Exercise 467Algebra › Equations and radicalsHigh schoolAdded 3 Nov 2026

A system by substitution

#system#substitution#nonlinear

Statement

Solve in $\mathbb{R}^2$ the system
$\begin{cases}x+y=5\\ xy=6\end{cases}$

Hint

$x$ and $y$ are roots of $t^2-5t+6=0$ .

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Method 1: substitution. $y=5-x$ , so $x(5-x)=6$ , i.e. $x^2-5x+6=0$ , giving $x=2$ or $x=3$ . Then $y=3$ or $y=2$ .

Method 2: Vieta. $x,y$ are roots of $t^2-St+P=0$ with $S=5$ , $P=6$ : $t^2-5t+6=0$ , roots $2$ and $3$ .

Conclusion. $(x,y)\in\{(2,3),(3,2)\}$ .