Algebraic Algebra IV

Algebra Level 2

Let x x and y y be positive integers satisfying

10 x + 6 y 2 x − 3 y \frac{10x + 6y}{2x - 3y} = 8

What are the values of x x and y y ?

Submit your answer as the smallest value of x + y x+y .

Please try my other problems in this series: Algebraic Algebra I , Algebraic Algebra II , Algebraic Algebra III


The answer is 6.

Rikhin Kavuru by Rikhin Kavuru , 90, USA

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1 solution

Elijah L
Dec 30, 2020

Rearrange:

10 x + 6 y 2 x − 3 y = 8 10 + 6 y = 8 ( 2 x − 3 y ) 10 x + 6 y = 16 x − 24 y 6 x = 30 y x = 5 y \begin{aligned} \dfrac{10x+6y}{2x-3y} &= 8\\ 10+6y &= 8(2x-3y)\\ 10x + 6y &= 16x -24y\\ 6x &= 30y\\ x &= 5y \end{aligned}

To minimize x + y x+y , it is clear to minimize x x . Because x x must be a positive integer, it is clear to take x = 1 x = 1 , y = 5 y=5 . Therefore, x + y = 6 x+y = \boxed{6} .

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