{"id":"cb_question_bank:40b2874d-223c-449f-a39e-a62239e9a36b","source_id":"cb_question_bank","source_item_id":"40b2874d-223c-449f-a39e-a62239e9a36b","external_id":"386cf16e-ddc8-4613-85b9-de02ba844834","ibn":null,"question_id":"f75bd744","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":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"4 x minus 6 y equals 10 y plus 2 t y equals one half plus 2 x In the given system of equations, t is a constant. If the system has no solution, what is the value of t ?","stimulus":"","stimulus_html":null,"stem_html":"

\n\t\n\t\t4\n\t\tx\n\t\n\t-\n\t\n\t\t6\n\t\ty\n\t\n\n=10y+2

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ty=12+2x

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In the given system of equations, t\n is a constant. If the system has no solution, what is the value of t\n?

","rationale_html":"

The correct answer is 8\n. The given system of equations can be solved using the elimination method. Multiplying both sides of the second equation in the given system by -2 yields -2ty=-1-4x, or -1-4x=-2ty. Adding this equation to the first equation in the given system, 4x-6y=10y+2, yields 4x-6y+-1-4x=10y+2+-2ty, or -1-6y=10y-2ty+2. Subtracting \n\t10\n\ty\n\n from both sides of this equation yields -1-6y-10y=10y-2ty+2-10y, or -1-16y=-2ty+2. If the given system has no solution, then the equation -1-16y=-2ty+2 has no solution. If this equation has no solution, the coefficients of y\n on each side of the equation, -16 and -2t, must be equal, which yields the equation -16=-2t. Dividing both sides of this equation by -2 yields 8=t. Thus, if the system has no solution, the value of t\n is 8\n.

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Alternate approach: A system of two linear equations in two variables, x\n and y\n, has no solution if the lines represented by the equations in the xy-plane are parallel and distinct. Lines represented by equations in the form Ax+By=C, where A\n, B\n, and C\n are constant terms, are parallel if the ratio of the x-coefficients is equal to the ratio of the y-coefficients, and distinct if the ratio of the x-coefficients are not equal to the ratio of the constant terms. Subtracting 10y from both sides of the first equation in the given system yields 4x-6y-10y=10y+2-10y, or 4x-16y=2. Subtracting 2x from both sides of the second equation in the given system yields ty-2x=12+2x-2x, or -2x+ty=12. The ratio of the x-coefficients for these equations is -24, or -12. The ratio of the y-coefficients for these equations is -t16. The ratio of the constant terms for these equations is 122, or 14. Since the ratio of the x-coefficients, -12, is not equal to the ratio of the constants, 14, the lines represented by the equations are distinct. Setting the ratio of the x-coefficients equal to the ratio of the y-coefficients yields -12=-t16. Multiplying both sides of this equation by -16 yields -12-16=-t16-16, or t=8. Therefore, when t=8, the lines represented by these equations are parallel. Thus, if the system has no solution, the value of t\n is 8\n.

","correct_answer":["8"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.817000+00:00","source_updated_at":"2023-08-02T20:25:59.817000+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}}