2x−3y/5=5y−2z/3=3z−5x/2=10x-15y/25=15y-6z/9=6z-10x/4=...+..+..../25+9+4=0/31=0

=> 2x=3y;  5y=2z ;  3z=5x => x/3=y/2; y/2=z/5

=> x/3=y/2 =z/5 = 12x/36=5y/10=3z/15= (12x+5y-3z)/31

      x/3 = 3y/6=2z/10 = (x-3y+2z)/7

=>  (12x+5y-3z)/ (x-3y+2z)=31/7

help me ai nhanh nhất mik tích cho

a) Ta có: \(\left(\dfrac{3}{4}\right)^{2021}>\left(\dfrac{3}{4}\right)^1=\dfrac{3}{4}\)

\(\Leftrightarrow\left(\dfrac{3}{4}\right)^{2021}+1>\dfrac{3}{4}+1\)

Áp dụng t/c dtsbn:

\(\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{4}=\dfrac{10x}{20}=\dfrac{3z}{12}=\dfrac{5y}{15}=\dfrac{3y}{9}=\dfrac{2z}{8}=\dfrac{10x+3z-5y}{17}=\dfrac{x+3y-2z}{3}\\ \Rightarrow P=\dfrac{10x+3z-5y}{x+3y-2z}=\dfrac{17}{3}\)

 đặt\(A=\dfrac{x^3}{2x+3y+5z}+\dfrac{y^3}{2y+3z+5x}+\dfrac{z^3}{2z+3x+5y}\)

\(=>A=\dfrac{x^4}{2x^2+3xy+5xz}+\dfrac{y^4}{2y^2+3yz+5xy}+\dfrac{z^4}{2z^2+3xz+5yz}\)

BBDT AM-GM 

\(=>A\ge\dfrac{\left(x^2+y^2+z^2\right)^2}{2\left(x^2+y^2+z^2\right)+8\left(xy+yz+xz\right)}\)

theo BDT AM -GM ta chứng minh được \(xy+yz+xz\le x^2+y^2+z^2\)

vì \(x^2+y^2\ge2xy\)

\(y^2+z^2\ge2yz\)

\(x^2+z^2\ge2xz\)

\(=>2\left(x^2+y^2+z^2\right)\ge2\left(xy+yz+xz\right)< =>xy+yz+xz\le x^2+y^2+z^2\)

\(=>2\left(x^2+y^2+z^2\right)+8\left(xy+yz+xz\right)\le10\left(x^2+y^2+z^2\right)\)

\(=>A\ge\dfrac{\left(x^2+y^2+z^2\right)^2}{10\left(x^2+y^2+z^2\right)}=\dfrac{x^2+y^2+z^2}{10}=\dfrac{\dfrac{1}{3}}{10}=\dfrac{1}{30}\left(đpcm\right)\)

dấu"=" xảy ra<=>x=y=z=1/3

ngu

Giải:
Ta có: \(\dfrac{3x-2y}{5}=\dfrac{5y-3z}{2}=\dfrac{2z-5x}{2}\)

\(\Rightarrow\dfrac{15x-10y}{25}=\dfrac{10y-6z}{4}=\dfrac{6z-15x}{6}\)

Áp dụng tính chất dãy tỉ số bằng nhau có:

\(\Rightarrow\dfrac{15x-10y}{25}=\dfrac{10y-6z}{4}=\dfrac{6z-15x}{6}=\dfrac{15x-10y+10y-6z+6z-15x}{25+4+6}=0\)

\(\Rightarrow\left\{{}\begin{matrix}15x-10y=0\\10y-6z=0\\6z-15x=0\end{matrix}\right.\Rightarrow15x=10y=6z\)

\(\Rightarrow\dfrac{15x}{30}=\dfrac{10y}{30}=\dfrac{6z}{30}\Rightarrow\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{5}\)

Áp dụng tính chất dãy tỉ số bằng nhau có:

\(\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{5}=\dfrac{10x}{20}=\dfrac{3y}{9}=\dfrac{2z}{10}=\dfrac{10x-3y-2z}{20-9-10}=\dfrac{5}{1}=5\)

\(\Rightarrow\left\{{}\begin{matrix}x=10\\y=15\\z=25\end{matrix}\right.\)

Vậy...

\(\dfrac{3x-2y}{5}=\dfrac{5y-3z}{2}=\dfrac{2z-5x}{2}\)

\(\Rightarrow\dfrac{5\left(3x-2y\right)}{25}=\dfrac{2\left(5y-3z\right)}{4}=\dfrac{3\left(2z-5x\right)}{6}\)

\(\Rightarrow\dfrac{15x-10y}{25}=\dfrac{10y-6z}{4}=\dfrac{6z-15x}{6}\)

Áp dụng tính chất dãy tỉ số bằng nhau ta có:

\(\dfrac{15x-10y}{25}=\dfrac{10y-6z}{4}=\dfrac{6z-15x}{6}\)

\(=\dfrac{15x-10y+10y-6z+6z-15x}{25+4+6}\)

\(=0\)

\(\Rightarrow\left\{{}\begin{matrix}3x=2y\Rightarrow\dfrac{x}{2}=\dfrac{y}{3}\\5y=3z\Rightarrow\dfrac{y}{3}=\dfrac{z}{5}\\2z=5x\Rightarrow\dfrac{z}{5}=\dfrac{x}{2}\end{matrix}\right.\)

\(\Rightarrow\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{5}\)

\(\Rightarrow\dfrac{10x}{20}=\dfrac{3y}{9}=\dfrac{2z}{10}\)

Áp dụng tính chất dãy tỉ số bằng nhau ta có:

\(\dfrac{10x}{20}=\dfrac{3y}{9}=\dfrac{2z}{10}=\dfrac{10x-3y-2z}{20-9-10}=\dfrac{5}{1}=5\)

\(\Rightarrow\left\{{}\begin{matrix}x=5.2=10\\y=5.3=15\\z=5.5=25\end{matrix}\right.\)

\(\dfrac{3x-2y}{5}=\dfrac{2z-5x}{3}=\dfrac{5y-3z}{2}\)

\(=\dfrac{15x-10y}{25}=\dfrac{6z-15x}{9}=\dfrac{10y-6z}{4}\)

Áp dụng tính chất của dãy tỉ số bằng nhau ta có :

\(\dfrac{3x-2y}{5}=\dfrac{2z-5x}{3}=\dfrac{5y-3z}{2}=\dfrac{15x-10y}{25}=\dfrac{6z-15x}{9}=\dfrac{10y-6z}{4}\)

\(=\dfrac{15x-10y+6z-15x+10y-6z}{25+9+4}=0\)

⇒\(3x=2y\)⇒\(\dfrac{x}{2}=\dfrac{y}{3}\)

⇒\(2z=5x\)⇒\(\dfrac{x}{2}=\dfrac{z}{5}\)

⇒\(\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{5}\)

Áp dụng tính chất của dãy tỉ số bằng nhau ta có :

\(\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{5}=\dfrac{2x}{6}=\dfrac{3y}{9}=\dfrac{5z}{25}\)\(=\dfrac{2x+3y-5z}{6+9-25}=\dfrac{-60}{-10}=6\)

⇒\(\dfrac{x}{2}=6\)⇒\(x=12\)

⇒\(\dfrac{y}{3}=6\)⇒\(y=18\)

⇒\(\dfrac{z}{5}=6\)⇒\(z=30\)

Vậy \(x=12;y=18;z=30\)