{"id":"cb_question_bank:0752e3f1-5acc-45cf-883e-5a93989a9531","source_id":"cb_question_bank","source_item_id":"0752e3f1-5acc-45cf-883e-5a93989a9531","external_id":"5c09d547-9da9-4b9a-a436-ddc8c0c70764","ibn":null,"question_id":"ff501705","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":"three halves y minus one fourth x equals two thirds minus three halves y one half x plus three halves equals p y plus nine halves In the given system of equations, p is a constant. If the system has no solution, what is the value of p ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"three halves y minus one fourth x equals two thirds minus three halves y\"><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mi>y</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mi>x</mi><mo>=</mo><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><mo>-</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mi>y</mi></math></p>\n<p style=\"text-align: center;\"><math alttext=\"one half x plus three halves equals p y plus nine halves\"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mi>x</mi><mo>+</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>=</mo><mi>p</mi><mi>y</mi><mo>+</mo><mrow><mfrac><mn>9</mn><mn>2</mn></mfrac></mrow></math></p>\n<p style=\"text-align: left;\">In the given system of equations, <math alttext=\"p\"><mi>p</mi>\n</math> is a constant. If the system has no solution, what is the value of <math alttext=\"p\"><mi>p</mi>\n</math>?</p>","rationale_html":"<p>The correct answer is <math alttext=\"6\"><mn>6</mn>\n</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 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>;&nbsp;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&nbsp;<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 first equation in the given system is <math alttext=\"three halves y minus one fourth x equals two thirds minus three halves y\"><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>y</mi><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>y</mi></math>. Multiplying each side of this equation by <math alttext=\"12\"><mn>12</mn>\n</math> yields <math alttext=\"18 y minus 3 x equals 8 minus 18 y\"><mn>18</mn><mi>y</mi><mo>-</mo><mn>3</mn><mi>x</mi><mo>=</mo><mn>8</mn><mo>-</mo><mn>18</mn><mi>y</mi></math>. Adding <math alttext=\"18 y\"><mrow>\n\t<mn>18</mn>\n\t<mi>y</mi>\n</mrow>\n</math> to each side of this equation yields <math alttext=\"36 y minus 3 x equals 8\"><mn>36</mn><mi>y</mi><mo>-</mo><mn>3</mn><mi>x</mi><mo>=</mo><mn>8</mn></math>, or <math alttext=\"minus 3 x plus 36 y equals 8\"><mo>-</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>36</mn><mi>y</mi><mo>=</mo><mn>8</mn></math>. The second equation in the given system is <math alttext=\"one half x plus three halves equals p y plus nine halves\"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>=</mo><mi>p</mi><mi>y</mi><mo>+</mo><mfrac><mn>9</mn><mn>2</mn></mfrac></math>. Multiplying each side of this equation by <math alttext=\"2\"><mn>2</mn>\n</math> yields <math alttext=\"x plus 3 equals 2 p y plus 9\"><mi>x</mi><mo>+</mo><mn>3</mn><mo>=</mo><mn>2</mn><mi>p</mi><mi>y</mi><mo>+</mo><mn>9</mn></math>. Subtracting <math alttext=\"2 p y\"><mrow>\n\t<mn>2</mn>\n\t<mi>p</mi>\n\t<mi>y</mi>\n</mrow>\n</math> from each side of this equation yields <math alttext=\"x plus 3 minus 2 p y equals 9\"><mi>x</mi><mo>+</mo><mn>3</mn><mo>-</mo><mn>2</mn><mi>p</mi><mi>y</mi><mo>=</mo><mn>9</mn></math>. Subtracting <math alttext=\"3\"><mn>3</mn>\n</math> from each side of this equation yields <math alttext=\"x minus 2 p y equals 6\"><mi>x</mi><mo>-</mo><mn>2</mn><mi>p</mi><mi>y</mi><mo>=</mo><mn>6</mn></math>. Therefore, the two equations in the given system, written in standard form, are&nbsp;<math alttext=\"minus 3 x plus 36 y equals 8\"><mo>-</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>36</mn><mi>y</mi><mo>=</mo><mn>8</mn></math> and<math alttext=\"x minus 2 p y equals 6\"><mo>&nbsp;</mo><mi>x</mi><mo>-</mo><mn>2</mn><mi>p</mi><mi>y</mi><mo>=</mo><mn>6</mn></math>. As previously stated, if this system has no solution, the lines represented by the equations in the <em>xy</em>-plane are parallel and distinct, meaning the proportion&nbsp;<math alttext=\"StartFraction 1 Over negative 3 EndFraction equals StartFraction minus 2 p Over 36 EndFraction\"><mfrac><mn>1</mn><mrow><mo>-</mo><mn>3</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mn>2</mn><mi>p</mi></mrow><mn>36</mn></mfrac></math>, or&nbsp;<math alttext=\"negative one third equals minus StartFraction p Over 18 EndFraction\"><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>=</mo><mo>-</mo><mfrac><mi>p</mi><mn>18</mn></mfrac></math>, is true and the proportion <math alttext=\"six eighths equals StartFraction 1 Over negative 3 EndFraction\"><mfrac><mn>6</mn><mn>8</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mo>-</mo><mn>3</mn></mrow></mfrac></math> is not true. The proportion <math alttext=\"six eighths equals StartFraction 1 Over negative 3 EndFraction\"><mfrac><mn>6</mn><mn>8</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mo>-</mo><mn>3</mn></mrow></mfrac></math> is not true. Multiplying each side of the true proportion, <math alttext=\"negative one third equals minus StartFraction p Over 18 EndFraction\"><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>=</mo><mo>-</mo><mfrac><mi>p</mi><mn>18</mn></mfrac></math>,&nbsp; by <math alttext=\"negative 18\"><mo>-</mo><mn>18</mn>\n</math> yields <math alttext=\"6 equals p\"><mn>6</mn><mo>=</mo><mi>p</mi></math>. Therefore, if the system has no solution, then the value of <math alttext=\"p\"><mi>p</mi>\n</math> is <math alttext=\"6\"><mn>6</mn>\n</math>.&nbsp;</p>","correct_answer":["6"],"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}}