Example of a symmetric system

Let us see the sum-product method at work on a fractional and symmetric system.

Example

$\begin{cases} \dfrac{1}{x}+\dfrac{1}{y}=\dfrac{12}{35} \\[6pt] \dfrac{x}{y}+\dfrac{y}{x}=\dfrac{74}{35} \end{cases} \qquad\text{Existence conditions: } x\neq 0,\, y\neq 0.$ The first equation gives $\dfrac{S}{P}=\dfrac{12}{35}$; the second $\dfrac{S^2-2P}{P}=\dfrac{74}{35}$.

From the first: $P=\dfrac{35S}{12}$. Substituting into the second: $35S^2 - 144P = 0 \implies 35S^2-144\cdot\frac{35S}{12}=0 \implies 35S(S-12)=0.$ $S=0$ is not acceptable (existence conditions), hence $S=12$, $P=35$. $t^2-12t+35=0 \implies (t-7)(t-5)=0 \implies \boxed{(x,y)=(7,5) \text{ or } (5,7)}$

Links

Topics: Second-degree equations
Concepts: Symmetric system · Sum and product of the roots
Methods: Sum and product of the roots
Skills: Solving systems