{"id":"cb_question_bank:5fcfd8be-26a3-41c3-ab2c-b9b2153fcc94","source_id":"cb_question_bank","source_item_id":"5fcfd8be-26a3-41c3-ab2c-b9b2153fcc94","external_id":null,"ibn":"05435-DC","question_id":"e1248a5c","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":"In the system of equations below, a and c are constants. Equation 1: one half x, plus one third y, equals one sixth. Equation 2: a, x, plus y equals c If the system of equations has an infinite number of solutions x comma y , what is the value of a ?","stimulus":"","stimulus_html":"<div class=\"stimulus_reference \">\n          <div class=\"passage \">\n            <div class=\"prose style:1 \">\n              <p class=\"passage_para \"></p>\n            </div>\n          </div>\n        </div>\n","stem_html":"<p class=\"stem_paragraph \">In the system of equations below, <span class=\"italic\">a</span> and <span class=\"italic\">c</span> are constants.<span class=\"math_expression \"></span></p><p class=\"stem_paragraph \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/54fee36b4927d3a5f29a7fb1c48e5d811de19a6510c297f5e11f43eadc6445bb\" alt=\"Equation 1: one half x, plus one third y, equals one sixth. Equation 2: a, x, plus y equals c\"></span></span>\n          </p><p class=\"stem_paragraph \">If the system of equations has an infinite number of solutions <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/338db7e3a31d3afebe84d6ca556e60cbe33b3baf49002acc912917c4ff25d31e\" alt=\"x comma y \"></span></span>, what is the value of <span class=\"italic\">a</span> ?</p><p></p>\n","rationale_html":"<p>Choice D is correct. A system of two linear equations has infinitely many solutions if one equation is equivalent to the other. This means that when the two equations are written in the same form, each coefficient or constant in one equation is equal to the corresponding coefficient or constant in the other equation multiplied by the same number. The equations in the given system of equations are written in the same form, with <span class=\"italic\">x</span> and <span class=\"italic\">y</span> on the left-hand side and a constant on the right-hand side of the equation. The coefficient of <span class=\"italic\">y</span> in the second equation is equal to the coefficient of <span class=\"italic\">y</span> in the first equation multiplied by 3. Therefore, <span class=\"italic\">a</span>, the coefficient of <span class=\"italic\">x</span> in the second equation, must be equal to 3 times the coefficient of <span class=\"italic\">x</span> in the first equation: <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d6e29b1044c0936b89188a0440f2a937e90929519deecb1a738fcd9360feca9f\" alt=\"a, equals, one half times 3\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/7b342e9ba7f6de4eb168da646f514255abd4f12175ab6b38eb003afb236ed079\" alt=\"a, equals three halves\"></span>.</p><p>Choices A, B, and C are incorrect. When <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f91a1d1089051c7b35af4ee36930881ac033eb94765d45c06d342c1209638cfe\" alt=\"a, equals negative one half\"></span>, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/9fde9ae15e275e0c5371efd5cfdecf4d737f63f1a7c30c4425b2b829a5f4ff05\" alt=\"a, equals 0\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/7cfda5bcc014a626b134da8a190436c1d8ec343f27599ff1527f9004a877e063\" alt=\"a, equals one half\"></span>, the given system of equations has one solution.</p><p></p>\n","correct_answer":["D"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.629000+00:00","source_updated_at":"2023-08-02T20:25:59.629000+00:00","options":[{"label":"A","content_html":"<span class=\"choice_cell align:right \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c32675bf7ded3a05413c4197ed13d278dd4cd18b26f7e63428e41d3fb6ed70bb\" alt=\"negative one half\"></span></span>\n          </span>\n","ord":0},{"label":"B","content_html":"<span class=\"choice_cell align:right \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e646b570dc3598aa1fe55743f6c749ef456a13deb8575aca77a309272b8cbc26\" alt=\"0\"></span></span>\n          </span>\n","ord":1},{"label":"C","content_html":"<span class=\"choice_cell align:right \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/acd304db4db3ee4a427462753769f86b9941b320104a3856da731c39db5d894c\" alt=\"one half\"></span></span>\n          </span>\n","ord":2},{"label":"D","content_html":"<span class=\"choice_cell align:right \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/544c1fbf41c46eecd241e072f3fef539763dd2bbcf0350489f405f9d1fa52c78\" alt=\"three halves\"></span></span>\n          </span>\n","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}}