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2023 AMC 12A Problem 10

Problem 10 of 25EasierAlgebra

Positive real numbers xx and yy satisfy y3=x2y^3=x^2 and (y−x)2=4y2.(y-x)^2=4y^2. What is x+y?x+y?

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Solution

From (y−x)2=4y2(y-x)^2=4y^2 we get y−x=±2y.y-x=\pm 2y. The choice y−x=2yy-x=2y gives x=−y<0,x=-y\lt 0, impossible, so y−x=−2y,y-x=-2y, meaning x=3y.x=3y. Substituting into y3=x2=9y2y^3=x^2=9y^2 gives y=9,y=9, hence x=27x=27 and x+y=36.x+y=36. Thus, the correct answer is D.
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Tagged: system of equations · substitution

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