x/y+z=y/z+x=z/x+y Tính giá trị mỗi phân thức đó
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TN
30 tháng 6 2018
\(\frac{x}{\left(x-y\right)\left(x-z\right)}\) \(+\frac{y}{\left(x-y\right)\left(y-z\right)}\)\(+\frac{z}{\left(y-z\right)\left(z-x\right)}\)
\(=\)\(\frac{x\left(y-z\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\) \(+\frac{y\left(x-z\right)}{\left(x-y\right)\left(y-z\right)\left(x-z\right)}-\)\(\frac{z\left(x-y\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(=\frac{x\left(y-z\right)+y\left(x-z\right)-z\left(x-y\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(=\)\(\frac{xy-xz+xy-yz-xz+yz}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(=\)\(\frac{2xy-2xz}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(=\frac{2x\left(y-z\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(=\)\(\frac{2x}{\left(x-y\right)\left(x-z\right)}\)
VT
1
AH
Akai Haruma
Giáo viên
15 tháng 7 2023
Lời giải:
Ta có:
$(x+y+z)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})=2023.\frac{2024}{2023}$
$\Leftrightarrow 1+\frac{x}{y}+\frac{x}{z}+\frac{y}{x}+1+\frac{y}{z}+\frac{z}{x}+\frac{z}{y}+1=2024$
$\Leftrightarrow 3+\frac{x+z}{y}+\frac{y+z}{x}+\frac{x+y}{z}=2024$
$\Leftrightarrow 3+B=2024$
$\Leftrightarrow B=2021$
13 tháng 11 2021
TH1: \(x+y+z+t\ne0\)
Áp dụng t/c dtsbn ta có:
\(\dfrac{x}{y+z+t}=\dfrac{y}{z+t+x}=\dfrac{z}{t+x+y}=\dfrac{t}{x+y+z}=\dfrac{x+y+z+t}{3\left(x+y+z+t\right)}=\dfrac{1}{3}\)\(\dfrac{x}{y+z+t}=\dfrac{1}{3}\Rightarrow3x=y+z+t\Rightarrow4x=x+y+z+t\\ \dfrac{y}{z+t+x}=\dfrac{1}{3}\Rightarrow3y=x+z+t\Rightarrow4y=x+y+z+t\\ \dfrac{z}{t+x+y}=\dfrac{1}{3}\Rightarrow3z=x+y+t\Rightarrow4z=x+y+z+t\\ \dfrac{t}{x+y+z}=\dfrac{1}{3}\Rightarrow3t=x+y+z\Rightarrow4t=x+y+z+t\)
\(\Rightarrow4x=4y=4z=4t\\
\Rightarrow x=y=z=t\)
\(P=\dfrac{x+y}{z+t}+\dfrac{y+z}{t+x}+\dfrac{z+t}{x+y}+\dfrac{t+x}{y+z}\\ =1+1+1+1\\ =4\)
TH1: \(x+y+z+t=0\)
\(\Rightarrow\left\{{}\begin{matrix}x+y=-\left(z+t\right)\\y+z=-\left(x+t\right)\\z+t=-\left(x+y\right)\\t+x=-\left(y+z\right)\end{matrix}\right.\)
\(P=\dfrac{x+y}{z+t}+\dfrac{y+z}{t+x}+\dfrac{z+t}{x+y}+\dfrac{t+x}{y+z}\\ =\dfrac{-\left(z+t\right)}{z+t}+\dfrac{-\left(t+x\right)}{t+x}+\dfrac{-\left(x+y\right)}{x+y}+\dfrac{-\left(y+z\right)}{y+z}\\ =-1-1-1-1\\ =-4\)
cht nhé bạn