{"id":"cb_question_bank:c5cca568-b415-4b8d-8424-3588ba0eaee3","source_id":"cb_question_bank","source_item_id":"c5cca568-b415-4b8d-8424-3588ba0eaee3","external_id":"9961525f-7f16-4e8c-9aef-754d7403a0cf","ibn":null,"question_id":"14360f84","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":"4 x minus 9 y equals 9 y plus 5 h y equals 2 plus 4 x In the given system of equations, h is a constant. If the system has no solution, what is the value of h ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"4 x minus 9 y equals 9 y plus 5\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>4</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>9</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>9</mn>\n\t\t\t<mi>y</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mn>5</mn>\n\t</mrow>\n</mrow>\n</math></p>\n<p style=\"text-align: center;\"><math alttext=\"h y equals 2 plus 4 x\"><mi>h</mi><mi>y</mi><mo>=</mo><mrow><mn>2</mn></mrow><mo>+</mo><mrow><mn>4</mn></mrow><mi>x</mi></math></p>\n<p style=\"text-align: left;\">In the given system of equations, <math alttext=\"h\"><mi>h</mi>\n</math> is a constant. If the system has no solution, what is the value of <math alttext=\"h\"><mi>h</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">Choice D 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 distinct and parallel. The graphs of two lines in the <em>xy</em>-plane represented by equations 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\"><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 the form&nbsp;<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> by subtracting <math alttext=\"9 y\"><mrow>\n\t<mn>9</mn>\n\t<mi>y</mi>\n</mrow>\n</math> from both sides of the equation to yield <math alttext=\"4 x minus 18 y equals 5\"><mn>4</mn><mi>x</mi><mo>-</mo><mn>18</mn><mi>y</mi><mo>=</mo><mn>5</mn></math>. The second equation in the given system can be written in the form&nbsp;<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> by subtracting <math alttext=\"4 x\"><mrow>\n\t<mn>4</mn>\n\t<mi>x</mi>\n</mrow>\n</math> from both sides of the equation to yield <math alttext=\"minus 4 x plus h y equals 2\"><mo>-</mo><mn>4</mn><mi>x</mi><mo>+</mo><mi>h</mi><mi>y</mi><mo>=</mo><mn>2</mn></math>. The coefficient of <math alttext=\"x\"><mi>x</mi>\n</math> in this second equation, <math alttext=\"negative 4\"><mo>-</mo><mn>4</mn></math>, is&nbsp;<math alttext=\"negative 1\"><mo>-</mo><mn>1</mn></math> times the coefficient of <math alttext=\"x\"><mi>x</mi>\n</math> in the first equation, <math alttext=\"4\"><mn>4</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=\"h\"><mi>h</mi>\n</math>, must also be <math alttext=\"negative 1\"><mo>-</mo><mn>1</mn></math> times the coefficient of <math alttext=\"y\"><mi>y</mi>\n</math> in the first equation, <math alttext=\"negative 18\"><mo>-</mo><mn>18</mn></math>. Thus, <math alttext=\"h equals minus 1 left parenthesis negative 18 right parenthesis\"><mi>h</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>(</mo><mo>-</mo><mn>18</mn><mo>)</mo></math>, or <math alttext=\"h equals 18\"><mi>h</mi><mo>=</mo><mn>18</mn></math>. Therefore, if the given system has no solution, the value of <math alttext=\"h\"><mi>h</mi>\n</math> is <math alttext=\"18\"><mn>18</mn>\n</math>.</p>\n<p style=\"text-align: left;\">Choice A is incorrect. If the value of <math alttext=\"h\"><mi>h</mi>\n</math> is <math alttext=\"negative 9\"><mo>-</mo><mn>9</mn>\n</math>, then the given system would have one solution, rather than no solution.</p>\n<p style=\"text-align: left;\">Choice B is incorrect. If the value of <math alttext=\"h\"><mi>h</mi>\n</math> is <math alttext=\"0\"><mn>0</mn>\n</math>, then the given system would have one solution, rather than no solution.</p>\n<p style=\"text-align: left;\">Choice C is incorrect. If the value of <math alttext=\"h\"><mi>h</mi>\n</math> is <math alttext=\"9\"><mn>9</mn>\n</math>, then the given system would have one solution, rather than no solution.</p>","correct_answer":["D"],"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=\"negative 9\"><mo>-</mo><mn>9</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"0\"><mn>0</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"9\"><mn>9</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"18\"><mn>18</mn>\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}}