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Algebra Difficulty 3.9 AMC 10/12 Find the answer United States

Problem:
If $x, y, z$ are real numbers such that $x y=6$ , $x-z=2$ , and $x+y+z=9$ , compute $\frac{x}{y}-\frac{z}{x}-\frac{z^{2}}{x y}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
Let $k=\frac{x}{y}-\frac{z}{x}-\frac{z^{2}}{x y}=\frac{x^{2}-y z-z^{2}}{x y}$ . We have
$k+1=\frac{x^{2}+x y-y z-z^{2}}{x y}=\frac{x^{2}-x z+x y-y z+z x-z^{2}}{x y}=\frac{(x+y+z)(x-z)}{x y}=\frac{9 \cdot 2}{6}=3,$
so $k=2$ .

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