A nonlinear system

#nonlinear system#elimination#symmetry

Statement

Solve in $\mathbb{R}^2$ the system
$\begin{cases}x^2+y^2=25\\ x+y=7\end{cases}$

Hint

Use $(x+y)^2=x^2+y^2+2xy$ .

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Compute $xy$ . $(x+y)^2=49=x^2+y^2+2xy=25+2xy$ , so $xy=12$ .

Vieta. $x,y$ are roots of $t^2-7t+12=0$ : $t=3,4$ .

Conclusion. $(x,y)\in\{(3,4),(4,3)\}$ .