{"id":"cb_practice_pdf:t6-math-m2-q27","source_id":"cb_practice_pdf","source_item_id":"t6-math-m2-q27","external_id":null,"ibn":null,"question_id":null,"program":"SAT","module":"math","domain_code":null,"domain":null,"skill_code":null,"skill":null,"difficulty":null,"score_band":null,"answer_type":"spr","answer_source":"rationale","layout_warning":"Part of this question was typeset outside the text flow — a fraction, subscript or similar — and PDF extraction re-inserted it in the wrong place. The wording is reliable; the symbols may be out of order.","form_id":"cb_practice_pdf:test-6","module_number":2,"position":27,"stem":"- - - - - -_48y_+24 - - - ~ 48x−64y = = −12x ry 1/8 In the given system of equations, r is a constant. If the system has no solution, what is the value of r? OP","stimulus":"","stimulus_html":null,"stem_html":"<p>- - - - - -_48y_+24 - - - ~ 48x−64y = = −12x ry 1/8 In the given system of equations, r is a constant. If the system has no solution, what is the value of r? OP</p>","rationale_html":"<p>-28 The correct answer is . A system of two linear equations in two variables, x and y, has no solution if the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by + = equations in the form Ax By C, where A, B, and C are constants, are parallel if the coefficients for x and y in one equation are proportional to the corresponding coefficients for x and y in the other equation. The first equation in 48 -64 =48 +24 + = the given system, x y y , can be written in the form Ax By C by 48 48 -112 =24 subtracting y from both sides of the equation to yield x y . The = 1 -12 second equation in the given system, ry 8 x, can be written in the form + = 12 12 + = 1 Ax By C by adding x to both sides of the equation to yield x ry 8. The 1 coefficient of x in the second equation is 4 times the coefficient of x in the first 48 1 =12 equation. That is, b 4 l . For the lines to be parallel, the coefficient of y in the 1 second equation must also be 4 times the coefficient of y in the first equation. -112 1 = -28= Therefore, b 4 l r, or r. Thus, if the given system has no solution, the -28 value of r is . 52 SAT PRACTICE TEST #6 ANSWER EXPLANATIONS</p>","correct_answer":["-28"],"has_media":false,"has_mathml":false,"has_table":false,"source_created_at":null,"source_updated_at":null,"options":[],"attribution":{"source_id":"cb_practice_pdf","source_name":"College Board full-length SAT practice tests (PDF)","rights_holder":"College Board","canonical_url":"https://satsuite.collegeboard.org/practice/practice-tests/paper","retrieved_from":"https://satsuite.collegeboard.org/practice/practice-tests/paper","attribution":"Full-length SAT practice tests 4–11 © College Board, published free at satsuite.collegeboard.org.","disclaimer":"Extracted from PDF layout: equations, figures and tables do not survive intact. Use the original PDFs for anything the text renders poorly. SAT® is a trademark registered by the College Board, which is not affiliated with and does not endorse this project.","license":"College Board copyright. Free to download; not licensed for redistribution.","redistributable":false,"item_count":905}}