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Proportion/Introductory

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Problem

Suppose \frac{1}{20} is either x or y in the following system: \begin{cases}xy=\frac{1}{k}\\x=ky\end{cases} Find the possible values of k.

Solution

If x=\frac{1}{20}, then

\frac{1}{20}=ky and
\frac{y}{20}=\frac{1}{k}

Solving gets us:

y=\frac{20}{k}
\frac{1}{20}=k\frac{20}{k}
\frac{1}{20}=20

Thus, there is no solution when x=\frac{1}{20}
If y=\frac{1}{20}, then

\frac{x}{20}=\frac{1}{k} \Longrightarrow xk=20
x=\frac{k}{20}
\left(\frac{k}{20}\right)\cdot k=20
k^2=400
k=\pm 20

Thus, the possible values of k are (20,-20).

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