Question

Rezolva următoarele sisteme de ecuații prin metoda substituției:

Answer 1

$\it g)\ \ \begin{cases} \it x-2y=-5\\ \\ \it 3x+y=6\ \Rightarrow\ y=6-3x\ \ \ \ \ \ (*)\ rela\c{\it t}ia\ de\ substitu\c{\it t}ie\end{cases}\\ \\ \\ x-2y=-5\ \stackrel{(*)}{\Longrightarrow}\ x-2(6-3x)=-5 \Rightarrow x-12+6x=-5 \Rightarrow \\ \\ \Rightarrow x+6x=-5+12 \Rightarrow 7x=7 \Rightarrow x=1$

$\it x=1\ \stackrel{(*)}{\Longrightarrow}\ y=6-3 \Rightarrow y=3$

Mulțimea soluțiilor sistemului este:

$\it S=\{(1,\ \ 3)\}$

$\it c)\ \ \begin{cases}\it 5x+y=-11\ \Rightarrow\ y=-11-5x\ \ \ \ \ (*)\\ \\ \it 2x+3y=-7\end{cases}\\ \\ \\ 2x+3y=-7\ \stackrel{(*)}{\Longrightarrow}\ 2x+3(-11-5x)=-7 \Rightarrow 2x-33-15x=-7 \Rightarrow \\ \\ \\ \Rightarrow 2x-15x=-7+33 \Rightarrow -13x=26|_{:(-13)} \Rightarrow x=-2$

$\it x=-2\ \stackrel{(*)}{\Longrightarrow}\ y=-11-5\cdot(-2)=-11+10=-1\\ \\ S=\{(-2,\ -1)\}$