1957 AHSME Problems/Problem 45

Problem

If two real numbers $x$ and $y$ satisfy the equation $\frac{x}{y} = x - y$, then:

$\textbf{(A)}\ {x \ge 4}\text{ or }{x \le 0}\qquad \\  \textbf{(B)}\ {y}\text{ can equal }{1}\qquad \\  \textbf{(C)}\ \text{both }{x}\text{ and }{y}\text{ must be irrational}\qquad\\  \textbf{(D)}\ {x}\text{ and }{y}\text{ cannot both be integers}\qquad\\  \textbf{(E)}\ \text{both }{x}\text{ and }{y}\text{ must be rational}$

Solution

From the initial equation $\tfrac x y = x-y$, we can solve for $y$ in terms of $x$ as follows:  xy=xyx=xyy2y2xy+x=0y=x±x24x2 Because $y$ is real, the discriminant of the above expression for $y$ must be $\geq 0$, so $x^2-4x=x(x-4) \geq 0$. This is true when $x \geq 4$ and $x \leq 0$, so we choose answer $\boxed{\textbf{(A)}}$.

See Also

1957 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 44
Followed by
Problem 46
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