If 5x - 13xy + 6y = 0, find x : y.
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A.
(2 : 1) only
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B.
(5 : 3) or (1 : 2)
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C.
(3 : 5) only
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D.
(3 : 5 ) or (2 : 1)
Correct Answer:
D. (3 : 5 ) or (2 : 1)
Explanation:
To find the ratio of x to y, we can solve the quadratic-style equation 5x² - 13xy + 6y² = 0 by factoring. We look for two numbers that multiply to 30 (5 * 6) and add up to -13, which are -10 and -3. Rewriting the middle term, we get 5x² - 10xy - 3xy + 6y² = 0. Grouping the terms gives 5x(x - 2y) - 3y(x - 2y) = 0, which factors into (5x - 3y)(x - 2y) = 0.
Setting each factor to zero, we find two possible relationships: 5x = 3y, which means x/y = 3/5, or x = 2y, which means x/y = 2/1. Therefore, the ratio x : y can be either 3 : 5 or 2 : 1.
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