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\(\left|2x-3y\right|+\left|2y+3z\right|+\left|x+y+z\right|=0\)
\(\Rightarrow\left\{{}\begin{matrix}2x-3y=0\\2y+3z=0\\x+y+z=0\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}2x=3y\\3z=-2y\\x+y+z=0\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}x=\dfrac{3y}{2}\\z=\dfrac{-2y}{3}\\x+y+z=0\end{matrix}\right.\)
\(\Rightarrow x=y=z=0\)
Ta có: \(\left|2x-3y\right|+\left|2y+3z\right|+\left|x+y+\frac{x}{z}\right|\ge0\left(\hept{\begin{cases}\forall x,y,z\\z\ne0\end{cases}}\right)\)
\(\Rightarrow\hept{\begin{cases}2x-3y=0\\2y+3z=0\\x+y+\frac{x}{z}=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=\frac{3}{2}y\\z=-\frac{2}{3}y\\\frac{3}{2}y-\frac{2}{3}y+\frac{\frac{3}{2}y}{-\frac{2}{3}y}=0\end{cases}}\)
\(\Rightarrow\hept{\begin{cases}x=\frac{3}{2}y\\z=-\frac{2}{3}y\\\frac{5}{6}y=\frac{9}{4}\end{cases}}\Rightarrow\hept{\begin{cases}x=\frac{3}{2}y=\frac{81}{20}\\y=\frac{27}{10}\\z=\frac{-9}{5}\end{cases}}\)
a) \(\frac{x}{6}=\frac{y}{-7};\frac{x}{3}=\frac{z}{-8}\Rightarrow\frac{y}{-21}=\frac{x}{18}=\frac{z}{-48}\)
Áp dụng tính chất dãy tỉ số bằng nhau
Ta có: \(\frac{2x}{36}=\frac{2y}{-42}=\frac{3z}{-144}=\frac{2x-2y+3z}{36-\left(-42\right)+\left(-144\right)}=\frac{56}{-66}=\frac{-28}{33}\)
\(\Rightarrow2x=\frac{28}{33}.36=\frac{-336}{11}\Rightarrow x=\frac{-168}{11}\)
\(2y=\frac{-28}{33}.\left(-42\right)=\frac{392}{11}\Rightarrow y=\frac{196}{11}\)
\(3z=\frac{-28}{33}.\left(-144\right)=\frac{1344}{11}\Rightarrow z=\frac{448}{11}\)
b) \(3x=-4y=2z\Rightarrow\frac{x}{-4}=\frac{y}{3};\frac{y}{2}=\frac{z}{-4}\Rightarrow\frac{x}{-8}=\frac{y}{6}=\frac{z}{-12}\)
\(\Rightarrow\frac{2x}{-16}=\frac{2y}{12}=\frac{3z}{-36}=\frac{2x-2y+3z}{-16-12+\left(-36\right)}=\frac{56}{-64}=\frac{-7}{8}\)
\(\Rightarrow2x=\frac{-7}{8}.\left(-16\right)=14\Rightarrow x=7\)
\(2y=\frac{-7}{8}.12=\frac{-21}{2}\Rightarrow y=\frac{-21}{4}\)
\(3z=\frac{-7}{8}.\left(-36\right)=\frac{63}{2}\Rightarrow z=\frac{21}{2}\)
c) Tương tự
