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áp dụng tính chất của dãy tỉ số bằng nhau ta có:\(\frac{ }{ }\)
y+z-x/x=z+x-y/y=x+y-z/z
=y+z-x+z+x-y+x+y-z/x+y+z
=(y-y)+(z-z)-(x-x)+z+x+y/x+y+z
=0+0+0+x+y+z/x+y+z=1
\(\Leftrightarrow\)x=y=z (*)
thay (*) vào B ta có:
B=(1+x/x)(1+x/x)(1+x/x)
=2.2.2=8
Áp dụng tính chất dãy tỉ số bằng nhau ta có :
\(...=\frac{y+z-x+z+x-y+x+y-z}{x+y+z}=\frac{x+y+z}{x+y+z}=1\)( vì x + y + z \(\ne\)0 )
\(\Rightarrow\hept{\begin{cases}\frac{y+z-x}{x}=1\\\frac{z+x-y}{y}=1\\\frac{x+y-z}{z}=1\end{cases}}\Rightarrow\hept{\begin{cases}y+z-x=x\\z+x-y=y\\x+y-z=z\end{cases}}\Rightarrow\hept{\begin{cases}y+z=2x\\z+x=2y\\x+y=2z\end{cases}}\Rightarrow x=y=z\)
Thế x = y = z vào B ta được :
\(B=\left(1+\frac{y}{y}\right)\left(1+\frac{x}{x}\right)\left(1+\frac{z}{z}\right)=\left(1+1\right)\left(1+1\right)\left(1+1\right)=2\cdot2\cdot2=8\)
làm lần lượt nhá,dài dòng quá khó coi.ahihihi!
\(\frac{1-\frac{1}{\sqrt{49}}+\frac{1}{49}-\frac{1}{7\left(\sqrt{7}\right)^2}}{\frac{\sqrt{64}}{2}-\frac{4}{7}+\left(\frac{2}{7}\right)^2-\frac{4}{343}}=\frac{1-\frac{1}{7}+\frac{1}{49}-\frac{1}{343}}{4-\frac{4}{7}+\frac{4}{49}-\frac{4}{343}}\)
\(=\frac{1-\frac{1}{7}+\frac{1}{49}-\frac{1}{343}}{4\left(1-\frac{1}{7}+\frac{1}{49}-\frac{1}{343}\right)}=\frac{1}{4}\)
\(\frac{y+z-x}{x}=\frac{z+x-y}{y}=\frac{x+y-z}{z}\)\(\Rightarrow\frac{y+z}{x}-1=\frac{z+x}{y}-1=\frac{x+y}{z}-1\)
\(\Rightarrow\frac{y+z}{x}=\frac{z+x}{y}=\frac{x+y}{z}\)
Áp dụng tính chất dãy tỉ số bằng nhau, ta có:
\(\frac{y+z}{x}=\frac{z+x}{y}=\frac{x+y}{z}=\frac{y+z+z+x+x+y}{x+y+z}=\frac{2\left(x+y+z\right)}{x+y+z}=2\)
Do đó: +) \(\frac{y+z}{x}=2\)\(\Rightarrow y+z=2x\)
+) \(\frac{z+x}{y}=2\)\(\Rightarrow z+x=2y\)
+) \(\frac{x+y}{z}=2\)\(\Rightarrow x+y=2z\)
Ta có: \(B=\left(1+\frac{x}{y}\right)\left(1+\frac{y}{z}\right)\left(1+\frac{z}{x}\right)=\frac{y+x}{y}.\frac{z+y}{z}.\frac{x+z}{x}=\frac{2z}{y}.\frac{2x}{z}.\frac{2y}{x}=2.2.2=8\)
Áp dụng tính chất của dãy tỉ số = nhau ta có:
\(\frac{y+z-x}{x}=\frac{z+x-y}{y}=\frac{x+y-z}{z}=\frac{\left(y+z-x\right)+\left(z+x-y\right)+\left(x+y-z\right)}{x+y+z}=\frac{x+y+z}{x+y+z}\) (1)
Xét 1 trường hợp:
- TH1: x + y + z = 0 \(\Rightarrow\begin{cases}x+y=-z\\y+z=-x\\z+x=-y\end{cases}\)
Ta có: \(\left(1+\frac{x}{y}\right)\left(1+\frac{y}{z}\right)\left(1+\frac{z}{x}\right)=\frac{x+y}{y}.\frac{y+z}{z}.\frac{z+x}{x}=\frac{-z}{y}.\frac{-x}{z}.\frac{-y}{x}=-1\)
- TH2: \(x+y+z\ne0\)
Từ (1) \(\Rightarrow\frac{y+z-x}{x}=\frac{z+x-y}{y}=\frac{x+y-z}{z}=\frac{x+y+z}{x+y+z}=1\)
\(\Rightarrow\begin{cases}y+z-x=x\\z+x-y=y\\x+y-z=z\end{cases}\)\(\Rightarrow\begin{cases}y+z=2x\\z+x=2y\\x+y=2z\end{cases}\)
Ta có: \(\left(1+\frac{x}{y}\right)\left(1+\frac{y}{z}\right)\left(1+\frac{z}{x}\right)=\frac{x+y}{y}.\frac{y+z}{z}.\frac{x+z}{x}=\frac{2z}{y}.\frac{2x}{z}.\frac{2y}{x}=2^3=8\)
ta có y+z-x/x=z+x-y/y=x+y-z/z=y+z-x+z+x-y+x+y-z/x+y+z=(2y-y)+(2x-x)+(2z-z)/x+y+z=y+x+z/x+y+z=1
=>y+z-x/x=1 =>z+x-y/y=1
z+x-y/y=1 x+y-z/z=1
=> y+z-x=x => z+x-y=y
z+x-y=y x+y-z=z
=>2y-2x=x-y =>2z-2y=y-z
3y-3x=0 3z-3y=0
y-x=0 z-y=0
=>x=y =>z=y
=>x=y=z
=> y+z-x/x+z+x-y/y+x+y-z/z= 0,(3)+0,(3)+0,(3)=1
=>x +y+z=0,(3)+0,(3)+0,(3)=1
thay vào b=(1+x/y). (1+y/z). (1+z/x)
b=(1+0,(3)/0,(3)).(1+0,(3)/0,(3)).(1+0,(3)/0,(3))
b=(1+1).(1+1).(1+1)
b=2.2.2
b=2^3
b=8
CÂU TRẢ LỜI TRƯỚC MK BẤM NHẦM
Ta có: \(\frac{x+y-3}{z}=\frac{y+z+1}{x}=\frac{z+x+2}{y}=\frac{1}{x+y+z}\)
\(\Rightarrow\frac{z}{x+y-3}=\frac{x}{y+z+1}=\frac{y}{z+x+2}=x+y+z\)
TH1: \(x+y+z=0\)
Áp dụng tính chất dãy tỉ số bằng nhau, ta có:
\(\frac{z}{x+y-3}=\frac{x}{y+z+1}=\frac{y}{z+x+2}=\frac{x+y+z}{x+y-3+y+z+1+z+x+2}\)
\(=\frac{x+y+z}{x+y+y+z+z+x}=\frac{x+y+z}{2\left(x+y+z\right)}=\frac{1}{2}\)
\(\Rightarrow x+y+z=\frac{1}{2}\)
\(\Rightarrow x+y=\frac{1}{2}-z\)
\(y+z=\frac{1}{2}-x\)
\(z+x=\frac{1}{2}-y\)
Thay \(x+y-3=\frac{1}{2}-z-3\)
\(\Rightarrow\frac{z}{\frac{1}{2}-z+3}=\frac{1}{2}\)
\(\Rightarrow2z=\frac{1}{2}-z-3\)
\(\Rightarrow2z+z=\frac{1}{2}-3\)
\(\Rightarrow3z=-\frac{5}{2}\Rightarrow z=-\frac{5}{6}\)
Thay \(y+z+1=\frac{1}{2}-x+1\)
\(\Rightarrow\frac{x}{\frac{1}{2}-x+1}=\frac{1}{2}\)
\(\Rightarrow2x=\frac{1}{2}-x+1\)
\(\Rightarrow2x+x=\frac{1}{2}+1\)
\(\Rightarrow3x=\frac{3}{2}\Rightarrow x=\frac{1}{2}\)
Thay \(z+x+2=\frac{1}{2}-y+2\)
\(\Rightarrow\frac{y}{\frac{1}{2}-y+2}=\frac{1}{2}\)
\(\Rightarrow2y=\frac{1}{2}-y+2\)
\(\Rightarrow2y+y=\frac{1}{2}+2\)
\(\Rightarrow3y=\frac{5}{2}\Rightarrow y=\frac{5}{6}\)
Ta có: \(A=\left(x+y+z-\frac{3}{2}\right)^{2019}\)
\(=\left(\frac{1}{2}+\frac{5}{6}+-\frac{5}{6}-\frac{3}{2}\right)^{2019}\)
\(=\left[\left(\frac{1}{2}-\frac{3}{2}\right)+\left(-\frac{5}{6}+\frac{5}{6}\right)\right]^{2019}\)
\(=\left(-1\right)^{2019}=-1\)
TH2: x + y + z = 0
\(\Rightarrow\frac{z}{x+y-3}=\frac{x}{y+z+1}=\frac{y}{z+x+2}=0\)
\(\Rightarrow x=y=z=0\)
\(A=\left(x+y+z-\frac{3}{2}\right)^{2019}\)
\(=\left(0-\frac{3}{2}\right)^{2019}=\left(-\frac{3}{2}\right)^{2019}\)
Ah! Mk nhầm chút. TH1 là khác 0 nhé!!!!!!