\n\t\t\t<mi>y</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mn>24</mn>\n\t</mrow>\n</mrow>\n</math></p>\n<p style=\"text-align: center;\"><math alttext=\"r y equals one sixth minus 16 x\"><mi>r</mi><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>6</mn></mrow></mfrac><mo>-</mo><mrow><mn>16</mn></mrow><mi>x</mi></math></p>\n<p style=\"text-align: left;\">In the given system of equations, <math alttext=\"r\"><mi>r</mi>\n</math> is a constant. If the system has no solution, what is the value of <math alttext=\"r\"><mi>r</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"negative 34\"><mo>-</mo><mn>34</mn></math>. 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 distinct and parallel. Two 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>, where <math alttext=\"upper A\"><mi>A</mi>\n</math>, <math alttext=\"upper B\"><mi>B</mi>\n</math>, and <math alttext=\"upper C\"><mi>C</mi>\n</math> are constants, 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. The first equation in the given system can be written in standard form by subtracting <math alttext=\"30 y\"><mrow>\n\t<mn>30</mn>\n\t<mi>y</mi>\n</mrow>\n</math> from both sides of the equation to yield <math alttext=\"48 x minus 102 y equals 24\"><mn>48</mn><mi>x</mi><mo>-</mo><mn>102</mn><mi>y</mi><mo>=</mo><mn>24</mn></math>. The second equation in the given system can be written in standard form by adding <math alttext=\"16 x\"><mn>16</mn><mi>x</mi></math> to both sides of the equation to yield <math alttext=\"16 x plus r y equals one sixth\"><mn>16</mn><mi>x</mi><mo>+</mo><mi>r</mi><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></math>. &nbsp;The coefficient of <math alttext=\"x\"><mi>x</mi>\n</math> in this second equation, <math alttext=\"16\"><mn>16</mn>\n</math>, is&nbsp;<math alttext=\"one third\"><mfrac><mn>1</mn><mn>3</mn></mfrac></math> times the coefficient of <math alttext=\"x\"><mi>x</mi>\n</math> in the first equation, <math alttext=\"48\"><mn>48</mn>\n</math>. For the lines to be parallel the coefficient of <math alttext=\"y\"><mi>y</mi>\n</math> in the second equation, <math alttext=\"r\"><mi>r</mi>\n</math>, must also be&nbsp;<math alttext=\"one third\"><mfrac><mn>1</mn><mn>3</mn></mfrac></math> times the coefficient of <math alttext=\"y\"><mi>y</mi>\n</math> in the first equation, <math alttext=\"negative 102\"><mo>-</mo><mn>102</mn></math>. Thus, <math alttext=\"r equals one third left parenthesis negative 102 right parenthesis\"><mi>r</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>(</mo><mo>-</mo><mn>102</mn><mo>)</mo></math>, or <math alttext=\"r equals negative 34\"><mi>r</mi><mo>=</mo><mo>-</mo><mn>34</mn></math>. Therefore, if the given system has no solution, the value of <math alttext=\"r\"><mi>r</mi>\n</math> is <math alttext=\"negative 34\"><mo>-</mo><mn>34</mn></math>.</p>","correct_answer":["-34"],"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":[],"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}}