Question

Resolva o sistema abaixo pelos 3 métodos conhecidos: adição,comparação,e substituição:

x+y=6
2x-y=24

Answer 1

$\begin{cases}x+y=6\\ 2x-y=24\end{cases}\\\\\\\\\\\\ Por~~Adi{\c c}\~ao.~~ Some~~ as ~~equa{\c c}\~oes~~e~~elimine~~y;\\\\3x=30\to~~x= \dfrac{30}{3} \to~~\boxed{x=10}\\\\ x+y = 6 \to~~10+y=6\to~~y=6-10\to~~\boxed{y=-4}$






$Por~~Compara{\c c}\~ao.~~Isole~~x~~ou~~y~~nas~~duas~~equa{\c c}\~oes.~~Tendo~~x\\igual~~a~~tal~~valor~~em~~ambas,~~igualamos ~~esses ~~valores~~afim\\de~~encontrar~~x,~~se~~~~tiver~~isolado~~y.~~Depois~~de~~encontrado\\o~~valor~~para~~x,~~substitua-o~~em~~x~~em~~uma~~das~~equa{\c c}\~oes\\afim~~de~~encontrar~~y.$



$x+y=6\to~~y=-x+6\\\\\\ 2x-y=24\to~~-y=-2x+24~(-1)\to~~y=2x-24\\\\\\\\\\ -x+6=2x-24\to~~-x-2x=-24-6\to~~-3x=-30~(-1)\to\\\\ 3x=30\to~~x= \dfrac{30}{3}\to~~ \boxed{x=10}\\\\\\ y=2x-24\to~~y=2.10-24\to~~y=20-24\to~~ \boxed{y=-4}$






$Por~~Substitui{\c c}\~ao. ~~Isolando~~x~~na ~~1\°~~equa{\c c}\~ao, ~~substitui~~o~~valor\\de~~x~~na ~~2\°~~equa{\c c}\~ao~~afim~~de~~encontrar~~y.\\Depois~~de~~encontrado~~y,~~volta~~na~~1\°~~equa{\c c}\~ao~~e~~substitui~~em\\y~~afim~~de~~encontrar~~x.\\\\\\\\ x+y=6\to~~x=-y+6\\\\\\\\ 2x-y=24\to~~2(-y+6)-y=24\to~~-2y+12-y=24\to\\\\-2y-y=24-12\to~~-3y=12~(-1)\to~~ 3y=-12\to~~y=- \dfrac{12}{3} \to\\\\ \boxed{y=-4}\\\\\\\\ x=-y+6\to~~x = -(-4)+6\to~~x=4+6\to~~\boxed{x=10}$




$\Large\boxed{\boxed{\boxed{S=\{10~;~-4\}}}}$