{"id":"cb_question_bank:3e6c99cd-5b2a-42b8-b07d-d2ce2b79a97a","source_id":"cb_question_bank","source_item_id":"3e6c99cd-5b2a-42b8-b07d-d2ce2b79a97a","external_id":null,"ibn":"029630-DC","question_id":"fdee0fbf","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.C.","skill":"Linear equations in two variables","difficulty":"H","score_band":6,"answer_type":"spr","answer_source":"rationale","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In the xy -plane, line k intersects the y -axis at the point with coordinates 0 comma negative 6 and passes through the point with coordinates 2 comma 2 . If the point with coordinates 20 comma w lies on line k , what is the value of w ?","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">In the <span class=\"italic\">xy</span>-plane, line <span class=\"italic\">k</span> intersects the <span class=\"italic\">y</span>-axis at the point <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/87c9001376bb3992fcc5902fea7d1aaf9beff3511a7b34cd7a522c804b97972d\" alt=\"with coordinates 0 comma negative 6\"></span></span> and passes through the point <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/132380b529bd43aff29b45cd5ac045ed90f5c7363a2e3f3f1787a6d11ee62070\" alt=\"with coordinates 2 comma 2\"></span></span>.<span class=\"formatted_line_break delivery_mode:Both \"></span> If the point <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bfcfc94d2030b8129863be7048a4a3b77ba64144b901a0ca284145dfecc8c0d4\" alt=\"with coordinates 20 comma w \"></span></span> lies on line <span class=\"italic\">k</span>, what is the value of <span class=\"italic\">w </span>?</p>\n","rationale_html":"<p>The correct answer is 74. The <span class=\"italic\">y</span>-intercept of a line in the <span class=\"italic\">xy</span>-plane is the ordered pair <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/29eda5c740cd27d857d88b32b6e6a8a1d263911ac678e319dd0387efa41b0e58\" alt=\"x comma y \"></span> of the point of intersection of the line with the <span class=\"italic\">y</span>-axis. Since line <span class=\"italic\">k</span> intersects the <span class=\"italic\">y</span>-axis at the point <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d71c174f73f36bef70f45844f7b7763c0d8a3a6eab2522ec80afbe64115071fb\" alt=\"with coordinates 0 comma negative 6\"></span>, it follows that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/164784256324d57bef8d9b9c6e9970f3ec098e8bb7b2345362c149d455e33307\" alt=\"the point with coordinates 0 comma negative 6 \"></span> is the <span class=\"italic\">y</span>-intercept of this line. An equation of any line in the <span class=\"italic\">xy</span>-plane can be written in the form <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 of the line and <span class=\"italic\">b</span> is the <span class=\"italic\">y</span>-coordinate of the <span class=\"italic\">y</span>-intercept. Therefore, the equation of line <span class=\"italic\">k</span> can be written as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4390ae46d8e77ea04f2fe4b5934fd0cd8b39cf48515300982ff7a13a2be5f38b\" alt=\"y equals, m x plus negative 6\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/26d9e34713209c58585fc4249e6dcac25b54214d2a0639424a03bd7564ff8cb0\" alt=\"y equals, m x minus 6\"></span>. The value of <span class=\"italic\">m</span> can be found by substituting the <span class=\"italic\">x</span>- and <span class=\"italic\">y</span>-coordinates from a point on the line, such as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e7875c30d2e723613a0de6c612b138bfce06c32b0b1afbdba263a3012da746d3\" alt=\"the point with coordinates 2 comma 2\"></span>, for <span class=\"italic\">x</span> and <span class=\"italic\">y,</span> respectively. This results in <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/9ae5e2d84e7b4de9e3352ca1c91766260e737238e9bea8e3fd770395a80c9cd7\" alt=\"2 equals, 2 m minus 6\"></span>. Solving this equation for <span class=\"italic\">m</span> gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2bc845b9cf290820bd628ea4e64a4fbe344b447b35ee08fdbbe16d967bd0f24f\" alt=\"m equals 4\"></span>. Therefore, an equation of line<span class=\"italic\"> k</span> is <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/cc044752a70df125cd3a5bbc7f00e8fc4b503bbfd5ce17c070aeafbc5be38cb2\" alt=\"y equals, 4 x minus 6\"></span>. The value of <span class=\"italic\">w</span> can be found by substituting the <span class=\"italic\">x</span>-coordinate, 20, for <span class=\"italic\">x</span> in the equation of line <span class=\"italic\">k</span> and solving this equation for <span class=\"italic\">y.</span> This gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/27a86b5c71cfb28959b688744f216b127ad9ce289297734a229c98bff8b905d9\" alt=\"y equals, 4 times 20, minus 6\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/24d5d809d77f5eb0ea423a27a093c7e3d03e90c39ebd5575573141f8ac0d5812\" alt=\"y equals 74\"></span>. Since <span class=\"italic\">w</span> is the <span class=\"italic\">y</span>-coordinate of this point, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e004840c779922d1f3a3f49977d7ce6fdd8fd53d4b0209e022e10b644d38ed4f\" alt=\"w equals 74\"></span>.</p>\n","correct_answer":["74"],"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":[],"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}}