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moneybird
If y / (x-z) = (x + y) / z = x / y for three distinct positive numbers x, y, and z, find x / y.
\[\frac{y}{x-z}=\frac{x+y}{z}=\frac{x}{y}\]
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\[a=\frac{x}{y}\]\[\frac{x+y}{z}=\frac{x}{y}\]\[z=\frac{y(x+y)}{x} \ \ \star\]plugging \(\star\) in\[\frac{y}{x-z}=\frac{x}{y}\]gives\[\frac{y}{x-\frac{y(x+y)}{x}}=\frac{x}{y}\]\[y^2=x^2-xy-y^2\]\[2\frac{y^2}{x^2}=1-\frac{y}{x}\]\[\frac{2}{a^2}=1-\frac{1}{a}\]\[a^2-a=2\]