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Cho \(\frac{3x-2y}{4}=\frac{2z-4x}{3}=\frac{4y-3z}{2}\)
Cmr : \(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\)
Giải:
Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\frac{3x-2y}{4}=\frac{2z-4x}{3}=\frac{4y-3z}{2}=\frac{12x-8y}{16}=\frac{6z-12x}{9}=\frac{8y-6z}{4}=\frac{12x-8y+6z-12x+8y-6z}{16+9+4}=\frac{0}{16+9+4}=0\)
\(\Rightarrow3x-2y=0\)
\(\Rightarrow2z-4x=0\)
\(\Rightarrow4y-3z=0\)
Ta có: \(3x-2y=0\Rightarrow3x=2y\Rightarrow\frac{x}{2}=\frac{y}{3}\) (1)
\(2z-4x=0\Rightarrow2z=4x\Rightarrow\frac{z}{4}=\frac{x}{2}\) (2)
Từ (1) và (2) suy ra \(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\)
\(\Rightarrowđpcm\)
\(\frac{3x-2y}{4}=\frac{2z-4x}{3}=\frac{4y-3z}{2}\)
\(\Rightarrow\frac{4\left(3x-2y\right)}{4.4}=\frac{3\left(2z-4x\right)}{3.3}=\frac{2\left(4y-3z\right)}{2.2}\)
\(\Rightarrow\frac{12x-8y}{16}=\frac{6z-12x}{9}=\frac{8y-6z}{4}\)
Áp dụng TCDTSBN ta có:
\(\frac{12x-8y}{16}=\frac{6z-12x}{9}=\frac{8y-6z}{4}=\frac{12x-8y+6z-12x+8y-6z}{16+9+4}=\frac{0}{29}=0\)
\(\Rightarrow\frac{3x-2y}{4}=0\Rightarrow3x=2y\Rightarrow\frac{x}{2}=\frac{y}{3}\left(1\right)\)
\(\frac{2z-4x}{3}=0\Rightarrow2z=4x\Rightarrow\frac{x}{2}=\frac{z}{4}\left(2\right)\)
Từ (1) và (2) => \(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\)
ta có:\(\frac{3x-2y}{4}\)=\(\frac{2z-4x}{3}\)=\(\frac{4y-3z}{2}\)
=\(\frac{12x-8y}{16}\)=\(\frac{6z-12x}{9}\)=\(\frac{8y-6z}{4}\)=\(\frac{12x-8y+6z-12x+8y-6z}{16+9+4}\)
=>\(\hept{\begin{cases}3x-2y=0\\2z-4x=0\\4y-3z=0\end{cases}}\)=>\(\hept{\begin{cases}3x=2y\\2z=4x\\4y-3z\end{cases}}\)=>\(\hept{\begin{cases}\frac{x}{2}=\frac{y}{3}\\\frac{z}{4}=\frac{x}{2}\\\frac{y}{3}=\frac{z}{4}\end{cases}}\)=>\(\hept{\begin{cases}x\\2\end{cases}}=\frac{y}{3}=\frac{z}{4}\)
\(\frac{3x-2y}{4}=\frac{2z-4x}{3}=\frac{4y-3z}{2}\)
\(\frac{12x-8y}{16}=\frac{6z-12x}{9}=\frac{8y-6z}{4}\)
\(\frac{12x-8y}{16}=\frac{6z-12x}{9}=\frac{8y-6z}{4}=\frac{\left(12x-8y\right)+\left(6z-12x\right)+\left(8y-6z\right)}{16+9+4}=\frac{0}{29}=0\)
\(\Rightarrow\frac{3x-2y}{4}=0\Rightarrow3x-2y=0\Rightarrow3x=2y\Rightarrow\frac{x}{2}=\frac{y}{3}\)( 1 )
\(\Rightarrow\frac{4y-3z}{2}=0\Rightarrow4y-3z=0\Rightarrow4y=3z\Rightarrow\frac{y}{3}=\frac{z}{4}\)( 2 )
Từ ( 1 ) và ( 2 ) \(\Rightarrow\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\)
\(\frac{3x-2y}{4}=\frac{2z-4x}{3}=\frac{4y-3z}{2}=\frac{12x-8y}{16}=\)\(\frac{6z-12x}{9}=\frac{8y-6z}{4}\)\(=\frac{12x-8y+6z-12x+8y-6z}{16+9+4}=\frac{0}{29}=0\)
\(\Rightarrow3x-2y=0\Rightarrow\frac{x}{2}=\frac{y}{3}\left(1\right)\)
\(\Rightarrow2z-4x=0\Rightarrow\frac{z}{4}=\frac{x}{2}\left(2\right)\)
Từ (1) và (2) => \(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\)
~~ học tốt ~~
\(\frac{3x-2y}{4}=\frac{2z-4x}{3}=\frac{4y-3z}{2}=\frac{4\cdot\left(3x-2y\right)}{16}=\frac{3\cdot\left(2z-4x\right)}{9}=\frac{2\cdot\left(4y-3z\right)}{4}=\)
\(=\frac{12x-8y}{16}=\frac{6z-12x}{9}=\frac{8y-6z}{4}=\frac{12x-8y+6z-12x+8y-6z}{16+9+4}=0\)
\(\Rightarrow3x-2y=0\Rightarrow\frac{x}{2}=\frac{y}{3}\left(1\right)\)
\(\Rightarrow2z-4x=0\Rightarrow z=2x\Rightarrow\frac{x}{2}=\frac{z}{4}\left(2\right)\)
Từ (1) và (2) => \(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\)đpcm
Note: Nếu bạn đã HỎI hãy có trách nhiệm khi được TRẢ LỜI
\(\frac{3x-2y}{4}=\frac{2z-4x}{3}=\frac{4y-3z}{2}\)
\(\Rightarrow\frac{12x-8y}{16}=\frac{6z-12x}{9}=\frac{4y-3z}{4}\)
\(=\frac{12x-8y+6z-12x+8y-6z}{16+9+4}\)\(=0\)
\(\Rightarrow\frac{12x-8y}{16}=0\Rightarrow12x=8y\Rightarrow\frac{x}{y}=\frac{2}{3}\Rightarrow\frac{x}{2}=\frac{y}{3}\left(1\right)\)
\(\Rightarrow\frac{6z-12x}{9}=0\Rightarrow6z=12x\Rightarrow\frac{x}{z}=\frac{2}{4}\Rightarrow\frac{x}{2}=\frac{z}{4}\left(2\right)\)
Từ (1)&(2)=>\(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\)
Ta có : \(\frac{3x-2y}{4}=\frac{2z-4x}{3}=\frac{4y-3z}{2}\)
=>\(\frac{4\left(3x-2y\right)}{16}=\frac{3\left(2z-4y\right)}{9}=\frac{2\left(4y-3z\right)}{4}\)
Hay \(\frac{12x-8y}{16}=\frac{6z-12y}{9}=\frac{8y-6z}{4}\)= \(\frac{12x-8y+6z-12y+8y-6z}{16+9+4}=0\)
+, \(\frac{12x-8y}{16}=0\)=>\(12x-8y=0\)=>\(12x=8y\Rightarrow3x=2y\Rightarrow\frac{x}{2}=\frac{y}{3}\left(1\right)\)
+, \(\frac{6z-12x}{9}=0\Rightarrow6z-12x=0\Rightarrow6z=12x\Rightarrow z=2x\Rightarrow\frac{z}{4}=\frac{x}{2}\left(2\right)\)
+, \(\frac{8y-6z}{4}=0\Rightarrow8y-6z=0\Rightarrow8y=6z\Rightarrow4y=3z\Rightarrow\frac{y}{3}=\frac{z}{4}\left(3\right)\)
Từ (1) , (2) và (3) ta suy ra : \(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\)(đpcm)
Ta có: \(\frac{3x-2y}{4}=\frac{2z-4x}{3}=\frac{4y-3z}{2}\)
=> \(\frac{4\left(3x-2y\right)}{4^2}=\frac{3\left(2z-4x\right)}{3^2}=\frac{2\left(4y-3z\right)}{2^2}\)
=> \(\frac{12x-8y}{16}=\frac{6z-12x}{9}=\frac{8y-6z}{4}\)
Áp dụng t/c của dãy tỉ số bằng nhau, ta có:
\(\frac{12x-8y}{16}=\frac{6z-12x}{9}=\frac{8y-6z}{4}=\frac{12x-8y+6z-12x+8y-6z}{16+9+4}=0\)
=> \(\hept{\begin{cases}\frac{3x-2y}{4}=0\\\frac{2z-4x}{3}=0\\\frac{4y-3z}{2}=0\end{cases}}\) => \(\hept{\begin{cases}3x-2y=0\\2z-4x=0\\4y-3z=0\end{cases}}\) => \(\hept{\begin{cases}3x=2y\\2z=4x\\4y=3z\end{cases}}\) => \(\hept{\begin{cases}\frac{x}{2}=\frac{y}{3}\\\frac{x}{2}=\frac{z}{4}\\\frac{y}{3}=\frac{z}{4}\end{cases}}\) => \(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\)
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