Tìm x biết: \(\frac{x+1}{x-1}=\frac{x-2019}{x+2019}\)
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1 tháng 5 2020
\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{x+y+z}\)=> \(\frac{xy+yz+zx}{xyz}=\frac{1}{x+y+z}\)
=> (x+y+z)(xy+yz+zx) = xyz
=> \(x^2y+xy^2+y^2z+yz^2+zx^2+z^2x+2xyz=0\)
=> (x+y)(y+z)(z+x) = 0
=> \(\left[{}\begin{matrix}x=-y\\y=-z\\z=-x\end{matrix}\right.\)
TH1: x = -y
=> \(\frac{1}{x^{2019}}+\frac{1}{y^{2019}}+\frac{1}{z^{2019}}=\frac{1}{\left(-y\right)^{2019}}+\frac{1}{y^{2019}}+\frac{1}{z^{2019}}=\frac{1}{z^{2019}}\)
=> \(\frac{1}{x^{2019}+y^{2019}+z^{2019}}=\frac{1}{\left(-y\right)^{2019}+y^{2019}+z^{2019}}=\frac{1}{z^{2019}}\)
=> ĐPCM
Tương tự với TH2 và TH3
13 tháng 11 2019
\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{x+y+z}\)
\(\Leftrightarrow\frac{1}{x}+\frac{1}{y}+\frac{1}{z}-\frac{1}{x+y+z}=0\)
\(\Leftrightarrow\frac{x+y}{xy}+\frac{x+y}{z\left(x+y+z\right)}=0\)
\(\Leftrightarrow\left(x+y\right)\cdot\frac{xy+z\left(x+y+z\right)}{xyz\left(x+y+z\right)}=0\)
\(\Leftrightarrow\left(x+y\right)\left(y+z\right)\left(z+x\right)=0\)
\(\Leftrightarrow x=-y\left(h\right)y=-z\left(h\right)z=-x\)
Nếu
\(x=-y\Rightarrow\frac{1}{x^{2019}}+\frac{1}{y^{2019}}+\frac{1}{z^{2019}}=\frac{1}{x^{2019}}-\frac{1}{x^{2019}}+\frac{1}{z^{2019}}=\frac{1}{z^{2019}}\)
\(\frac{1}{x^{2019}+y^{2019}+z^{2019}}=\frac{1}{x^{2019}-x^{2019}+z^{2019}}=\frac{1}{z^{2019}}\)
Tương tự các TH còn lại nha!
P/S:Có 1 bài chặt hơn ntnày:
Cho \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{x+y+z}\) thì \(\frac{1}{x^n}+\frac{1}{y^n}+\frac{1}{z^n}=\frac{1}{x^n+y^n+z^n}\) với n lẻ.
19 tháng 6 2017
Ta có : \(\frac{x-1}{2017}+\frac{x-2}{2018}-\frac{x-3}{2019}=\frac{x-4}{2020}\)
\(\Rightarrow\frac{x-1}{2017}+\frac{x-2}{2018}=\frac{x-4}{2020}+\frac{x-3}{2019}\)
\(\Rightarrow1+\frac{x-1}{2017}+1+\frac{x-2}{2018}=1+\frac{x-4}{2020}+1+\frac{x-3}{2019}\)
\(\Rightarrow\frac{2016+x}{2017}+\frac{2016+x}{2018}=\frac{2016+x}{2020}+\frac{2016+x}{2019}\)
\(\Rightarrow\frac{2016+x}{2017}+\frac{2016+x}{2018}-\frac{2016+x}{2019}-\frac{2016+x}{2020}=0\)
\(\Rightarrow\left(2016+x\right)\left(\frac{1}{2017}+\frac{1}{2018}-\frac{1}{2019}-\frac{1}{2020}\right)=0\)
\(\text{Mà :
}\)\(\frac{1}{2017}+\frac{1}{2018}-\frac{1}{2019}-\frac{1}{2020}\ne0\)
\(\text{Nên : }\) \(2016+x=0\)
\(\Rightarrow x=-2016\)
\(ĐKXĐ:\hept{\begin{cases}x-1\ne0\\x+2019\ne0\end{cases}}\Leftrightarrow\hept{\begin{cases}x\ne1\\x\Leftrightarrow-2019\end{cases}}\)
\(\frac{x+1}{x-1}=\frac{x-2019}{x+2019}\Leftrightarrow\frac{x+1}{x-1}-\frac{x-2019}{x+2019}=0\)
\(\Leftrightarrow\frac{x+1}{x-1}+\frac{2019-x}{x+2019}=0\Leftrightarrow\frac{\left(x+1\right)\left(x+2019\right)+\left(x-1\right)\left(2019-x\right)}{\left(x-1\right)\left(x+2019\right)}=0\)
\(\Leftrightarrow\frac{x^2+2020x+2019+2020x-x^2-2019}{\left(x-1\right)\left(x+2019\right)}=0\)
\(\Leftrightarrow\frac{4040x}{\left(x-1\right)\left(x+2019\right)}=0\Leftrightarrow4040x=0\Leftrightarrow x=0\)
Vậy \(x=0\)