Question

Система уравнений
2ху-xz+yz=2xyz
1/x-1/z=3
1/y+1/z=2

Answer 1

$left { {{2xy-xz+yz=2xyz} atop { frac{1}{x} - frac{1}{z} =3} ||*xyz (x neq 0, y neq 0, z  neq 0} atop { frac{1}{y}+frac{1}{z}=2 ||*xyz right. \ left { {{2xy-xz+yz=2xyz} atop { yz - xy =3xyz} ||*(-1)} atop { xz+xy=2xyz right. \left { {{2xy-xz+yz=2xyz} atop {xy - yz =-3xyz}} atop { xy+xz=2xyz right\left { {{2xy+xy+xy-xz+xz+yz-yz=2xyz-3xyz+2xyz} atop {xy - yz =-3xyz}} atop { xy+xz=2xyz right \ left { {{4xy=xyz ||/xy} atop {xy - yz =-3xyz}} atop { xy+xz=2xyz right$
$left { {{z=4} atop {xy - yz =-3xyz} ||/y} atop { xy+xz=2xyz ||/x}right\ left { {{z=4} atop {x - z =-3xz}} atop { y+z=2yzright \ left { {{z=4} atop {x - 4 =-3x*4}} atop { y+4=2y*4right\ left { {{4=z} atop {x+12x=4}} atop { y-8y=-4right\ left { {{z=4} atop {x= frac{4}{13} }} atop { y= frac{4}{7} right$
Проверка.
32/7*13-16/13+16/7=128/13*7; 32-16*7+16*13=128; 32+16*6=128; 128=128.
13/4-1/4=3; 3=3.
7/4+1/4=2; 2=2.
Ответ: $x= frac{4}{13}; y= frac{4}{7}; z=4.$