Question

Considere o sistema de equaçoes e determine os valores de x e y que são soluçoes desse sistema {3x-y=9}
{xy=12}

Answer 1

$\begin{cases}3x-y=9\\ xy=12\end{cases}\\\\\\\\ 3x-y=9\to~~-y=-3x+9~(-1)\to~~ y=3x-9\\\\ xy=12\to~~ x.(3x-9)=12\to~~3x^2-9x-12=0\\\\ a=3;~b=-9;~c=-12\\\\\\ \Delta=b^2-4ac\to \Delta=(-9)^2-4.3.(-12)\to \Delta= 81+144\to \boxed{\Delta=225} \\\\ x' \neq x''$



$x= \dfrac{-b\pm \sqrt{\Delta} }{2a} \to ~~ x= \dfrac{-(-9)\pm \sqrt{225} }{2.3} \to ~~ x= \dfrac{9\pm 15 }{6} \to\\\\\\ x'= \dfrac{9+ 15 }{6} \to~~ x'= \dfrac{24 }{6} \to~~ \boxed{x'=4} \\\\\\ x''= \dfrac{9-15 }{6} \to~~ x''= \dfrac{-6 }{6} \to~~ \boxed{x''=-1}$



$y=3x-9\\\\\\ y'=3.4-9\to ~~ y'=12-9\to~~ \boxed{y'=3}\\\\\\y''=3.(-1)-9\to ~~ y''=-3-9\to~~ \boxed{y''=-12}\\\\\\\\ \Large\boxed{\boxed{S=\left\{ (4;3)~~;~~(-1;-12)\right\}}}$