{"id":"cb_question_bank:867aef05-ad80-4096-a118-879493bc0fc9","source_id":"cb_question_bank","source_item_id":"867aef05-ad80-4096-a118-879493bc0fc9","external_id":"ae003425-4a05-45a2-b10f-6a51a4a26073","ibn":null,"question_id":"895628b5","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.B.","skill":"Nonlinear equations in one variable and systems of equations in two variables","difficulty":"M","score_band":4,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"y equals left parenthesis x minus 2 right parenthesis left parenthesis x plus 4 right parenthesis y equals 6 x minus 12 Which ordered pair left parenthesis x comma y right parenthesis is the solution to the given system of equations?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"y equals left parenthesis x minus 2 right parenthesis left parenthesis x plus 4 right parenthesis\"><mi>y</mi><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mrow><mn>2</mn></mrow></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mrow><mn>4</mn></mrow></mrow></mfenced></math></p>\n<p style=\"text-align: center;\"><math alttext=\"y equals 6 x minus 12\"><mi>y</mi><mo>=</mo><mrow><mn>6</mn></mrow><mi>x</mi><mo>-</mo><mrow><mn>12</mn></mrow></math></p>\n<p style=\"text-align: left;\">Which ordered pair <math alttext=\"left parenthesis x comma y right parenthesis\"><mfenced><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></math> is the solution to the given system of equations?</p>","rationale_html":"<p>Choice C is correct. The second equation in the given system of equations is <math alttext=\"y equals 6 x minus 12\"><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>12</mn>\n\t</mrow>\n</mrow>\n</math>. Substituting <math alttext=\"6 x minus 12\"><mrow>\n\t<mrow>\n\t\t<mn>6</mn>\n\t\t<mi>x</mi>\n\t</mrow>\n\t<mo>-</mo>\n\t<mn>12</mn>\n</mrow>\n</math> for <math alttext=\"y\"><mi>y</mi>\n</math> in the first equation of the given system yields <math alttext=\"6 x minus 12 equals left parenthesis x minus 2 right parenthesis left parenthesis x plus 4 right parenthesis\"><mn>6</mn><mi>x</mi><mo>-</mo><mn>12</mn><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced></math>. Factoring <math alttext=\"6\"><mn>6</mn>\n</math> out of the left-hand side of this equation yields <math alttext=\"6 left parenthesis x minus 2 right parenthesis equals left parenthesis x minus 2 right parenthesis left parenthesis x plus 4 right parenthesis\"><mn>6</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced></math>. An expression with a factor of the form <math alttext=\"left parenthesis x minus a right parenthesis\"><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math> is equal to zero when <math alttext=\"x equals a\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mi>a</mi>\n</mrow>\n</math>. Each side of this equation has a factor of <math alttext=\"left parenthesis x minus 2 right parenthesis\"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math>, so each side of the equation is equal to zero when <math alttext=\"x equals 2\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>2</mn>\n</mrow>\n</math>. Substituting <math alttext=\"2\"><mn>2</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> into the equation <math alttext=\"6 left parenthesis x minus 2 right parenthesis equals left parenthesis x minus 2 right parenthesis left parenthesis x plus 4 right parenthesis\"><mn>6</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced></math> yields <math alttext=\"6 left parenthesis 2 minus 2 right parenthesis equals left parenthesis 2 minus 2 right parenthesis left parenthesis 2 plus 4 right parenthesis\"><mn>6</mn><mfenced><mrow><mn>2</mn><mo>-</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mfenced><mrow><mn>2</mn><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mo>+</mo><mn>4</mn></mrow></mfenced></math>, or <math alttext=\"0 equals 0\"><mn>0</mn><mo>=</mo><mn>0</mn></math>, which is true. Substituting <math alttext=\"2\"><mn>2</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> into the second equation in the given system of equations yields <math alttext=\"y equals 6 left parenthesis 2 right parenthesis minus 12\"><mi>y</mi><mo>=</mo><mn>6</mn><mfenced><mn>2</mn></mfenced><mo>-</mo><mn>12</mn></math>, or <math alttext=\"y equals 0\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math>. Therefore, the solution to the system of equations is the ordered pair <math alttext=\"left parenthesis 2 comma 0 right parenthesis\"><mfenced><mrow><mn>2</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math>.</p>\n<p>Choice A is incorrect and may result from switching the order of the solutions for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"y\"><mi>y</mi>\n</math>.</p>\n<p>Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["C"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.823000+00:00","source_updated_at":"2023-08-02T20:25:59.823000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"left parenthesis 0 comma 2 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mrow><mn>2</mn></mrow></mrow></mfenced></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"left parenthesis negative 4 comma 2 right parenthesis\"><mfenced><mrow><mo>-</mo><mrow><mn>4</mn></mrow><mo>,</mo><mrow><mn>2</mn></mrow></mrow></mfenced></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"left parenthesis 2 comma 0 right parenthesis\"><mfenced><mrow><mrow><mn>2</mn></mrow><mo>,</mo><mn>0</mn></mrow></mfenced></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"left parenthesis 2 comma negative 4 right parenthesis\"><mfenced><mrow><mrow><mn>2</mn></mrow><mo>,</mo><mo>-</mo><mrow><mn>4</mn></mrow></mrow></mfenced></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}}