Question

[Lê Thành Đạt]

Tìm các số hữu tỉ x, y, z biết rằng:

\(\frac{x}{3}=\frac{y}{5}=\frac{z}{7}\)

\(\left(x+y\right)xyz^2+\left(y+z\right)yzx^2+\left(z+x\right)zxy^2=477120\)

Answer 1

[Hoàng Lê Bảo Ngọc]

Đặt \(\frac{x}{3}=\frac{y}{5}=\frac{z}{7}=k\Rightarrow x=3k;y=5k;z=7k\)

\(xy+yz+zx=3k.5k+5k.7k+7k.3k=k^2\left(15+35+21\right)=71k^2;xyz=3k.5k.7k=105k^3\)

Ta có :  \(xyz\left(xz+yz+xy+xz+yz+xy\right)=477120\)

\(\Rightarrow xyz\left(xz+yz+xy\right)=238560\)\(\Rightarrow105k^3.71k^2=238560\Rightarrow k^5=32=2^5\Rightarrow k=2\)

Vậy : x= 6 ; y = 10 ; z = 14