Question

Rezolvați sistemul:

$2|x| - |y| = 1 \\ |x| - 2 |y| = - 4$


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Answer 1

Răspuns:

{2lxl-lyl=1

{lxl-2lyl= -4

Inmultesti prima ecuatie cu -2 si o scazi din a 2-a

lxl-2lyl-2(2lxl-lyl)= -4-2

lxl-2lyl-4lxl+2lyl= -6

-3lxl= -6

lxl=-6/(-3)

lxl=2

x=+/-2

Inlocuiesti in prima  ecuatie

2l-2l-lyl=1

2*2-lyl=1

-lyl=1-4

-lyl= -3

lyl=3

y=+/3

x= 2

2*l2l-lyl=1

2*2-lyl=1

4-lyl=1

-lyl=1-4

-lyl= -3

lyl=3

y=+/3

Solutii

(x,y)={(-2,-3) ;(-2,3)(2,-3),(2,3)}

Explicație pas cu pas:

Answer 2

$\it Not\breve am\ |x|=a,\ \ |y|=b,\ iar\ sistemul\ devine:\\ \\ \begin{cases} \it 2a-b=1\\ \\ \it a-2b=-4 \Rightarrow a=2b-4\ \ \ \ \ (*)\end{cases}\\ \\ 2a-b=1\ \stackrel{(*)}{\Longrightarrow}\ 2(2b-4)-b=1 \Rightarrow 4b-8-b=1 \Rightarrow 3b=9 \Rightarrow b=3\\ \\ b=3\ \stackrel{(*)}{\Longrightarrow}\ a=2$

$\it \begin{cases} \it a=2 \Rightarrow |x|=2 \Rightarrow x=\pm2\\ \\ \it b=3 \Rightarrow |y|=3 \Rightarrow y=\pm3\end{cases}\\ \\ \\ S=\{(-2,\ -3),\ (-2,\ 3),\ (2,\ -3),\ (2,\ 3)\}$