{"id":"cb_question_bank:708b54e9-855c-44e4-9908-380185be33cf","source_id":"cb_question_bank","source_item_id":"708b54e9-855c-44e4-9908-380185be33cf","external_id":null,"ibn":"029477-DC","question_id":"0b46bad5","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.C.","skill":"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 equation above, a and b are constants and 0 is less than a, which is less than b . Which of the following could represent the graph of the equation in the xy -plane?","stimulus":"a, x, plus b y, equals b","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/7f07a560b161a7fc3f970ef86dbe2c28636836b08a7411909af944c949728e03\" alt=\"a, x, plus b y, equals b\"></span></span>\n        </p>\n      </div>\n","stem_html":"<p class=\"stem_paragraph \">In the equation above, <span class=\"italic\">a</span> and <span class=\"italic\">b</span> are constants and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6c391ef72eaa243b31a677688e650541ec779680f9a36f88d48388aa6ec38665\" alt=\"0 is less than a, which is less than b\"></span>. Which of the following could represent the graph of the equation in the <span class=\"italic \">xy</span>-plane?</p>\n","rationale_html":"<p>Choice C is correct. The given equation <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/7f07a560b161a7fc3f970ef86dbe2c28636836b08a7411909af944c949728e03\" alt=\"a, x plus b y, equals b\"></span> can be rewritten in slope-intercept form, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/98cc65bff6363d038c2c0246e66fa8d21d086ba4de032406eb99b65bb1982bb7\" alt=\"y equals, m x plus k\"></span>, where <span class=\"italic\">m</span> represents the slope of the line represented by the equation, and <span class=\"italic\">k</span> represents the <span class=\"italic\">y</span>-coordinate of the <span class=\"italic\">y</span>-intercept of the line. Subtracting <span class=\"italic\">ax</span> from both sides of the equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bf8d88bf76ff7b202681b4936c5b87dbdbdbafc8d7508cb247da5b717ccfb02b\" alt=\"b y equals, negative a, x plus b\"></span>, and dividing both sides of this equation by <span class=\"italic\">b</span> yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/8a87b10d3a5f6e612d7fd6f6db9ff3efa3252c32d35000044cc5ffd09a5a3de2\" alt=\"y equals, the negative of the fraction a over b, end fraction, times x, plus, the fraction b over b, end fraction\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/90590a8466b71fe604697134a28b17df2f2551a9a7783e6986e06b08d5f756ef\" alt=\"y equals, the negative of the fraction a over b, end fraction, times x, plus 1\"></span>. With the equation now in slope-intercept form, it shows that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b3255ad683d5c4cb093fd5c29b514fe9ae9b8f33e078620bb34024d9fb7b7424\" alt=\"k equals 1\"></span>, which means the <span class=\"italic\">y</span>-coordinate of the <span class=\"italic\">y</span>-intercept is 1. It’s given that <span class=\"italic\">a</span> and <span class=\"italic\">b</span> are both greater than 0 (positive) and that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c9b6789fc0ea5a150727265baa52b1798e05e5a5a2db5d416aa344ec401f53b7\" alt=\"a is less than b\"></span>. Since <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/cbd5b8c0392b08b7568702b4510947ea2503822d4b537857f36acc82b84ce64f\" alt=\"m equals, the negative of the fraction a over b\"></span>, the slope of the line must be a value between <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c1e573ec087416273e06563cee97890abeeed317436562b06a92c8920d4c759d\" alt=\"negative 1\"></span> and 0. Choice C is the only graph of a line that has a <span class=\"italic\">y</span>-value of the <span class=\"italic\">y</span>-intercept that is 1 and a slope that is between <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c1e573ec087416273e06563cee97890abeeed317436562b06a92c8920d4c759d\" alt=\"negative 1\"></span> and 0.<span class=\"italic\"></span></p><p>Choices A, B, and D are incorrect because the slopes of the lines in these graphs aren’t between <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c1e573ec087416273e06563cee97890abeeed317436562b06a92c8920d4c759d\" alt=\"negative 1\"></span> and 0.</p><p></p>\n","correct_answer":["C"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.626000+00:00","source_updated_at":"2023-08-02T20:25:59.626000+00:00","options":[{"label":"A","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"inline_image \">\n            </span>\n          </p><p>\n          <img src=\"/api/assets/8777bc00bec0ada981fd846d81ba4bd1b8152d66ef922a5701cd80482358f88d\" alt=\"The answer choice presents the graph of a line in the x y-plane. The number 1 is indicated on both axes. The line slants downward and to the right, crossing the y axis at 1, and crossing the x axis at 1.\" width=\"82\" height=\"89\"></p><p></p>\n","ord":0},{"label":"B","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"inline_image \">\n            </span>\n          </p><p>\n          <img src=\"/api/assets/1ada8bc18358dcaa945e39d6c90d02945157772d9a80dd0750ef7b43c76b5ca0\" alt=\"The answer choice presents the graph of a line in the x y-plane. The number 1 is indicated on both axes. The line slants downward and to the right, crossing the x axis at negative 1, and crossing the y axis at negative 1.\" width=\"82\" height=\"89\"></p><p></p>\n","ord":1},{"label":"C","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"inline_image \">\n            </span>\n          </p><p>\n          <img src=\"/api/assets/17de2800653a0338ef6d8c4d15e8f65ad09df246c568e7c34d826b7a03036a3a\" alt=\"The answer choice presents the graph of a line in the x y-plane. The number 1 is indicated on both axes. The line slants downward and to the right, crossing the y axis at 1, and crossing the x axis at 2.\" width=\"82\" height=\"89\"></p><p></p>\n","ord":2},{"label":"D","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"inline_image \">\n            </span>\n          </p><p>\n          <img src=\"/api/assets/197199096e18ddc33f462dbfee2e378bd5212a5ec1afdedbf90ee1d7a43acb9a\" alt=\"The answer choice presents the graph of a line in the x y-plane. The number 1 is indicated on both axes. The line slants downward and to the right, crossing the y axis at 1, and crossing the x axis at 0 point 5.\" width=\"82\" height=\"89\"></p><p></p>\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}}