Lời giải:
HPT \(\Leftrightarrow \left\{\begin{matrix} \frac{1}{x}+\frac{1}{y}=\frac{3}{8}\\ \frac{1}{y}+\frac{1}{z}=\frac{3}{4}\\ \frac{1}{z}+\frac{1}{x}=\frac{5}{6}\end{matrix}\right.\Rightarrow 2(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})=\frac{3}{8}+\frac{3}{4}+\frac{5}{6}\)

\(\Leftrightarrow \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{47}{48}\)

\(\Rightarrow \left\{\begin{matrix} \frac{1}{z}=\frac{47}{48}-\frac{3}{8}\\ \frac{1}{x}=\frac{47}{48}-\frac{3}{4}\\ \frac{1}{y}=\frac{47}{48}-\frac{5}{6}\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x=\frac{48}{29}\\ y=\frac{48}{11}\\ z=\frac{48}{7}\end{matrix}\right.\)

\(\left\{{}\begin{matrix}3x^2+xz-yz+y^2=2\left(1\right)\\y^2+xy-yz+z^2=0\left(2\right)\\x^2-xy-xz-z^2=2\left(3\right)\end{matrix}\right.\)

Lấy (2) cộng (3) ta được

\(x^2+y^2-yz-zx=2\) (4)

Lấy (1) - (4) ta được

\(2x\left(x+z\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=-z\end{matrix}\right.\)

Xét 2 TH rồi thay vào tìm được y và z

1. \(\left\{{}\begin{matrix}6xy=5\left(x+y\right)\\3yz=2\left(y+z\right)\\7zx=10\left(z+x\right)\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{x+y}{xy}=\dfrac{6}{5}\\\dfrac{y+z}{yz}=\dfrac{3}{2}\\\dfrac{z+x}{zx}=\dfrac{7}{10}\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{6}{5}\\\dfrac{1}{y}+\dfrac{1}{z}=\dfrac{3}{2}\\\dfrac{1}{z}+\dfrac{1}{x}=\dfrac{7}{10}\end{matrix}\right.\)

Đến đây thì dễ rồi nhé

2. \(\left\{{}\begin{matrix}\left(xy-x\right)-\left(y-1\right)=6\\\left(yz-y\right)-\left(z-1\right)=12\\\left(zx-z\right)-\left(x-1\right)=8\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(x-1\right)\left(y-1\right)=6\\\left(y-1\right)\left(z-1\right)=12\\\left(z-1\right)\left(x-1\right)=8\end{matrix}\right.\)

Đến đây dễ rồi

\(\Leftrightarrow\left\{{}\begin{matrix}xy+x+y+1=2\\yz+y+z+1=5\\zx+z+x+1=10\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(x+1\right)\left(y+1\right)=2\\\left(y+1\right)\left(z+1\right)=5\\\left(z+1\right)\left(x+1\right)=10\end{matrix}\right.\) (1)

Nhân vế với vế: \(\left[\left(x+1\right)\left(y+1\right)\left(z+1\right)\right]^2=100\)

\(\Leftrightarrow\left(x+1\right)\left(y+1\right)\left(z+1\right)=10\) (2)

Chia vế cho vế của (2) cho từng pt của (1):

\(\Rightarrow\left\{{}\begin{matrix}z+1=5\\x+1=2\\y+1=1\end{matrix}\right.\) \(\Rightarrow\left(x;y;z\right)=\left(1;0;4\right)\) (loại)

Hệ vô nghiệm do \(y>0\)

Lời giải:

\(\text{HPT}\Leftrightarrow \left\{\begin{matrix} \frac{1}{a}=\frac{1}{x}-\frac{1}{y}-\frac{1}{z}(1)\\ \frac{1}{b}=\frac{1}{y}-\frac{1}{z}-\frac{1}{x}(2)\\ \frac{1}{c}=\frac{1}{z}-\frac{1}{x}-\frac{1}{y}(3)\end{matrix}\right.\Rightarrow -\left [\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right]=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}(4)\)

Lấy \((1),(2),(3)+(4)\Rightarrow \left\{\begin{matrix} -\left(\frac{1}{b}+\frac{1}{c}\right)=\frac{2}{x}\\ -\left(\frac{1}{a}+\frac{1}{c}\right)=\frac{2}{y}\\ -\left(\frac{1}{a}+\frac{1}{b}\right)=\frac{2}{z}\end{matrix}\right.\rightarrow \left\{\begin{matrix} x=\frac{-2bc}{b+c}\\ y=\frac{-2ac}{a+c}\\ z=\frac{-2ab}{a+b}\end{matrix}\right.\)

Vậy nghiệm của HPT là \((x,y,z)=\left(\frac{-2bc}{b+c},\frac{-2ac}{a+c},\frac{-2ab}{a+b}\right)\)

PT (1) \(\Leftrightarrow2\left(x^2+y^2+z^2\right)-2\left(xy+yz+xz\right)=0\)

\(\Leftrightarrow\left(x^2-2xy+y^2\right)+\left(y^2-2yz+z^2\right)+\left(z^2-2zx+x^2\right)=0\)

\(\Leftrightarrow\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2=0\)

Nhận thấy VT\(\ge\)0 với mọi x,y,z

Dấu = xảy ra <=> x=y=z

Thay x=y=z vào pt (2) ta được:

\(3x^{2021}=3^{2022}\) \(\Leftrightarrow x^{2021}=3^{2021}\) \(\Leftrightarrow x=3\)

\(\Rightarrow x=y=z=3\)

Vậy (x;y;z)=(3;3;3)

Lời giải:

HPT \(\Leftrightarrow \left\{\begin{matrix} x(x+y+z)=2\\ y(y+z+x)=3\\ z(z+x+y)=4\end{matrix}\right.(*)\).

Dễ thấy $x+y+z\neq 0$. Khi đó ta có:

\(\frac{x}{y}=\frac{x(x+y+z)}{y(y+z+x)}=\frac{2}{3}(1)\)

\(\frac{y}{z}=\frac{y(y+z+x)}{z(z+x+y)}=\frac{3}{4}(2)\)

Từ \((1);(2)\Rightarrow \frac{x}{2}=\frac{y}{3}=\frac{z}{4}\) .

Đặt \(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}=k\Rightarrow x=2k; y=3k; z=4k\)

Thay vào PT thứ nhất của $(*)$ suy ra:

\(2k(2k+3k+4k)=2\)

\(\Leftrightarrow 18k^2=2\Rightarrow k=\pm \frac{1}{3}\)

Nếu \(k=\frac{1}{3}\Rightarrow (x,y,z)=(2k,3k,4k)=(\frac{2}{3}; 1; \frac{4}{3})\)

Nếu \(k=\frac{-1}{3}\Rightarrow (x,y,z)=(2k,3k,4k)=(\frac{-2}{3}; -1; \frac{-4}{3})\)