{"id":"cb_question_bank:afc2e63a-5445-4300-83ec-cdaa29176791","source_id":"cb_question_bank","source_item_id":"afc2e63a-5445-4300-83ec-cdaa29176791","external_id":null,"ibn":"022887-DC","question_id":"e2e3942f","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":"In the system of equations above, a is a constant. If the system of equations has no solution, what is the value of a ?","stimulus":"Equation 1: y equals, 2 x plus 1. Equation 2: y equals, a, x minus 8.","stimulus_html":"<div class=\"stimulus_reference \">\n        <p class=\"standalone_statement style:1 \">\n          <span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/a70365fdbc531280c542b1fc42432f9ec0ab825e4c6582ca25000428e7dd6c86\" alt=\"Equation 1: y equals, 2 x plus 1. Equation 2: y equals, a, x minus 8.\"></span></span>\n        </p>\n      </div>\n","stem_html":"<p class=\"stem_paragraph \">In the system of equations above, <span class=\"italic\">a</span>&nbsp;is a constant. If the system of equations has no solution, what is the value&nbsp;of&nbsp;<span class=\"italic\">a</span>&nbsp;?</p>\n","rationale_html":"<p>Choice D is correct. A system of two linear equations has no solution when the graphs of the equations have the same slope and different <span class=\"italic\">y-</span>coordinates of the <span class=\"italic\">y</span>-intercepts. Each of the given equations is written in the slope-intercept form of a linear equation, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/39dda01984e771026b6a8a741d3ebb66f1a29f1e81583b67f8d50f6e2dbb783e\" alt=\"y equals, m x plus b\"></span>, where <span class=\"italic\">m</span> is the slope and <span class=\"italic\">b</span> is the <span class=\"italic\">y</span>-coordinate of the <span class=\"italic\">y</span>-intercept of the graph of the equation. For these two linear equations, the <span class=\"italic\">y-</span>coordinates of the <span class=\"italic\">y</span>-intercepts are different: <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2f1494c799ca1decbcb94ac7e70aeb91da2c02dfba7cb54dae8e89aee537a8e1\" alt=\"1\"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0074a4418aa88cd673760d477e8b81bfb5e2a1a048bed34062ed94e05b6de763\" alt=\"negative 8\"></span>. Thus, if the system of equations has no solution, the slopes of the two linear equations must be the same. The slope of the first linear equation is 2. Therefore, for the system of equations to have no solution, the value of <span class=\"italic\">a</span> must be 2.</p><p>Choices A, B, and C are incorrect and may result from conceptual and computational errors.</p><p></p>\n","correct_answer":["D"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.618000+00:00","source_updated_at":"2023-08-02T20:25:59.618000+00:00","options":[{"label":"A","content_html":"<span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c32675bf7ded3a05413c4197ed13d278dd4cd18b26f7e63428e41d3fb6ed70bb\" alt=\"negative one half\"></span>\n","ord":0},{"label":"B","content_html":"<span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e646b570dc3598aa1fe55743f6c749ef456a13deb8575aca77a309272b8cbc26\" alt=\"0\"></span>\n","ord":1},{"label":"C","content_html":"<span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2f1494c799ca1decbcb94ac7e70aeb91da2c02dfba7cb54dae8e89aee537a8e1\" alt=\"1\"></span>\n","ord":2},{"label":"D","content_html":"<span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d870b3a6c89096490c39cc291dea5fd94193a953487d36d45d7f8548ac0b8f4e\" alt=\"2\"></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}}