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Algebra Difficulty 4.9 AIME Prove it North Macedonia

For which values of xx and yy, the difference of 2x+158\frac{2x+15}{8} and 113(y1)1\frac{1}{3} \cdot (y-1) will be 3 times smaller than 2(52y)2 \cdot (5-2y) and x+52\frac{x+5}{2} will be for 0.1250.125 bigger than 3y3y?

Solution

From the condition in the problem we get the equations
2x+158113(y1)=2(52y)3 \frac{2x+15}{8} - 1\frac{1}{3} \cdot (y-1) = \frac{2(5-2y)}{3}
and
x+52=3y+0.125. \frac{x+5}{2} = 3y + 0.125.
They are equivalent to
6x=3 6x = 3
and
4x24y=22. 4x - 24y = -22.
The solution to this system is
x=12 x = \frac{1}{2}
and
y=1. y = 1.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.