a/ Đảo ngược lại rồi đặc \(\frac{1}{x}=a;\frac{1}{y}=b;\frac{1}{z}=c\)

b/ Dễ thấy vai trò x, y, z như nhau nên ta chỉ cần xét 1 trường hợp tiêu biểu thôi.

Xét \(x>y>z\)

\(\Rightarrow\frac{1}{x}< \frac{1}{y}< \frac{1}{z}\)

\(\Rightarrow x+\frac{1}{y}>z+\frac{1}{x}\)(trái giả thuyết)

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

\(\Rightarrow x+\frac{1}{x}=2\)

\(\Leftrightarrow x=1\)

a) \(\hept{\begin{cases}\left(x-1\right)\left(2x+y\right)=0\\\left(y+1\right)\left(2y-x\right)=0\end{cases}}\)
\(\cdot x=1\Rightarrow\hept{\begin{cases}0=0\\\left(y+1\right)\left(2y-1\right)=0\end{cases}}\Leftrightarrow\hept{\begin{cases}0=0\\y=-1;y=\frac{1}{2}\end{cases}}\)
\(\cdot y=-1\Rightarrow\hept{\begin{cases}\left(x-1\right)\left(2x-1\right)=0\\0=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=1;x=\frac{1}{2}\\0=0\end{cases}}\)
\(\cdot x=2y\Rightarrow\hept{\begin{cases}\left(2y-1\right)5y=0\\0=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}y=0\Rightarrow x=0\\y=\frac{1}{2}\Rightarrow x=1\end{cases}}\)
\(y=-2x\Rightarrow\hept{\begin{cases}0=0\\\left(1-2x\right)5x=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=\frac{1}{2}\Rightarrow y=-1\\x=0\Rightarrow y=0\end{cases}}\)

b) \(\hept{\begin{cases}x+y=\frac{21}{8}\\\frac{x}{y}+\frac{y}{x}=\frac{37}{6}\end{cases}\Leftrightarrow\hept{\begin{cases}x=\frac{21}{8}-y\\\left(\frac{21}{8}-y\right)^2+y^2=\frac{37}{6}y\left(\frac{21}{8}-y\right)\end{cases}}}\)

\(\Leftrightarrow\hept{\begin{cases}x=\frac{21}{8}-y\\2y^2-\frac{21}{4}y+\frac{441}{64}=-\frac{37}{6}y^2+\frac{259}{16}y\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}x=\frac{21}{8}-y\\1568y^2-4116y+1323=0\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}x=\frac{3}{8}\\y=\frac{9}{4}\end{cases}}hay\hept{\begin{cases}x=\frac{9}{4}\\y=\frac{3}{8}\end{cases}}\)

c) \(\hept{\begin{cases}\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2\\\frac{2}{xy}-\frac{1}{z^2}=4\end{cases}\Leftrightarrow\hept{\begin{cases}\frac{1}{z^2}=\left(2-\frac{1}{x}-\frac{1}{y}\right)^2\\\frac{1}{z^2}=\frac{2}{xy}-4\end{cases}}}\)\(\Leftrightarrow\hept{\begin{cases}\left(2xy-x-y\right)^2=-4x^2y^2+2xy\\\frac{1}{z^2}=\frac{2}{xy}-4\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}8x^2y^2-4x^2y-4xy^2+x^2+y^2-2xy+2xy=0\\\frac{1}{z^2}=\frac{2}{xy}-4\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}4x^2y^2-4x^2y+x^2+4x^2y^2-4xy^2+y^2=0\\\frac{1}{z^2}=\frac{2}{xy}-4\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}\left(2xy-x\right)^2+\left(2xy-y\right)^2=0\\\frac{1}{z^2}=\frac{2}{xy}-4\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}x=y=\frac{1}{2}\\z=\frac{-1}{2}\end{cases}}\)
d) \(\hept{\begin{cases}xy+x+y=71\\x^2y+xy^2=880\end{cases}}\). Đặt \(\hept{\begin{cases}x+y=S\\xy=P\end{cases}}\), ta có: \(\hept{\begin{cases}S+P=71\\SP=880\end{cases}}\Leftrightarrow\hept{\begin{cases}S=71-P\\P\left(71-P\right)=880\end{cases}}\Leftrightarrow\hept{\begin{cases}S=71-P\\P^2-71P+880=0\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}S=16\\P=55\end{cases}}hay\hept{\begin{cases}S=55\\P=16\end{cases}}\)
\(\cdot\hept{\begin{cases}S=16\\P=55\end{cases}}\Leftrightarrow\hept{\begin{cases}x+y=16\\xy=55\end{cases}}\Leftrightarrow\hept{\begin{cases}x=16-y\\y\left(16-y\right)=55\end{cases}}\Leftrightarrow\hept{\begin{cases}x=16-y\\y^2-16y+55=0\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}x=5\\y=11\end{cases}}hay\hept{\begin{cases}x=11\\y=5\end{cases}}\)

\(\cdot\hept{\begin{cases}S=55\\P=16\end{cases}}\Leftrightarrow\hept{\begin{cases}x+y=55\\xy=16\end{cases}}\Leftrightarrow\hept{\begin{cases}x=55-y\\y\left(55-y\right)=16\end{cases}}\Leftrightarrow\hept{\begin{cases}x=55-y\\y^2-55y+16=0\end{cases}}\)

\(\Leftrightarrow\hept{\begin{cases}x=\frac{55-3\sqrt{329}}{2}\\y=\frac{55+3\sqrt{329}}{2}\end{cases}}hay\hept{\begin{cases}x=\frac{55+3\sqrt{329}}{2}\\y=\frac{55-3\sqrt{329}}{2}\end{cases}}\)

e) \(\hept{\begin{cases}x\sqrt{y}+y\sqrt{x}=12\\x\sqrt{x}+y\sqrt{y}=28\end{cases}}\). Đặt \(\hept{\begin{cases}S=\sqrt{x}+\sqrt{y}\\P=\sqrt{xy}\end{cases}}\), ta có \(\hept{\begin{cases}SP=12\\P\left(S^2-2P\right)=28\end{cases}}\Leftrightarrow\hept{\begin{cases}S=\frac{12}{P}\\P\left(\frac{144}{P^2}-2P\right)=28\end{cases}}\Leftrightarrow\hept{\begin{cases}S=\frac{12}{P}\\2P^4+28P^2-144P=0\end{cases}}\)
Tự làm tiếp nhá! Đuối lắm luôn

Bài b nhé bạn!

\(\hept{\begin{cases}\frac{xyz}{x+y}=2\\\frac{xyz}{y+z}=\frac{6}{5}\\\frac{xyz}{x+z}=\frac{3}{2}\end{cases}\Leftrightarrow\hept{\begin{cases}\frac{x+y}{xyz}=\frac{1}{2}\\\frac{y+z}{xyz}=\frac{5}{6}\\\frac{x+z}{xyz}=\frac{2}{3}\end{cases}}}\)\(\Leftrightarrow\hept{\begin{cases}\frac{1}{yz}+\frac{1}{xz}=\frac{1}{2}\\\frac{1}{xz}+\frac{1}{xy}=\frac{5}{6}\\\frac{1}{xy}+\frac{1}{yz}=\frac{2}{3}\end{cases}}\Rightarrow\frac{1}{xy}+\frac{1}{yz}+\frac{1}{xz}=\frac{\frac{1}{2}+\frac{5}{6}+\frac{2}{3}}{2}=1\)

Trừ lại từng phương trình trong hệ:

\(\hept{\begin{cases}\frac{1}{xy}=\frac{1}{2}\\\frac{1}{yz}=\frac{1}{6}\\\frac{1}{xz}=\frac{1}{3}\end{cases}}\Leftrightarrow\hept{\begin{cases}xy=2\\yz=6\\xz=3\end{cases}\Rightarrow xyz=\sqrt{2.6.3}=6}\)

Chia lại từng phương trình trong hệ mới, được:

\(\hept{\begin{cases}z=3\\x=1\\y=2\end{cases}}\)

Vậy \(\left(x;y;z\right)=\left(1;2;3\right)\)

Xong rồi đó!!!

\(\hept{\begin{cases}x+y+z=3\left(1\right)\\\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{3}\left(2\right)\\x^2+y^2+z^2=17\left(3\right)\end{cases}}\left(DK:x,y,z\ne0\right)\)

Ta co:

\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\ge\frac{9}{x+y+z}=3>\frac{1}{3}\)

\(\Rightarrow\frac{1}{x}+\frac{1}{y}+\frac{1}{z}>\frac{1}{3}\)

Vay HPT vo nghiem

Bài làm:

Ta có: \(x+\frac{1}{y}+y+\frac{1}{z}+z+\frac{1}{x}=\left(x+\frac{1}{x}\right)+\left(y+\frac{1}{y}\right)+\left(z+\frac{1}{z}\right)\)

\(\ge2\sqrt{x.\frac{1}{x}}+2\sqrt{y.\frac{1}{y}}+2\sqrt{z.\frac{1}{z}}=2+2+2=6\)

Mà theo đề bài: \(x+\frac{1}{y}+y+\frac{1}{z}+z+\frac{1}{x}=6\)

Nên dấu "=" xảy ra khi: \(x=\frac{1}{x}\) ; \(y=\frac{1}{y}\) ; \(z=\frac{1}{z}\)

\(\Rightarrow x=y=z=1\)