Question

x y z sunt invers proportionale cu 3 8 5 daca x+4y-3z=14, afla x y z​

Answer 1

Răspuns:

Explicație pas cu pas:

Metoda 1:

3x = 8y = 5z

x+4y-3z=14 I x3

3x + 12y - 9z = 42

8y + 12y -9z = 42

8y = 5z, ultimele doua relatii cu acolada de sistem.

20y - 9z = 42

4y = 5z/2, 20y = 25z/2

25z/2 - 9z = 42

25z - 18z = 84

7z = 84

z = 12

x = 5*12/3 = 5*4 = 20

x = 20

y = 3x/8 = 3x20/8 = 3x5/2 = 15/2 = 7,5

Metoda a 2-a:

3x = 8y = 5z = k

x = k/3

y = k/8

z = k/5

x+4y-3z=14,

k/3 + k/2 - 3k/5 = 14

10k + 15k - 18k = 30*14

7k = 30*14

k = 60

x = 20

y = 60/8 = 15/2 = 7,5

z = 12

Verificarea este evidenta:

3x=8y=5z=60.

Answer 2

$\it \{x,\ y,\ z\}\ i.\ p.\ \{3,\ 8,\ 5\} \Rightarrow 3x=8y=5z \Rightarrow \begin{cases} \it 3x=5z|_{:5} \Rightarrow z=0,6x\\ \\ \it 3x=8y|_{:8} \Rightarrow y=0,375x \end{cases}\ \ \ (*)\\ \\ \\ x+4y-3z=14\ \stackrel{(*)}{\Longrightarrow}\ x+4\cdot0,375x-3\cdot0,6x=14 \Rightarrow x+1,5x-1,8x=14 \Rightarrow \\ \\ \Rightarrow 0,7x=14|_{\cdot10} \Rightarrow 7x=140|_{:7} \Rightarrow x=20$

$\it y=0,375\cdot20 \Rightarrow y=7,5\\ \\ z=0,6\cdot20 \Rightarrow z=12$

Numerele cerute sunt:   x=20;   y=7,5;   z=12.