Question

Va rog mult de tot sa ma ajutați.
x/y=15/14
y/z=15/14
x + y - 2z = 129
Aflați x, y, z.

Answer 1

Folosim produsul mezilor este egal cu produsul extremilor.
Il scoatem pe y in functie de x: 
$\frac{x}{y}=\frac{15}{14}\Leftrightarrow 14x=15y\Leftrightarrow y=\frac{14}{15}x$
Il scoatem pe y in functie de z:
[tex] \frac{y}{z}=\frac{15}{14}\Leftrightarrow 14y=15z\Leftrightarrow y=\frac{15}{14}z[/tex]
Egalam cele doua:
[tex] \left \{ {{y=\frac{14}{15}x} \atop {y=\frac{15}{14}z} \right. \Leftrightarrow \frac{14}{15}x=\frac{15}{14}z\Leftrightarrow 14\cdot14x=15\cdot15z\Leftrightarrow 196x=225z\\ \Leftrightarrow z=\frac{196}{225}x[/tex]
Inlocuim y si z si obtinem o ecuatie cu o singura necunoscuta:
[tex]x+y-2z=129\\ x+\frac{14}{15}x+2\frac{196}{225}x=129\\[/tex]
Aducem la acelasi numitor, acesta fiind 225:
[tex]\frac{225}{225}x+\frac{14\cdot15}{15\cdot15}x-2\frac{196}{225}x=\frac{129\cdot225}{225}\Leftrightarrow \frac{225x+210x-392x}{225}\\ \\ =\frac{129\cdot225}{225}\Leftrightarrow \frac{43}{225}x=\frac{129\cdot225}{225}\\ 43x=129\cdot225\\ x=\frac{129\cdot225}{43}=675\\ \\ y=\frac{14}{15}x=\frac{14}{15}\cdot675=630\\ \\ z=\frac{196}{255}x=\frac{196}{225}\cdot675=588[/tex]