{"id":"cb_question_bank:d882d810-5adf-43e2-870e-cb596101d850","source_id":"cb_question_bank","source_item_id":"d882d810-5adf-43e2-870e-cb596101d850","external_id":"99370f6d-8788-4824-a8aa-20c268f5a86c","ibn":null,"question_id":"75012ee7","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":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"2 x plus 3 y equals 7 10 x plus 15 y equals 35 For each real number r , which of the following points lies on the graph of each equation in the xy -plane for the given system?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"2 x plus 3 y equals 7\"><mrow><mn>2</mn></mrow><mi>x</mi><mo>+</mo><mrow><mn>3</mn></mrow><mi>y</mi><mo>=</mo><mrow><mn>7</mn></mrow></math><br><math alttext=\"10 x plus 15 y equals 35\"><mrow><mn>10</mn></mrow><mi>x</mi><mo>+</mo><mrow><mn>15</mn></mrow><mi>y</mi><mo>=</mo><mrow><mn>35</mn></mrow></math></p>\n<p style=\"text-align: left;\">For each real number <math alttext=\"r\"><mi>r</mi>\n</math>, which of the following points lies on the graph of each equation in the <em>xy</em>-plane for the given system?</p>","rationale_html":"<p style=\"text-align: left;\">Choice B is correct. The two given equations are equivalent because the second equation can be obtained from the first equation by multiplying each side of the equation by <math alttext=\"5\"><mn>5</mn>\n</math>. Thus, the graphs of the equations are coincident, so if a point lies on the graph of one of the equations, it also lies on the graph of the other equation. A point <math alttext=\"left parenthesis x comma y right parenthesis\"><mfenced><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></math> lies on the graph of an equation in the <em>xy</em>-plane if and only if this point represents a solution to the equation. It is sufficient, therefore, to find the point that represents a solution to the first given equation. Substituting the <em>x</em>- and <em>y</em>-coordinates of choice B, <math alttext=\"minus StartFraction 3 r Over 2 EndFraction plus seven halves\"><mo>-</mo><mfrac><mrow><mn>3</mn><mi>r</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></math> and <math alttext=\"r\"><mi>r</mi>\n</math>, for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"y\"><mi>y</mi>\n</math>, respectively, in the first equation yields <math alttext=\"2 left parenthesis minus StartFraction 3 r Over 2 EndFraction plus seven halves right parenthesis plus 3 r equals 7\"><mn>2</mn><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>3</mn><mi>r</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow></mfenced><mo>+</mo><mn>3</mn><mi>r</mi><mo>=</mo><mn>7</mn></math>, which is equivalent to&nbsp;<math alttext=\"minus 3 r plus 7 plus 3 r equals 7\"><mo>-</mo><mn>3</mn><mi>r</mi><mo>+</mo><mn>7</mn><mo>+</mo><mn>3</mn><mi>r</mi><mo>=</mo><mn>7</mn></math>, or&nbsp;<math alttext=\"7 equals 7\"><mn>7</mn><mo>=</mo><mn>7</mn></math>. Therefore, the point <math alttext=\"left parenthesis minus StartFraction 3 r Over 2 EndFraction plus seven halves comma r right parenthesis\"><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>3</mn><mi>r</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mo>,</mo><mi>r</mi></mrow></mfenced></math> represents a solution to the first equation and thus lies on the graph of each equation in the <em>xy</em>-plane for the given system.</p>\n<p style=\"text-align: left;\">Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice C is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["B"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.818000+00:00","source_updated_at":"2023-08-02T20:25:59.818000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"left parenthesis StartFraction r Over 5 EndFraction plus 7 comma minus StartFraction r Over 5 EndFraction plus 35 right parenthesis\"><mfenced><mrow><mfrac><mi>r</mi><mn>5</mn></mfrac><mo>+</mo><mrow><mn>7</mn></mrow><mo>,</mo><mo>-</mo><mfrac><mi>r</mi><mn>5</mn></mfrac><mo>+</mo><mrow><mn>35</mn></mrow></mrow></mfenced></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"left parenthesis minus StartFraction 3 r Over 2 EndFraction plus seven halves comma r right parenthesis\"><mfenced><mrow><mo>-</mo><mfrac><mrow><mrow><mn>3</mn></mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>7</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mo>&nbsp;</mo><mi>r</mi></mrow></mfenced></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"left parenthesis r comma StartFraction 2 r Over 3 EndFraction plus seven thirds right parenthesis\"><mfenced><mrow><mi>r</mi><mo>,</mo><mo>&nbsp;</mo><mfrac><mrow><mrow><mn>2</mn></mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>+</mo><mrow><mfrac><mn>7</mn><mn>3</mn></mfrac></mrow></mrow></mfenced></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"left parenthesis r comma minus StartFraction 3 r Over 2 EndFraction plus seven halves right parenthesis\"><mfenced><mrow><mi>r</mi><mo>,</mo><mo>-</mo><mfrac><mrow><mrow><mn>3</mn></mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>7</mn></mrow><mrow><mn>2</mn></mrow></mfrac></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}}