{"id":"cb_question_bank:09398c0c-b40b-4309-9ea8-a451e1342d9d","source_id":"cb_question_bank","source_item_id":"09398c0c-b40b-4309-9ea8-a451e1342d9d","external_id":"7e535b93-6192-4963-87af-547e39bbdfcf","ibn":null,"question_id":"9bbce683","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":"x y 18 130 23 160 26 178 For line h , the table shows three values of x and their corresponding values of y . Line k is the result of translating line h down 5 units in the xy -plane. What is the x -intercept of line k ?","stimulus":"","stimulus_html":null,"stem_html":"
\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
x\ny\n
18\n130\n
23\n160\n
26\n178\n
\n

For line h\n, the table shows three values of x\n and their corresponding values of y\n. Line k\n is the result of translating line h\n down 5\n units in the xy-plane. What is the x-intercept of line k\n?

","rationale_html":"

Choice D is correct. The equation of line h\n can be written in slope-intercept form \n\ty\n\t=\n\t\n\t\t\n\t\t\tm\n\t\t\tx\n\t\t\n\t\t+\n\t\tb\n\t\n\n, where m\n is the slope of the line and 0,b is the y-intercept of the line. It’s given that line h\n contains the points 18,130, 23,160, and 26,178. Therefore, its slope m\n can be found as 160-13023-18, or 6\n. Substituting 6\n for m\n in the equation \n\ty\n\t=\n\t\n\t\t\n\t\t\tm\n\t\t\tx\n\t\t\n\t\t+\n\t\tb\n\t\n\n yields \n\ty\n\t=\n\t\n\t\t\n\t\t\t6\n\t\t\tx\n\t\t\n\t\t+\n\t\tb\n\t\n\n. Substituting 130\n for y\n and 18\n for x\n in this equation yields 130=6(18)+b, or 130=108+b. Subtracting 108\n from both sides of this equation yields \n\t22\n\t=\n\tb\n\n. Substituting 22\n for b\n in \n\ty\n\t=\n\t\n\t\t\n\t\t\t6\n\t\t\tx\n\t\t\n\t\t+\n\t\tb\n\t\n\n yields \n\ty\n\t=\n\t\n\t\t\n\t\t\t6\n\t\t\tx\n\t\t\n\t\t+\n\t\t22\n\t\n\n. Since line k\n is the result of translating line h\n down 5\n units, an equation of line k\n is y=6x+22-5, or \n\ty\n\t=\n\t\n\t\t\n\t\t\t6\n\t\t\tx\n\t\t\n\t\t+\n\t\t17\n\t\n\n. Substituting 0\n for y\n in this equation yields \n\t0\n\t=\n\t\n\t\t\n\t\t\t6\n\t\t\tx\n\t\t\n\t\t+\n\t\t17\n\t\n\n. Solving this equation for x\n yields x=-176. Therefore, the x-intercept of line k\n is -176,0.

\n

Choice A is incorrect and may result from conceptual or calculation errors.

\n

Choice B is incorrect and may result from conceptual or calculation errors.

\n

Choice C is incorrect and may result from conceptual or calculation errors.

","correct_answer":["D"],"has_media":false,"has_mathml":true,"has_table":true,"source_created_at":"2023-08-02T20:25:59.815000+00:00","source_updated_at":"2023-08-02T20:25:59.815000+00:00","options":[{"label":"A","content_html":"

-263,0

","ord":0},{"label":"B","content_html":"

-92,0

","ord":1},{"label":"C","content_html":"

-113,0

","ord":2},{"label":"D","content_html":"

-176,0

","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}}