The sum-product method

A system is said to be symmetric when the unknowns $x$ and $y$ appear interchangeably: swapping $x$ with $y$ leaves the system unchanged. In these cases it is convenient to change unknowns, passing to the sum $S=x+y$ and the product $P=xy$.

Property — Sum-product substitution

When a system involves the unknowns $x,y$ only through their sum $S=x+y$ and their product $P=xy$, one substitutes $S$ and $P$, solves, and then recovers $x,y$ from the equation: $t^2-St+P=0.$

The equation $t^2-St+P=0$ is precisely the one that has $x$ and $y$ as its solutions: this is seen from Vieta’s formulas, since the sum of its roots is $S$ and the product is $P$.

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